TPTP Problem File: ITP219

TPTP Problem File: ITP219_1.p

%------------------------------------------------------------------------------
% File     : ITP219_1 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_List_Assn 00252_007651
% 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    : 0062_VEBT_List_Assn_00252_007651 [Des22]

% Status   : Theorem
% Rating   : 0.33 v8.2.0, 0.20 v8.1.0
% Syntax   : Number of formulae    : 7867 (1616 unt;2002 typ;   0 def)
%            Number of atoms       : 15460 (5122 equ)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 11430 (1835   ~; 341   |;1091   &)
%                                         (1573 <=>;6590  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   6 avg)
%            Maximal term depth    :   38 (   2 avg)
%            Number of types       :  507 ( 506 usr)
%            Number of type conns  : 2199 (1210   >; 989   *;   0   +;   0  <<)
%            Number of predicates  :  178 ( 176 usr;   1 prp; 0-4 aty)
%            Number of functors    : 1320 (1320 usr; 286 con; 0-5 aty)
%            Number of variables   : 16662 (16280   !; 382   ?;16662   :)
% SPC      : TF0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 17:06:46.389
%------------------------------------------------------------------------------
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tff(ty_n_t__List__Olist_It__HOL__Obool_J,type,
    list_bool: $tType ).

tff(ty_n_t__fun_Itf__d_Mt__HOL__Obool_J,type,
    fun_d_bool: $tType ).

tff(ty_n_t__fun_Itf__c_Mt__HOL__Obool_J,type,
    fun_c_bool: $tType ).

tff(ty_n_t__List__Olist_It__Rat__Orat_J,type,
    list_rat: $tType ).

tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
    list_num: $tType ).

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

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

tff(ty_n_t__Set__Oset_It__HOL__Obool_J,type,
    set_bool: $tType ).

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

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

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

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

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

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

tff(ty_n_t__List__Olist_Itf__d_J,type,
    list_d: $tType ).

tff(ty_n_t__List__Olist_Itf__c_J,type,
    list_c: $tType ).

tff(ty_n_t__Set__Oset_Itf__d_J,type,
    set_d: $tType ).

tff(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

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

tff(ty_n_t__HOL__Obool,type,
    bool: $tType ).

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

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

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

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

tff(ty_n_tf__d,type,
    d: $tType ).

tff(ty_n_tf__c,type,
    c: $tType ).

% Explicit typings (1496)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_Pr4329608150637261639at_nat > fun_Pr1607443439640623155t_bool ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: set_int > fun_int_bool ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_nat > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_Pr1261947904930325089at_nat > fun_Pr7757163693017731481t_bool ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_fu6080891106736489294t_bool > fun_fu9160419385450930796l_bool ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: set_nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: num > fun_in1696471567231710611nt_int ).

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

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_int ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: code_integer > fun_Co3025358997185726883r_bool ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: nat > fun_na3408937105440547634t_bool ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_na4786964484885125043t_bool ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_nat_bool > fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_nat_bool > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_nat_bool ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_Pr1308109350235310159nt_int ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: 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: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: nat > fun_nat_fun_nat_bool ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_Pr6001797050664311790t_bool ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: assn > fun_as4100195077546555551t_bool ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: assn > fun_as4100195077546555551t_bool ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: assn > fun_Pr6001797050664311790t_bool ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_Pr6001797050664311790t_bool > fun_fu7398481913972163267t_bool ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_Pr6001797050664311790t_bool > fun_fu7398481913972163267t_bool ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_fu7398481913972163267t_bool ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: code_natural > fun_Co9168753396221475450atural ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: produc7822875418678951345atural > fun_Co1890133979025130057atural ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: code_natural > fun_Co1105491363547889651atural ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: heap_e7401611519738050253t_unit > fun_he5875662316245821270t_bool ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: assn > fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: nat > fun_he5234484606919320760it_nat ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_fun_nat_bool ).

tff(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: fun_fu1264473476461866025l_assn ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: fun_as4100195077546555551t_bool ).

tff(sy_c_Assertions_Oentails,type,
    entails: fun_as7903617535776176121n_bool ).

tff(sy_c_Assertions_Oentailst,type,
    entailst: assn > fun_assn_bool ).

tff(sy_c_Assertions_Oin__range,type,
    in_range: fun_Pr6001797050664311790t_bool ).

tff(sy_c_Assertions_Ois__pure__assn,type,
    is_pure_assn: fun_assn_bool ).

tff(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: fun_Pr6001797050664311790t_bool ).

tff(sy_c_Assertions_Oproper,type,
    proper: fun_fu9160419385450930796l_bool ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: bool > 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: bool > fun_Pr6001797050664311790t_bool ).

tff(sy_c_Assertions_OrelH,type,
    relH: ( set_nat * heap_e7401611519738050253t_unit * heap_e7401611519738050253t_unit ) > $o ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: fun_Pr6001797050664311790t_bool > fun_fu7398481913972163267t_bool ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun_Pr6803231168968006865t_bool ).

tff(sy_c_Assertions_Owand__assn,type,
    wand_assn: fun_as3983628580203722482n_assn ).

tff(sy_c_Assertions_Owand__raw,type,
    wand_raw: fun_Pr6001797050664311790t_bool > fun_fu7398481913972163267t_bool ).

tff(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: fun_Pr6803231168968006865t_bool ).

tff(sy_c_Automation_OFI,type,
    fi: ( list_P8527749157015355191n_assn * assn * assn * assn * assn ) > fun_assn_bool ).

tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: ( assn * assn ) > fun_assn_bool ).

tff(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: ( list_P8527749157015355191n_assn * assn * assn ) > fun_assn_bool ).

tff(sy_c_Automation_OSLN,type,
    sln: assn ).

tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Pr8621012493645324487l_bool ).

tff(sy_c_BNF__Cardinal__Arithmetic_Oczero_001t__HOL__Obool,type,
    bNF_Ca1500629177841160323o_bool: set_Pr8621012493645324487l_bool ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
    bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001t__HOL__Obool,type,
    bNF_Ca3869711322664823701n_bool: ( set_bool * set_Pr8621012493645324487l_bool ) > $o ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re6250860962936578807nt_int: ( fun_int_fun_int_bool * fun_Pr3099820731930151597t_bool ) > fun_fu2194082084716825453t_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( fun_int_fun_int_bool * fun_Pr3145601080792579042t_bool ) > fun_fu8159772421396582028t_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re4808202430149235895nt_int: ( fun_int_fun_int_bool * fun_fu2194082084716825453t_bool * fun_in1696471567231710611nt_int * fun_in1696471567231710611nt_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__fun_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
    bNF_re2486238095283335024nt_rat: ( fun_int_fun_int_bool * fun_fu8159772421396582028t_bool * fun_in1696471567231710611nt_int * fun_int_fun_int_rat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re6223507645442431167l_bool: ( fun_nat_fun_nat_bool * fun_bo1841576699660343469l_bool ) > fun_fu5827305053654988013l_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( fun_nat_fun_nat_bool * fun_Pr5932607294251420486t_bool * fun_na1476630412504255114at_nat * fun_nat_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J_001t__fun_It__Nat__Onat_Mt__HOL__Obool_J,type,
    bNF_re1480548845665215743t_bool: ( fun_nat_fun_nat_bool * fun_fu5827305053654988013l_bool * fun_nat_fun_nat_bool * fun_nat_fun_nat_bool ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( fun_Pr3099820731930151597t_bool * fun_Pr3099820731930151597t_bool ) > fun_fu2754956555950464963t_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re4734134812813710757nt_int: ( fun_Pr3099820731930151597t_bool * fun_fu2754956555950464963t_bool * fun_Pr1308109350235310159nt_int * fun_Pr1308109350235310159nt_int ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( fun_Pr3145601080792579042t_bool * fun_Pr3145601080792579042t_bool ) > fun_fu2749353902725170073t_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re6323452265856231590at_rat: ( fun_Pr3145601080792579042t_bool * fun_fu2749353902725170073t_bool * fun_Pr1308109350235310159nt_int * fun_rat_fun_rat_rat ) > $o ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__HOL__Obool_001t__HOL__Obool,type,
    bNF_re1498568104593929552l_bool: ( fun_Pr5932607294251420486t_bool * fun_bo1841576699660343469l_bool ) > fun_fu813382667476268716l_bool ).

tff(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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tff(sy_c_List_OremoveAll_001t__Int__Oint,type,
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tff(sy_c_Misc_Omerge_001t__Int__Oint,type,
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tff(sy_c_Misc_Omerge_001t__Nat__Onat,type,
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tff(sy_c_Misc_Omerge_001t__Num__Onum,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Int__Oint,type,
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tff(sy_c_Misc_Omergesort__remdups_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oslice_001t__Int__Oint,type,
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tff(sy_c_Misc_Oslice_001t__Nat__Onat,type,
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tff(sy_c_Misc_Oslice_001tf__c,type,
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tff(sy_c_Misc_Oslice_001tf__d,type,
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tff(sy_c_Misc_Ozipf_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Ozipf_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_Misc_Ozipf_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Misc_Ozipf_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Misc_Ozipf_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Misc_Ozipf_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    zipf_nat_int_nat: ( fun_nat_fun_int_nat * list_nat * list_int ) > list_nat ).

tff(sy_c_Misc_Ozipf_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Misc_Ozipf_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Multiset_Oms__weak,type,
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tff(sy_c_Multiset_Opw__leq,type,
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tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

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    size_size_list_bool: list_bool > nat ).

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    size_size_list_int: list_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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    nat_prod_decode_aux: ( nat * nat ) > product_prod_nat_nat ).

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tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

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tff(sy_c_Num_Onat__of__num,type,
    nat_of_num: num > nat ).

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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Rat_OFract,type,
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tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__HOL__Obool_J_Mt__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__HOL__Obool_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member2818932950976722259et_nat: ( produc1542783094787327164et_nat * set_Pr9222568342844664434et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__HOL__Obool_J_Mt__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__HOL__Obool_J_J,type,
    member9182180520733025616t_bool: ( produc1469476007717053223t_bool * set_Pr2055239900342597383t_bool ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member690604328116145115nt_int: ( produc4441460161354545860nt_int * set_Pr233528076387193594nt_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Int__Oint_J,type,
    member4017226226790010994_c_int: ( product_prod_c_int * set_Pr4413417460947882513_c_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mt__Nat__Onat_J,type,
    member8195077246299207702_c_nat: ( product_prod_c_nat * set_Pr6903500605879609269_c_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__c_J,type,
    member5074992359041316560od_c_c: ( product_prod_c_c * set_Product_prod_c_c ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__c_Mtf__d_J,type,
    member5074992363344545361od_c_d: ( product_prod_c_d * set_Product_prod_c_d ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__d_Mt__Int__Oint_J,type,
    member5252670555746405491_d_int: ( product_prod_d_int * set_Pr786264039281237394_d_int ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__d_Mt__Nat__Onat_J,type,
    member207149538400826391_d_nat: ( product_prod_d_nat * set_Pr3276347184212964150_d_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__d_Mtf__c_J,type,
    member2287536777068640527od_d_c: ( product_prod_d_c * set_Product_prod_d_c ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_Itf__d_Mtf__d_J,type,
    member2287536781371869328od_d_d: ( product_prod_d_d * set_Product_prod_d_d ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: ( set_Pr1261947904930325089at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_member_001t__fun_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member7723273712176309973e_term: ( fun_in6429348457993719614e_term * set_fu6991988656872959732e_term ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member4565687819040644368e_term: ( fun_Pr272337284026993017e_term * set_fu2883028153058477999e_term ) > $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_Mt__HOL__Obool_J,type,
    member4125802036136091927t_bool: ( fun_Pr6001797050664311790t_bool * set_fu6080891106736489294t_bool ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__fun_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__HOL__Obool_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member2948136515907314945e_term: ( fun_Pr9164807099015279978e_term * set_fu8544203214900410144e_term ) > $o ).

tff(sy_c_member_001tf__c,type,
    member_c: ( c * set_c ) > $o ).

tff(sy_c_member_001tf__d,type,
    member_d: ( d * set_d ) > $o ).

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

tff(sy_v_A,type,
    a: fun_c_fun_d_assn ).

tff(sy_v_F,type,
    f: assn ).

tff(sy_v_I,type,
    i: set_nat ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: assn ).

tff(sy_v_i,type,
    i2: nat ).

tff(sy_v_x,type,
    x: c ).

tff(sy_v_xi,type,
    xi: d ).

tff(sy_v_xs,type,
    xs: list_c ).

tff(sy_v_xsi,type,
    xsi: list_d ).

% Relevant facts (5827)
tff(fact_0_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H))
         => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),H)) )
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ).

% entailsI
tff(fact_1__092_060open_062P_A_092_060Longrightarrow_062_092_060_094sub_062A_Afalse_092_060close_062,axiom,
    pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,p),bot_bot_assn)) ).

% \<open>P \<Longrightarrow>\<^sub>A false\<close>
tff(fact_2_assms_I3_J,axiom,
    member_nat(i2,i) ).

% assms(3)
tff(fact_3_ent__iffI,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),B))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),A))
       => ( A = B ) ) ) ).

% ent_iffI
tff(fact_4_ent__refl,axiom,
    ! [P: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P)) ).

% ent_refl
tff(fact_5_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Q),R))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R)) ) ) ).

% ent_trans
tff(fact_6_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ).

% is_entails
tff(fact_7_ent__false,axiom,
    ! [P: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,bot_bot_assn),P)) ).

% ent_false
tff(fact_8_ent__fwd,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat,Q: assn] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
       => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),H2)) ) ) ).

% ent_fwd
tff(fact_9_entailsD,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
       => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),H2)) ) ) ).

% entailsD
tff(fact_10_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
    <=> ! [H3: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H3))
         => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),H3)) ) ) ).

% entails_def
tff(fact_11_ent__false__iff,axiom,
    ! [P: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),bot_bot_assn))
    <=> ! [H3: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H3)) ) ).

% ent_false_iff
tff(fact_12_assms_I4_J,axiom,
    pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L824818263033719201sn_c_d(i,a,list_update_c(xs,i2,x),list_update_d(xsi,i2,xi))),f)),q)) ).

% assms(4)
tff(fact_13_ent__imp__entt,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => pp(aa_assn_bool(entailst(P),Q)) ) ).

% ent_imp_entt
tff(fact_14_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_ass8975924923219239432t_bool(rep_assn,X) = aa_ass8975924923219239432t_bool(rep_assn,Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_15_entt__refl,axiom,
    ! [A: assn] : pp(aa_assn_bool(entailst(A),A)) ).

% entt_refl
tff(fact_16_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_17_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_18_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C)),B2) ).

% assn_aci(10)
tff(fact_19_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C)) ).

% star_aci(3)
tff(fact_20_star__aci_I2_J,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),A2) ).

% star_aci(2)
tff(fact_21_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,bot_bot_assn),H2)) ).

% mod_false
tff(fact_22_mod__starD,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H2))
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,A),H1))
          & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,B),H22)) ) ) ).

% mod_starD
tff(fact_23_mod__starE,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H2))
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,A2),X_1))
         => ! [H_2: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,B2),H_2)) ) ) ).

% mod_starE
tff(fact_24_entt__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(entailst(A),B))
     => ( pp(aa_assn_bool(entailst(B),C2))
       => pp(aa_assn_bool(entailst(A),C2)) ) ) ).

% entt_trans
tff(fact_25_star__assoc,axiom,
    ! [A2: assn,B2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C)) ).

% star_assoc
tff(fact_26_entt__fr__drop,axiom,
    ! [F: assn,F2: assn,A: assn] :
      ( pp(aa_assn_bool(entailst(F),F2))
     => pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F),A)),F2)) ) ).

% entt_fr_drop
tff(fact_27_entt__fr__refl,axiom,
    ! [F: assn,F2: assn,A: assn] :
      ( pp(aa_assn_bool(entailst(F),F2))
     => pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F),A)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F2),A))) ) ).

% entt_fr_refl
tff(fact_28_entt__frame__fwd,axiom,
    ! [P: assn,Q: assn,A: assn,F: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(P),Q))
     => ( pp(aa_assn_bool(entailst(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F)))
       => ( pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F)),B))
         => pp(aa_assn_bool(entailst(A),B)) ) ) ) ).

% entt_frame_fwd
tff(fact_29_entt__star__mono,axiom,
    ! [A: assn,B: assn,C2: assn,D: assn] :
      ( pp(aa_assn_bool(entailst(A),B))
     => ( pp(aa_assn_bool(entailst(C2),D))
       => pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),D))) ) ) ).

% entt_star_mono
tff(fact_30_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P) ).

% assn_times_comm
tff(fact_31_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),R)) ).

% assn_times_assoc
tff(fact_32_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P: assn,R: assn,F: assn] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Ps),H2))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R))
       => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F)))
         => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F)),H2)) ) ) ) ).

% mod_frame_fwd
tff(fact_33_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),F)))
       => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,R),F)),Q))
         => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Ps),Q)) ) ) ) ).

% ent_frame_fwd
tff(fact_34_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),P2))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Q),Q2))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P2),Q2))) ) ) ).

% ent_star_mono
tff(fact_35_fr__rot__rhs,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),B))) ) ).

% fr_rot_rhs
tff(fact_36_fr__refl,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),B))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2))) ) ).

% fr_refl
tff(fact_37_fr__rot,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C2))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A)),C2)) ) ).

% fr_rot
tff(fact_38_assms_I1_J,axiom,
    pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,p),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_d_assn(aa_c_fun_d_assn(a,x),xi)),vEBT_L824818263033719201sn_c_d(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,i),insert_nat2(i2,bot_bot_set_nat)),a,xs,xsi))),f))) ).

% assms(1)
tff(fact_39_list__update__overwrite,axiom,
    ! [Xs: list_c,I: nat,X: c,Y: c] : list_update_c(list_update_c(Xs,I,X),I,Y) = list_update_c(Xs,I,Y) ).

% list_update_overwrite
tff(fact_40_list__update__overwrite,axiom,
    ! [Xs: list_d,I: nat,X: d,Y: d] : list_update_d(list_update_d(Xs,I,X),I,Y) = list_update_d(Xs,I,Y) ).

% list_update_overwrite
tff(fact_41_False,axiom,
    size_size_list_c(xs) != size_size_list_d(xsi) ).

% False
tff(fact_42_assms_I2_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,i2),size_size_list_c(xs))) ).

% assms(2)
tff(fact_43_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( pp(aa_assn_bool(fI_QUERY(P,Q),F))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F))) ) ).

% frame_inference_init
tff(fact_44_mem__Collect__eq,axiom,
    ! [A2: produc3843707927480180839at_nat,P: fun_Pr1607443439640623155t_bool] :
      ( member8757157785044589968at_nat(A2,collec6321179662152712658at_nat(P))
    <=> pp(aa_Pro1809015098595223084t_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_45_mem__Collect__eq,axiom,
    ! [A2: int,P: fun_int_bool] :
      ( member_int(A2,collect_int(P))
    <=> pp(aa_int_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_46_mem__Collect__eq,axiom,
    ! [A2: nat,P: fun_nat_bool] :
      ( member_nat(A2,collect_nat(P))
    <=> pp(aa_nat_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_47_mem__Collect__eq,axiom,
    ! [A2: product_prod_nat_nat,P: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),collec3392354462482085612at_nat(P)))
    <=> pp(aa_Pro2232559814647311250t_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_48_mem__Collect__eq,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,P: fun_fu9160419385450930796l_bool] :
      ( member4125802036136091927t_bool(A2,collec3007137136614167897t_bool(P))
    <=> pp(aa_fun5588795374382764133l_bool(P,A2)) ) ).

% mem_Collect_eq
tff(fact_49_Collect__mem__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : collec6321179662152712658at_nat(aTP_Lamm_a(A)) = A ).

% Collect_mem_eq
tff(fact_50_Collect__mem__eq,axiom,
    ! [A: set_int] : collect_int(aTP_Lamm_aa(A)) = A ).

% Collect_mem_eq
tff(fact_51_Collect__mem__eq,axiom,
    ! [A: set_nat] : collect_nat(aTP_Lamm_ab(A)) = A ).

% Collect_mem_eq
tff(fact_52_Collect__mem__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_ac(A)) = A ).

% Collect_mem_eq
tff(fact_53_Collect__mem__eq,axiom,
    ! [A: set_fu6080891106736489294t_bool] : collec3007137136614167897t_bool(aTP_Lamm_ad(A)) = A ).

% Collect_mem_eq
tff(fact_54_Collect__cong,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_nat_bool(P,X2))
        <=> pp(aa_nat_bool(Q,X2)) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_55_Collect__cong,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( ! [X2: product_prod_nat_nat] :
          ( pp(aa_Pro2232559814647311250t_bool(P,X2))
        <=> pp(aa_Pro2232559814647311250t_bool(Q,X2)) )
     => ( collec3392354462482085612at_nat(P) = collec3392354462482085612at_nat(Q) ) ) ).

% Collect_cong
tff(fact_56_Collect__cong,axiom,
    ! [P: fun_fu9160419385450930796l_bool,Q: fun_fu9160419385450930796l_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool] :
          ( pp(aa_fun5588795374382764133l_bool(P,X2))
        <=> pp(aa_fun5588795374382764133l_bool(Q,X2)) )
     => ( collec3007137136614167897t_bool(P) = collec3007137136614167897t_bool(Q) ) ) ).

% Collect_cong
tff(fact_57_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( pp(aa_assn_bool(fI_QUERY(P,Q),F))
    <=> pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),F))) ) ).

% FI_QUERY_def
tff(fact_58_ent__wand__frameI,axiom,
    ! [Q: assn,R: assn,F: assn,S: assn,P: assn,X3: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)),F)),S))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,F),X3)))
       => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),X3)),R))
         => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),S)) ) ) ) ).

% ent_wand_frameI
tff(fact_59_ent__wandI,axiom,
    ! [Q: assn,P: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),P)),R))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R))) ) ).

% ent_wandI
tff(fact_60_ent__mp,axiom,
    ! [P: assn,Q: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,P),Q))),Q)) ).

% ent_mp
tff(fact_61_length__list__update,axiom,
    ! [Xs: list_c,I: nat,X: c] : size_size_list_c(list_update_c(Xs,I,X)) = size_size_list_c(Xs) ).

% length_list_update
tff(fact_62_length__list__update,axiom,
    ! [Xs: list_d,I: nat,X: d] : size_size_list_d(list_update_d(Xs,I,X)) = size_size_list_d(Xs) ).

% length_list_update
tff(fact_63_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : size_size_list_nat(list_update_nat(Xs,I,X)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_64_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] : size_size_list_int(list_update_int(Xs,I,X)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_65_order__less__imp__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_66_order__less__imp__not__less,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ~ pp(aa_assn_bool(ord_less_assn(Y),X)) ) ).

% order_less_imp_not_less
tff(fact_67_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_68_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_69_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_imp_not_less
tff(fact_70_order__less__imp__not__eq2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_71_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_72_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_73_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_74_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
tff(fact_75_order__less__imp__not__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_76_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_77_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_78_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_79_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% order_less_imp_not_eq
tff(fact_80_linorder__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
      | ( X = Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).

% linorder_less_linear
tff(fact_81_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      | ( X = Y )
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_less_linear
tff(fact_82_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      | ( X = Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_less_linear
tff(fact_83_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      | ( X = Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_less_linear
tff(fact_84_order__less__imp__triv,axiom,
    ! [X: code_integer,Y: code_integer,P: bool] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_85_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,P: bool] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_assn(Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_86_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: bool] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_87_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_88_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
       => pp(P) ) ) ).

% order_less_imp_triv
tff(fact_89_order__less__not__sym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).

% order_less_not_sym
tff(fact_90_order__less__not__sym,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ~ pp(aa_assn_bool(ord_less_assn(Y),X)) ) ).

% order_less_not_sym
tff(fact_91_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_not_sym
tff(fact_92_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_not_sym
tff(fact_93_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_not_sym
tff(fact_94_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Co8499238673658658103nteger,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,X2)),aa_Cod1783974652533396728nteger(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_95_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Co4490034186337852034r_assn,C: assn] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,X2)),aa_Code_integer_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_96_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_num,C: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,X2)),aa_Code_integer_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_97_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_nat,C: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F3,X2)),aa_Code_integer_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_98_order__less__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_int,C: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F3,X2)),aa_Code_integer_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_99_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_as9138348466139350018nteger,C: code_integer] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_100_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_101_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_num,C: num] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_102_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_103_order__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_int,C: int] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,A2)),C)) ) ) ) ).

% order_less_subst2
tff(fact_104_order__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_Co8499238673658658103nteger,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(F3,B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,X2)),aa_Cod1783974652533396728nteger(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_105_order__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_as9138348466139350018nteger,B2: assn,C: assn] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_106_order__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_num_Code_integer,B2: num,C: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_num_Code_integer(F3,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => ( ! [X2: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F3,X2)),aa_num_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_num_Code_integer(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_107_order__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_nat_Code_integer,B2: nat,C: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,X2)),aa_nat_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_108_order__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_int_Code_integer,B2: int,C: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_int_Code_integer(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F3,X2)),aa_int_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_int_Code_integer(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_109_order__less__subst1,axiom,
    ! [A2: assn,F3: fun_Co4490034186337852034r_assn,B2: code_integer,C: code_integer] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_Code_integer_assn(F3,B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,X2)),aa_Code_integer_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_Code_integer_assn(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_110_order__less__subst1,axiom,
    ! [A2: assn,F3: fun_assn_assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_assn_assn(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_assn_assn(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_111_order__less__subst1,axiom,
    ! [A2: assn,F3: fun_num_assn,B2: num,C: num] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_num_assn(F3,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => ( ! [X2: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_num_assn(F3,X2)),aa_num_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_num_assn(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_112_order__less__subst1,axiom,
    ! [A2: assn,F3: fun_nat_assn,B2: nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_nat_assn(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_nat_assn(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_113_order__less__subst1,axiom,
    ! [A2: assn,F3: fun_int_assn,B2: int,C: int] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_int_assn(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_int_assn(F3,X2)),aa_int_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_int_assn(F3,C))) ) ) ) ).

% order_less_subst1
tff(fact_114_order__less__irrefl,axiom,
    ! [X: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),X)) ).

% order_less_irrefl
tff(fact_115_order__less__irrefl,axiom,
    ! [X: assn] : ~ pp(aa_assn_bool(ord_less_assn(X),X)) ).

% order_less_irrefl
tff(fact_116_order__less__irrefl,axiom,
    ! [X: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X)) ).

% order_less_irrefl
tff(fact_117_order__less__irrefl,axiom,
    ! [X: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X)) ).

% order_less_irrefl
tff(fact_118_order__less__irrefl,axiom,
    ! [X: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X)) ).

% order_less_irrefl
tff(fact_119_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Co8499238673658658103nteger,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( ( aa_Cod1783974652533396728nteger(F3,B2) = C )
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,X2)),aa_Cod1783974652533396728nteger(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_120_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Co4490034186337852034r_assn,C: assn] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( ( aa_Code_integer_assn(F3,B2) = C )
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,X2)),aa_Code_integer_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_121_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_num,C: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( ( aa_Code_integer_num(F3,B2) = C )
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,X2)),aa_Code_integer_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_122_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_nat,C: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( ( aa_Code_integer_nat(F3,B2) = C )
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F3,X2)),aa_Code_integer_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_123_ord__less__eq__subst,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_int,C: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( ( aa_Code_integer_int(F3,B2) = C )
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F3,X2)),aa_Code_integer_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_124_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_as9138348466139350018nteger,C: code_integer] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( ( aa_assn_Code_integer(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_125_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( ( aa_assn_assn(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_126_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_num,C: num] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( ( aa_assn_num(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_127_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( ( aa_assn_nat(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_128_ord__less__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_int,C: int] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( ( aa_assn_int(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,A2)),C)) ) ) ) ).

% ord_less_eq_subst
tff(fact_129_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F3: fun_Co8499238673658658103nteger,B2: code_integer,C: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(F3,B2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,X2)),aa_Cod1783974652533396728nteger(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_130_ord__eq__less__subst,axiom,
    ! [A2: assn,F3: fun_Co4490034186337852034r_assn,B2: code_integer,C: code_integer] :
      ( ( A2 = aa_Code_integer_assn(F3,B2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_Code_integer_assn(F3,X2)),aa_Code_integer_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_Code_integer_assn(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_131_ord__eq__less__subst,axiom,
    ! [A2: num,F3: fun_Code_integer_num,B2: code_integer,C: code_integer] :
      ( ( A2 = aa_Code_integer_num(F3,B2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,X2)),aa_Code_integer_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_Code_integer_num(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_132_ord__eq__less__subst,axiom,
    ! [A2: nat,F3: fun_Code_integer_nat,B2: code_integer,C: code_integer] :
      ( ( A2 = aa_Code_integer_nat(F3,B2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(F3,X2)),aa_Code_integer_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_Code_integer_nat(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_133_ord__eq__less__subst,axiom,
    ! [A2: int,F3: fun_Code_integer_int,B2: code_integer,C: code_integer] :
      ( ( A2 = aa_Code_integer_int(F3,B2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Code_integer_int(F3,X2)),aa_Code_integer_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_Code_integer_int(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_134_ord__eq__less__subst,axiom,
    ! [A2: code_integer,F3: fun_as9138348466139350018nteger,B2: assn,C: assn] :
      ( ( A2 = aa_assn_Code_integer(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_135_ord__eq__less__subst,axiom,
    ! [A2: assn,F3: fun_assn_assn,B2: assn,C: assn] :
      ( ( A2 = aa_assn_assn(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_assn_assn(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_136_ord__eq__less__subst,axiom,
    ! [A2: num,F3: fun_assn_num,B2: assn,C: assn] :
      ( ( A2 = aa_assn_num(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_assn_num(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_137_ord__eq__less__subst,axiom,
    ! [A2: nat,F3: fun_assn_nat,B2: assn,C: assn] :
      ( ( A2 = aa_assn_nat(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_assn_nat(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_138_ord__eq__less__subst,axiom,
    ! [A2: int,F3: fun_assn_int,B2: assn,C: assn] :
      ( ( A2 = aa_assn_int(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_assn_int(F3,C))) ) ) ) ).

% ord_eq_less_subst
tff(fact_139_order__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z)) ) ) ).

% order_less_trans
tff(fact_140_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_assn(Y),Z))
       => pp(aa_assn_bool(ord_less_assn(X),Z)) ) ) ).

% order_less_trans
tff(fact_141_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_trans
tff(fact_142_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_trans
tff(fact_143_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_trans
tff(fact_144_order__less__asym_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ).

% order_less_asym'
tff(fact_145_order__less__asym_H,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ~ pp(aa_assn_bool(ord_less_assn(B2),A2)) ) ).

% order_less_asym'
tff(fact_146_order__less__asym_H,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2)) ) ).

% order_less_asym'
tff(fact_147_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2)) ) ).

% order_less_asym'
tff(fact_148_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% order_less_asym'
tff(fact_149_linorder__neq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_150_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_151_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_152_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neq_iff
tff(fact_153_order__less__asym,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).

% order_less_asym
tff(fact_154_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ~ pp(aa_assn_bool(ord_less_assn(Y),X)) ) ).

% order_less_asym
tff(fact_155_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% order_less_asym
tff(fact_156_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% order_less_asym
tff(fact_157_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% order_less_asym
tff(fact_158_linorder__neqE,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).

% linorder_neqE
tff(fact_159_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_neqE
tff(fact_160_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE
tff(fact_161_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE
tff(fact_162_neq__if__length__neq,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( size_size_list_c(Xs) != size_size_list_c(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_163_neq__if__length__neq,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( size_size_list_d(Xs) != size_size_list_d(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_164_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_165_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_166_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_c] : size_size_list_c(Xs2) = N ).

% Ex_list_of_length
tff(fact_167_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_d] : size_size_list_d(Xs2) = N ).

% Ex_list_of_length
tff(fact_168_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_169_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_170_length__induct,axiom,
    ! [P: fun_list_c_bool,Xs: list_c] :
      ( ! [Xs2: list_c] :
          ( ! [Ys2: list_c] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_c(Ys2)),size_size_list_c(Xs2)))
             => pp(aa_list_c_bool(P,Ys2)) )
         => pp(aa_list_c_bool(P,Xs2)) )
     => pp(aa_list_c_bool(P,Xs)) ) ).

% length_induct
tff(fact_171_length__induct,axiom,
    ! [P: fun_list_d_bool,Xs: list_d] :
      ( ! [Xs2: list_d] :
          ( ! [Ys2: list_d] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_d(Ys2)),size_size_list_d(Xs2)))
             => pp(aa_list_d_bool(P,Ys2)) )
         => pp(aa_list_d_bool(P,Xs2)) )
     => pp(aa_list_d_bool(P,Xs)) ) ).

% length_induct
tff(fact_172_length__induct,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2)))
             => pp(aa_list_nat_bool(P,Ys2)) )
         => pp(aa_list_nat_bool(P,Xs2)) )
     => pp(aa_list_nat_bool(P,Xs)) ) ).

% length_induct
tff(fact_173_length__induct,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2)))
             => pp(aa_list_int_bool(P,Ys2)) )
         => pp(aa_list_int_bool(P,Xs2)) )
     => pp(aa_list_int_bool(P,Xs)) ) ).

% length_induct
tff(fact_174_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_175_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_176_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_177_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_178_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_179_order_Ostrict__implies__not__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_180_order_Ostrict__implies__not__eq,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_181_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_182_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_183_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_184_dual__order_Ostrict__trans,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_185_dual__order_Ostrict__trans,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ( pp(aa_assn_bool(ord_less_assn(C),B2))
       => pp(aa_assn_bool(ord_less_assn(C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_186_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_187_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_188_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% dual_order.strict_trans
tff(fact_189_not__less__iff__gr__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_190_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_191_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_192_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_193_order_Ostrict__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C)) ) ) ).

% order.strict_trans
tff(fact_194_order_Ostrict__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => pp(aa_assn_bool(ord_less_assn(A2),C)) ) ) ).

% order.strict_trans
tff(fact_195_order_Ostrict__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% order.strict_trans
tff(fact_196_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% order.strict_trans
tff(fact_197_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% order.strict_trans
tff(fact_198_linorder__less__wlog,axiom,
    ! [P: fun_Co8310303586393106575r_bool,A2: code_integer,B2: code_integer] :
      ( ! [A3: code_integer,B3: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A3),B3))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A3),B3)) )
     => ( ! [A3: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A3),A3))
       => ( ! [A3: code_integer,B3: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,B3),A3))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A3),B3)) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_199_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_bool,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A3),B3))
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
     => ( ! [A3: num] : pp(aa_num_bool(aa_num_fun_num_bool(P,A3),A3))
       => ( ! [A3: num,B3: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(P,B3),A3))
             => pp(aa_num_bool(aa_num_fun_num_bool(P,A3),B3)) )
         => pp(aa_num_bool(aa_num_fun_num_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_200_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),A3))
       => ( ! [A3: nat,B3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_201_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int] : pp(aa_int_bool(aa_int_fun_int_bool(P,A3),A3))
       => ( ! [A3: int,B3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
             => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) ) ) ) ).

% linorder_less_wlog
tff(fact_202_exists__least__iff,axiom,
    ! [P: fun_nat_bool] :
      ( ? [X_12: nat] : pp(aa_nat_bool(P,X_12))
    <=> ? [N2: nat] :
          ( pp(aa_nat_bool(P,N2))
          & ! [M: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
             => ~ pp(aa_nat_bool(P,M)) ) ) ) ).

% exists_least_iff
tff(fact_203_dual__order_Oirrefl,axiom,
    ! [A2: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),A2)) ).

% dual_order.irrefl
tff(fact_204_dual__order_Oirrefl,axiom,
    ! [A2: assn] : ~ pp(aa_assn_bool(ord_less_assn(A2),A2)) ).

% dual_order.irrefl
tff(fact_205_dual__order_Oirrefl,axiom,
    ! [A2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),A2)) ).

% dual_order.irrefl
tff(fact_206_dual__order_Oirrefl,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A2)) ).

% dual_order.irrefl
tff(fact_207_dual__order_Oirrefl,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),A2)) ).

% dual_order.irrefl
tff(fact_208_dual__order_Oasym,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% dual_order.asym
tff(fact_209_dual__order_Oasym,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ~ pp(aa_assn_bool(ord_less_assn(A2),B2)) ) ).

% dual_order.asym
tff(fact_210_dual__order_Oasym,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% dual_order.asym
tff(fact_211_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% dual_order.asym
tff(fact_212_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% dual_order.asym
tff(fact_213_linorder__cases,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( ( X != Y )
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).

% linorder_cases
tff(fact_214_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( ( X != Y )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ) ).

% linorder_cases
tff(fact_215_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( ( X != Y )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_cases
tff(fact_216_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( ( X != Y )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_cases
tff(fact_217_antisym__conv3,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_218_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_219_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_220_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv3
tff(fact_221_less__induct,axiom,
    ! [P: fun_nat_bool,A2: nat] :
      ( ! [X2: nat] :
          ( ! [Y3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),X2))
             => pp(aa_nat_bool(P,Y3)) )
         => pp(aa_nat_bool(P,X2)) )
     => pp(aa_nat_bool(P,A2)) ) ).

% less_induct
tff(fact_222_ord__less__eq__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( ( B2 = C )
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_223_ord__less__eq__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( ( B2 = C )
       => pp(aa_assn_bool(ord_less_assn(A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_224_ord__less__eq__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( ( B2 = C )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_225_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( ( B2 = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_226_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( ( B2 = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% ord_less_eq_trans
tff(fact_227_ord__eq__less__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 = B2 )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_228_ord__eq__less__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( A2 = B2 )
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => pp(aa_assn_bool(ord_less_assn(A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_229_ord__eq__less__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( A2 = B2 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_230_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_231_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% ord_eq_less_trans
tff(fact_232_order_Oasym,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ).

% order.asym
tff(fact_233_order_Oasym,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ~ pp(aa_assn_bool(ord_less_assn(B2),A2)) ) ).

% order.asym
tff(fact_234_order_Oasym,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2)) ) ).

% order.asym
tff(fact_235_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2)) ) ).

% order.asym
tff(fact_236_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% order.asym
tff(fact_237_less__imp__neq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_238_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_239_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_240_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_241_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( X != Y ) ) ).

% less_imp_neq
tff(fact_242_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X_1)) ).

% gt_ex
tff(fact_243_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X_1)) ).

% gt_ex
tff(fact_244_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y2),X)) ).

% lt_ex
tff(fact_245_bot_Onot__eq__extremum,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( A2 != bot_bo228742789529271731at_nat )
    <=> ord_le2604355607129572851at_nat(bot_bo228742789529271731at_nat,A2) ) ).

% bot.not_eq_extremum
tff(fact_246_bot_Onot__eq__extremum,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,bot_bo2099793752762293965at_nat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_247_bot_Onot__eq__extremum,axiom,
    ! [A2: set_bool] :
      ( ( A2 != bot_bot_set_bool )
    <=> ord_less_set_bool(bot_bot_set_bool,A2) ) ).

% bot.not_eq_extremum
tff(fact_248_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
    <=> ord_less_set_nat(bot_bot_set_nat,A2) ) ).

% bot.not_eq_extremum
tff(fact_249_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
    <=> ord_less_set_int(bot_bot_set_int,A2) ) ).

% bot.not_eq_extremum
tff(fact_250_bot_Onot__eq__extremum,axiom,
    ! [A2: assn] :
      ( ( A2 != bot_bot_assn )
    <=> pp(aa_assn_bool(ord_less_assn(bot_bot_assn),A2)) ) ).

% bot.not_eq_extremum
tff(fact_251_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,bot_bot_nat),A2)) ) ).

% bot.not_eq_extremum
tff(fact_252_bot_Oextremum__strict,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ~ ord_le2604355607129572851at_nat(A2,bot_bo228742789529271731at_nat) ).

% bot.extremum_strict
tff(fact_253_bot_Oextremum__strict,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),bot_bo2099793752762293965at_nat)) ).

% bot.extremum_strict
tff(fact_254_bot_Oextremum__strict,axiom,
    ! [A2: set_bool] : ~ ord_less_set_bool(A2,bot_bot_set_bool) ).

% bot.extremum_strict
tff(fact_255_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ ord_less_set_nat(A2,bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_256_bot_Oextremum__strict,axiom,
    ! [A2: set_int] : ~ ord_less_set_int(A2,bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_257_bot_Oextremum__strict,axiom,
    ! [A2: assn] : ~ pp(aa_assn_bool(ord_less_assn(A2),bot_bot_assn)) ).

% bot.extremum_strict
tff(fact_258_bot_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),bot_bot_nat)) ).

% bot.extremum_strict
tff(fact_259_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_c,A: fun_c_fun_c_assn] :
      ( ( size_size_list_c(L) != size_size_list_c(Li) )
     => ( vEBT_L7968352277243653187sn_c_c(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_260_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_d,A: fun_c_fun_d_assn] :
      ( ( size_size_list_c(L) != size_size_list_d(Li) )
     => ( vEBT_L7968352277243653188sn_c_d(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_261_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_nat,A: fun_c_fun_nat_assn] :
      ( ( size_size_list_c(L) != size_size_list_nat(Li) )
     => ( vEBT_L4839309063161974733_c_nat(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_262_list__assn__aux__ineq__len,axiom,
    ! [L: list_c,Li: list_int,A: fun_c_fun_int_assn] :
      ( ( size_size_list_c(L) != size_size_list_int(Li) )
     => ( vEBT_L4836818592652924457_c_int(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_263_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_c,A: fun_d_fun_c_assn] :
      ( ( size_size_list_d(L) != size_size_list_c(Li) )
     => ( vEBT_L5180896695270977154sn_d_c(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_264_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_d,A: fun_d_fun_d_assn] :
      ( ( size_size_list_d(L) != size_size_list_d(Li) )
     => ( vEBT_L5180896695270977155sn_d_d(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_265_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_nat,A: fun_d_fun_nat_assn] :
      ( ( size_size_list_d(L) != size_size_list_nat(Li) )
     => ( vEBT_L6074753392118369230_d_nat(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_266_list__assn__aux__ineq__len,axiom,
    ! [L: list_d,Li: list_int,A: fun_d_fun_int_assn] :
      ( ( size_size_list_d(L) != size_size_list_int(Li) )
     => ( vEBT_L6072262921609318954_d_int(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_267_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li: list_c,A: fun_nat_fun_c_assn] :
      ( ( size_size_list_nat(L) != size_size_list_c(Li) )
     => ( vEBT_L6400351906427472171_nat_c(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_268_list__assn__aux__ineq__len,axiom,
    ! [L: list_nat,Li: list_d,A: fun_nat_fun_d_assn] :
      ( ( size_size_list_nat(L) != size_size_list_d(Li) )
     => ( vEBT_L6400351906427472172_nat_d(A,L,Li) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_269_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_c,X: c,X4: c] :
      ( ( I != I2 )
     => ( list_update_c(list_update_c(Xs,I,X),I2,X4) = list_update_c(list_update_c(Xs,I2,X4),I,X) ) ) ).

% list_update_swap
tff(fact_270_list__update__swap,axiom,
    ! [I: nat,I2: nat,Xs: list_d,X: d,X4: d] :
      ( ( I != I2 )
     => ( list_update_d(list_update_d(Xs,I,X),I2,X4) = list_update_d(list_update_d(Xs,I2,X4),I,X) ) ) ).

% list_update_swap
tff(fact_271_insert__Diff__single,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat))) = insert9069300056098147895at_nat(A2,A) ).

% insert_Diff_single
tff(fact_272_insert__Diff__single,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) = insert8211810215607154385at_nat(A2,A) ).

% insert_Diff_single
tff(fact_273_insert__Diff__single,axiom,
    ! [A2: bool,A: set_bool] : insert_bool2(A2,minus_minus_set_bool(A,insert_bool2(A2,bot_bot_set_bool))) = insert_bool2(A2,A) ).

% insert_Diff_single
tff(fact_274_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] : insert_int2(A2,minus_minus_set_int(A,insert_int2(A2,bot_bot_set_int))) = insert_int2(A2,A) ).

% insert_Diff_single
tff(fact_275_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] : insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(A2,bot_bot_set_nat))) = insert_nat2(A2,A) ).

% insert_Diff_single
tff(fact_276_insert__Diff1,axiom,
    ! [X: bool,B: set_bool,A: set_bool] :
      ( member_bool(X,B)
     => ( minus_minus_set_bool(insert_bool2(X,A),B) = minus_minus_set_bool(A,B) ) ) ).

% insert_Diff1
tff(fact_277_insert__Diff1,axiom,
    ! [X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(X,B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A)),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) ) ) ).

% insert_Diff1
tff(fact_278_insert__Diff1,axiom,
    ! [X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A)),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) ) ) ).

% insert_Diff1
tff(fact_279_insert__Diff1,axiom,
    ! [X: int,B: set_int,A: set_int] :
      ( member_int(X,B)
     => ( minus_minus_set_int(insert_int2(X,A),B) = minus_minus_set_int(A,B) ) ) ).

% insert_Diff1
tff(fact_280_insert__Diff1,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(X,B)
     => ( minus_5708602904570203829t_bool(insert969342481230318014t_bool(X,A),B) = minus_5708602904570203829t_bool(A,B) ) ) ).

% insert_Diff1
tff(fact_281_insert__Diff1,axiom,
    ! [X: nat,B: set_nat,A: set_nat] :
      ( member_nat(X,B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) ) ) ).

% insert_Diff1
tff(fact_282_Diff__insert0,axiom,
    ! [X: bool,A: set_bool,B: set_bool] :
      ( ~ member_bool(X,A)
     => ( minus_minus_set_bool(A,insert_bool2(X,B)) = minus_minus_set_bool(A,B) ) ) ).

% Diff_insert0
tff(fact_283_Diff__insert0,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(X,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(X,B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) ) ) ).

% Diff_insert0
tff(fact_284_Diff__insert0,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(X,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) ) ) ).

% Diff_insert0
tff(fact_285_Diff__insert0,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ member_int(X,A)
     => ( minus_minus_set_int(A,insert_int2(X,B)) = minus_minus_set_int(A,B) ) ) ).

% Diff_insert0
tff(fact_286_Diff__insert0,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(X,A)
     => ( minus_5708602904570203829t_bool(A,insert969342481230318014t_bool(X,B)) = minus_5708602904570203829t_bool(A,B) ) ) ).

% Diff_insert0
tff(fact_287_Diff__insert0,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ member_nat(X,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(X,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) ) ) ).

% Diff_insert0
tff(fact_288_Diff__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),bot_bo228742789529271731at_nat) = A ).

% Diff_empty
tff(fact_289_Diff__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),bot_bo2099793752762293965at_nat) = A ).

% Diff_empty
tff(fact_290_Diff__empty,axiom,
    ! [A: set_bool] : minus_minus_set_bool(A,bot_bot_set_bool) = A ).

% Diff_empty
tff(fact_291_Diff__empty,axiom,
    ! [A: set_int] : minus_minus_set_int(A,bot_bot_set_int) = A ).

% Diff_empty
tff(fact_292_Diff__empty,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),bot_bot_set_nat) = A ).

% Diff_empty
tff(fact_293_empty__Diff,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,bot_bo228742789529271731at_nat),A) = bot_bo228742789529271731at_nat ).

% empty_Diff
tff(fact_294_empty__Diff,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,bot_bo2099793752762293965at_nat),A) = bot_bo2099793752762293965at_nat ).

% empty_Diff
tff(fact_295_empty__Diff,axiom,
    ! [A: set_bool] : minus_minus_set_bool(bot_bot_set_bool,A) = bot_bot_set_bool ).

% empty_Diff
tff(fact_296_empty__Diff,axiom,
    ! [A: set_int] : minus_minus_set_int(bot_bot_set_int,A) = bot_bot_set_int ).

% empty_Diff
tff(fact_297_empty__Diff,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,bot_bot_set_nat),A) = bot_bot_set_nat ).

% empty_Diff
tff(fact_298_Diff__cancel,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),A) = bot_bo228742789529271731at_nat ).

% Diff_cancel
tff(fact_299_Diff__cancel,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),A) = bot_bo2099793752762293965at_nat ).

% Diff_cancel
tff(fact_300_Diff__cancel,axiom,
    ! [A: set_bool] : minus_minus_set_bool(A,A) = bot_bot_set_bool ).

% Diff_cancel
tff(fact_301_Diff__cancel,axiom,
    ! [A: set_int] : minus_minus_set_int(A,A) = bot_bot_set_int ).

% Diff_cancel
tff(fact_302_Diff__cancel,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),A) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_303_singletonI,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool] : member4125802036136091927t_bool(A2,insert969342481230318014t_bool(A2,bot_bo2714889297266208186t_bool)) ).

% singletonI
tff(fact_304_singletonI,axiom,
    ! [A2: produc3843707927480180839at_nat] : member8757157785044589968at_nat(A2,insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat)) ).

% singletonI
tff(fact_305_singletonI,axiom,
    ! [A2: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) ).

% singletonI
tff(fact_306_singletonI,axiom,
    ! [A2: bool] : member_bool(A2,insert_bool2(A2,bot_bot_set_bool)) ).

% singletonI
tff(fact_307_singletonI,axiom,
    ! [A2: nat] : member_nat(A2,insert_nat2(A2,bot_bot_set_nat)) ).

% singletonI
tff(fact_308_singletonI,axiom,
    ! [A2: int] : member_int(A2,insert_int2(A2,bot_bot_set_int)) ).

% singletonI
tff(fact_309_Diff__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(A2,B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)),insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat)) ).

% Diff_insert
tff(fact_310_Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(A2,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)) ).

% Diff_insert
tff(fact_311_Diff__insert,axiom,
    ! [A: set_bool,A2: bool,B: set_bool] : minus_minus_set_bool(A,insert_bool2(A2,B)) = minus_minus_set_bool(minus_minus_set_bool(A,B),insert_bool2(A2,bot_bot_set_bool)) ).

% Diff_insert
tff(fact_312_Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] : minus_minus_set_int(A,insert_int2(A2,B)) = minus_minus_set_int(minus_minus_set_int(A,B),insert_int2(A2,bot_bot_set_int)) ).

% Diff_insert
tff(fact_313_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(A2,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),insert_nat2(A2,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_314_insert__Diff,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,A)
     => ( insert969342481230318014t_bool(A2,minus_5708602904570203829t_bool(A,insert969342481230318014t_bool(A2,bot_bo2714889297266208186t_bool))) = A ) ) ).

% insert_Diff
tff(fact_315_insert__Diff,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,A)
     => ( insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat))) = A ) ) ).

% insert_Diff
tff(fact_316_insert__Diff,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
     => ( insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))) = A ) ) ).

% insert_Diff
tff(fact_317_insert__Diff,axiom,
    ! [A2: bool,A: set_bool] :
      ( member_bool(A2,A)
     => ( insert_bool2(A2,minus_minus_set_bool(A,insert_bool2(A2,bot_bot_set_bool))) = A ) ) ).

% insert_Diff
tff(fact_318_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( member_int(A2,A)
     => ( insert_int2(A2,minus_minus_set_int(A,insert_int2(A2,bot_bot_set_int))) = A ) ) ).

% insert_Diff
tff(fact_319_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( member_nat(A2,A)
     => ( insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(A2,bot_bot_set_nat))) = A ) ) ).

% insert_Diff
tff(fact_320_Diff__insert2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(A2,B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat))),B) ).

% Diff_insert2
tff(fact_321_Diff__insert2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(A2,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat))),B) ).

% Diff_insert2
tff(fact_322_Diff__insert2,axiom,
    ! [A: set_bool,A2: bool,B: set_bool] : minus_minus_set_bool(A,insert_bool2(A2,B)) = minus_minus_set_bool(minus_minus_set_bool(A,insert_bool2(A2,bot_bot_set_bool)),B) ).

% Diff_insert2
tff(fact_323_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B: set_int] : minus_minus_set_int(A,insert_int2(A2,B)) = minus_minus_set_int(minus_minus_set_int(A,insert_int2(A2,bot_bot_set_int)),B) ).

% Diff_insert2
tff(fact_324_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(A2,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(A2,bot_bot_set_nat))),B) ).

% Diff_insert2
tff(fact_325_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A)),insert9069300056098147895at_nat(Y,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(Y,bot_bo228742789529271731at_nat))) ) ) ).

% insert_minus_eq
tff(fact_326_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A)),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(Y,bot_bo2099793752762293965at_nat))) ) ) ).

% insert_minus_eq
tff(fact_327_insert__minus__eq,axiom,
    ! [X: bool,Y: bool,A: set_bool] :
      ( ~ ( pp(X)
        <=> pp(Y) )
     => ( minus_minus_set_bool(insert_bool2(X,A),insert_bool2(Y,bot_bot_set_bool)) = insert_bool2(X,minus_minus_set_bool(A,insert_bool2(Y,bot_bot_set_bool))) ) ) ).

% insert_minus_eq
tff(fact_328_insert__minus__eq,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( X != Y )
     => ( minus_minus_set_int(insert_int2(X,A),insert_int2(Y,bot_bot_set_int)) = insert_int2(X,minus_minus_set_int(A,insert_int2(Y,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_329_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( X != Y )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A)),insert_nat2(Y,bot_bot_set_nat)) = insert_nat2(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(Y,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_330_Diff__insert__absorb,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(X,A)
     => ( minus_5708602904570203829t_bool(insert969342481230318014t_bool(X,A),insert969342481230318014t_bool(X,bot_bo2714889297266208186t_bool)) = A ) ) ).

% Diff_insert_absorb
tff(fact_331_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(X,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = A ) ) ).

% Diff_insert_absorb
tff(fact_332_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = A ) ) ).

% Diff_insert_absorb
tff(fact_333_Diff__insert__absorb,axiom,
    ! [X: bool,A: set_bool] :
      ( ~ member_bool(X,A)
     => ( minus_minus_set_bool(insert_bool2(X,A),insert_bool2(X,bot_bot_set_bool)) = A ) ) ).

% Diff_insert_absorb
tff(fact_334_Diff__insert__absorb,axiom,
    ! [X: int,A: set_int] :
      ( ~ member_int(X,A)
     => ( minus_minus_set_int(insert_int2(X,A),insert_int2(X,bot_bot_set_int)) = A ) ) ).

% Diff_insert_absorb
tff(fact_335_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ member_nat(X,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A)),insert_nat2(X,bot_bot_set_nat)) = A ) ) ).

% Diff_insert_absorb
tff(fact_336_empty__Collect__eq,axiom,
    ! [P: fun_fu9160419385450930796l_bool] :
      ( ( bot_bo2714889297266208186t_bool = collec3007137136614167897t_bool(P) )
    <=> ! [X5: fun_Pr6001797050664311790t_bool] : ~ pp(aa_fun5588795374382764133l_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_337_empty__Collect__eq,axiom,
    ! [P: fun_Pr1607443439640623155t_bool] :
      ( ( bot_bo228742789529271731at_nat = collec6321179662152712658at_nat(P) )
    <=> ! [X5: produc3843707927480180839at_nat] : ~ pp(aa_Pro1809015098595223084t_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_338_empty__Collect__eq,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ( bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(P) )
    <=> ! [X5: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_339_empty__Collect__eq,axiom,
    ! [P: fun_bool_bool] :
      ( ( bot_bot_set_bool = collect_bool(P) )
    <=> ! [X5: bool] : ~ pp(aa_bool_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_340_empty__Collect__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X5: nat] : ~ pp(aa_nat_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_341_empty__Collect__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X5: int] : ~ pp(aa_int_bool(P,X5)) ) ).

% empty_Collect_eq
tff(fact_342_Collect__empty__eq,axiom,
    ! [P: fun_fu9160419385450930796l_bool] :
      ( ( collec3007137136614167897t_bool(P) = bot_bo2714889297266208186t_bool )
    <=> ! [X5: fun_Pr6001797050664311790t_bool] : ~ pp(aa_fun5588795374382764133l_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_343_Collect__empty__eq,axiom,
    ! [P: fun_Pr1607443439640623155t_bool] :
      ( ( collec6321179662152712658at_nat(P) = bot_bo228742789529271731at_nat )
    <=> ! [X5: produc3843707927480180839at_nat] : ~ pp(aa_Pro1809015098595223084t_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_344_Collect__empty__eq,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ! [X5: product_prod_nat_nat] : ~ pp(aa_Pro2232559814647311250t_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_345_Collect__empty__eq,axiom,
    ! [P: fun_bool_bool] :
      ( ( collect_bool(P) = bot_bot_set_bool )
    <=> ! [X5: bool] : ~ pp(aa_bool_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_346_Collect__empty__eq,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X5: nat] : ~ pp(aa_nat_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_347_Collect__empty__eq,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X5: int] : ~ pp(aa_int_bool(P,X5)) ) ).

% Collect_empty_eq
tff(fact_348_all__not__in__conv,axiom,
    ! [A: set_fu6080891106736489294t_bool] :
      ( ! [X5: fun_Pr6001797050664311790t_bool] : ~ member4125802036136091927t_bool(X5,A)
    <=> ( A = bot_bo2714889297266208186t_bool ) ) ).

% all_not_in_conv
tff(fact_349_all__not__in__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ! [X5: produc3843707927480180839at_nat] : ~ member8757157785044589968at_nat(X5,A)
    <=> ( A = bot_bo228742789529271731at_nat ) ) ).

% all_not_in_conv
tff(fact_350_all__not__in__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A))
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_351_all__not__in__conv,axiom,
    ! [A: set_bool] :
      ( ! [X5: bool] : ~ member_bool(X5,A)
    <=> ( A = bot_bot_set_bool ) ) ).

% all_not_in_conv
tff(fact_352_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ! [X5: nat] : ~ member_nat(X5,A)
    <=> ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_353_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ! [X5: int] : ~ member_int(X5,A)
    <=> ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_354_empty__iff,axiom,
    ! [C: fun_Pr6001797050664311790t_bool] : ~ member4125802036136091927t_bool(C,bot_bo2714889297266208186t_bool) ).

% empty_iff
tff(fact_355_empty__iff,axiom,
    ! [C: produc3843707927480180839at_nat] : ~ member8757157785044589968at_nat(C,bot_bo228742789529271731at_nat) ).

% empty_iff
tff(fact_356_empty__iff,axiom,
    ! [C: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),bot_bo2099793752762293965at_nat)) ).

% empty_iff
tff(fact_357_empty__iff,axiom,
    ! [C: bool] : ~ member_bool(C,bot_bot_set_bool) ).

% empty_iff
tff(fact_358_empty__iff,axiom,
    ! [C: nat] : ~ member_nat(C,bot_bot_set_nat) ).

% empty_iff
tff(fact_359_empty__iff,axiom,
    ! [C: int] : ~ member_int(C,bot_bot_set_int) ).

% empty_iff
tff(fact_360_insertCI,axiom,
    ! [A2: bool,B: set_bool,B2: bool] :
      ( ( ~ member_bool(A2,B)
       => ( pp(A2)
        <=> pp(B2) ) )
     => member_bool(A2,insert_bool2(B2,B)) ) ).

% insertCI
tff(fact_361_insertCI,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ~ member8757157785044589968at_nat(A2,B)
       => ( A2 = B2 ) )
     => member8757157785044589968at_nat(A2,insert9069300056098147895at_nat(B2,B)) ) ).

% insertCI
tff(fact_362_insertCI,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B))
       => ( A2 = B2 ) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B2,B))) ) ).

% insertCI
tff(fact_363_insertCI,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( ~ member_nat(A2,B)
       => ( A2 = B2 ) )
     => member_nat(A2,insert_nat2(B2,B)) ) ).

% insertCI
tff(fact_364_insertCI,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( ~ member_int(A2,B)
       => ( A2 = B2 ) )
     => member_int(A2,insert_int2(B2,B)) ) ).

% insertCI
tff(fact_365_insertCI,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool,B2: fun_Pr6001797050664311790t_bool] :
      ( ( ~ member4125802036136091927t_bool(A2,B)
       => ( A2 = B2 ) )
     => member4125802036136091927t_bool(A2,insert969342481230318014t_bool(B2,B)) ) ).

% insertCI
tff(fact_366_insert__iff,axiom,
    ! [A2: bool,B2: bool,A: set_bool] :
      ( member_bool(A2,insert_bool2(B2,A))
    <=> ( ( pp(A2)
        <=> pp(B2) )
        | member_bool(A2,A) ) ) ).

% insert_iff
tff(fact_367_insert__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,insert9069300056098147895at_nat(B2,A))
    <=> ( ( A2 = B2 )
        | member8757157785044589968at_nat(A2,A) ) ) ).

% insert_iff
tff(fact_368_insert__iff,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B2,A)))
    <=> ( ( A2 = B2 )
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A)) ) ) ).

% insert_iff
tff(fact_369_insert__iff,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( member_nat(A2,insert_nat2(B2,A))
    <=> ( ( A2 = B2 )
        | member_nat(A2,A) ) ) ).

% insert_iff
tff(fact_370_insert__iff,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( member_int(A2,insert_int2(B2,A))
    <=> ( ( A2 = B2 )
        | member_int(A2,A) ) ) ).

% insert_iff
tff(fact_371_insert__iff,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,insert969342481230318014t_bool(B2,A))
    <=> ( ( A2 = B2 )
        | member4125802036136091927t_bool(A2,A) ) ) ).

% insert_iff
tff(fact_372_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] : insert_nat2(X,insert_nat2(X,A)) = insert_nat2(X,A) ).

% insert_absorb2
tff(fact_373_insert__absorb2,axiom,
    ! [X: int,A: set_int] : insert_int2(X,insert_int2(X,A)) = insert_int2(X,A) ).

% insert_absorb2
tff(fact_374_insert__absorb2,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(X,A)) = insert8211810215607154385at_nat(X,A) ).

% insert_absorb2
tff(fact_375_insert__absorb2,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(X,A)) = insert9069300056098147895at_nat(X,A) ).

% insert_absorb2
tff(fact_376_insert__absorb2,axiom,
    ! [X: bool,A: set_bool] : insert_bool2(X,insert_bool2(X,A)) = insert_bool2(X,A) ).

% insert_absorb2
tff(fact_377_DiffI,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,A)
     => ( ~ member8757157785044589968at_nat(C,B)
       => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)) ) ) ).

% DiffI
tff(fact_378_DiffI,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B))) ) ) ).

% DiffI
tff(fact_379_DiffI,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,A)
     => ( ~ member_int(C,B)
       => member_int(C,minus_minus_set_int(A,B)) ) ) ).

% DiffI
tff(fact_380_DiffI,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,A)
     => ( ~ member4125802036136091927t_bool(C,B)
       => member4125802036136091927t_bool(C,minus_5708602904570203829t_bool(A,B)) ) ) ).

% DiffI
tff(fact_381_DiffI,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,A)
     => ( ~ member_nat(C,B)
       => member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)) ) ) ).

% DiffI
tff(fact_382_Diff__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B))
    <=> ( member8757157785044589968at_nat(C,A)
        & ~ member8757157785044589968at_nat(C,B) ) ) ).

% Diff_iff
tff(fact_383_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
        & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% Diff_iff
tff(fact_384_Diff__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,minus_minus_set_int(A,B))
    <=> ( member_int(C,A)
        & ~ member_int(C,B) ) ) ).

% Diff_iff
tff(fact_385_Diff__iff,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,minus_5708602904570203829t_bool(A,B))
    <=> ( member4125802036136091927t_bool(C,A)
        & ~ member4125802036136091927t_bool(C,B) ) ) ).

% Diff_iff
tff(fact_386_Diff__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B))
    <=> ( member_nat(C,A)
        & ~ member_nat(C,B) ) ) ).

% Diff_iff
tff(fact_387_Diff__idemp,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) ).

% Diff_idemp
tff(fact_388_not__psubset__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ~ ord_le2604355607129572851at_nat(A,bot_bo228742789529271731at_nat) ).

% not_psubset_empty
tff(fact_389_not__psubset__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A),bot_bo2099793752762293965at_nat)) ).

% not_psubset_empty
tff(fact_390_not__psubset__empty,axiom,
    ! [A: set_bool] : ~ ord_less_set_bool(A,bot_bot_set_bool) ).

% not_psubset_empty
tff(fact_391_not__psubset__empty,axiom,
    ! [A: set_nat] : ~ ord_less_set_nat(A,bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_392_not__psubset__empty,axiom,
    ! [A: set_int] : ~ ord_less_set_int(A,bot_bot_set_int) ).

% not_psubset_empty
tff(fact_393_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(A,B)
     => ? [B3: produc3843707927480180839at_nat] : member8757157785044589968at_nat(B3,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),A)) ) ).

% psubset_imp_ex_mem
tff(fact_394_psubset__imp__ex__mem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A),B))
     => ? [B3: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),A))) ) ).

% psubset_imp_ex_mem
tff(fact_395_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
     => ? [B3: int] : member_int(B3,minus_minus_set_int(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_396_psubset__imp__ex__mem,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le3815294503984575994t_bool(A,B)
     => ? [B3: fun_Pr6001797050664311790t_bool] : member4125802036136091927t_bool(B3,minus_5708602904570203829t_bool(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_397_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_set_nat(A,B)
     => ? [B3: nat] : member_nat(B3,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A)) ) ).

% psubset_imp_ex_mem
tff(fact_398_remove__subset,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,S: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(X,S)
     => ord_le3815294503984575994t_bool(minus_5708602904570203829t_bool(S,insert969342481230318014t_bool(X,bot_bo2714889297266208186t_bool)),S) ) ).

% remove_subset
tff(fact_399_remove__subset,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(X,S)
     => ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)),S) ) ).

% remove_subset
tff(fact_400_remove__subset,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),S))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),S)) ) ).

% remove_subset
tff(fact_401_remove__subset,axiom,
    ! [X: bool,S: set_bool] :
      ( member_bool(X,S)
     => ord_less_set_bool(minus_minus_set_bool(S,insert_bool2(X,bot_bot_set_bool)),S) ) ).

% remove_subset
tff(fact_402_remove__subset,axiom,
    ! [X: int,S: set_int] :
      ( member_int(X,S)
     => ord_less_set_int(minus_minus_set_int(S,insert_int2(X,bot_bot_set_int)),S) ) ).

% remove_subset
tff(fact_403_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( member_nat(X,S)
     => ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat2(X,bot_bot_set_nat)),S) ) ).

% remove_subset
tff(fact_404_memb__imp__not__empty,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,S: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(X,S)
     => ( S != bot_bo2714889297266208186t_bool ) ) ).

% memb_imp_not_empty
tff(fact_405_memb__imp__not__empty,axiom,
    ! [X: produc3843707927480180839at_nat,S: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(X,S)
     => ( S != bot_bo228742789529271731at_nat ) ) ).

% memb_imp_not_empty
tff(fact_406_memb__imp__not__empty,axiom,
    ! [X: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),S))
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% memb_imp_not_empty
tff(fact_407_memb__imp__not__empty,axiom,
    ! [X: bool,S: set_bool] :
      ( member_bool(X,S)
     => ( S != bot_bot_set_bool ) ) ).

% memb_imp_not_empty
tff(fact_408_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( member_nat(X,S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_409_memb__imp__not__empty,axiom,
    ! [X: int,S: set_int] :
      ( member_int(X,S)
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_410_set__notEmptyE,axiom,
    ! [S: set_fu6080891106736489294t_bool] :
      ( ( S != bot_bo2714889297266208186t_bool )
     => ~ ! [X2: fun_Pr6001797050664311790t_bool] : ~ member4125802036136091927t_bool(X2,S) ) ).

% set_notEmptyE
tff(fact_411_set__notEmptyE,axiom,
    ! [S: set_Pr4329608150637261639at_nat] :
      ( ( S != bot_bo228742789529271731at_nat )
     => ~ ! [X2: produc3843707927480180839at_nat] : ~ member8757157785044589968at_nat(X2,S) ) ).

% set_notEmptyE
tff(fact_412_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X2: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),S)) ) ).

% set_notEmptyE
tff(fact_413_set__notEmptyE,axiom,
    ! [S: set_bool] :
      ( ( S != bot_bot_set_bool )
     => ~ ! [X2: bool] : ~ member_bool(X2,S) ) ).

% set_notEmptyE
tff(fact_414_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X2: nat] : ~ member_nat(X2,S) ) ).

% set_notEmptyE
tff(fact_415_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X2: int] : ~ member_int(X2,S) ) ).

% set_notEmptyE
tff(fact_416_bot__set__def,axiom,
    bot_bo2714889297266208186t_bool = collec3007137136614167897t_bool(bot_bo8489179610771909376l_bool) ).

% bot_set_def
tff(fact_417_bot__set__def,axiom,
    bot_bo228742789529271731at_nat = collec6321179662152712658at_nat(bot_bo1984517246807695559t_bool) ).

% bot_set_def
tff(fact_418_bot__set__def,axiom,
    bot_bo2099793752762293965at_nat = collec3392354462482085612at_nat(bot_bo3782518310583830573t_bool) ).

% bot_set_def
tff(fact_419_bot__set__def,axiom,
    bot_bot_set_bool = collect_bool(bot_bo4756488196683355415l_bool) ).

% bot_set_def
tff(fact_420_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_bool) ).

% bot_set_def
tff(fact_421_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_bool) ).

% bot_set_def
tff(fact_422_ex__in__conv,axiom,
    ! [A: set_fu6080891106736489294t_bool] :
      ( ? [X5: fun_Pr6001797050664311790t_bool] : member4125802036136091927t_bool(X5,A)
    <=> ( A != bot_bo2714889297266208186t_bool ) ) ).

% ex_in_conv
tff(fact_423_ex__in__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ? [X5: produc3843707927480180839at_nat] : member8757157785044589968at_nat(X5,A)
    <=> ( A != bot_bo228742789529271731at_nat ) ) ).

% ex_in_conv
tff(fact_424_ex__in__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ? [X5: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A))
    <=> ( A != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_425_ex__in__conv,axiom,
    ! [A: set_bool] :
      ( ? [X5: bool] : member_bool(X5,A)
    <=> ( A != bot_bot_set_bool ) ) ).

% ex_in_conv
tff(fact_426_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ? [X5: nat] : member_nat(X5,A)
    <=> ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_427_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ? [X5: int] : member_int(X5,A)
    <=> ( A != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_428_equals0I,axiom,
    ! [A: set_fu6080891106736489294t_bool] :
      ( ! [Y2: fun_Pr6001797050664311790t_bool] : ~ member4125802036136091927t_bool(Y2,A)
     => ( A = bot_bo2714889297266208186t_bool ) ) ).

% equals0I
tff(fact_429_equals0I,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ! [Y2: produc3843707927480180839at_nat] : ~ member8757157785044589968at_nat(Y2,A)
     => ( A = bot_bo228742789529271731at_nat ) ) ).

% equals0I
tff(fact_430_equals0I,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [Y2: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),A))
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_431_equals0I,axiom,
    ! [A: set_bool] :
      ( ! [Y2: bool] : ~ member_bool(Y2,A)
     => ( A = bot_bot_set_bool ) ) ).

% equals0I
tff(fact_432_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y2: nat] : ~ member_nat(Y2,A)
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_433_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y2: int] : ~ member_int(Y2,A)
     => ( A = bot_bot_set_int ) ) ).

% equals0I
tff(fact_434_equals0D,axiom,
    ! [A: set_fu6080891106736489294t_bool,A2: fun_Pr6001797050664311790t_bool] :
      ( ( A = bot_bo2714889297266208186t_bool )
     => ~ member4125802036136091927t_bool(A2,A) ) ).

% equals0D
tff(fact_435_equals0D,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ( A = bot_bo228742789529271731at_nat )
     => ~ member8757157785044589968at_nat(A2,A) ) ).

% equals0D
tff(fact_436_equals0D,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( ( A = bot_bo2099793752762293965at_nat )
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A)) ) ).

% equals0D
tff(fact_437_equals0D,axiom,
    ! [A: set_bool,A2: bool] :
      ( ( A = bot_bot_set_bool )
     => ~ member_bool(A2,A) ) ).

% equals0D
tff(fact_438_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ member_nat(A2,A) ) ).

% equals0D
tff(fact_439_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ member_int(A2,A) ) ).

% equals0D
tff(fact_440_emptyE,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool] : ~ member4125802036136091927t_bool(A2,bot_bo2714889297266208186t_bool) ).

% emptyE
tff(fact_441_emptyE,axiom,
    ! [A2: produc3843707927480180839at_nat] : ~ member8757157785044589968at_nat(A2,bot_bo228742789529271731at_nat) ).

% emptyE
tff(fact_442_emptyE,axiom,
    ! [A2: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),bot_bo2099793752762293965at_nat)) ).

% emptyE
tff(fact_443_emptyE,axiom,
    ! [A2: bool] : ~ member_bool(A2,bot_bot_set_bool) ).

% emptyE
tff(fact_444_emptyE,axiom,
    ! [A2: nat] : ~ member_nat(A2,bot_bot_set_nat) ).

% emptyE
tff(fact_445_emptyE,axiom,
    ! [A2: int] : ~ member_int(A2,bot_bot_set_int) ).

% emptyE
tff(fact_446_insertE,axiom,
    ! [A2: bool,B2: bool,A: set_bool] :
      ( member_bool(A2,insert_bool2(B2,A))
     => ( ( pp(A2)
        <=> ~ pp(B2) )
       => member_bool(A2,A) ) ) ).

% insertE
tff(fact_447_insertE,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,insert9069300056098147895at_nat(B2,A))
     => ( ( A2 != B2 )
       => member8757157785044589968at_nat(A2,A) ) ) ).

% insertE
tff(fact_448_insertE,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B2,A)))
     => ( ( A2 != B2 )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A)) ) ) ).

% insertE
tff(fact_449_insertE,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( member_nat(A2,insert_nat2(B2,A))
     => ( ( A2 != B2 )
       => member_nat(A2,A) ) ) ).

% insertE
tff(fact_450_insertE,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( member_int(A2,insert_int2(B2,A))
     => ( ( A2 != B2 )
       => member_int(A2,A) ) ) ).

% insertE
tff(fact_451_insertE,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,insert969342481230318014t_bool(B2,A))
     => ( ( A2 != B2 )
       => member4125802036136091927t_bool(A2,A) ) ) ).

% insertE
tff(fact_452_insertI1,axiom,
    ! [A2: bool,B: set_bool] : member_bool(A2,insert_bool2(A2,B)) ).

% insertI1
tff(fact_453_insertI1,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] : member8757157785044589968at_nat(A2,insert9069300056098147895at_nat(A2,B)) ).

% insertI1
tff(fact_454_insertI1,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(A2,B))) ).

% insertI1
tff(fact_455_insertI1,axiom,
    ! [A2: nat,B: set_nat] : member_nat(A2,insert_nat2(A2,B)) ).

% insertI1
tff(fact_456_insertI1,axiom,
    ! [A2: int,B: set_int] : member_int(A2,insert_int2(A2,B)) ).

% insertI1
tff(fact_457_insertI1,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool] : member4125802036136091927t_bool(A2,insert969342481230318014t_bool(A2,B)) ).

% insertI1
tff(fact_458_insertI2,axiom,
    ! [A2: bool,B: set_bool,B2: bool] :
      ( member_bool(A2,B)
     => member_bool(A2,insert_bool2(B2,B)) ) ).

% insertI2
tff(fact_459_insertI2,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( member8757157785044589968at_nat(A2,B)
     => member8757157785044589968at_nat(A2,insert9069300056098147895at_nat(B2,B)) ) ).

% insertI2
tff(fact_460_insertI2,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),insert8211810215607154385at_nat(B2,B))) ) ).

% insertI2
tff(fact_461_insertI2,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( member_nat(A2,B)
     => member_nat(A2,insert_nat2(B2,B)) ) ).

% insertI2
tff(fact_462_insertI2,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( member_int(A2,B)
     => member_int(A2,insert_int2(B2,B)) ) ).

% insertI2
tff(fact_463_insertI2,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool,B2: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(A2,B)
     => member4125802036136091927t_bool(A2,insert969342481230318014t_bool(B2,B)) ) ).

% insertI2
tff(fact_464_Set_Oset__insert,axiom,
    ! [X: bool,A: set_bool] :
      ( member_bool(X,A)
     => ~ ! [B4: set_bool] :
            ( ( A = insert_bool2(X,B4) )
           => member_bool(X,B4) ) ) ).

% Set.set_insert
tff(fact_465_Set_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(X,A)
     => ~ ! [B4: set_Pr4329608150637261639at_nat] :
            ( ( A = insert9069300056098147895at_nat(X,B4) )
           => member8757157785044589968at_nat(X,B4) ) ) ).

% Set.set_insert
tff(fact_466_Set_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
     => ~ ! [B4: set_Pr1261947904930325089at_nat] :
            ( ( A = insert8211810215607154385at_nat(X,B4) )
           => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B4)) ) ) ).

% Set.set_insert
tff(fact_467_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( member_nat(X,A)
     => ~ ! [B4: set_nat] :
            ( ( A = insert_nat2(X,B4) )
           => member_nat(X,B4) ) ) ).

% Set.set_insert
tff(fact_468_Set_Oset__insert,axiom,
    ! [X: int,A: set_int] :
      ( member_int(X,A)
     => ~ ! [B4: set_int] :
            ( ( A = insert_int2(X,B4) )
           => member_int(X,B4) ) ) ).

% Set.set_insert
tff(fact_469_Set_Oset__insert,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(X,A)
     => ~ ! [B4: set_fu6080891106736489294t_bool] :
            ( ( A = insert969342481230318014t_bool(X,B4) )
           => member4125802036136091927t_bool(X,B4) ) ) ).

% Set.set_insert
tff(fact_470_insert__ident,axiom,
    ! [X: bool,A: set_bool,B: set_bool] :
      ( ~ member_bool(X,A)
     => ( ~ member_bool(X,B)
       => ( ( insert_bool2(X,A) = insert_bool2(X,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_471_insert__ident,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(X,A)
     => ( ~ member8757157785044589968at_nat(X,B)
       => ( ( insert9069300056098147895at_nat(X,A) = insert9069300056098147895at_nat(X,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_472_insert__ident,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
       => ( ( insert8211810215607154385at_nat(X,A) = insert8211810215607154385at_nat(X,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_473_insert__ident,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ member_nat(X,A)
     => ( ~ member_nat(X,B)
       => ( ( insert_nat2(X,A) = insert_nat2(X,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_474_insert__ident,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ member_int(X,A)
     => ( ~ member_int(X,B)
       => ( ( insert_int2(X,A) = insert_int2(X,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_475_insert__ident,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(X,A)
     => ( ~ member4125802036136091927t_bool(X,B)
       => ( ( insert969342481230318014t_bool(X,A) = insert969342481230318014t_bool(X,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_476_insert__absorb,axiom,
    ! [A2: bool,A: set_bool] :
      ( member_bool(A2,A)
     => ( insert_bool2(A2,A) = A ) ) ).

% insert_absorb
tff(fact_477_insert__absorb,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,A)
     => ( insert9069300056098147895at_nat(A2,A) = A ) ) ).

% insert_absorb
tff(fact_478_insert__absorb,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
     => ( insert8211810215607154385at_nat(A2,A) = A ) ) ).

% insert_absorb
tff(fact_479_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( member_nat(A2,A)
     => ( insert_nat2(A2,A) = A ) ) ).

% insert_absorb
tff(fact_480_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( member_int(A2,A)
     => ( insert_int2(A2,A) = A ) ) ).

% insert_absorb
tff(fact_481_insert__absorb,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,A)
     => ( insert969342481230318014t_bool(A2,A) = A ) ) ).

% insert_absorb
tff(fact_482_insert__eq__iff,axiom,
    ! [A2: bool,A: set_bool,B2: bool,B: set_bool] :
      ( ~ member_bool(A2,A)
     => ( ~ member_bool(B2,B)
       => ( ( insert_bool2(A2,A) = insert_bool2(B2,B) )
        <=> ( ( ( pp(A2)
              <=> pp(B2) )
             => ( A = B ) )
            & ( ( pp(A2)
              <=> ~ pp(B2) )
             => ? [C3: set_bool] :
                  ( ( A = insert_bool2(B2,C3) )
                  & ~ member_bool(B2,C3)
                  & ( B = insert_bool2(A2,C3) )
                  & ~ member_bool(A2,C3) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_483_insert__eq__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(A2,A)
     => ( ~ member8757157785044589968at_nat(B2,B)
       => ( ( insert9069300056098147895at_nat(A2,A) = insert9069300056098147895at_nat(B2,B) )
        <=> ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_Pr4329608150637261639at_nat] :
                  ( ( A = insert9069300056098147895at_nat(B2,C3) )
                  & ~ member8757157785044589968at_nat(B2,C3)
                  & ( B = insert9069300056098147895at_nat(A2,C3) )
                  & ~ member8757157785044589968at_nat(A2,C3) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_484_insert__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B2),B))
       => ( ( insert8211810215607154385at_nat(A2,A) = insert8211810215607154385at_nat(B2,B) )
        <=> ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_Pr1261947904930325089at_nat] :
                  ( ( A = insert8211810215607154385at_nat(B2,C3) )
                  & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B2),C3))
                  & ( B = insert8211810215607154385at_nat(A2,C3) )
                  & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C3)) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_485_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
      ( ~ member_nat(A2,A)
     => ( ~ member_nat(B2,B)
       => ( ( insert_nat2(A2,A) = insert_nat2(B2,B) )
        <=> ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_nat] :
                  ( ( A = insert_nat2(B2,C3) )
                  & ~ member_nat(B2,C3)
                  & ( B = insert_nat2(A2,C3) )
                  & ~ member_nat(A2,C3) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_486_insert__eq__iff,axiom,
    ! [A2: int,A: set_int,B2: int,B: set_int] :
      ( ~ member_int(A2,A)
     => ( ~ member_int(B2,B)
       => ( ( insert_int2(A2,A) = insert_int2(B2,B) )
        <=> ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_int] :
                  ( ( A = insert_int2(B2,C3) )
                  & ~ member_int(B2,C3)
                  & ( B = insert_int2(A2,C3) )
                  & ~ member_int(A2,C3) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_487_insert__eq__iff,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B2: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(A2,A)
     => ( ~ member4125802036136091927t_bool(B2,B)
       => ( ( insert969342481230318014t_bool(A2,A) = insert969342481230318014t_bool(B2,B) )
        <=> ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_fu6080891106736489294t_bool] :
                  ( ( A = insert969342481230318014t_bool(B2,C3) )
                  & ~ member4125802036136091927t_bool(B2,C3)
                  & ( B = insert969342481230318014t_bool(A2,C3) )
                  & ~ member4125802036136091927t_bool(A2,C3) ) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_488_insert__commute,axiom,
    ! [X: nat,Y: nat,A: set_nat] : insert_nat2(X,insert_nat2(Y,A)) = insert_nat2(Y,insert_nat2(X,A)) ).

% insert_commute
tff(fact_489_insert__commute,axiom,
    ! [X: int,Y: int,A: set_int] : insert_int2(X,insert_int2(Y,A)) = insert_int2(Y,insert_int2(X,A)) ).

% insert_commute
tff(fact_490_insert__commute,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(X,insert8211810215607154385at_nat(Y,A)) = insert8211810215607154385at_nat(Y,insert8211810215607154385at_nat(X,A)) ).

% insert_commute
tff(fact_491_insert__commute,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(X,insert9069300056098147895at_nat(Y,A)) = insert9069300056098147895at_nat(Y,insert9069300056098147895at_nat(X,A)) ).

% insert_commute
tff(fact_492_insert__commute,axiom,
    ! [X: bool,Y: bool,A: set_bool] : insert_bool2(X,insert_bool2(Y,A)) = insert_bool2(Y,insert_bool2(X,A)) ).

% insert_commute
tff(fact_493_mk__disjoint__insert,axiom,
    ! [A2: bool,A: set_bool] :
      ( member_bool(A2,A)
     => ? [B4: set_bool] :
          ( ( A = insert_bool2(A2,B4) )
          & ~ member_bool(A2,B4) ) ) ).

% mk_disjoint_insert
tff(fact_494_mk__disjoint__insert,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,A)
     => ? [B4: set_Pr4329608150637261639at_nat] :
          ( ( A = insert9069300056098147895at_nat(A2,B4) )
          & ~ member8757157785044589968at_nat(A2,B4) ) ) ).

% mk_disjoint_insert
tff(fact_495_mk__disjoint__insert,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
     => ? [B4: set_Pr1261947904930325089at_nat] :
          ( ( A = insert8211810215607154385at_nat(A2,B4) )
          & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B4)) ) ) ).

% mk_disjoint_insert
tff(fact_496_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( member_nat(A2,A)
     => ? [B4: set_nat] :
          ( ( A = insert_nat2(A2,B4) )
          & ~ member_nat(A2,B4) ) ) ).

% mk_disjoint_insert
tff(fact_497_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( member_int(A2,A)
     => ? [B4: set_int] :
          ( ( A = insert_int2(A2,B4) )
          & ~ member_int(A2,B4) ) ) ).

% mk_disjoint_insert
tff(fact_498_mk__disjoint__insert,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,A)
     => ? [B4: set_fu6080891106736489294t_bool] :
          ( ( A = insert969342481230318014t_bool(A2,B4) )
          & ~ member4125802036136091927t_bool(A2,B4) ) ) ).

% mk_disjoint_insert
tff(fact_499_DiffE,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B))
     => ~ ( member8757157785044589968at_nat(C,A)
         => member8757157785044589968at_nat(C,B) ) ) ).

% DiffE
tff(fact_500_DiffE,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)))
     => ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% DiffE
tff(fact_501_DiffE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,minus_minus_set_int(A,B))
     => ~ ( member_int(C,A)
         => member_int(C,B) ) ) ).

% DiffE
tff(fact_502_DiffE,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,minus_5708602904570203829t_bool(A,B))
     => ~ ( member4125802036136091927t_bool(C,A)
         => member4125802036136091927t_bool(C,B) ) ) ).

% DiffE
tff(fact_503_DiffE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B))
     => ~ ( member_nat(C,A)
         => member_nat(C,B) ) ) ).

% DiffE
tff(fact_504_DiffD1,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B))
     => member8757157785044589968at_nat(C,A) ) ).

% DiffD1
tff(fact_505_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A)) ) ).

% DiffD1
tff(fact_506_DiffD1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,minus_minus_set_int(A,B))
     => member_int(C,A) ) ).

% DiffD1
tff(fact_507_DiffD1,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,minus_5708602904570203829t_bool(A,B))
     => member4125802036136091927t_bool(C,A) ) ).

% DiffD1
tff(fact_508_DiffD1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B))
     => member_nat(C,A) ) ).

% DiffD1
tff(fact_509_DiffD2,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B))
     => ~ member8757157785044589968at_nat(C,B) ) ).

% DiffD2
tff(fact_510_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)))
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ).

% DiffD2
tff(fact_511_DiffD2,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,minus_minus_set_int(A,B))
     => ~ member_int(C,B) ) ).

% DiffD2
tff(fact_512_DiffD2,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,minus_5708602904570203829t_bool(A,B))
     => ~ member4125802036136091927t_bool(C,B) ) ).

% DiffD2
tff(fact_513_DiffD2,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B))
     => ~ member_nat(C,B) ) ).

% DiffD2
tff(fact_514_singleton__inject,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_515_singleton__inject,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_516_singleton__inject,axiom,
    ! [A2: bool,B2: bool] :
      ( ( insert_bool2(A2,bot_bot_set_bool) = insert_bool2(B2,bot_bot_set_bool) )
     => ( pp(A2)
      <=> pp(B2) ) ) ).

% singleton_inject
tff(fact_517_singleton__inject,axiom,
    ! [A2: nat,B2: nat] :
      ( ( insert_nat2(A2,bot_bot_set_nat) = insert_nat2(B2,bot_bot_set_nat) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_518_singleton__inject,axiom,
    ! [A2: int,B2: int] :
      ( ( insert_int2(A2,bot_bot_set_int) = insert_int2(B2,bot_bot_set_int) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_519_insert__not__empty,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A2,A) != bot_bo228742789529271731at_nat ).

% insert_not_empty
tff(fact_520_insert__not__empty,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,A) != bot_bo2099793752762293965at_nat ).

% insert_not_empty
tff(fact_521_insert__not__empty,axiom,
    ! [A2: bool,A: set_bool] : insert_bool2(A2,A) != bot_bot_set_bool ).

% insert_not_empty
tff(fact_522_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] : insert_nat2(A2,A) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_523_insert__not__empty,axiom,
    ! [A2: int,A: set_int] : insert_int2(A2,A) != bot_bot_set_int ).

% insert_not_empty
tff(fact_524_doubleton__eq__iff,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D2: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A2,insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(C,insert9069300056098147895at_nat(D2,bot_bo228742789529271731at_nat)) )
    <=> ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_525_doubleton__eq__iff,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,C: product_prod_nat_nat,D2: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A2,insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(C,insert8211810215607154385at_nat(D2,bot_bo2099793752762293965at_nat)) )
    <=> ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_526_doubleton__eq__iff,axiom,
    ! [A2: bool,B2: bool,C: bool,D2: bool] :
      ( ( insert_bool2(A2,insert_bool2(B2,bot_bot_set_bool)) = insert_bool2(C,insert_bool2(D2,bot_bot_set_bool)) )
    <=> ( ( ( pp(A2)
          <=> pp(C) )
          & ( pp(B2)
          <=> pp(D2) ) )
        | ( ( pp(A2)
          <=> pp(D2) )
          & ( pp(B2)
          <=> pp(C) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_527_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( insert_nat2(A2,insert_nat2(B2,bot_bot_set_nat)) = insert_nat2(C,insert_nat2(D2,bot_bot_set_nat)) )
    <=> ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_528_doubleton__eq__iff,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( insert_int2(A2,insert_int2(B2,bot_bot_set_int)) = insert_int2(C,insert_int2(D2,bot_bot_set_int)) )
    <=> ( ( ( A2 = C )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
tff(fact_529_singleton__iff,axiom,
    ! [B2: fun_Pr6001797050664311790t_bool,A2: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(B2,insert969342481230318014t_bool(A2,bot_bo2714889297266208186t_bool))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_530_singleton__iff,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat] :
      ( member8757157785044589968at_nat(B2,insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_531_singleton__iff,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B2),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_532_singleton__iff,axiom,
    ! [B2: bool,A2: bool] :
      ( member_bool(B2,insert_bool2(A2,bot_bot_set_bool))
    <=> ( pp(B2)
      <=> pp(A2) ) ) ).

% singleton_iff
tff(fact_533_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( member_nat(B2,insert_nat2(A2,bot_bot_set_nat))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_534_singleton__iff,axiom,
    ! [B2: int,A2: int] :
      ( member_int(B2,insert_int2(A2,bot_bot_set_int))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_535_singletonD,axiom,
    ! [B2: fun_Pr6001797050664311790t_bool,A2: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(B2,insert969342481230318014t_bool(A2,bot_bo2714889297266208186t_bool))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_536_singletonD,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat] :
      ( member8757157785044589968at_nat(B2,insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_537_singletonD,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B2),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_538_singletonD,axiom,
    ! [B2: bool,A2: bool] :
      ( member_bool(B2,insert_bool2(A2,bot_bot_set_bool))
     => ( pp(B2)
      <=> pp(A2) ) ) ).

% singletonD
tff(fact_539_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( member_nat(B2,insert_nat2(A2,bot_bot_set_nat))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_540_singletonD,axiom,
    ! [B2: int,A2: int] :
      ( member_int(B2,insert_int2(A2,bot_bot_set_int))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_541_insert__Diff__if,axiom,
    ! [X: bool,B: set_bool,A: set_bool] :
      ( ( member_bool(X,B)
       => ( minus_minus_set_bool(insert_bool2(X,A),B) = minus_minus_set_bool(A,B) ) )
      & ( ~ member_bool(X,B)
       => ( minus_minus_set_bool(insert_bool2(X,A),B) = insert_bool2(X,minus_minus_set_bool(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_542_insert__Diff__if,axiom,
    ! [X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat(X,B)
       => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A)),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) ) )
      & ( ~ member8757157785044589968at_nat(X,B)
       => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,insert9069300056098147895at_nat(X,A)),B) = insert9069300056098147895at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)) ) ) ) ).

% insert_Diff_if
tff(fact_543_insert__Diff__if,axiom,
    ! [X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A)),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,insert8211810215607154385at_nat(X,A)),B) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)) ) ) ) ).

% insert_Diff_if
tff(fact_544_insert__Diff__if,axiom,
    ! [X: int,B: set_int,A: set_int] :
      ( ( member_int(X,B)
       => ( minus_minus_set_int(insert_int2(X,A),B) = minus_minus_set_int(A,B) ) )
      & ( ~ member_int(X,B)
       => ( minus_minus_set_int(insert_int2(X,A),B) = insert_int2(X,minus_minus_set_int(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_545_insert__Diff__if,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool,A: set_fu6080891106736489294t_bool] :
      ( ( member4125802036136091927t_bool(X,B)
       => ( minus_5708602904570203829t_bool(insert969342481230318014t_bool(X,A),B) = minus_5708602904570203829t_bool(A,B) ) )
      & ( ~ member4125802036136091927t_bool(X,B)
       => ( minus_5708602904570203829t_bool(insert969342481230318014t_bool(X,A),B) = insert969342481230318014t_bool(X,minus_5708602904570203829t_bool(A,B)) ) ) ) ).

% insert_Diff_if
tff(fact_546_insert__Diff__if,axiom,
    ! [X: nat,B: set_nat,A: set_nat] :
      ( ( member_nat(X,B)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) ) )
      & ( ~ member_nat(X,B)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,insert_nat2(X,A)),B) = insert_nat2(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)) ) ) ) ).

% insert_Diff_if
tff(fact_547_set__minus__singleton__eq,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,X3: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(X,X3)
     => ( minus_5708602904570203829t_bool(X3,insert969342481230318014t_bool(X,bot_bo2714889297266208186t_bool)) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_548_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(X,X3)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,X3),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_549_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X3: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),X3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X3),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_550_set__minus__singleton__eq,axiom,
    ! [X: bool,X3: set_bool] :
      ( ~ member_bool(X,X3)
     => ( minus_minus_set_bool(X3,insert_bool2(X,bot_bot_set_bool)) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_551_set__minus__singleton__eq,axiom,
    ! [X: int,X3: set_int] :
      ( ~ member_int(X,X3)
     => ( minus_minus_set_int(X3,insert_int2(X,bot_bot_set_int)) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_552_set__minus__singleton__eq,axiom,
    ! [X: nat,X3: set_nat] :
      ( ~ member_nat(X,X3)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X3),insert_nat2(X,bot_bot_set_nat)) = X3 ) ) ).

% set_minus_singleton_eq
tff(fact_553_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_c_fun_nat_assn,Xs: list_c,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_c_fun_nat_assn(A,nth_c(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L6585172037124793648_c_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6585172037124793648_c_nat(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_554_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_c_fun_int_assn,Xs: list_c,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_c_fun_int_assn(A,nth_c(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L6582681566615743372_c_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6582681566615743372_c_int(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_555_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_d_fun_nat_assn,Xs: list_d,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_d_fun_nat_assn(A,nth_d(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L7820616366081188145_d_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7820616366081188145_d_nat(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_556_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_d_fun_int_assn,Xs: list_d,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_d_fun_int_assn(A,nth_d(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L7818125895572137869_d_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7818125895572137869_d_int(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_557_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_nat_fun_nat_assn,Xs: list_nat,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L7892069108621897922at_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7892069108621897922at_nat(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_558_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_nat_fun_int_assn,Xs: list_nat,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L7889578638112847646at_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L7889578638112847646at_int(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_559_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_int_fun_nat_assn,Xs: list_int,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8891422820522952478nt_nat(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_560_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_int_fun_int_assn,Xs: list_int,I: nat,Xsi: list_int,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L8888932350013902202nt_int(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_561_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_c_fun_d_assn,Xs: list_c,I: nat,Xsi: list_d,I3: set_nat,F: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_d_assn(aa_c_fun_d_assn(A,nth_c(Xs,I)),nth_d(Xsi,I))),vEBT_L824818263033719201sn_c_d(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi))),F)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( member_nat(I,I3)
         => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L824818263033719201sn_c_d(I3,A,Xs,Xsi)),F)),Q))
           => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_562_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] : the_el221668144340439132at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) = X ).

% the_elem_eq
tff(fact_563_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] : the_el2281957884133575798at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) = X ).

% the_elem_eq
tff(fact_564_the__elem__eq,axiom,
    ! [X: bool] :
      ( pp(the_elem_bool(insert_bool2(X,bot_bot_set_bool)))
    <=> pp(X) ) ).

% the_elem_eq
tff(fact_565_the__elem__eq,axiom,
    ! [X: nat] : the_elem_nat(insert_nat2(X,bot_bot_set_nat)) = X ).

% the_elem_eq
tff(fact_566_the__elem__eq,axiom,
    ! [X: int] : the_elem_int(insert_int2(X,bot_bot_set_int)) = X ).

% the_elem_eq
tff(fact_567_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: fun_c_fun_nat_assn,Xsi: list_nat] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( vEBT_L6585172037124793648_c_nat(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_c_fun_nat_assn(A,nth_c(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L6585172037124793648_c_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_568_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: fun_c_fun_int_assn,Xsi: list_int] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( vEBT_L6582681566615743372_c_int(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_c_fun_int_assn(A,nth_c(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L6582681566615743372_c_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_569_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: fun_d_fun_nat_assn,Xsi: list_nat] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( vEBT_L7820616366081188145_d_nat(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_d_fun_nat_assn(A,nth_d(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L7820616366081188145_d_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_570_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: fun_d_fun_int_assn,Xsi: list_int] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( vEBT_L7818125895572137869_d_int(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_d_fun_int_assn(A,nth_d(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L7818125895572137869_d_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_571_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_nat_fun_nat_assn,Xsi: list_nat] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L7892069108621897922at_nat(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L7892069108621897922at_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_572_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_nat_fun_int_assn,Xsi: list_int] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L7889578638112847646at_int(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L7889578638112847646at_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_573_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: fun_int_fun_nat_assn,Xsi: list_nat] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( vEBT_L8891422820522952478nt_nat(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_574_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: fun_int_fun_int_assn,Xsi: list_int] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( vEBT_L8888932350013902202nt_int(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_575_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: fun_c_fun_d_assn,Xsi: list_d] :
      ( member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( vEBT_L824818263033719201sn_c_d(I3,A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_d_assn(aa_c_fun_d_assn(A,nth_c(Xs,I)),nth_d(Xsi,I))),vEBT_L824818263033719201sn_c_d(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,I3),insert_nat2(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_576_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : is_sin2937591304547752795at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% is_singletonI
tff(fact_577_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))) ).

% is_singletonI
tff(fact_578_is__singletonI,axiom,
    ! [X: bool] : is_singleton_bool(insert_bool2(X,bot_bot_set_bool)) ).

% is_singletonI
tff(fact_579_is__singletonI,axiom,
    ! [X: nat] : is_singleton_nat(insert_nat2(X,bot_bot_set_nat)) ).

% is_singletonI
tff(fact_580_is__singletonI,axiom,
    ! [X: int] : is_singleton_int(insert_int2(X,bot_bot_set_int)) ).

% is_singletonI
tff(fact_581_inf__period_I2_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,D: code_integer,Q: fun_Co8822979462182099027r_bool] :
      ( ! [X2: code_integer,K: code_integer] :
          ( pp(aa_Code_integer_bool(P,X2))
        <=> pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),D)))) )
     => ( ! [X2: code_integer,K: code_integer] :
            ( pp(aa_Code_integer_bool(Q,X2))
          <=> pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),D)))) )
       => ! [X6: code_integer,K2: code_integer] :
            ( ( pp(aa_Code_integer_bool(P,X6))
              | pp(aa_Code_integer_bool(Q,X6)) )
          <=> ( pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X6),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D))))
              | pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X6),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D)))) ) ) ) ) ).

% inf_period(2)
tff(fact_582_inf__period_I2_J,axiom,
    ! [P: fun_rat_bool,D: rat,Q: fun_rat_bool] :
      ( ! [X2: rat,K: rat] :
          ( pp(aa_rat_bool(P,X2))
        <=> pp(aa_rat_bool(P,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D)))) )
     => ( ! [X2: rat,K: rat] :
            ( pp(aa_rat_bool(Q,X2))
          <=> pp(aa_rat_bool(Q,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D)))) )
       => ! [X6: rat,K2: rat] :
            ( ( pp(aa_rat_bool(P,X6))
              | pp(aa_rat_bool(Q,X6)) )
          <=> ( pp(aa_rat_bool(P,minus_minus_rat(X6,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D))))
              | pp(aa_rat_bool(Q,minus_minus_rat(X6,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D)))) ) ) ) ) ).

% inf_period(2)
tff(fact_583_inf__period_I2_J,axiom,
    ! [P: fun_int_bool,D: int,Q: fun_int_bool] :
      ( ! [X2: int,K: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
     => ( ! [X2: int,K: int] :
            ( pp(aa_int_bool(Q,X2))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ! [X6: int,K2: int] :
            ( ( pp(aa_int_bool(P,X6))
              | pp(aa_int_bool(Q,X6)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))))
              | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) ) ) ) ) ).

% inf_period(2)
tff(fact_584_inf__period_I1_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,D: code_integer,Q: fun_Co8822979462182099027r_bool] :
      ( ! [X2: code_integer,K: code_integer] :
          ( pp(aa_Code_integer_bool(P,X2))
        <=> pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),D)))) )
     => ( ! [X2: code_integer,K: code_integer] :
            ( pp(aa_Code_integer_bool(Q,X2))
          <=> pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),D)))) )
       => ! [X6: code_integer,K2: code_integer] :
            ( ( pp(aa_Code_integer_bool(P,X6))
              & pp(aa_Code_integer_bool(Q,X6)) )
          <=> ( pp(aa_Code_integer_bool(P,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X6),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D))))
              & pp(aa_Code_integer_bool(Q,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X6),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D)))) ) ) ) ) ).

% inf_period(1)
tff(fact_585_inf__period_I1_J,axiom,
    ! [P: fun_rat_bool,D: rat,Q: fun_rat_bool] :
      ( ! [X2: rat,K: rat] :
          ( pp(aa_rat_bool(P,X2))
        <=> pp(aa_rat_bool(P,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D)))) )
     => ( ! [X2: rat,K: rat] :
            ( pp(aa_rat_bool(Q,X2))
          <=> pp(aa_rat_bool(Q,minus_minus_rat(X2,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),D)))) )
       => ! [X6: rat,K2: rat] :
            ( ( pp(aa_rat_bool(P,X6))
              & pp(aa_rat_bool(Q,X6)) )
          <=> ( pp(aa_rat_bool(P,minus_minus_rat(X6,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D))))
              & pp(aa_rat_bool(Q,minus_minus_rat(X6,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K2),D)))) ) ) ) ) ).

% inf_period(1)
tff(fact_586_inf__period_I1_J,axiom,
    ! [P: fun_int_bool,D: int,Q: fun_int_bool] :
      ( ! [X2: int,K: int] :
          ( pp(aa_int_bool(P,X2))
        <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
     => ( ! [X2: int,K: int] :
            ( pp(aa_int_bool(Q,X2))
          <=> pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
       => ! [X6: int,K2: int] :
            ( ( pp(aa_int_bool(P,X6))
              & pp(aa_int_bool(Q,X6)) )
          <=> ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))))
              & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D)))) ) ) ) ) ).

% inf_period(1)
tff(fact_587_left__diff__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ).

% left_diff_distrib
tff(fact_588_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A2,B2)),C) = 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,B2),C)) ).

% left_diff_distrib
tff(fact_589_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) = aa_int_int(aa_int_fun_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,B2),C)) ).

% left_diff_distrib
tff(fact_590_right__diff__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) ).

% right_diff_distrib
tff(fact_591_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% right_diff_distrib
tff(fact_592_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% right_diff_distrib
tff(fact_593_left__diff__distrib_H,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),C)),A2) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)) ).

% left_diff_distrib'
tff(fact_594_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B2,C)),A2) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)) ).

% left_diff_distrib'
tff(fact_595_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)) ).

% left_diff_distrib'
tff(fact_596_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)) ).

% left_diff_distrib'
tff(fact_597_right__diff__distrib_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),C)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) ).

% right_diff_distrib'
tff(fact_598_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% right_diff_distrib'
tff(fact_599_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) ).

% right_diff_distrib'
tff(fact_600_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% right_diff_distrib'
tff(fact_601_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(D2),C))
       => pp(aa_rat_bool(ord_less_rat(minus_minus_rat(A2,C)),minus_minus_rat(B2,D2))) ) ) ).

% diff_strict_mono
tff(fact_602_diff__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,D2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,D2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),C)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),D2))) ) ) ).

% diff_strict_mono
tff(fact_603_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,D2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D2))) ) ) ).

% diff_strict_mono
tff(fact_604_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_605_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_606_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_c,X: c] :
      ( ( I != J )
     => ( nth_c(list_update_c(Xs,I,X),J) = nth_c(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_607_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_d,X: d] :
      ( ( I != J )
     => ( nth_d(list_update_d(Xs,I,X),J) = nth_d(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_608_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_609_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_610_list__update__id,axiom,
    ! [Xs: list_c,I: nat] : list_update_c(Xs,I,nth_c(Xs,I)) = Xs ).

% list_update_id
tff(fact_611_list__update__id,axiom,
    ! [Xs: list_d,I: nat] : list_update_d(Xs,I,nth_d(Xs,I)) = Xs ).

% list_update_id
tff(fact_612_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_c,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( nth_c(list_update_c(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_613_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_d,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( nth_d(list_update_d(Xs,I,X),I) = X ) ) ).

% nth_list_update_eq
tff(fact_614_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_615_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_616_nth__update__invalid,axiom,
    ! [I: nat,L: list_c,J: nat,X: c] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L)))
     => ( nth_c(list_update_c(L,J,X),I) = nth_c(L,I) ) ) ).

% nth_update_invalid
tff(fact_617_nth__update__invalid,axiom,
    ! [I: nat,L: list_d,J: nat,X: d] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L)))
     => ( nth_d(list_update_d(L,J,X),I) = nth_d(L,I) ) ) ).

% nth_update_invalid
tff(fact_618_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,X)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_619_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,X: int] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( aa_nat_int(nth_int(list_update_int(L,J,X)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_620_psubsetD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ord_le2604355607129572851at_nat(A,B)
     => ( member8757157785044589968at_nat(C,A)
       => member8757157785044589968at_nat(C,B) ) ) ).

% psubsetD
tff(fact_621_psubsetD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A),B))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% psubsetD
tff(fact_622_psubsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ord_less_set_nat(A,B)
     => ( member_nat(C,A)
       => member_nat(C,B) ) ) ).

% psubsetD
tff(fact_623_psubsetD,axiom,
    ! [A: set_int,B: set_int,C: int] :
      ( ord_less_set_int(A,B)
     => ( member_int(C,A)
       => member_int(C,B) ) ) ).

% psubsetD
tff(fact_624_psubsetD,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool,C: fun_Pr6001797050664311790t_bool] :
      ( ord_le3815294503984575994t_bool(A,B)
     => ( member4125802036136091927t_bool(C,A)
       => member4125802036136091927t_bool(C,B) ) ) ).

% psubsetD
tff(fact_625_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_c(Xs)))
           => ( nth_c(Xs,I4) = nth_c(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_626_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_d(Xs)))
           => ( nth_d(Xs,I4) = nth_d(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_627_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_628_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_629_Skolem__list__nth,axiom,
    ! [K3: nat,P: fun_nat_fun_c_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
         => ? [X_12: c] : pp(aa_c_bool(aa_nat_fun_c_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_c] :
          ( ( size_size_list_c(Xs3) = K3 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
             => pp(aa_c_bool(aa_nat_fun_c_bool(P,I4),nth_c(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_630_Skolem__list__nth,axiom,
    ! [K3: nat,P: fun_nat_fun_d_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
         => ? [X_12: d] : pp(aa_d_bool(aa_nat_fun_d_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_d] :
          ( ( size_size_list_d(Xs3) = K3 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
             => pp(aa_d_bool(aa_nat_fun_d_bool(P,I4),nth_d(Xs3,I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_631_Skolem__list__nth,axiom,
    ! [K3: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
         => ? [X_12: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K3 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I4),aa_nat_nat(nth_nat(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_632_Skolem__list__nth,axiom,
    ! [K3: nat,P: fun_nat_fun_int_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
         => ? [X_12: int] : pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),X_12)) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K3 )
          & ! [I4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),K3))
             => pp(aa_int_bool(aa_nat_fun_int_bool(P,I4),aa_nat_int(nth_int(Xs3),I4))) ) ) ) ).

% Skolem_list_nth
tff(fact_633_nth__equalityI,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
           => ( nth_c(Xs,I5) = nth_c(Ys,I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_634_nth__equalityI,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
     => ( ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
           => ( nth_d(Xs,I5) = nth_d(Ys,I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_635_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
           => ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Ys),I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_636_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_int(Xs)))
           => ( aa_nat_int(nth_int(Xs),I5) = aa_nat_int(nth_int(Ys),I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_637_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_c_fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
         => ? [Li2: c] : pp(aa_nat_bool(aa_c_fun_nat_bool(P,Li2),I5)) )
     => ~ ! [L2: list_c] :
            ( ( size_size_list_c(L2) = N )
           => ~ ! [I6: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I6),N))
                 => pp(aa_nat_bool(aa_c_fun_nat_bool(P,nth_c(L2,I6)),I6)) ) ) ) ).

% obtain_list_from_elements
tff(fact_638_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_d_fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
         => ? [Li2: d] : pp(aa_nat_bool(aa_d_fun_nat_bool(P,Li2),I5)) )
     => ~ ! [L2: list_d] :
            ( ( size_size_list_d(L2) = N )
           => ~ ! [I6: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I6),N))
                 => pp(aa_nat_bool(aa_d_fun_nat_bool(P,nth_d(L2,I6)),I6)) ) ) ) ).

% obtain_list_from_elements
tff(fact_639_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_nat_fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
         => ? [Li2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,Li2),I5)) )
     => ~ ! [L2: list_nat] :
            ( ( size_size_list_nat(L2) = N )
           => ~ ! [I6: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I6),N))
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(L2),I6)),I6)) ) ) ) ).

% obtain_list_from_elements
tff(fact_640_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_int_fun_nat_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
         => ? [Li2: int] : pp(aa_nat_bool(aa_int_fun_nat_bool(P,Li2),I5)) )
     => ~ ! [L2: list_int] :
            ( ( size_size_list_int(L2) = N )
           => ~ ! [I6: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I6),N))
                 => pp(aa_nat_bool(aa_int_fun_nat_bool(P,aa_nat_int(nth_int(L2),I6)),I6)) ) ) ) ).

% obtain_list_from_elements
tff(fact_641_is__singleton__the__elem,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( is_sin2937591304547752795at_nat(A)
    <=> ( A = insert9069300056098147895at_nat(the_el221668144340439132at_nat(A),bot_bo228742789529271731at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_642_is__singleton__the__elem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A))
    <=> ( A = insert8211810215607154385at_nat(the_el2281957884133575798at_nat(A),bot_bo2099793752762293965at_nat) ) ) ).

% is_singleton_the_elem
tff(fact_643_is__singleton__the__elem,axiom,
    ! [A: set_bool] :
      ( is_singleton_bool(A)
    <=> ( A = insert_bool2(the_elem_bool(A),bot_bot_set_bool) ) ) ).

% is_singleton_the_elem
tff(fact_644_is__singleton__the__elem,axiom,
    ! [A: set_nat] :
      ( is_singleton_nat(A)
    <=> ( A = insert_nat2(the_elem_nat(A),bot_bot_set_nat) ) ) ).

% is_singleton_the_elem
tff(fact_645_is__singleton__the__elem,axiom,
    ! [A: set_int] :
      ( is_singleton_int(A)
    <=> ( A = insert_int2(the_elem_int(A),bot_bot_set_int) ) ) ).

% is_singleton_the_elem
tff(fact_646_is__singletonI_H,axiom,
    ! [A: set_fu6080891106736489294t_bool] :
      ( ( A != bot_bo2714889297266208186t_bool )
     => ( ! [X2: fun_Pr6001797050664311790t_bool,Y2: fun_Pr6001797050664311790t_bool] :
            ( member4125802036136091927t_bool(X2,A)
           => ( member4125802036136091927t_bool(Y2,A)
             => ( X2 = Y2 ) ) )
       => is_sin8604298300380296930t_bool(A) ) ) ).

% is_singletonI'
tff(fact_647_is__singletonI_H,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( A != bot_bo228742789529271731at_nat )
     => ( ! [X2: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X2,A)
           => ( member8757157785044589968at_nat(Y2,A)
             => ( X2 = Y2 ) ) )
       => is_sin2937591304547752795at_nat(A) ) ) ).

% is_singletonI'
tff(fact_648_is__singletonI_H,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
     => ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A))
           => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),A))
             => ( X2 = Y2 ) ) )
       => pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A)) ) ) ).

% is_singletonI'
tff(fact_649_is__singletonI_H,axiom,
    ! [A: set_bool] :
      ( ( A != bot_bot_set_bool )
     => ( ! [X2: bool,Y2: bool] :
            ( member_bool(X2,A)
           => ( member_bool(Y2,A)
             => ( pp(X2)
              <=> pp(Y2) ) ) )
       => is_singleton_bool(A) ) ) ).

% is_singletonI'
tff(fact_650_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X2: nat,Y2: nat] :
            ( member_nat(X2,A)
           => ( member_nat(Y2,A)
             => ( X2 = Y2 ) ) )
       => is_singleton_nat(A) ) ) ).

% is_singletonI'
tff(fact_651_is__singletonI_H,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X2: int,Y2: int] :
            ( member_int(X2,A)
           => ( member_int(Y2,A)
             => ( X2 = Y2 ) ) )
       => is_singleton_int(A) ) ) ).

% is_singletonI'
tff(fact_652_nth__list__update,axiom,
    ! [I: nat,Xs: list_c,J: nat,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( ( ( I = J )
         => ( nth_c(list_update_c(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_c(list_update_c(Xs,I,X),J) = nth_c(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_653_nth__list__update,axiom,
    ! [I: nat,Xs: list_d,J: nat,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( ( ( I = J )
         => ( nth_d(list_update_d(Xs,I,X),J) = X ) )
        & ( ( I != J )
         => ( nth_d(list_update_d(Xs,I,X),J) = nth_d(Xs,J) ) ) ) ) ).

% nth_list_update
tff(fact_654_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,X)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_655_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( ( I = J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = X ) )
        & ( ( I != J )
         => ( aa_nat_int(nth_int(list_update_int(Xs,I,X)),J) = aa_nat_int(nth_int(Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_656_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_c,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( ( list_update_c(Xs,I,X) = Xs )
      <=> ( nth_c(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_657_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_d,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( ( list_update_d(Xs,I,X) = Xs )
      <=> ( nth_d(Xs,I) = X ) ) ) ).

% list_update_same_conv
tff(fact_658_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( ( list_update_nat(Xs,I,X) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_659_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( ( list_update_int(Xs,I,X) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_660_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_c,X: c] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L))) )
       => ( nth_c(list_update_c(L,I,X),J) = X ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L))) )
       => ( nth_c(list_update_c(L,I,X),J) = nth_c(L,J) ) ) ) ).

% nth_list_update'
tff(fact_661_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_d,X: d] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L))) )
       => ( nth_d(list_update_d(L,I,X),J) = X ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L))) )
       => ( nth_d(list_update_d(L,I,X),J) = nth_d(L,J) ) ) ) ).

% nth_list_update'
tff(fact_662_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_nat,X: nat] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X)),J) = X ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L))) )
       => ( aa_nat_nat(nth_nat(list_update_nat(L,I,X)),J) = aa_nat_nat(nth_nat(L),J) ) ) ) ).

% nth_list_update'
tff(fact_663_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L: list_int,X: int] :
      ( ( ( ( I = J )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L))) )
       => ( aa_nat_int(nth_int(list_update_int(L,I,X)),J) = X ) )
      & ( ~ ( ( I = J )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L))) )
       => ( aa_nat_int(nth_int(list_update_int(L,I,X)),J) = aa_nat_int(nth_int(L),J) ) ) ) ).

% nth_list_update'
tff(fact_664_linorder__neqE__linordered__idom,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_665_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_666_pinf_I1_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q2: fun_Co8822979462182099027r_bool] :
      ( ? [Z2: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X2))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z2: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X2))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q2,X2)) ) )
       => ? [Z3: code_integer] :
          ! [X6: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
           => ( ( pp(aa_Code_integer_bool(P,X6))
                & pp(aa_Code_integer_bool(Q,X6)) )
            <=> ( pp(aa_Code_integer_bool(P2,X6))
                & pp(aa_Code_integer_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_667_pinf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z2: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z2: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q2,X2)) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
           => ( ( pp(aa_num_bool(P,X6))
                & pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P2,X6))
                & pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_668_pinf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z2: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X2))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z2: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q2,X2)) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
           => ( ( pp(aa_nat_bool(P,X6))
                & pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P2,X6))
                & pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_669_pinf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z2: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X2))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q2,X2)) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P2,X6))
                & pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(1)
tff(fact_670_pinf_I2_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q2: fun_Co8822979462182099027r_bool] :
      ( ? [Z2: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X2))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z2: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z2),X2))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q2,X2)) ) )
       => ? [Z3: code_integer] :
          ! [X6: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
           => ( ( pp(aa_Code_integer_bool(P,X6))
                | pp(aa_Code_integer_bool(Q,X6)) )
            <=> ( pp(aa_Code_integer_bool(P2,X6))
                | pp(aa_Code_integer_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_671_pinf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z2: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z2: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z2),X2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q2,X2)) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
           => ( ( pp(aa_num_bool(P,X6))
                | pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P2,X6))
                | pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_672_pinf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z2: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X2))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z2: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z2),X2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q2,X2)) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
           => ( ( pp(aa_nat_bool(P,X6))
                | pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P2,X6))
                | pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_673_pinf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z2: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X2))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q2,X2)) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P2,X6))
                | pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% pinf(2)
tff(fact_674_pinf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_675_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_676_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_677_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(3)
tff(fact_678_pinf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_679_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_680_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_681_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
     => ( X6 != T ) ) ).

% pinf(4)
tff(fact_682_pinf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),T)) ) ).

% pinf(5)
tff(fact_683_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),T)) ) ).

% pinf(5)
tff(fact_684_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),T)) ) ).

% pinf(5)
tff(fact_685_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T)) ) ).

% pinf(5)
tff(fact_686_pinf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,T),X6)) ) ).

% pinf(7)
tff(fact_687_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X6)) ) ).

% pinf(7)
tff(fact_688_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X6)) ) ).

% pinf(7)
tff(fact_689_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6)) ) ).

% pinf(7)
tff(fact_690_minf_I1_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q2: fun_Co8822979462182099027r_bool] :
      ( ? [Z2: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z2))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z2: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z2))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q2,X2)) ) )
       => ? [Z3: code_integer] :
          ! [X6: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
           => ( ( pp(aa_Code_integer_bool(P,X6))
                & pp(aa_Code_integer_bool(Q,X6)) )
            <=> ( pp(aa_Code_integer_bool(P2,X6))
                & pp(aa_Code_integer_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_691_minf_I1_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z2: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z2: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q2,X2)) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
           => ( ( pp(aa_num_bool(P,X6))
                & pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P2,X6))
                & pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_692_minf_I1_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z2: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z2))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z2: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q2,X2)) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
           => ( ( pp(aa_nat_bool(P,X6))
                & pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P2,X6))
                & pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_693_minf_I1_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z2: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z2))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q2,X2)) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P2,X6))
                & pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% minf(1)
tff(fact_694_minf_I2_J,axiom,
    ! [P: fun_Co8822979462182099027r_bool,P2: fun_Co8822979462182099027r_bool,Q: fun_Co8822979462182099027r_bool,Q2: fun_Co8822979462182099027r_bool] :
      ( ? [Z2: code_integer] :
        ! [X2: code_integer] :
          ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z2))
         => ( pp(aa_Code_integer_bool(P,X2))
          <=> pp(aa_Code_integer_bool(P2,X2)) ) )
     => ( ? [Z2: code_integer] :
          ! [X2: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Z2))
           => ( pp(aa_Code_integer_bool(Q,X2))
            <=> pp(aa_Code_integer_bool(Q2,X2)) ) )
       => ? [Z3: code_integer] :
          ! [X6: code_integer] :
            ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
           => ( ( pp(aa_Code_integer_bool(P,X6))
                | pp(aa_Code_integer_bool(Q,X6)) )
            <=> ( pp(aa_Code_integer_bool(P2,X6))
                | pp(aa_Code_integer_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_695_minf_I2_J,axiom,
    ! [P: fun_num_bool,P2: fun_num_bool,Q: fun_num_bool,Q2: fun_num_bool] :
      ( ? [Z2: num] :
        ! [X2: num] :
          ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z2))
         => ( pp(aa_num_bool(P,X2))
          <=> pp(aa_num_bool(P2,X2)) ) )
     => ( ? [Z2: num] :
          ! [X2: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Z2))
           => ( pp(aa_num_bool(Q,X2))
            <=> pp(aa_num_bool(Q2,X2)) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
           => ( ( pp(aa_num_bool(P,X6))
                | pp(aa_num_bool(Q,X6)) )
            <=> ( pp(aa_num_bool(P2,X6))
                | pp(aa_num_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_696_minf_I2_J,axiom,
    ! [P: fun_nat_bool,P2: fun_nat_bool,Q: fun_nat_bool,Q2: fun_nat_bool] :
      ( ? [Z2: nat] :
        ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z2))
         => ( pp(aa_nat_bool(P,X2))
          <=> pp(aa_nat_bool(P2,X2)) ) )
     => ( ? [Z2: nat] :
          ! [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Z2))
           => ( pp(aa_nat_bool(Q,X2))
            <=> pp(aa_nat_bool(Q2,X2)) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
           => ( ( pp(aa_nat_bool(P,X6))
                | pp(aa_nat_bool(Q,X6)) )
            <=> ( pp(aa_nat_bool(P2,X6))
                | pp(aa_nat_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_697_minf_I2_J,axiom,
    ! [P: fun_int_bool,P2: fun_int_bool,Q: fun_int_bool,Q2: fun_int_bool] :
      ( ? [Z2: int] :
        ! [X2: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z2))
         => ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P2,X2)) ) )
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z2))
           => ( pp(aa_int_bool(Q,X2))
            <=> pp(aa_int_bool(Q2,X2)) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
            <=> ( pp(aa_int_bool(P2,X6))
                | pp(aa_int_bool(Q2,X6)) ) ) ) ) ) ).

% minf(2)
tff(fact_698_minf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_699_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_700_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_701_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
     => ( X6 != T ) ) ).

% minf(3)
tff(fact_702_minf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_703_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_704_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_705_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
     => ( X6 != T ) ) ).

% minf(4)
tff(fact_706_minf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),T)) ) ).

% minf(5)
tff(fact_707_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),T)) ) ).

% minf(5)
tff(fact_708_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),T)) ) ).

% minf(5)
tff(fact_709_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T)) ) ).

% minf(5)
tff(fact_710_minf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,T),X6)) ) ).

% minf(7)
tff(fact_711_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,T),X6)) ) ).

% minf(7)
tff(fact_712_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,T),X6)) ) ).

% minf(7)
tff(fact_713_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6)) ) ).

% minf(7)
tff(fact_714_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_715_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_716_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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,B2),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_717_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,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,B2),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_718_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,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,B2),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_719_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_720_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_721_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_722_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_723_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_724_mult_Oassoc,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ).

% mult.assoc
tff(fact_725_mult_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C)) ).

% mult.assoc
tff(fact_726_mult_Oassoc,axiom,
    ! [A2: rat,B2: 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),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% mult.assoc
tff(fact_727_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% mult.assoc
tff(fact_728_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% mult.assoc
tff(fact_729_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_c,Xs4: list_c,Xsi: list_c,Xsi2: list_c,A: fun_c_fun_c_assn,A4: fun_c_fun_c_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_c(Xs) = size_size_list_c(Xs4) )
       => ( ( size_size_list_c(Xsi) = size_size_list_c(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
                 => ( ( size_size_list_c(Xs) = size_size_list_c(Xsi) )
                   => ( ( nth_c(Xs,I5) = nth_c(Xs4,I5) )
                      & ( nth_c(Xsi,I5) = nth_c(Xsi2,I5) )
                      & ( aa_c_assn(aa_c_fun_c_assn(A,nth_c(Xs,I5)),nth_c(Xsi,I5)) = aa_c_assn(aa_c_fun_c_assn(A4,nth_c(Xs4,I5)),nth_c(Xsi2,I5)) ) ) ) ) )
           => ( vEBT_L824818263033719200sn_c_c(I3,A,Xs,Xsi) = vEBT_L824818263033719200sn_c_c(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_730_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_c,Xs4: list_c,Xsi: list_nat,Xsi2: list_nat,A: fun_c_fun_nat_assn,A4: fun_c_fun_nat_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_c(Xs) = size_size_list_c(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
                 => ( ( size_size_list_c(Xs) = size_size_list_nat(Xsi) )
                   => ( ( nth_c(Xs,I5) = nth_c(Xs4,I5) )
                      & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) )
                      & ( aa_nat_assn(aa_c_fun_nat_assn(A,nth_c(Xs,I5)),aa_nat_nat(nth_nat(Xsi),I5)) = aa_nat_assn(aa_c_fun_nat_assn(A4,nth_c(Xs4,I5)),aa_nat_nat(nth_nat(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L6585172037124793648_c_nat(I3,A,Xs,Xsi) = vEBT_L6585172037124793648_c_nat(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_731_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_c,Xs4: list_c,Xsi: list_int,Xsi2: list_int,A: fun_c_fun_int_assn,A4: fun_c_fun_int_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_c(Xs) = size_size_list_c(Xs4) )
       => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
                 => ( ( size_size_list_c(Xs) = size_size_list_int(Xsi) )
                   => ( ( nth_c(Xs,I5) = nth_c(Xs4,I5) )
                      & ( aa_nat_int(nth_int(Xsi),I5) = aa_nat_int(nth_int(Xsi2),I5) )
                      & ( aa_int_assn(aa_c_fun_int_assn(A,nth_c(Xs,I5)),aa_nat_int(nth_int(Xsi),I5)) = aa_int_assn(aa_c_fun_int_assn(A4,nth_c(Xs4,I5)),aa_nat_int(nth_int(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L6582681566615743372_c_int(I3,A,Xs,Xsi) = vEBT_L6582681566615743372_c_int(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_732_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_c,Xsi2: list_c,A: fun_d_fun_c_assn,A4: fun_d_fun_c_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
       => ( ( size_size_list_c(Xsi) = size_size_list_c(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                 => ( ( size_size_list_d(Xs) = size_size_list_c(Xsi) )
                   => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                      & ( nth_c(Xsi,I5) = nth_c(Xsi2,I5) )
                      & ( aa_c_assn(aa_d_fun_c_assn(A,nth_d(Xs,I5)),nth_c(Xsi,I5)) = aa_c_assn(aa_d_fun_c_assn(A4,nth_d(Xs4,I5)),nth_c(Xsi2,I5)) ) ) ) ) )
           => ( vEBT_L7260734717915818975sn_d_c(I3,A,Xs,Xsi) = vEBT_L7260734717915818975sn_d_c(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_733_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_d,Xsi2: list_d,A: fun_d_fun_d_assn,A4: fun_d_fun_d_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
       => ( ( size_size_list_d(Xsi) = size_size_list_d(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                 => ( ( size_size_list_d(Xs) = size_size_list_d(Xsi) )
                   => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                      & ( nth_d(Xsi,I5) = nth_d(Xsi2,I5) )
                      & ( aa_d_assn(aa_d_fun_d_assn(A,nth_d(Xs,I5)),nth_d(Xsi,I5)) = aa_d_assn(aa_d_fun_d_assn(A4,nth_d(Xs4,I5)),nth_d(Xsi2,I5)) ) ) ) ) )
           => ( vEBT_L7260734717915818976sn_d_d(I3,A,Xs,Xsi) = vEBT_L7260734717915818976sn_d_d(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_734_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_nat,Xsi2: list_nat,A: fun_d_fun_nat_assn,A4: fun_d_fun_nat_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                 => ( ( size_size_list_d(Xs) = size_size_list_nat(Xsi) )
                   => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                      & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) )
                      & ( aa_nat_assn(aa_d_fun_nat_assn(A,nth_d(Xs,I5)),aa_nat_nat(nth_nat(Xsi),I5)) = aa_nat_assn(aa_d_fun_nat_assn(A4,nth_d(Xs4,I5)),aa_nat_nat(nth_nat(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L7820616366081188145_d_nat(I3,A,Xs,Xsi) = vEBT_L7820616366081188145_d_nat(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_735_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_d,Xs4: list_d,Xsi: list_int,Xsi2: list_int,A: fun_d_fun_int_assn,A4: fun_d_fun_int_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
       => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                 => ( ( size_size_list_d(Xs) = size_size_list_int(Xsi) )
                   => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                      & ( aa_nat_int(nth_int(Xsi),I5) = aa_nat_int(nth_int(Xsi2),I5) )
                      & ( aa_int_assn(aa_d_fun_int_assn(A,nth_d(Xs,I5)),aa_nat_int(nth_int(Xsi),I5)) = aa_int_assn(aa_d_fun_int_assn(A4,nth_d(Xs4,I5)),aa_nat_int(nth_int(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L7818125895572137869_d_int(I3,A,Xs,Xsi) = vEBT_L7818125895572137869_d_int(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_736_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_nat,Xs4: list_nat,Xsi: list_c,Xsi2: list_c,A: fun_nat_fun_c_assn,A4: fun_nat_fun_c_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xs4) )
       => ( ( size_size_list_c(Xsi) = size_size_list_c(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
                 => ( ( size_size_list_nat(Xs) = size_size_list_c(Xsi) )
                   => ( ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Xs4),I5) )
                      & ( nth_c(Xsi,I5) = nth_c(Xsi2,I5) )
                      & ( aa_c_assn(aa_nat_fun_c_assn(A,aa_nat_nat(nth_nat(Xs),I5)),nth_c(Xsi,I5)) = aa_c_assn(aa_nat_fun_c_assn(A4,aa_nat_nat(nth_nat(Xs4),I5)),nth_c(Xsi2,I5)) ) ) ) ) )
           => ( vEBT_L8146214880390291086_nat_c(I3,A,Xs,Xsi) = vEBT_L8146214880390291086_nat_c(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_737_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_nat,Xs4: list_nat,Xsi: list_d,Xsi2: list_d,A: fun_nat_fun_d_assn,A4: fun_nat_fun_d_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xs4) )
       => ( ( size_size_list_d(Xsi) = size_size_list_d(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
                 => ( ( size_size_list_nat(Xs) = size_size_list_d(Xsi) )
                   => ( ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Xs4),I5) )
                      & ( nth_d(Xsi,I5) = nth_d(Xsi2,I5) )
                      & ( aa_d_assn(aa_nat_fun_d_assn(A,aa_nat_nat(nth_nat(Xs),I5)),nth_d(Xsi,I5)) = aa_d_assn(aa_nat_fun_d_assn(A4,aa_nat_nat(nth_nat(Xs4),I5)),nth_d(Xsi2,I5)) ) ) ) ) )
           => ( vEBT_L8146214880390291087_nat_d(I3,A,Xs,Xsi) = vEBT_L8146214880390291087_nat_d(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_738_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat,A: fun_nat_fun_nat_assn,A4: fun_nat_fun_nat_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I5: nat] :
                ( member_nat(I5,I3)
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
                 => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xsi) )
                   => ( ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Xs4),I5) )
                      & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) )
                      & ( aa_nat_assn(aa_nat_fun_nat_assn(A,aa_nat_nat(nth_nat(Xs),I5)),aa_nat_nat(nth_nat(Xsi),I5)) = aa_nat_assn(aa_nat_fun_nat_assn(A4,aa_nat_nat(nth_nat(Xs4),I5)),aa_nat_nat(nth_nat(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L7892069108621897922at_nat(I3,A,Xs,Xsi) = vEBT_L7892069108621897922at_nat(I7,A4,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_739_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_c_fun_c_assn,A4: fun_c_fun_c_assn,Xs: list_c,Xs4: list_c,Xsi: list_c,Xsi2: list_c] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_c(Xs) = size_size_list_c(Xs4) )
         => ( ( size_size_list_c(Xsi) = size_size_list_c(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
                   => ( ( size_size_list_c(Xs) = size_size_list_c(Xsi) )
                     => ( ( nth_c(Xs,I5) = nth_c(Xs4,I5) )
                        & ( nth_c(Xsi,I5) = nth_c(Xsi2,I5) ) ) ) ) )
             => ( vEBT_L824818263033719200sn_c_c(I3,A,Xs,Xsi) = vEBT_L824818263033719200sn_c_c(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_740_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_c_fun_nat_assn,A4: fun_c_fun_nat_assn,Xs: list_c,Xs4: list_c,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_c(Xs) = size_size_list_c(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
                   => ( ( size_size_list_c(Xs) = size_size_list_nat(Xsi) )
                     => ( ( nth_c(Xs,I5) = nth_c(Xs4,I5) )
                        & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L6585172037124793648_c_nat(I3,A,Xs,Xsi) = vEBT_L6585172037124793648_c_nat(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_741_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_c_fun_int_assn,A4: fun_c_fun_int_assn,Xs: list_c,Xs4: list_c,Xsi: list_int,Xsi2: list_int] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_c(Xs) = size_size_list_c(Xs4) )
         => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
                   => ( ( size_size_list_c(Xs) = size_size_list_int(Xsi) )
                     => ( ( nth_c(Xs,I5) = nth_c(Xs4,I5) )
                        & ( aa_nat_int(nth_int(Xsi),I5) = aa_nat_int(nth_int(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L6582681566615743372_c_int(I3,A,Xs,Xsi) = vEBT_L6582681566615743372_c_int(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_742_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_d_fun_c_assn,A4: fun_d_fun_c_assn,Xs: list_d,Xs4: list_d,Xsi: list_c,Xsi2: list_c] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
         => ( ( size_size_list_c(Xsi) = size_size_list_c(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                   => ( ( size_size_list_d(Xs) = size_size_list_c(Xsi) )
                     => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                        & ( nth_c(Xsi,I5) = nth_c(Xsi2,I5) ) ) ) ) )
             => ( vEBT_L7260734717915818975sn_d_c(I3,A,Xs,Xsi) = vEBT_L7260734717915818975sn_d_c(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_743_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_d_fun_d_assn,A4: fun_d_fun_d_assn,Xs: list_d,Xs4: list_d,Xsi: list_d,Xsi2: list_d] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
         => ( ( size_size_list_d(Xsi) = size_size_list_d(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                   => ( ( size_size_list_d(Xs) = size_size_list_d(Xsi) )
                     => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                        & ( nth_d(Xsi,I5) = nth_d(Xsi2,I5) ) ) ) ) )
             => ( vEBT_L7260734717915818976sn_d_d(I3,A,Xs,Xsi) = vEBT_L7260734717915818976sn_d_d(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_744_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_d_fun_nat_assn,A4: fun_d_fun_nat_assn,Xs: list_d,Xs4: list_d,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                   => ( ( size_size_list_d(Xs) = size_size_list_nat(Xsi) )
                     => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                        & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L7820616366081188145_d_nat(I3,A,Xs,Xsi) = vEBT_L7820616366081188145_d_nat(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_745_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_d_fun_int_assn,A4: fun_d_fun_int_assn,Xs: list_d,Xs4: list_d,Xsi: list_int,Xsi2: list_int] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_d(Xs) = size_size_list_d(Xs4) )
         => ( ( size_size_list_int(Xsi) = size_size_list_int(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
                   => ( ( size_size_list_d(Xs) = size_size_list_int(Xsi) )
                     => ( ( nth_d(Xs,I5) = nth_d(Xs4,I5) )
                        & ( aa_nat_int(nth_int(Xsi),I5) = aa_nat_int(nth_int(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L7818125895572137869_d_int(I3,A,Xs,Xsi) = vEBT_L7818125895572137869_d_int(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_746_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_nat_fun_c_assn,A4: fun_nat_fun_c_assn,Xs: list_nat,Xs4: list_nat,Xsi: list_c,Xsi2: list_c] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xs4) )
         => ( ( size_size_list_c(Xsi) = size_size_list_c(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
                   => ( ( size_size_list_nat(Xs) = size_size_list_c(Xsi) )
                     => ( ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Xs4),I5) )
                        & ( nth_c(Xsi,I5) = nth_c(Xsi2,I5) ) ) ) ) )
             => ( vEBT_L8146214880390291086_nat_c(I3,A,Xs,Xsi) = vEBT_L8146214880390291086_nat_c(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_747_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_nat_fun_d_assn,A4: fun_nat_fun_d_assn,Xs: list_nat,Xs4: list_nat,Xsi: list_d,Xsi2: list_d] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xs4) )
         => ( ( size_size_list_d(Xsi) = size_size_list_d(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
                   => ( ( size_size_list_nat(Xs) = size_size_list_d(Xsi) )
                     => ( ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Xs4),I5) )
                        & ( nth_d(Xsi,I5) = nth_d(Xsi2,I5) ) ) ) ) )
             => ( vEBT_L8146214880390291087_nat_d(I3,A,Xs,Xsi) = vEBT_L8146214880390291087_nat_d(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_748_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_nat_fun_nat_assn,A4: fun_nat_fun_nat_assn,Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A4 )
       => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I5: nat] :
                  ( member_nat(I5,I3)
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
                   => ( ( size_size_list_nat(Xs) = size_size_list_nat(Xsi) )
                     => ( ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Xs4),I5) )
                        & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L7892069108621897922at_nat(I3,A,Xs,Xsi) = vEBT_L7892069108621897922at_nat(I7,A4,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_749_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C: rat,B2: rat] : minus_minus_rat(minus_minus_rat(A2,C),B2) = minus_minus_rat(minus_minus_rat(A2,B2),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_750_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_751_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_752_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( minus_minus_rat(A2,B2) = minus_minus_rat(C,D2) )
     => ( ( A2 = B2 )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_753_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( ( A2 = B2 )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_754_is__singletonE,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( is_sin2937591304547752795at_nat(A)
     => ~ ! [X2: produc3843707927480180839at_nat] : A != insert9069300056098147895at_nat(X2,bot_bo228742789529271731at_nat) ) ).

% is_singletonE
tff(fact_755_is__singletonE,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A))
     => ~ ! [X2: product_prod_nat_nat] : A != insert8211810215607154385at_nat(X2,bot_bo2099793752762293965at_nat) ) ).

% is_singletonE
tff(fact_756_is__singletonE,axiom,
    ! [A: set_bool] :
      ( is_singleton_bool(A)
     => ~ ! [X2: bool] : A != insert_bool2(X2,bot_bot_set_bool) ) ).

% is_singletonE
tff(fact_757_is__singletonE,axiom,
    ! [A: set_nat] :
      ( is_singleton_nat(A)
     => ~ ! [X2: nat] : A != insert_nat2(X2,bot_bot_set_nat) ) ).

% is_singletonE
tff(fact_758_is__singletonE,axiom,
    ! [A: set_int] :
      ( is_singleton_int(A)
     => ~ ! [X2: int] : A != insert_int2(X2,bot_bot_set_int) ) ).

% is_singletonE
tff(fact_759_is__singleton__def,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( is_sin2937591304547752795at_nat(A)
    <=> ? [X5: produc3843707927480180839at_nat] : A = insert9069300056098147895at_nat(X5,bot_bo228742789529271731at_nat) ) ).

% is_singleton_def
tff(fact_760_is__singleton__def,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_sin2850979758926227957at_nat,A))
    <=> ? [X5: product_prod_nat_nat] : A = insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat) ) ).

% is_singleton_def
tff(fact_761_is__singleton__def,axiom,
    ! [A: set_bool] :
      ( is_singleton_bool(A)
    <=> ? [X5: bool] : A = insert_bool2(X5,bot_bot_set_bool) ) ).

% is_singleton_def
tff(fact_762_is__singleton__def,axiom,
    ! [A: set_nat] :
      ( is_singleton_nat(A)
    <=> ? [X5: nat] : A = insert_nat2(X5,bot_bot_set_nat) ) ).

% is_singleton_def
tff(fact_763_is__singleton__def,axiom,
    ! [A: set_int] :
      ( is_singleton_int(A)
    <=> ? [X5: int] : A = insert_int2(X5,bot_bot_set_int) ) ).

% is_singleton_def
tff(fact_764_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: fun_c_fun_nat_assn,Xsi: list_nat] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( vEBT_L6585172037124793648_c_nat(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_c_fun_nat_assn(A,nth_c(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L6585172037124793648_c_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_765_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: fun_c_fun_int_assn,Xsi: list_int] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( vEBT_L6582681566615743372_c_int(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_c_fun_int_assn(A,nth_c(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L6582681566615743372_c_int(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_766_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: fun_d_fun_nat_assn,Xsi: list_nat] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( vEBT_L7820616366081188145_d_nat(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_d_fun_nat_assn(A,nth_d(Xs,I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L7820616366081188145_d_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_767_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_d,A: fun_d_fun_int_assn,Xsi: list_int] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( vEBT_L7818125895572137869_d_int(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_d_fun_int_assn(A,nth_d(Xs,I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L7818125895572137869_d_int(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_768_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_nat_fun_nat_assn,Xsi: list_nat] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L7892069108621897922at_nat(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L7892069108621897922at_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_769_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_nat_fun_int_assn,Xsi: list_int] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( vEBT_L7889578638112847646at_int(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L7889578638112847646at_int(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_770_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: fun_int_fun_nat_assn,Xsi: list_nat] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( vEBT_L8891422820522952478nt_nat(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(A,aa_nat_int(nth_int(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8891422820522952478nt_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_771_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_int,A: fun_int_fun_int_assn,Xsi: list_int] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( vEBT_L8888932350013902202nt_int(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(A,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xsi),I))),vEBT_L8888932350013902202nt_int(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_772_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_c,A: fun_c_fun_d_assn,Xsi: list_d] :
      ( ~ member_nat(I,I3)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( vEBT_L824818263033719201sn_c_d(insert_nat2(I,I3),A,Xs,Xsi) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_d_assn(aa_c_fun_d_assn(A,nth_c(Xs,I)),nth_d(Xsi,I))),vEBT_L824818263033719201sn_c_d(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_773_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => pp(aa_rat_bool(ord_less_rat(minus_minus_rat(A2,C)),minus_minus_rat(B2,C))) ) ).

% diff_strict_right_mono
tff(fact_774_diff__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),C)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),C))) ) ).

% diff_strict_right_mono
tff(fact_775_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C))) ) ).

% diff_strict_right_mono
tff(fact_776_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(B2),A2))
     => pp(aa_rat_bool(ord_less_rat(minus_minus_rat(C,A2)),minus_minus_rat(C,B2))) ) ).

% diff_strict_left_mono
tff(fact_777_diff__strict__left__mono,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),A2)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B2))) ) ).

% diff_strict_left_mono
tff(fact_778_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))) ) ).

% diff_strict_left_mono
tff(fact_779_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( minus_minus_rat(A2,B2) = minus_minus_rat(C,D2) )
     => ( pp(aa_rat_bool(ord_less_rat(A2),B2))
      <=> pp(aa_rat_bool(ord_less_rat(C),D2)) ) ) ).

% diff_eq_diff_less
tff(fact_780_diff__eq__diff__less,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),D2) )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D2)) ) ) ).

% diff_eq_diff_less
tff(fact_781_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D2)) ) ) ).

% diff_eq_diff_less
tff(fact_782_bot__empty__eq,axiom,
    ! [X6: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(bot_bo8489179610771909376l_bool,X6))
    <=> member4125802036136091927t_bool(X6,bot_bo2714889297266208186t_bool) ) ).

% bot_empty_eq
tff(fact_783_bot__empty__eq,axiom,
    ! [X6: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(bot_bo1984517246807695559t_bool,X6))
    <=> member8757157785044589968at_nat(X6,bot_bo228742789529271731at_nat) ) ).

% bot_empty_eq
tff(fact_784_bot__empty__eq,axiom,
    ! [X6: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(bot_bo3782518310583830573t_bool,X6))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X6),bot_bo2099793752762293965at_nat)) ) ).

% bot_empty_eq
tff(fact_785_bot__empty__eq,axiom,
    ! [X6: bool] :
      ( pp(aa_bool_bool(bot_bo4756488196683355415l_bool,X6))
    <=> member_bool(X6,bot_bot_set_bool) ) ).

% bot_empty_eq
tff(fact_786_bot__empty__eq,axiom,
    ! [X6: nat] :
      ( pp(aa_nat_bool(bot_bot_fun_nat_bool,X6))
    <=> member_nat(X6,bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_787_bot__empty__eq,axiom,
    ! [X6: int] :
      ( pp(aa_int_bool(bot_bot_fun_int_bool,X6))
    <=> member_int(X6,bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_788_Collect__empty__eq__bot,axiom,
    ! [P: fun_fu9160419385450930796l_bool] :
      ( ( collec3007137136614167897t_bool(P) = bot_bo2714889297266208186t_bool )
    <=> ( P = bot_bo8489179610771909376l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_789_Collect__empty__eq__bot,axiom,
    ! [P: fun_Pr1607443439640623155t_bool] :
      ( ( collec6321179662152712658at_nat(P) = bot_bo228742789529271731at_nat )
    <=> ( P = bot_bo1984517246807695559t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_790_Collect__empty__eq__bot,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( ( collec3392354462482085612at_nat(P) = bot_bo2099793752762293965at_nat )
    <=> ( P = bot_bo3782518310583830573t_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_791_Collect__empty__eq__bot,axiom,
    ! [P: fun_bool_bool] :
      ( ( collect_bool(P) = bot_bot_set_bool )
    <=> ( P = bot_bo4756488196683355415l_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_792_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_bool] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_793_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_bool] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_bool ) ) ).

% Collect_empty_eq_bot
tff(fact_794_remove__def,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] : remove5300788332298244428at_nat(X,A) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% remove_def
tff(fact_795_remove__def,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] : remove6466555014256735590at_nat(X,A) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% remove_def
tff(fact_796_remove__def,axiom,
    ! [X: bool,A: set_bool] : remove_bool(X,A) = minus_minus_set_bool(A,insert_bool2(X,bot_bot_set_bool)) ).

% remove_def
tff(fact_797_remove__def,axiom,
    ! [X: int,A: set_int] : remove_int(X,A) = minus_minus_set_int(A,insert_int2(X,bot_bot_set_int)) ).

% remove_def
tff(fact_798_remove__def,axiom,
    ! [X: nat,A: set_nat] : remove_nat(X,A) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(X,bot_bot_set_nat)) ).

% remove_def
tff(fact_799_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s5460976970255530739at_nat(L)))
     => ( insert8211810215607154385at_nat(nth_Pr7617993195940197384at_nat(L,I),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(L,I,X))) = insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(L)) ) ) ).

% insert_swap_set_eq
tff(fact_800_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1588839187904960729at_nat(L)))
     => ( insert9069300056098147895at_nat(nth_Pr1239524528086697198at_nat(L,I),set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(L,I,X))) = insert9069300056098147895at_nat(X,set_Pr3765526544606949372at_nat(L)) ) ) ).

% insert_swap_set_eq
tff(fact_801_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_bool,X: bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_bool(L)))
     => ( insert_bool2(aa_nat_bool(nth_bool(L),I),set_bool2(list_update_bool(L,I,X))) = insert_bool2(X,set_bool2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_802_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6829681357464350627n_assn(L)))
     => ( insert5290817439147925377n_assn(nth_Pr1769885009046257848n_assn(L,I),set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,X))) = insert5290817439147925377n_assn(X,set_Pr1139785259514867910n_assn(L)) ) ) ).

% insert_swap_set_eq
tff(fact_803_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_c,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L)))
     => ( insert_c(nth_c(L,I),set_c2(list_update_c(L,I,X))) = insert_c(X,set_c2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_804_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_d,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L)))
     => ( insert_d(nth_d(L,I),set_d2(list_update_d(L,I,X))) = insert_d(X,set_d2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_805_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( insert_nat2(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,X))) = insert_nat2(X,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_806_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( insert_int2(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,X))) = insert_int2(X,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_807_set__swap,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6829681357464350627n_assn(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s6829681357464350627n_assn(Xs)))
       => ( set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(list_u4534839942911652127n_assn(Xs,I,nth_Pr1769885009046257848n_assn(Xs,J)),J,nth_Pr1769885009046257848n_assn(Xs,I))) = set_Pr1139785259514867910n_assn(Xs) ) ) ) ).

% set_swap
tff(fact_808_set__swap,axiom,
    ! [I: nat,Xs: list_c,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_c(Xs)))
       => ( set_c2(list_update_c(list_update_c(Xs,I,nth_c(Xs,J)),J,nth_c(Xs,I))) = set_c2(Xs) ) ) ) ).

% set_swap
tff(fact_809_set__swap,axiom,
    ! [I: nat,Xs: list_d,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_d(Xs)))
       => ( set_d2(list_update_d(list_update_d(Xs,I,nth_d(Xs,J)),J,nth_d(Xs,I))) = set_d2(Xs) ) ) ) ).

% set_swap
tff(fact_810_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_811_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_812_Set_Ois__empty__def,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( is_emp7572232645145209145at_nat(A)
    <=> ( A = bot_bo228742789529271731at_nat ) ) ).

% Set.is_empty_def
tff(fact_813_Set_Ois__empty__def,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(is_emp1662574758705540307at_nat,A))
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% Set.is_empty_def
tff(fact_814_Set_Ois__empty__def,axiom,
    ! [A: set_bool] :
      ( is_empty_bool(A)
    <=> ( A = bot_bot_set_bool ) ) ).

% Set.is_empty_def
tff(fact_815_Set_Ois__empty__def,axiom,
    ! [A: set_nat] :
      ( is_empty_nat(A)
    <=> ( A = bot_bot_set_nat ) ) ).

% Set.is_empty_def
tff(fact_816_Set_Ois__empty__def,axiom,
    ! [A: set_int] :
      ( is_empty_int(A)
    <=> ( A = bot_bot_set_int ) ) ).

% Set.is_empty_def
tff(fact_817_distinct__swap,axiom,
    ! [I: nat,Xs: list_c,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_c(Xs)))
       => ( pp(aa_list_c_bool(distinct_c,list_update_c(list_update_c(Xs,I,nth_c(Xs,J)),J,nth_c(Xs,I))))
        <=> pp(aa_list_c_bool(distinct_c,Xs)) ) ) ) ).

% distinct_swap
tff(fact_818_distinct__swap,axiom,
    ! [I: nat,Xs: list_d,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_d(Xs)))
       => ( pp(aa_list_d_bool(distinct_d,list_update_d(list_update_d(Xs,I,nth_d(Xs,J)),J,nth_d(Xs,I))))
        <=> pp(aa_list_d_bool(distinct_d,Xs)) ) ) ) ).

% distinct_swap
tff(fact_819_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
       => ( pp(aa_list_nat_bool(distinct_nat,list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))))
        <=> pp(aa_list_nat_bool(distinct_nat,Xs)) ) ) ) ).

% distinct_swap
tff(fact_820_distinct__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
       => ( pp(aa_list_int_bool(distinct_int,list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))))
        <=> pp(aa_list_int_bool(distinct_int,Xs)) ) ) ) ).

% distinct_swap
tff(fact_821_list__ex__length,axiom,
    ! [P: fun_c_bool,Xs: list_c] :
      ( pp(aa_list_c_bool(list_ex_c(P),Xs))
    <=> ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_c(Xs)))
          & pp(aa_c_bool(P,nth_c(Xs,N2))) ) ) ).

% list_ex_length
tff(fact_822_list__ex__length,axiom,
    ! [P: fun_d_bool,Xs: list_d] :
      ( pp(aa_list_d_bool(list_ex_d(P),Xs))
    <=> ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_d(Xs)))
          & pp(aa_d_bool(P,nth_d(Xs,N2))) ) ) ).

% list_ex_length
tff(fact_823_list__ex__length,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( pp(aa_list_nat_bool(list_ex_nat(P),Xs))
    <=> ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
          & pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N2))) ) ) ).

% list_ex_length
tff(fact_824_list__ex__length,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( pp(aa_list_int_bool(list_ex_int(P),Xs))
    <=> ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_int(Xs)))
          & pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N2))) ) ) ).

% list_ex_length
tff(fact_825_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: bool] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),pure_assn(B2))))
    <=> ( ! [H3: produc3658429121746597890et_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H3))
           => pp(B2) )
        & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ).

% ent_pure_post_iff
tff(fact_826_pure__assn__eq__conv,axiom,
    ! [P: bool,Q: bool] :
      ( ( pure_assn(P) = pure_assn(Q) )
    <=> ( pp(P)
      <=> pp(Q) ) ) ).

% pure_assn_eq_conv
tff(fact_827_member__remove,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(X,remove5300788332298244428at_nat(Y,A))
    <=> ( member8757157785044589968at_nat(X,A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_828_member__remove,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),remove6466555014256735590at_nat(Y,A)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_829_member__remove,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( member_nat(X,remove_nat(Y,A))
    <=> ( member_nat(X,A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_830_member__remove,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( member_int(X,remove_int(Y,A))
    <=> ( member_int(X,A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_831_member__remove,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Y: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(X,remove7940366566026015955t_bool(Y,A))
    <=> ( member4125802036136091927t_bool(X,A)
        & ( X != Y ) ) ) ).

% member_remove
tff(fact_832_merge__pure__star,axiom,
    ! [A2: bool,B2: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,pure_assn(A2)),pure_assn(B2)) = pure_assn(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,A2),B2)) ).

% merge_pure_star
tff(fact_833_pure__false,axiom,
    pure_assn(fFalse) = bot_bot_assn ).

% pure_false
tff(fact_834_pure__assn__eq__false__iff,axiom,
    ! [P: bool] :
      ( ( pure_assn(P) = bot_bot_assn )
    <=> ~ pp(P) ) ).

% pure_assn_eq_false_iff
tff(fact_835_mod__pure__star__dist,axiom,
    ! [P: assn,B2: bool,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),pure_assn(B2))),H2))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
        & pp(B2) ) ) ).

% mod_pure_star_dist
tff(fact_836_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: bool,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),pure_assn(B2))),Q))
    <=> ( pp(B2)
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ) ).

% ent_pure_pre_iff
tff(fact_837_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
        & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),produc7507926704131184380et_nat(H2,bot_bot_set_nat))) ) ) ).

% mod_h_bot_iff(5)
tff(fact_838_mod__h__bot__iff_I1_J,axiom,
    ! [B2: bool,H2: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,pure_assn(B2)),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
    <=> pp(B2) ) ).

% mod_h_bot_iff(1)
tff(fact_839_list__ex__cong,axiom,
    ! [Xs: list_P5464809261938338413at_nat,Ys: list_P5464809261938338413at_nat,F3: fun_Pr1607443439640623155t_bool,G: fun_Pr1607443439640623155t_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X2,set_Pr3765526544606949372at_nat(Ys))
           => ( pp(aa_Pro1809015098595223084t_bool(F3,X2))
            <=> pp(aa_Pro1809015098595223084t_bool(G,X2)) ) )
       => ( list_e5181628221095382443at_nat(F3,Xs)
        <=> list_e5181628221095382443at_nat(G,Ys) ) ) ) ).

% list_ex_cong
tff(fact_840_list__ex__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,F3: fun_Pr7757163693017731481t_bool,G: fun_Pr7757163693017731481t_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),set_Pr5648618587558075414at_nat(Ys)))
           => ( pp(aa_Pro2232559814647311250t_bool(F3,X2))
            <=> pp(aa_Pro2232559814647311250t_bool(G,X2)) ) )
       => ( list_e7689525607045846085at_nat(F3,Xs)
        <=> list_e7689525607045846085at_nat(G,Ys) ) ) ) ).

% list_ex_cong
tff(fact_841_list__ex__cong,axiom,
    ! [Xs: list_f5857023252961307252t_bool,Ys: list_f5857023252961307252t_bool,F3: fun_fu9160419385450930796l_bool,G: fun_fu9160419385450930796l_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: fun_Pr6001797050664311790t_bool] :
            ( member4125802036136091927t_bool(X2,set_fu1437410266168021891t_bool(Ys))
           => ( pp(aa_fun5588795374382764133l_bool(F3,X2))
            <=> pp(aa_fun5588795374382764133l_bool(G,X2)) ) )
       => ( list_e3146091502832970034t_bool(F3,Xs)
        <=> list_e3146091502832970034t_bool(G,Ys) ) ) ) ).

% list_ex_cong
tff(fact_842_list__ex__cong,axiom,
    ! [Xs: list_int,Ys: list_int,F3: fun_int_bool,G: fun_int_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: int] :
            ( member_int(X2,set_int2(Ys))
           => ( pp(aa_int_bool(F3,X2))
            <=> pp(aa_int_bool(G,X2)) ) )
       => ( pp(aa_list_int_bool(list_ex_int(F3),Xs))
        <=> pp(aa_list_int_bool(list_ex_int(G),Ys)) ) ) ) ).

% list_ex_cong
tff(fact_843_list__ex__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F3: fun_nat_bool,G: fun_nat_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: nat] :
            ( member_nat(X2,set_nat2(Ys))
           => ( pp(aa_nat_bool(F3,X2))
            <=> pp(aa_nat_bool(G,X2)) ) )
       => ( pp(aa_list_nat_bool(list_ex_nat(F3),Xs))
        <=> pp(aa_list_nat_bool(list_ex_nat(G),Ys)) ) ) ) ).

% list_ex_cong
tff(fact_844_list__ex__cong,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,F3: fun_Pr8546840118223405289n_bool,G: fun_Pr8546840118223405289n_bool] :
      ( ( Xs = Ys )
     => ( ! [X2: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Ys))
           => ( pp(aa_Pro3608213857613363938n_bool(F3,X2))
            <=> pp(aa_Pro3608213857613363938n_bool(G,X2)) ) )
       => ( pp(aa_lis5212129917091495004n_bool(list_e1087220904399170805n_assn(F3),Xs))
        <=> pp(aa_lis5212129917091495004n_bool(list_e1087220904399170805n_assn(G),Ys)) ) ) ) ).

% list_ex_cong
tff(fact_845_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => pp(aa_list_int_bool(distinct_int,Xs)) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_846_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => pp(aa_list_nat_bool(distinct_nat,Xs)) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_847_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] : X != produc7507926704131184380et_nat(H,As) ).

% one_assn_raw.cases
tff(fact_848_bex2I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat,P: fun_se1534075524646452397t_bool] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),S)
     => ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),S)
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,A2),B2)) )
       => ? [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
            ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A3,B3),S)
            & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_849_bex2I,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),S))
     => ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),S))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) )
       => ? [A3: nat,B3: nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3)),S))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_850_bex2I,axiom,
    ! [A2: fun_Pr272337284026993017e_term,B2: produc8923325533196201883nteger,S: set_Pr5663546166630115725nteger,P: fun_fu1674507805476766837r_bool] :
      ( member6423679177046879214nteger(produc7384524951051686153nteger(A2,B2),S)
     => ( ( member6423679177046879214nteger(produc7384524951051686153nteger(A2,B2),S)
         => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(P,A2),B2)) )
       => ? [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] :
            ( member6423679177046879214nteger(produc7384524951051686153nteger(A3,B3),S)
            & pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_851_bex2I,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: produc3658429121746597890et_nat,S: set_Pr1106803562331729365et_nat,P: fun_fu7398481913972163267t_bool] :
      ( member8615881266069945910et_nat(produc8148480564485311129et_nat(A2,B2),S)
     => ( ( member8615881266069945910et_nat(produc8148480564485311129et_nat(A2,B2),S)
         => pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(P,A2),B2)) )
       => ? [A3: fun_Pr6001797050664311790t_bool,B3: produc3658429121746597890et_nat] :
            ( member8615881266069945910et_nat(produc8148480564485311129et_nat(A3,B3),S)
            & pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_852_bex2I,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: produc6108883039684432031et_nat,S: set_Pr9222568342844664434et_nat,P: fun_fu5758690788630904038t_bool] :
      ( member2818932950976722259et_nat(produc3083477295402903094et_nat(A2,B2),S)
     => ( ( member2818932950976722259et_nat(produc3083477295402903094et_nat(A2,B2),S)
         => pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(P,A2),B2)) )
       => ? [A3: fun_Pr6001797050664311790t_bool,B3: produc6108883039684432031et_nat] :
            ( member2818932950976722259et_nat(produc3083477295402903094et_nat(A3,B3),S)
            & pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_853_bex2I,axiom,
    ! [A2: fun_Pr9164807099015279978e_term,B2: product_prod_int_int,S: set_Pr233528076387193594nt_int,P: fun_fu4196931294019812808t_bool] :
      ( member690604328116145115nt_int(produc1560874099174258678nt_int(A2,B2),S)
     => ( ( member690604328116145115nt_int(produc1560874099174258678nt_int(A2,B2),S)
         => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(P,A2),B2)) )
       => ? [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] :
            ( member690604328116145115nt_int(produc1560874099174258678nt_int(A3,B3),S)
            & pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_854_bex2I,axiom,
    ! [A2: fun_in6429348457993719614e_term,B2: product_prod_int_int,S: set_Pr5033440245338114598nt_int,P: fun_fu5117548769634986908t_bool] :
      ( member500491123954006279nt_int(produc4852315488734385442nt_int(A2,B2),S)
     => ( ( member500491123954006279nt_int(produc4852315488734385442nt_int(A2,B2),S)
         => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(P,A2),B2)) )
       => ? [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] :
            ( member500491123954006279nt_int(produc4852315488734385442nt_int(A3,B3),S)
            & pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(P,A3),B3)) ) ) ) ).

% bex2I
tff(fact_855_pairself_Ocases,axiom,
    ! [X: produc6768802789813714928nt_int] :
      ~ ! [F4: fun_in6429348457993719614e_term,A3: int,B3: int] : X != produc4852315488734385442nt_int(F4,aa_int7462565939943104719nt_int(product_Pair_int_int(A3),B3)) ).

% pairself.cases
tff(fact_856_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A: fun_int_fun_int_assn,A4: fun_int_fun_int_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: int,Xi: int] :
              ( member_int(X2,set_int2(Xs4))
             => ( member_int(Xi,set_int2(Xsi2))
               => ( aa_int_assn(aa_int_fun_int_assn(A,X2),Xi) = aa_int_assn(aa_int_fun_int_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L74593716426352029nt_int(A,Xs,Xsi) = vEBT_L74593716426352029nt_int(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_857_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_nat,Xsi2: list_nat,A: fun_int_fun_nat_assn,A4: fun_int_fun_nat_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: int,Xi: nat] :
              ( member_int(X2,set_int2(Xs4))
             => ( member_nat(Xi,set_nat2(Xsi2))
               => ( aa_nat_assn(aa_int_fun_nat_assn(A,X2),Xi) = aa_nat_assn(aa_int_fun_nat_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L77084186935402305nt_nat(A,Xs,Xsi) = vEBT_L77084186935402305nt_nat(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_858_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_int,Xsi2: list_int,A: fun_nat_fun_int_assn,A4: fun_nat_fun_int_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: int] :
              ( member_nat(X2,set_nat2(Xs4))
             => ( member_int(Xi,set_int2(Xsi2))
               => ( aa_int_assn(aa_nat_fun_int_assn(A,X2),Xi) = aa_int_assn(aa_nat_fun_int_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L8298612041380073281at_int(A,Xs,Xsi) = vEBT_L8298612041380073281at_int(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_859_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_nat,Xsi2: list_nat,A: fun_nat_fun_nat_assn,A4: fun_nat_fun_nat_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: nat] :
              ( member_nat(X2,set_nat2(Xs4))
             => ( member_nat(Xi,set_nat2(Xsi2))
               => ( aa_nat_assn(aa_nat_fun_nat_assn(A,X2),Xi) = aa_nat_assn(aa_nat_fun_nat_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L8301102511889123557at_nat(A,Xs,Xsi) = vEBT_L8301102511889123557at_nat(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_860_list__assn__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Xsi: list_int,Xsi2: list_int,A: fun_Pr1050453709768890981t_assn,A4: fun_Pr1050453709768890981t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: product_prod_nat_nat,Xi: int] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),set_Pr5648618587558075414at_nat(Xs4)))
             => ( member_int(Xi,set_int2(Xsi2))
               => ( aa_int_assn(aa_Pro7338363084393913366t_assn(A,X2),Xi) = aa_int_assn(aa_Pro7338363084393913366t_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L6295016686126367274at_int(A,Xs,Xsi) = vEBT_L6295016686126367274at_int(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_861_list__assn__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Xsi: list_nat,Xsi2: list_nat,A: fun_Pr5858031453276514825t_assn,A4: fun_Pr5858031453276514825t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: product_prod_nat_nat,Xi: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),set_Pr5648618587558075414at_nat(Xs4)))
             => ( member_nat(Xi,set_nat2(Xsi2))
               => ( aa_nat_assn(aa_Pro6362827619187735994t_assn(A,X2),Xi) = aa_nat_assn(aa_Pro6362827619187735994t_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L6297507156635417550at_nat(A,Xs,Xsi) = vEBT_L6297507156635417550at_nat(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_862_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_P6011104703257516679at_nat,Xsi2: list_P6011104703257516679at_nat,A: fun_in6852455514944667241t_assn,A4: fun_in6852455514944667241t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: int,Xi: product_prod_nat_nat] :
              ( member_int(X2,set_int2(Xs4))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xi),set_Pr5648618587558075414at_nat(Xsi2)))
               => ( aa_Pro525162024677957869t_assn(aa_int7018050035326381266t_assn(A,X2),Xi) = aa_Pro525162024677957869t_assn(aa_int7018050035326381266t_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L3472024597287490640at_nat(A,Xs,Xsi) = vEBT_L3472024597287490640at_nat(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_863_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_P8527749157015355191n_assn,Xsi2: list_P8527749157015355191n_assn,A: fun_in2917358019923704601n_assn,A4: fun_in2917358019923704601n_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: int,Xi: produc6575502325842934193n_assn] :
              ( member_int(X2,set_int2(Xs4))
             => ( member7957490590177025114n_assn(Xi,set_Pr1139785259514867910n_assn(Xsi2))
               => ( aa_Pro2979818503705714077n_assn(aa_int4086841622967611778n_assn(A,X2),Xi) = aa_Pro2979818503705714077n_assn(aa_int4086841622967611778n_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L1620274777744359680n_assn(A,Xs,Xsi) = vEBT_L1620274777744359680n_assn(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_864_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_P6011104703257516679at_nat,Xsi2: list_P6011104703257516679at_nat,A: fun_na5930564906747369413t_assn,A4: fun_na5930564906747369413t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: product_prod_nat_nat] :
              ( member_nat(X2,set_nat2(Xs4))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xi),set_Pr5648618587558075414at_nat(Xsi2)))
               => ( aa_Pro525162024677957869t_assn(aa_nat6096159427129083438t_assn(A,X2),Xi) = aa_Pro525162024677957869t_assn(aa_nat6096159427129083438t_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L434182512823356844at_nat(A,Xs,Xsi) = vEBT_L434182512823356844at_nat(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_865_list__assn__cong,axiom,
    ! [Xs: list_nat,Xs4: list_nat,Xsi: list_P8527749157015355191n_assn,Xsi2: list_P8527749157015355191n_assn,A: fun_na2667955736958379125n_assn,A4: fun_na2667955736958379125n_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: nat,Xi: produc6575502325842934193n_assn] :
              ( member_nat(X2,set_nat2(Xs4))
             => ( member7957490590177025114n_assn(Xi,set_Pr1139785259514867910n_assn(Xsi2))
               => ( aa_Pro2979818503705714077n_assn(aa_nat3837439340002286302n_assn(A,X2),Xi) = aa_Pro2979818503705714077n_assn(aa_nat3837439340002286302n_assn(A4,X2),Xi) ) ) )
         => ( vEBT_L9083610150846024284n_assn(A,Xs,Xsi) = vEBT_L9083610150846024284n_assn(A4,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_866_distinct__Ex1,axiom,
    ! [Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( distin5355857245021852384at_nat(Xs)
     => ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_s1588839187904960729at_nat(Xs)))
            & ( nth_Pr1239524528086697198at_nat(Xs,X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_s1588839187904960729at_nat(Xs)))
                  & ( nth_Pr1239524528086697198at_nat(Xs,Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_867_distinct__Ex1,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( distin6923225563576452346at_nat(Xs)
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_s5460976970255530739at_nat(Xs)))
            & ( nth_Pr7617993195940197384at_nat(Xs,X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_s5460976970255530739at_nat(Xs)))
                  & ( nth_Pr7617993195940197384at_nat(Xs,Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_868_distinct__Ex1,axiom,
    ! [Xs: list_f5857023252961307252t_bool,X: fun_Pr6001797050664311790t_bool] :
      ( distin2825057960173885031t_bool(Xs)
     => ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_s1066996506189871328t_bool(Xs)))
            & ( nth_fu2663317684723071093t_bool(Xs,X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_s1066996506189871328t_bool(Xs)))
                  & ( nth_fu2663317684723071093t_bool(Xs,Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_869_distinct__Ex1,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,Xs))
     => ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_s6829681357464350627n_assn(Xs)))
            & ( nth_Pr1769885009046257848n_assn(Xs,X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_s6829681357464350627n_assn(Xs)))
                  & ( nth_Pr1769885009046257848n_assn(Xs,Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_870_distinct__Ex1,axiom,
    ! [Xs: list_c,X: c] :
      ( pp(aa_list_c_bool(distinct_c,Xs))
     => ( member_c(X,set_c2(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_size_list_c(Xs)))
            & ( nth_c(Xs,X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_size_list_c(Xs)))
                  & ( nth_c(Xs,Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_871_distinct__Ex1,axiom,
    ! [Xs: list_d,X: d] :
      ( pp(aa_list_d_bool(distinct_d,Xs))
     => ( member_d(X,set_d2(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_size_list_d(Xs)))
            & ( nth_d(Xs,X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_size_list_d(Xs)))
                  & ( nth_d(Xs,Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_872_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( member_nat(X,set_nat2(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_size_list_nat(Xs)))
            & ( aa_nat_nat(nth_nat(Xs),X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_size_list_nat(Xs)))
                  & ( aa_nat_nat(nth_nat(Xs),Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_873_distinct__Ex1,axiom,
    ! [Xs: list_int,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( member_int(X,set_int2(Xs))
       => ? [X2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),size_size_list_int(Xs)))
            & ( aa_nat_int(nth_int(Xs),X2) = X )
            & ! [Y3: nat] :
                ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y3),size_size_list_int(Xs)))
                  & ( aa_nat_int(nth_int(Xs),Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_874_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
    <=> pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H4,bot_bot_set_nat))) ) ).

% mod_h_bot_indep
tff(fact_875_distinct__list__update,axiom,
    ! [Xs: list_f5857023252961307252t_bool,A2: fun_Pr6001797050664311790t_bool,I: nat] :
      ( distin2825057960173885031t_bool(Xs)
     => ( ~ member4125802036136091927t_bool(A2,minus_5708602904570203829t_bool(set_fu1437410266168021891t_bool(Xs),insert969342481230318014t_bool(nth_fu2663317684723071093t_bool(Xs,I),bot_bo2714889297266208186t_bool)))
       => distin2825057960173885031t_bool(list_u922235751406726748t_bool(Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_876_distinct__list__update,axiom,
    ! [Xs: list_P8527749157015355191n_assn,A2: produc6575502325842934193n_assn,I: nat] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,Xs))
     => ( ~ member7957490590177025114n_assn(A2,minus_2301194108115264760n_assn(set_Pr1139785259514867910n_assn(Xs),insert5290817439147925377n_assn(nth_Pr1769885009046257848n_assn(Xs,I),bot_bo1176836662018730877n_assn)))
       => pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,list_u4534839942911652127n_assn(Xs,I,A2))) ) ) ).

% distinct_list_update
tff(fact_877_distinct__list__update,axiom,
    ! [Xs: list_c,A2: c,I: nat] :
      ( pp(aa_list_c_bool(distinct_c,Xs))
     => ( ~ member_c(A2,minus_minus_set_c(set_c2(Xs),insert_c(nth_c(Xs,I),bot_bot_set_c)))
       => pp(aa_list_c_bool(distinct_c,list_update_c(Xs,I,A2))) ) ) ).

% distinct_list_update
tff(fact_878_distinct__list__update,axiom,
    ! [Xs: list_d,A2: d,I: nat] :
      ( pp(aa_list_d_bool(distinct_d,Xs))
     => ( ~ member_d(A2,minus_minus_set_d(set_d2(Xs),insert_d(nth_d(Xs,I),bot_bot_set_d)))
       => pp(aa_list_d_bool(distinct_d,list_update_d(Xs,I,A2))) ) ) ).

% distinct_list_update
tff(fact_879_distinct__list__update,axiom,
    ! [Xs: list_P5464809261938338413at_nat,A2: produc3843707927480180839at_nat,I: nat] :
      ( distin5355857245021852384at_nat(Xs)
     => ( ~ member8757157785044589968at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,set_Pr3765526544606949372at_nat(Xs)),insert9069300056098147895at_nat(nth_Pr1239524528086697198at_nat(Xs,I),bot_bo228742789529271731at_nat)))
       => distin5355857245021852384at_nat(list_u4696772448584712917at_nat(Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_880_distinct__list__update,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A2: product_prod_nat_nat,I: nat] :
      ( distin6923225563576452346at_nat(Xs)
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,set_Pr5648618587558075414at_nat(Xs)),insert8211810215607154385at_nat(nth_Pr7617993195940197384at_nat(Xs,I),bot_bo2099793752762293965at_nat))))
       => distin6923225563576452346at_nat(list_u6180841689913720943at_nat(Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_881_distinct__list__update,axiom,
    ! [Xs: list_bool,A2: bool,I: nat] :
      ( distinct_bool(Xs)
     => ( ~ member_bool(A2,minus_minus_set_bool(set_bool2(Xs),insert_bool2(aa_nat_bool(nth_bool(Xs),I),bot_bot_set_bool)))
       => distinct_bool(list_update_bool(Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_882_distinct__list__update,axiom,
    ! [Xs: list_int,A2: int,I: nat] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( ~ member_int(A2,minus_minus_set_int(set_int2(Xs),insert_int2(aa_nat_int(nth_int(Xs),I),bot_bot_set_int)))
       => pp(aa_list_int_bool(distinct_int,list_update_int(Xs,I,A2))) ) ) ).

% distinct_list_update
tff(fact_883_distinct__list__update,axiom,
    ! [Xs: list_nat,A2: nat,I: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( ~ member_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat2(aa_nat_nat(nth_nat(Xs),I),bot_bot_set_nat)))
       => pp(aa_list_nat_bool(distinct_nat,list_update_nat(Xs,I,A2))) ) ) ).

% distinct_list_update
tff(fact_884_distinct__conv__nth,axiom,
    ! [Xs: list_c] :
      ( pp(aa_list_c_bool(distinct_c,Xs))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_c(Xs)))
         => ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_c(Xs)))
             => ( ( I4 != J2 )
               => ( nth_c(Xs,I4) != nth_c(Xs,J2) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_885_distinct__conv__nth,axiom,
    ! [Xs: list_d] :
      ( pp(aa_list_d_bool(distinct_d,Xs))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_d(Xs)))
         => ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_d(Xs)))
             => ( ( I4 != J2 )
               => ( nth_d(Xs,I4) != nth_d(Xs,J2) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_886_distinct__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_nat(Xs)))
             => ( ( I4 != J2 )
               => ( aa_nat_nat(nth_nat(Xs),I4) != aa_nat_nat(nth_nat(Xs),J2) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_887_distinct__conv__nth,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
         => ! [J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
             => ( ( I4 != J2 )
               => ( aa_nat_int(nth_int(Xs),I4) != aa_nat_int(nth_int(Xs),J2) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_888_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_c,I: nat,J: nat] :
      ( pp(aa_list_c_bool(distinct_c,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_c(Xs)))
         => ( ( nth_c(Xs,I) = nth_c(Xs,J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_889_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_d,I: nat,J: nat] :
      ( pp(aa_list_d_bool(distinct_d,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_d(Xs)))
         => ( ( nth_d(Xs,I) = nth_d(Xs,J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_890_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
         => ( ( aa_nat_nat(nth_nat(Xs),I) = aa_nat_nat(nth_nat(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_891_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
         => ( ( aa_nat_int(nth_int(Xs),I) = aa_nat_int(nth_int(Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_892_all__set__conv__nth,axiom,
    ! [L: list_P8527749157015355191n_assn,P: fun_Pr8546840118223405289n_bool] :
      ( ! [X5: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X5,set_Pr1139785259514867910n_assn(L))
         => pp(aa_Pro3608213857613363938n_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6829681357464350627n_assn(L)))
         => pp(aa_Pro3608213857613363938n_bool(P,nth_Pr1769885009046257848n_assn(L,I4))) ) ) ).

% all_set_conv_nth
tff(fact_893_all__set__conv__nth,axiom,
    ! [L: list_c,P: fun_c_bool] :
      ( ! [X5: c] :
          ( member_c(X5,set_c2(L))
         => pp(aa_c_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_c(L)))
         => pp(aa_c_bool(P,nth_c(L,I4))) ) ) ).

% all_set_conv_nth
tff(fact_894_all__set__conv__nth,axiom,
    ! [L: list_d,P: fun_d_bool] :
      ( ! [X5: d] :
          ( member_d(X5,set_d2(L))
         => pp(aa_d_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_d(L)))
         => pp(aa_d_bool(P,nth_d(L,I4))) ) ) ).

% all_set_conv_nth
tff(fact_895_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( member_nat(X5,set_nat2(L))
         => pp(aa_nat_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(L)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_896_all__set__conv__nth,axiom,
    ! [L: list_int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( member_int(X5,set_int2(L))
         => pp(aa_int_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(L)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(L),I4))) ) ) ).

% all_set_conv_nth
tff(fact_897_nth__mem,axiom,
    ! [N: nat,Xs: list_P5464809261938338413at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s1588839187904960729at_nat(Xs)))
     => member8757157785044589968at_nat(nth_Pr1239524528086697198at_nat(Xs,N),set_Pr3765526544606949372at_nat(Xs)) ) ).

% nth_mem
tff(fact_898_nth__mem,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,N)),set_Pr5648618587558075414at_nat(Xs))) ) ).

% nth_mem
tff(fact_899_nth__mem,axiom,
    ! [N: nat,Xs: list_f5857023252961307252t_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s1066996506189871328t_bool(Xs)))
     => member4125802036136091927t_bool(nth_fu2663317684723071093t_bool(Xs,N),set_fu1437410266168021891t_bool(Xs)) ) ).

% nth_mem
tff(fact_900_nth__mem,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6829681357464350627n_assn(Xs)))
     => member7957490590177025114n_assn(nth_Pr1769885009046257848n_assn(Xs,N),set_Pr1139785259514867910n_assn(Xs)) ) ).

% nth_mem
tff(fact_901_nth__mem,axiom,
    ! [N: nat,Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(Xs)))
     => member_c(nth_c(Xs,N),set_c2(Xs)) ) ).

% nth_mem
tff(fact_902_nth__mem,axiom,
    ! [N: nat,Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(Xs)))
     => member_d(nth_d(Xs,N),set_d2(Xs)) ) ).

% nth_mem
tff(fact_903_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => member_nat(aa_nat_nat(nth_nat(Xs),N),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_904_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => member_int(aa_nat_int(nth_int(Xs),N),set_int2(Xs)) ) ).

% nth_mem
tff(fact_905_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,P: fun_Pr8546840118223405289n_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6829681357464350627n_assn(Xs)))
     => ( ! [X2: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
           => pp(aa_Pro3608213857613363938n_bool(P,X2)) )
       => pp(aa_Pro3608213857613363938n_bool(P,nth_Pr1769885009046257848n_assn(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_906_list__ball__nth,axiom,
    ! [N: nat,Xs: list_c,P: fun_c_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(Xs)))
     => ( ! [X2: c] :
            ( member_c(X2,set_c2(Xs))
           => pp(aa_c_bool(P,X2)) )
       => pp(aa_c_bool(P,nth_c(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_907_list__ball__nth,axiom,
    ! [N: nat,Xs: list_d,P: fun_d_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(Xs)))
     => ( ! [X2: d] :
            ( member_d(X2,set_d2(Xs))
           => pp(aa_d_bool(P,X2)) )
       => pp(aa_d_bool(P,nth_d(Xs,N))) ) ) ).

% list_ball_nth
tff(fact_908_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( ! [X2: nat] :
            ( member_nat(X2,set_nat2(Xs))
           => pp(aa_nat_bool(P,X2)) )
       => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_909_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( ! [X2: int] :
            ( member_int(X2,set_int2(Xs))
           => pp(aa_int_bool(P,X2)) )
       => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),N))) ) ) ).

% list_ball_nth
tff(fact_910_in__set__conv__nth,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s1588839187904960729at_nat(Xs)))
          & ( nth_Pr1239524528086697198at_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_911_in__set__conv__nth,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s5460976970255530739at_nat(Xs)))
          & ( nth_Pr7617993195940197384at_nat(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_912_in__set__conv__nth,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s1066996506189871328t_bool(Xs)))
          & ( nth_fu2663317684723071093t_bool(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_913_in__set__conv__nth,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6829681357464350627n_assn(Xs)))
          & ( nth_Pr1769885009046257848n_assn(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_914_in__set__conv__nth,axiom,
    ! [X: c,Xs: list_c] :
      ( member_c(X,set_c2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_c(Xs)))
          & ( nth_c(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_915_in__set__conv__nth,axiom,
    ! [X: d,Xs: list_d] :
      ( member_d(X,set_d2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_d(Xs)))
          & ( nth_d(Xs,I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_916_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
          & ( aa_nat_nat(nth_nat(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_917_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
    <=> ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
          & ( aa_nat_int(nth_int(Xs),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_918_all__nth__imp__all__set,axiom,
    ! [Xs: list_P5464809261938338413at_nat,P: fun_Pr1607443439640623155t_bool,X: produc3843707927480180839at_nat] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s1588839187904960729at_nat(Xs)))
         => pp(aa_Pro1809015098595223084t_bool(P,nth_Pr1239524528086697198at_nat(Xs,I5))) )
     => ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
       => pp(aa_Pro1809015098595223084t_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_919_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr7757163693017731481t_bool,X: product_prod_nat_nat] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s5460976970255530739at_nat(Xs)))
         => pp(aa_Pro2232559814647311250t_bool(P,nth_Pr7617993195940197384at_nat(Xs,I5))) )
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => pp(aa_Pro2232559814647311250t_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_920_all__nth__imp__all__set,axiom,
    ! [Xs: list_f5857023252961307252t_bool,P: fun_fu9160419385450930796l_bool,X: fun_Pr6001797050664311790t_bool] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s1066996506189871328t_bool(Xs)))
         => pp(aa_fun5588795374382764133l_bool(P,nth_fu2663317684723071093t_bool(Xs,I5))) )
     => ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
       => pp(aa_fun5588795374382764133l_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_921_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr8546840118223405289n_bool,X: produc6575502325842934193n_assn] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_s6829681357464350627n_assn(Xs)))
         => pp(aa_Pro3608213857613363938n_bool(P,nth_Pr1769885009046257848n_assn(Xs,I5))) )
     => ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
       => pp(aa_Pro3608213857613363938n_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_922_all__nth__imp__all__set,axiom,
    ! [Xs: list_c,P: fun_c_bool,X: c] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_c(Xs)))
         => pp(aa_c_bool(P,nth_c(Xs,I5))) )
     => ( member_c(X,set_c2(Xs))
       => pp(aa_c_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_923_all__nth__imp__all__set,axiom,
    ! [Xs: list_d,P: fun_d_bool,X: d] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_d(Xs)))
         => pp(aa_d_bool(P,nth_d(Xs,I5))) )
     => ( member_d(X,set_d2(Xs))
       => pp(aa_d_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_924_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_bool,X: nat] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I5))) )
     => ( member_nat(X,set_nat2(Xs))
       => pp(aa_nat_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_925_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_bool,X: int] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I5))) )
     => ( member_int(X,set_int2(Xs))
       => pp(aa_int_bool(P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_926_all__set__conv__all__nth,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr8546840118223405289n_bool] :
      ( ! [X5: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X5,set_Pr1139785259514867910n_assn(Xs))
         => pp(aa_Pro3608213857613363938n_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_s6829681357464350627n_assn(Xs)))
         => pp(aa_Pro3608213857613363938n_bool(P,nth_Pr1769885009046257848n_assn(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_927_all__set__conv__all__nth,axiom,
    ! [Xs: list_c,P: fun_c_bool] :
      ( ! [X5: c] :
          ( member_c(X5,set_c2(Xs))
         => pp(aa_c_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_c(Xs)))
         => pp(aa_c_bool(P,nth_c(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_928_all__set__conv__all__nth,axiom,
    ! [Xs: list_d,P: fun_d_bool] :
      ( ! [X5: d] :
          ( member_d(X5,set_d2(Xs))
         => pp(aa_d_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_d(Xs)))
         => pp(aa_d_bool(P,nth_d(Xs,I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_929_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( member_nat(X5,set_nat2(Xs))
         => pp(aa_nat_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_930_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_bool] :
      ( ! [X5: int] :
          ( member_int(X5,set_int2(Xs))
         => pp(aa_int_bool(P,X5)) )
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),size_size_list_int(Xs)))
         => pp(aa_int_bool(P,aa_nat_int(nth_int(Xs),I4))) ) ) ).

% all_set_conv_all_nth
tff(fact_931_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1588839187904960729at_nat(L)))
     => ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: produc3843707927480180839at_nat] : member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_932_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s5460976970255530739at_nat(L)))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(L,I,Y))))
      <=> ( ( X = Y )
          | ! [Y4: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(L,I,Y4)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_933_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_f5857023252961307252t_bool,X: fun_Pr6001797050664311790t_bool,Y: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1066996506189871328t_bool(L)))
     => ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: fun_Pr6001797050664311790t_bool] : member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_934_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6829681357464350627n_assn(L)))
     => ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: produc6575502325842934193n_assn] : member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_935_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_c,X: c,Y: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L)))
     => ( member_c(X,set_c2(list_update_c(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: c] : member_c(X,set_c2(list_update_c(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_936_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_d,X: d,Y: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L)))
     => ( member_d(X,set_d2(list_update_d(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: d] : member_d(X,set_d2(list_update_d(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_937_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( member_nat(X,set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: nat] : member_nat(X,set_nat2(list_update_nat(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_938_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( member_int(X,set_int2(list_update_int(L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: int] : member_int(X,set_int2(list_update_int(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_939_in__set__upd__cases,axiom,
    ! [X: produc3843707927480180839at_nat,L: list_P5464809261938338413at_nat,I: nat,Y: produc3843707927480180839at_nat] :
      ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1588839187904960729at_nat(L)))
         => ( X != Y ) )
       => member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(L)) ) ) ).

% in_set_upd_cases
tff(fact_940_in__set__upd__cases,axiom,
    ! [X: product_prod_nat_nat,L: list_P6011104703257516679at_nat,I: nat,Y: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(L,I,Y))))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s5460976970255530739at_nat(L)))
         => ( X != Y ) )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(L))) ) ) ).

% in_set_upd_cases
tff(fact_941_in__set__upd__cases,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,L: list_f5857023252961307252t_bool,I: nat,Y: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1066996506189871328t_bool(L)))
         => ( X != Y ) )
       => member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(L)) ) ) ).

% in_set_upd_cases
tff(fact_942_in__set__upd__cases,axiom,
    ! [X: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn,I: nat,Y: produc6575502325842934193n_assn] :
      ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6829681357464350627n_assn(L)))
         => ( X != Y ) )
       => member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(L)) ) ) ).

% in_set_upd_cases
tff(fact_943_in__set__upd__cases,axiom,
    ! [X: c,L: list_c,I: nat,Y: c] :
      ( member_c(X,set_c2(list_update_c(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L)))
         => ( X != Y ) )
       => member_c(X,set_c2(L)) ) ) ).

% in_set_upd_cases
tff(fact_944_in__set__upd__cases,axiom,
    ! [X: d,L: list_d,I: nat,Y: d] :
      ( member_d(X,set_d2(list_update_d(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L)))
         => ( X != Y ) )
       => member_d(X,set_d2(L)) ) ) ).

% in_set_upd_cases
tff(fact_945_in__set__upd__cases,axiom,
    ! [X: nat,L: list_nat,I: nat,Y: nat] :
      ( member_nat(X,set_nat2(list_update_nat(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
         => ( X != Y ) )
       => member_nat(X,set_nat2(L)) ) ) ).

% in_set_upd_cases
tff(fact_946_in__set__upd__cases,axiom,
    ! [X: int,L: list_int,I: nat,Y: int] :
      ( member_int(X,set_int2(list_update_int(L,I,Y)))
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
         => ( X != Y ) )
       => member_int(X,set_int2(L)) ) ) ).

% in_set_upd_cases
tff(fact_947_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1588839187904960729at_nat(L)))
     => ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(L,I,Y)))
      <=> ( ( X = Y )
          | ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(L))
            & ! [Y4: produc3843707927480180839at_nat] : member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_948_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s5460976970255530739at_nat(L)))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(L,I,Y))))
      <=> ( ( X = Y )
          | ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(L)))
            & ! [Y4: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(L,I,Y4)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_949_in__set__upd__eq,axiom,
    ! [I: nat,L: list_f5857023252961307252t_bool,X: fun_Pr6001797050664311790t_bool,Y: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s1066996506189871328t_bool(L)))
     => ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(L,I,Y)))
      <=> ( ( X = Y )
          | ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(L))
            & ! [Y4: fun_Pr6001797050664311790t_bool] : member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_950_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s6829681357464350627n_assn(L)))
     => ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
      <=> ( ( X = Y )
          | ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(L))
            & ! [Y4: produc6575502325842934193n_assn] : member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_951_in__set__upd__eq,axiom,
    ! [I: nat,L: list_c,X: c,Y: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(L)))
     => ( member_c(X,set_c2(list_update_c(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_c(X,set_c2(L))
            & ! [Y4: c] : member_c(X,set_c2(list_update_c(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_952_in__set__upd__eq,axiom,
    ! [I: nat,L: list_d,X: d,Y: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(L)))
     => ( member_d(X,set_d2(list_update_d(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_d(X,set_d2(L))
            & ! [Y4: d] : member_d(X,set_d2(list_update_d(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_953_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
     => ( member_nat(X,set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_nat(X,set_nat2(L))
            & ! [Y4: nat] : member_nat(X,set_nat2(list_update_nat(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_954_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
     => ( member_int(X,set_int2(list_update_int(L,I,Y)))
      <=> ( ( X = Y )
          | ( member_int(X,set_int2(L))
            & ! [Y4: int] : member_int(X,set_int2(list_update_int(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_955_set__update__memI,axiom,
    ! [N: nat,Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s1588839187904960729at_nat(Xs)))
     => member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,N,X))) ) ).

% set_update_memI
tff(fact_956_set__update__memI,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,N,X)))) ) ).

% set_update_memI
tff(fact_957_set__update__memI,axiom,
    ! [N: nat,Xs: list_f5857023252961307252t_bool,X: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s1066996506189871328t_bool(Xs)))
     => member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(Xs,N,X))) ) ).

% set_update_memI
tff(fact_958_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6829681357464350627n_assn(Xs)))
     => member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,N,X))) ) ).

% set_update_memI
tff(fact_959_set__update__memI,axiom,
    ! [N: nat,Xs: list_c,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(Xs)))
     => member_c(X,set_c2(list_update_c(Xs,N,X))) ) ).

% set_update_memI
tff(fact_960_set__update__memI,axiom,
    ! [N: nat,Xs: list_d,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(Xs)))
     => member_d(X,set_d2(list_update_d(Xs,N,X))) ) ).

% set_update_memI
tff(fact_961_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => member_nat(X,set_nat2(list_update_nat(Xs,N,X))) ) ).

% set_update_memI
tff(fact_962_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => member_int(X,set_int2(list_update_int(Xs,N,X))) ) ).

% set_update_memI
tff(fact_963_set__update__distinct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat,X: produc6575502325842934193n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s6829681357464350627n_assn(Xs)))
       => ( set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,N,X)) = insert5290817439147925377n_assn(X,minus_2301194108115264760n_assn(set_Pr1139785259514867910n_assn(Xs),insert5290817439147925377n_assn(nth_Pr1769885009046257848n_assn(Xs,N),bot_bo1176836662018730877n_assn))) ) ) ) ).

% set_update_distinct
tff(fact_964_set__update__distinct,axiom,
    ! [Xs: list_c,N: nat,X: c] :
      ( pp(aa_list_c_bool(distinct_c,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(Xs)))
       => ( set_c2(list_update_c(Xs,N,X)) = insert_c(X,minus_minus_set_c(set_c2(Xs),insert_c(nth_c(Xs,N),bot_bot_set_c))) ) ) ) ).

% set_update_distinct
tff(fact_965_set__update__distinct,axiom,
    ! [Xs: list_d,N: nat,X: d] :
      ( pp(aa_list_d_bool(distinct_d,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(Xs)))
       => ( set_d2(list_update_d(Xs,N,X)) = insert_d(X,minus_minus_set_d(set_d2(Xs),insert_d(nth_d(Xs,N),bot_bot_set_d))) ) ) ) ).

% set_update_distinct
tff(fact_966_set__update__distinct,axiom,
    ! [Xs: list_P5464809261938338413at_nat,N: nat,X: produc3843707927480180839at_nat] :
      ( distin5355857245021852384at_nat(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s1588839187904960729at_nat(Xs)))
       => ( set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,N,X)) = insert9069300056098147895at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,set_Pr3765526544606949372at_nat(Xs)),insert9069300056098147895at_nat(nth_Pr1239524528086697198at_nat(Xs,N),bot_bo228742789529271731at_nat))) ) ) ) ).

% set_update_distinct
tff(fact_967_set__update__distinct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( distin6923225563576452346at_nat(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s5460976970255530739at_nat(Xs)))
       => ( set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,N,X)) = insert8211810215607154385at_nat(X,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,set_Pr5648618587558075414at_nat(Xs)),insert8211810215607154385at_nat(nth_Pr7617993195940197384at_nat(Xs,N),bot_bo2099793752762293965at_nat))) ) ) ) ).

% set_update_distinct
tff(fact_968_set__update__distinct,axiom,
    ! [Xs: list_bool,N: nat,X: bool] :
      ( distinct_bool(Xs)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_bool(Xs)))
       => ( set_bool2(list_update_bool(Xs,N,X)) = insert_bool2(X,minus_minus_set_bool(set_bool2(Xs),insert_bool2(aa_nat_bool(nth_bool(Xs),N),bot_bot_set_bool))) ) ) ) ).

% set_update_distinct
tff(fact_969_set__update__distinct,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => ( set_int2(list_update_int(Xs,N,X)) = insert_int2(X,minus_minus_set_int(set_int2(Xs),insert_int2(aa_nat_int(nth_int(Xs),N),bot_bot_set_int))) ) ) ) ).

% set_update_distinct
tff(fact_970_set__update__distinct,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => ( set_nat2(list_update_nat(Xs,N,X)) = insert_nat2(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat2(aa_nat_nat(nth_nat(Xs),N),bot_bot_set_nat))) ) ) ) ).

% set_update_distinct
tff(fact_971_prod_Oinject,axiom,
    ! [X1: fun_Pr272337284026993017e_term,X22: produc8923325533196201883nteger,Y1: fun_Pr272337284026993017e_term,Y22: produc8923325533196201883nteger] :
      ( ( produc7384524951051686153nteger(X1,X22) = produc7384524951051686153nteger(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_972_prod_Oinject,axiom,
    ! [X1: fun_Pr6001797050664311790t_bool,X22: produc3658429121746597890et_nat,Y1: fun_Pr6001797050664311790t_bool,Y22: produc3658429121746597890et_nat] :
      ( ( produc8148480564485311129et_nat(X1,X22) = produc8148480564485311129et_nat(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_973_prod_Oinject,axiom,
    ! [X1: fun_Pr6001797050664311790t_bool,X22: produc6108883039684432031et_nat,Y1: fun_Pr6001797050664311790t_bool,Y22: produc6108883039684432031et_nat] :
      ( ( produc3083477295402903094et_nat(X1,X22) = produc3083477295402903094et_nat(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_974_prod_Oinject,axiom,
    ! [X1: fun_Pr9164807099015279978e_term,X22: product_prod_int_int,Y1: fun_Pr9164807099015279978e_term,Y22: product_prod_int_int] :
      ( ( produc1560874099174258678nt_int(X1,X22) = produc1560874099174258678nt_int(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_975_prod_Oinject,axiom,
    ! [X1: fun_in6429348457993719614e_term,X22: product_prod_int_int,Y1: fun_in6429348457993719614e_term,Y22: product_prod_int_int] :
      ( ( produc4852315488734385442nt_int(X1,X22) = produc4852315488734385442nt_int(Y1,Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_976_old_Oprod_Oinject,axiom,
    ! [A2: fun_Pr272337284026993017e_term,B2: produc8923325533196201883nteger,A5: fun_Pr272337284026993017e_term,B5: produc8923325533196201883nteger] :
      ( ( produc7384524951051686153nteger(A2,B2) = produc7384524951051686153nteger(A5,B5) )
    <=> ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
tff(fact_977_old_Oprod_Oinject,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: produc3658429121746597890et_nat,A5: fun_Pr6001797050664311790t_bool,B5: produc3658429121746597890et_nat] :
      ( ( produc8148480564485311129et_nat(A2,B2) = produc8148480564485311129et_nat(A5,B5) )
    <=> ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
tff(fact_978_old_Oprod_Oinject,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: produc6108883039684432031et_nat,A5: fun_Pr6001797050664311790t_bool,B5: produc6108883039684432031et_nat] :
      ( ( produc3083477295402903094et_nat(A2,B2) = produc3083477295402903094et_nat(A5,B5) )
    <=> ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
tff(fact_979_old_Oprod_Oinject,axiom,
    ! [A2: fun_Pr9164807099015279978e_term,B2: product_prod_int_int,A5: fun_Pr9164807099015279978e_term,B5: product_prod_int_int] :
      ( ( produc1560874099174258678nt_int(A2,B2) = produc1560874099174258678nt_int(A5,B5) )
    <=> ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
tff(fact_980_old_Oprod_Oinject,axiom,
    ! [A2: fun_in6429348457993719614e_term,B2: product_prod_int_int,A5: fun_in6429348457993719614e_term,B5: product_prod_int_int] :
      ( ( produc4852315488734385442nt_int(A2,B2) = produc4852315488734385442nt_int(A5,B5) )
    <=> ( ( A2 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
tff(fact_981_distinct__union,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(distinct_int,union_int(Xs,Ys)))
    <=> pp(aa_list_int_bool(distinct_int,Ys)) ) ).

% distinct_union
tff(fact_982_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,union_nat(Xs,Ys)))
    <=> pp(aa_list_nat_bool(distinct_nat,Ys)) ) ).

% distinct_union
tff(fact_983_set__remove1__eq,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,Xs))
     => ( set_Pr1139785259514867910n_assn(remove1670527618125605709n_assn(X,Xs)) = minus_2301194108115264760n_assn(set_Pr1139785259514867910n_assn(Xs),insert5290817439147925377n_assn(X,bot_bo1176836662018730877n_assn)) ) ) ).

% set_remove1_eq
tff(fact_984_set__remove1__eq,axiom,
    ! [Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( distin5355857245021852384at_nat(Xs)
     => ( set_Pr3765526544606949372at_nat(remove8895909992688781315at_nat(X,Xs)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,set_Pr3765526544606949372at_nat(Xs)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ) ) ).

% set_remove1_eq
tff(fact_985_set__remove1__eq,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( distin6923225563576452346at_nat(Xs)
     => ( set_Pr5648618587558075414at_nat(remove5766854097993331357at_nat(X,Xs)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,set_Pr5648618587558075414at_nat(Xs)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ) ) ).

% set_remove1_eq
tff(fact_986_set__remove1__eq,axiom,
    ! [Xs: list_bool,X: bool] :
      ( distinct_bool(Xs)
     => ( set_bool2(remove1_bool(X,Xs)) = minus_minus_set_bool(set_bool2(Xs),insert_bool2(X,bot_bot_set_bool)) ) ) ).

% set_remove1_eq
tff(fact_987_set__remove1__eq,axiom,
    ! [Xs: list_int,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( set_int2(remove1_int(X,Xs)) = minus_minus_set_int(set_int2(Xs),insert_int2(X,bot_bot_set_int)) ) ) ).

% set_remove1_eq
tff(fact_988_set__remove1__eq,axiom,
    ! [Xs: list_nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( set_nat2(remove1_nat(X,Xs)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat2(X,bot_bot_set_nat)) ) ) ).

% set_remove1_eq
tff(fact_989_is__empty__set,axiom,
    ! [Xs: list_int] :
      ( is_empty_int(set_int2(Xs))
    <=> pp(aa_list_int_bool(null_int,Xs)) ) ).

% is_empty_set
tff(fact_990_is__empty__set,axiom,
    ! [Xs: list_nat] :
      ( is_empty_nat(set_nat2(Xs))
    <=> pp(aa_list_nat_bool(null_nat,Xs)) ) ).

% is_empty_set
tff(fact_991_is__empty__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( is_emp7749065762491207043n_assn(set_Pr1139785259514867910n_assn(Xs))
    <=> pp(aa_lis5212129917091495004n_bool(null_P7596217980778516671n_assn,Xs)) ) ).

% is_empty_set
tff(fact_992_set__removeAll,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(remove1763081955827946941n_assn(X,Xs)) = minus_2301194108115264760n_assn(set_Pr1139785259514867910n_assn(Xs),insert5290817439147925377n_assn(X,bot_bo1176836662018730877n_assn)) ).

% set_removeAll
tff(fact_993_set__removeAll,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] : set_Pr3765526544606949372at_nat(remove4908499291543849587at_nat(X,Xs)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,set_Pr3765526544606949372at_nat(Xs)),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)) ).

% set_removeAll
tff(fact_994_set__removeAll,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(remove3673390508374433037at_nat(X,Xs)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,set_Pr5648618587558075414at_nat(Xs)),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)) ).

% set_removeAll
tff(fact_995_set__removeAll,axiom,
    ! [X: bool,Xs: list_bool] : set_bool2(removeAll_bool(X,Xs)) = minus_minus_set_bool(set_bool2(Xs),insert_bool2(X,bot_bot_set_bool)) ).

% set_removeAll
tff(fact_996_set__removeAll,axiom,
    ! [X: int,Xs: list_int] : set_int2(removeAll_int(X,Xs)) = minus_minus_set_int(set_int2(Xs),insert_int2(X,bot_bot_set_int)) ).

% set_removeAll
tff(fact_997_set__removeAll,axiom,
    ! [X: nat,Xs: list_nat] : set_nat2(removeAll_nat(X,Xs)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_nat2(Xs)),insert_nat2(X,bot_bot_set_nat)) ).

% set_removeAll
tff(fact_998_listrel1__iff__update,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Ys),listre8180072825634266816at_nat(R2))
    <=> ? [Y4: set_Pr1261947904930325089at_nat,N2: nat] :
          ( member8757157785044589968at_nat(produc2922128104949294807at_nat(nth_se5679702624988424552at_nat(Xs,N2),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_s8736152011456118867at_nat(Xs)))
          & ( Ys = list_u8444657558853818831at_nat(Xs,N2,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_999_listrel1__iff__update,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(Xs,Ys),listrel1_c(R2))
    <=> ? [Y4: c,N2: nat] :
          ( member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,nth_c(Xs,N2)),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_c(Xs)))
          & ( Ys = list_update_c(Xs,N2,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1000_listrel1__iff__update,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(Xs,Ys),listrel1_d(R2))
    <=> ? [Y4: d,N2: nat] :
          ( member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,nth_d(Xs,N2)),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_d(Xs)))
          & ( Ys = list_update_d(Xs,N2,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1001_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
    <=> ? [Y4: nat,N2: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N2)),Y4)),R2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
          & ( Ys = list_update_nat(Xs,N2,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1002_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
    <=> ? [Y4: int,N2: nat] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),N2)),Y4),R2)
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_int(Xs)))
          & ( Ys = list_update_int(Xs,N2,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_1003_List_Oset__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(insert8944034826898310173at_nat(X,Xs)) = insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(Xs)) ).

% List.set_insert
tff(fact_1004_List_Oset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] : set_Pr3765526544606949372at_nat(insert4149901806602606979at_nat(X,Xs)) = insert9069300056098147895at_nat(X,set_Pr3765526544606949372at_nat(Xs)) ).

% List.set_insert
tff(fact_1005_List_Oset__insert,axiom,
    ! [X: bool,Xs: list_bool] : set_bool2(insert_bool(X,Xs)) = insert_bool2(X,set_bool2(Xs)) ).

% List.set_insert
tff(fact_1006_List_Oset__insert,axiom,
    ! [X: int,Xs: list_int] : set_int2(insert_int(X,Xs)) = insert_int2(X,set_int2(Xs)) ).

% List.set_insert
tff(fact_1007_List_Oset__insert,axiom,
    ! [X: nat,Xs: list_nat] : set_nat2(insert_nat(X,Xs)) = insert_nat2(X,set_nat2(Xs)) ).

% List.set_insert
tff(fact_1008_List_Oset__insert,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(insert3246601298802261197n_assn(X,Xs)) = insert5290817439147925377n_assn(X,set_Pr1139785259514867910n_assn(Xs)) ).

% List.set_insert
tff(fact_1009_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(Xs,Ys),listrel_c_c(R2))
    <=> ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_c(Xs)))
           => member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,nth_c(Xs,N2)),nth_c(Ys,N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1010_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_d,R2: set_Product_prod_c_d] :
      ( member7083281593699519441list_d(produc5515445822533511192list_d(Xs,Ys),listrel_c_d(R2))
    <=> ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_c(Xs)))
           => member5074992363344545361od_c_d(aa_d_P6917246732237330211od_c_d(aa_c_f7904789294214393838od_c_d(product_Pair_c_d,nth_c(Xs,N2)),nth_d(Ys,N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1011_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_nat,R2: set_Pr6903500605879609269_c_nat] :
      ( member6184937725913850668st_nat(produc6126748735280710287st_nat(Xs,Ys),listrel_c_nat(R2))
    <=> ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_c(Xs)))
           => member8195077246299207702_c_nat(aa_nat2369501448413299853_c_nat(aa_c_f4242068757942246672_c_nat(product_Pair_c_nat,nth_c(Xs,N2)),aa_nat_nat(nth_nat(Ys),N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1012_listrel__iff__nth,axiom,
    ! [Xs: list_c,Ys: list_int,R2: set_Pr4413417460947882513_c_int] :
      ( member3694854580982123912st_int(produc1948897715771513579st_int(Xs,Ys),listrel_c_int(R2))
    <=> ( ( size_size_list_c(Xs) = size_size_list_int(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_c(Xs)))
           => member4017226226790010994_c_int(aa_int4739195317257975621_c_int(aa_c_f8299530501364392392_c_int(product_Pair_c_int,nth_c(Xs,N2)),aa_nat_int(nth_int(Ys),N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1013_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_c,R2: set_Product_prod_d_c] :
      ( member1739767152416986127list_c(produc242965416994095062list_c(Xs,Ys),listrel_d_c(R2))
    <=> ( ( size_size_list_d(Xs) = size_size_list_c(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_d(Xs)))
           => member2287536777068640527od_d_c(aa_c_P4769231446293241506od_d_c(aa_d_f479461486583208174od_d_c(product_Pair_d_c,nth_d(Xs,N2)),nth_c(Ys,N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1014_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(Xs,Ys),listrel_d_d(R2))
    <=> ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_d(Xs)))
           => member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,nth_d(Xs,N2)),nth_d(Ys,N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1015_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_nat,R2: set_Pr3276347184212964150_d_nat] :
      ( member6851837201453759021st_nat(produc6793648210820618640st_nat(Xs,Ys),listrel_d_nat(R2))
    <=> ( ( size_size_list_d(Xs) = size_size_list_nat(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_d(Xs)))
           => member207149538400826391_d_nat(aa_nat3604945777369694350_d_nat(aa_d_f7443782029664826770_d_nat(product_Pair_d_nat,nth_d(Xs,N2)),aa_nat_nat(nth_nat(Ys),N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1016_listrel__iff__nth,axiom,
    ! [Xs: list_d,Ys: list_int,R2: set_Pr786264039281237394_d_int] :
      ( member4361754056522032265st_int(produc2615797191311421932st_int(Xs,Ys),listrel_d_int(R2))
    <=> ( ( size_size_list_d(Xs) = size_size_list_int(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_d(Xs)))
           => member5252670555746405491_d_int(aa_int5974639646214370118_d_int(aa_d_f2277871736232196682_d_int(product_Pair_d_int,nth_d(Xs,N2)),aa_nat_int(nth_int(Ys),N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1017_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_c,R2: set_Pr4335409928928841947_nat_c] :
      ( member6074218473364765234list_c(produc7723716018658481613list_c(Xs,Ys),listrel_nat_c(R2))
    <=> ( ( size_size_list_nat(Xs) = size_size_list_c(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
           => member8962352060716553276_nat_c(aa_c_P1942785857434746857_nat_c(aa_nat1477143957913088426_nat_c(product_Pair_nat_c,aa_nat_nat(nth_nat(Xs),N2)),nth_c(Ys,N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1018_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_d,R2: set_Pr4406443968975187932_nat_d] :
      ( member6145252513411111219list_d(produc7723716022961710414list_d(Xs,Ys),listrel_nat_d(R2))
    <=> ( ( size_size_list_nat(Xs) = size_size_list_d(Ys) )
        & ! [N2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N2),size_size_list_nat(Xs)))
           => member8962352065019782077_nat_d(aa_d_P9049729840602276459_nat_d(aa_nat8655121976823735212_nat_d(product_Pair_nat_d,aa_nat_nat(nth_nat(Xs),N2)),nth_d(Ys,N2)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_1019_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: bool] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),pure_assn(B2)))
    <=> ( ! [H3: produc3658429121746597890et_nat] :
            ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H3))
           => pp(B2) )
        & pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),one_one_assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_1020_mult__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A2) = A2 ).

% mult_1
tff(fact_1021_mult__1,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A2) = A2 ).

% mult_1
tff(fact_1022_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_1023_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_1024_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_1025_mult_Oright__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),one_one_Code_integer) = A2 ).

% mult.right_neutral
tff(fact_1026_mult_Oright__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),one_one_assn) = A2 ).

% mult.right_neutral
tff(fact_1027_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_1028_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_1029_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_1030_in__set__remove1,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( A2 != B2 )
     => ( member8757157785044589968at_nat(A2,set_Pr3765526544606949372at_nat(remove8895909992688781315at_nat(B2,Xs)))
      <=> member8757157785044589968at_nat(A2,set_Pr3765526544606949372at_nat(Xs)) ) ) ).

% in_set_remove1
tff(fact_1031_in__set__remove1,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( A2 != B2 )
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),set_Pr5648618587558075414at_nat(remove5766854097993331357at_nat(B2,Xs))))
      <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),set_Pr5648618587558075414at_nat(Xs))) ) ) ).

% in_set_remove1
tff(fact_1032_in__set__remove1,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ( A2 != B2 )
     => ( member4125802036136091927t_bool(A2,set_fu1437410266168021891t_bool(remove7684436675009651594t_bool(B2,Xs)))
      <=> member4125802036136091927t_bool(A2,set_fu1437410266168021891t_bool(Xs)) ) ) ).

% in_set_remove1
tff(fact_1033_in__set__remove1,axiom,
    ! [A2: int,B2: int,Xs: list_int] :
      ( ( A2 != B2 )
     => ( member_int(A2,set_int2(remove1_int(B2,Xs)))
      <=> member_int(A2,set_int2(Xs)) ) ) ).

% in_set_remove1
tff(fact_1034_in__set__remove1,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( ( A2 != B2 )
     => ( member_nat(A2,set_nat2(remove1_nat(B2,Xs)))
      <=> member_nat(A2,set_nat2(Xs)) ) ) ).

% in_set_remove1
tff(fact_1035_in__set__remove1,axiom,
    ! [A2: produc6575502325842934193n_assn,B2: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( A2 != B2 )
     => ( member7957490590177025114n_assn(A2,set_Pr1139785259514867910n_assn(remove1670527618125605709n_assn(B2,Xs)))
      <=> member7957490590177025114n_assn(A2,set_Pr1139785259514867910n_assn(Xs)) ) ) ).

% in_set_remove1
tff(fact_1036_removeAll__id,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ( remove4908499291543849587at_nat(X,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_1037_removeAll__id,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( remove3673390508374433037at_nat(X,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_1038_removeAll__id,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ( remove7324150401486087674t_bool(X,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_1039_removeAll__id,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ member_int(X,set_int2(Xs))
     => ( removeAll_int(X,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_1040_removeAll__id,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ member_nat(X,set_nat2(Xs))
     => ( removeAll_nat(X,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_1041_removeAll__id,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ( remove1763081955827946941n_assn(X,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_1042_pure__true,axiom,
    pure_assn(fTrue) = one_one_assn ).

% pure_true
tff(fact_1043_pure__assn__eq__emp__iff,axiom,
    ! [P: bool] :
      ( ( pure_assn(P) = one_one_assn )
    <=> pp(P) ) ).

% pure_assn_eq_emp_iff
tff(fact_1044_entt__emp,axiom,
    ! [A: assn] : pp(aa_assn_bool(entailst(A),one_one_assn)) ).

% entt_emp
tff(fact_1045_in__set__insert,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ( insert4149901806602606979at_nat(X,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_1046_in__set__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( insert8944034826898310173at_nat(X,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_1047_in__set__insert,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ( insert451209248500086026t_bool(X,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_1048_in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
     => ( insert_int(X,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_1049_in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(Xs))
     => ( insert_nat(X,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_1050_in__set__insert,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ( insert3246601298802261197n_assn(X,Xs) = Xs ) ) ).

% in_set_insert
tff(fact_1051_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_1052_distinct__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,insert_int(X,Xs)))
    <=> pp(aa_list_int_bool(distinct_int,Xs)) ) ).

% distinct_insert
tff(fact_1053_distinct__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,insert_nat(X,Xs)))
    <=> pp(aa_list_nat_bool(distinct_nat,Xs)) ) ).

% distinct_insert
tff(fact_1054_ent__pure__pre__iff__sng,axiom,
    ! [B2: bool,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,pure_assn(B2)),Q))
    <=> ( pp(B2)
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,one_one_assn),Q)) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_1055_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
    <=> ( X = one_one_Code_integer ) ) ).

% one_reorient
tff(fact_1056_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
    <=> ( X = one_one_assn ) ) ).

% one_reorient
tff(fact_1057_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_1058_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_1059_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_1060_times__assn__raw_Ocases,axiom,
    ! [X: produc1542783094787327164et_nat] :
      ~ ! [P3: fun_Pr6001797050664311790t_bool,Q3: fun_Pr6001797050664311790t_bool,H: heap_e7401611519738050253t_unit,As: set_nat] : X != produc3083477295402903094et_nat(P3,produc8148480564485311129et_nat(Q3,produc7507926704131184380et_nat(H,As))) ).

% times_assn_raw.cases
tff(fact_1061_listrel1__eq__len,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(Xs,Ys),listrel1_c(R2))
     => ( size_size_list_c(Xs) = size_size_list_c(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1062_listrel1__eq__len,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(Xs,Ys),listrel1_d(R2))
     => ( size_size_list_d(Xs) = size_size_list_d(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1063_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
     => ( size_size_list_nat(Xs) = size_size_list_nat(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1064_listrel1__eq__len,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
     => ( size_size_list_int(Xs) = size_size_list_int(Ys) ) ) ).

% listrel1_eq_len
tff(fact_1065_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(Xs,Ys),listrel_c_c(R2))
     => ( size_size_list_c(Xs) = size_size_list_c(Ys) ) ) ).

% listrel_eq_len
tff(fact_1066_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_d,R2: set_Product_prod_c_d] :
      ( member7083281593699519441list_d(produc5515445822533511192list_d(Xs,Ys),listrel_c_d(R2))
     => ( size_size_list_c(Xs) = size_size_list_d(Ys) ) ) ).

% listrel_eq_len
tff(fact_1067_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_nat,R2: set_Pr6903500605879609269_c_nat] :
      ( member6184937725913850668st_nat(produc6126748735280710287st_nat(Xs,Ys),listrel_c_nat(R2))
     => ( size_size_list_c(Xs) = size_size_list_nat(Ys) ) ) ).

% listrel_eq_len
tff(fact_1068_listrel__eq__len,axiom,
    ! [Xs: list_c,Ys: list_int,R2: set_Pr4413417460947882513_c_int] :
      ( member3694854580982123912st_int(produc1948897715771513579st_int(Xs,Ys),listrel_c_int(R2))
     => ( size_size_list_c(Xs) = size_size_list_int(Ys) ) ) ).

% listrel_eq_len
tff(fact_1069_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_c,R2: set_Product_prod_d_c] :
      ( member1739767152416986127list_c(produc242965416994095062list_c(Xs,Ys),listrel_d_c(R2))
     => ( size_size_list_d(Xs) = size_size_list_c(Ys) ) ) ).

% listrel_eq_len
tff(fact_1070_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(Xs,Ys),listrel_d_d(R2))
     => ( size_size_list_d(Xs) = size_size_list_d(Ys) ) ) ).

% listrel_eq_len
tff(fact_1071_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_nat,R2: set_Pr3276347184212964150_d_nat] :
      ( member6851837201453759021st_nat(produc6793648210820618640st_nat(Xs,Ys),listrel_d_nat(R2))
     => ( size_size_list_d(Xs) = size_size_list_nat(Ys) ) ) ).

% listrel_eq_len
tff(fact_1072_listrel__eq__len,axiom,
    ! [Xs: list_d,Ys: list_int,R2: set_Pr786264039281237394_d_int] :
      ( member4361754056522032265st_int(produc2615797191311421932st_int(Xs,Ys),listrel_d_int(R2))
     => ( size_size_list_d(Xs) = size_size_list_int(Ys) ) ) ).

% listrel_eq_len
tff(fact_1073_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_c,R2: set_Pr4335409928928841947_nat_c] :
      ( member6074218473364765234list_c(produc7723716018658481613list_c(Xs,Ys),listrel_nat_c(R2))
     => ( size_size_list_nat(Xs) = size_size_list_c(Ys) ) ) ).

% listrel_eq_len
tff(fact_1074_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_d,R2: set_Pr4406443968975187932_nat_d] :
      ( member6145252513411111219list_d(produc7723716022961710414list_d(Xs,Ys),listrel_nat_d(R2))
     => ( size_size_list_nat(Xs) = size_size_list_d(Ys) ) ) ).

% listrel_eq_len
tff(fact_1075_distinct__remove1__removeAll,axiom,
    ! [Xs: list_int,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( remove1_int(X,Xs) = removeAll_int(X,Xs) ) ) ).

% distinct_remove1_removeAll
tff(fact_1076_distinct__remove1__removeAll,axiom,
    ! [Xs: list_nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( remove1_nat(X,Xs) = removeAll_nat(X,Xs) ) ) ).

% distinct_remove1_removeAll
tff(fact_1077_mult_Ocomm__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),one_one_Code_integer) = A2 ).

% mult.comm_neutral
tff(fact_1078_mult_Ocomm__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),one_one_assn) = A2 ).

% mult.comm_neutral
tff(fact_1079_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_1080_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_1081_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_1082_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_1083_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_1084_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_1085_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_1086_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_1087_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),P) = P ).

% assn_one_left
tff(fact_1088_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A2) = A2 ).

% norm_assertion_simps(1)
tff(fact_1089_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),one_one_assn) = A2 ).

% norm_assertion_simps(2)
tff(fact_1090_remove1__idem,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ( remove8895909992688781315at_nat(X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_1091_remove1__idem,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( remove5766854097993331357at_nat(X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_1092_remove1__idem,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ( remove7684436675009651594t_bool(X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_1093_remove1__idem,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ member_int(X,set_int2(Xs))
     => ( remove1_int(X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_1094_remove1__idem,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ member_nat(X,set_nat2(Xs))
     => ( remove1_nat(X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_1095_remove1__idem,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ( remove1670527618125605709n_assn(X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_1096_notin__set__remove1,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat,Y: produc3843707927480180839at_nat] :
      ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(remove8895909992688781315at_nat(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_1097_notin__set__remove1,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(remove5766854097993331357at_nat(Y,Xs)))) ) ).

% notin_set_remove1
tff(fact_1098_notin__set__remove1,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool,Y: fun_Pr6001797050664311790t_bool] :
      ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(remove7684436675009651594t_bool(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_1099_notin__set__remove1,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ~ member_int(X,set_int2(Xs))
     => ~ member_int(X,set_int2(remove1_int(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_1100_notin__set__remove1,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ~ member_nat(X,set_nat2(Xs))
     => ~ member_nat(X,set_nat2(remove1_nat(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_1101_notin__set__remove1,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(remove1670527618125605709n_assn(Y,Xs))) ) ).

% notin_set_remove1
tff(fact_1102_distinct__remove1,axiom,
    ! [Xs: list_int,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => pp(aa_list_int_bool(distinct_int,remove1_int(X,Xs))) ) ).

% distinct_remove1
tff(fact_1103_distinct__remove1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => pp(aa_list_nat_bool(distinct_nat,remove1_nat(X,Xs))) ) ).

% distinct_remove1
tff(fact_1104_distinct__removeAll,axiom,
    ! [Xs: list_int,X: int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => pp(aa_list_int_bool(distinct_int,removeAll_int(X,Xs))) ) ).

% distinct_removeAll
tff(fact_1105_distinct__removeAll,axiom,
    ! [Xs: list_nat,X: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => pp(aa_list_nat_bool(distinct_nat,removeAll_nat(X,Xs))) ) ).

% distinct_removeAll
tff(fact_1106_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( pp(aa_rat_bool(ord_less_rat(one_one_rat),M2))
     => ( pp(aa_rat_bool(ord_less_rat(one_one_rat),N))
       => pp(aa_rat_bool(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M2),N))) ) ) ).

% less_1_mult
tff(fact_1107_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),M2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),N))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M2),N))) ) ) ).

% less_1_mult
tff(fact_1108_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% less_1_mult
tff(fact_1109_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M2),N))) ) ) ).

% less_1_mult
tff(fact_1110_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(fI_QUERY(P,Q),one_one_assn))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q)) ) ).

% entails_solve_init(2)
tff(fact_1111_prod__induct4,axiom,
    ! [P: fun_Pr229822445725053876t_bool,X: produc1542783094787327164et_nat] :
      ( ! [A3: fun_Pr6001797050664311790t_bool,B3: fun_Pr6001797050664311790t_bool,C4: heap_e7401611519738050253t_unit,D3: set_nat] : pp(aa_Pro6092667153363046813t_bool(P,produc3083477295402903094et_nat(A3,produc8148480564485311129et_nat(B3,produc7507926704131184380et_nat(C4,D3)))))
     => pp(aa_Pro6092667153363046813t_bool(P,X)) ) ).

% prod_induct4
tff(fact_1112_prod__induct3,axiom,
    ! [P: fun_Pr5991498150630465241r_bool,X: produc1616345218596728535nteger] :
      ( ! [A3: fun_Pr272337284026993017e_term,B3: code_integer,C4: code_integer] : pp(aa_Pro3640759179027349698r_bool(P,produc7384524951051686153nteger(A3,produc1086072967326762835nteger(B3,C4))))
     => pp(aa_Pro3640759179027349698r_bool(P,X)) ) ).

% prod_induct3
tff(fact_1113_prod__induct3,axiom,
    ! [P: fun_Pr3073222722562426257t_bool,X: produc6108883039684432031et_nat] :
      ( ! [A3: fun_Pr6001797050664311790t_bool,B3: heap_e7401611519738050253t_unit,C4: set_nat] : pp(aa_Pro7454463142485379962t_bool(P,produc8148480564485311129et_nat(A3,produc7507926704131184380et_nat(B3,C4))))
     => pp(aa_Pro7454463142485379962t_bool(P,X)) ) ).

% prod_induct3
tff(fact_1114_prod__induct3,axiom,
    ! [P: fun_Pr229822445725053876t_bool,X: produc1542783094787327164et_nat] :
      ( ! [A3: fun_Pr6001797050664311790t_bool,B3: fun_Pr6001797050664311790t_bool,C4: produc3658429121746597890et_nat] : pp(aa_Pro6092667153363046813t_bool(P,produc3083477295402903094et_nat(A3,produc8148480564485311129et_nat(B3,C4))))
     => pp(aa_Pro6092667153363046813t_bool(P,X)) ) ).

% prod_induct3
tff(fact_1115_prod__induct3,axiom,
    ! [P: fun_Pr6019674992625747500t_bool,X: produc4441460161354545860nt_int] :
      ( ! [A3: fun_Pr9164807099015279978e_term,B3: int,C4: int] : pp(aa_Pro5031108791230793749t_bool(P,produc1560874099174258678nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C4))))
     => pp(aa_Pro5031108791230793749t_bool(P,X)) ) ).

% prod_induct3
tff(fact_1116_prod__induct3,axiom,
    ! [P: fun_Pr5504123824928244736t_bool,X: produc6768802789813714928nt_int] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: int,C4: int] : pp(aa_Pro7591400126937138665t_bool(P,produc4852315488734385442nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C4))))
     => pp(aa_Pro7591400126937138665t_bool(P,X)) ) ).

% prod_induct3
tff(fact_1117_prod__cases4,axiom,
    ! [Y: produc1542783094787327164et_nat] :
      ~ ! [A3: fun_Pr6001797050664311790t_bool,B3: fun_Pr6001797050664311790t_bool,C4: heap_e7401611519738050253t_unit,D3: set_nat] : Y != produc3083477295402903094et_nat(A3,produc8148480564485311129et_nat(B3,produc7507926704131184380et_nat(C4,D3))) ).

% prod_cases4
tff(fact_1118_prod__cases3,axiom,
    ! [Y: produc1616345218596728535nteger] :
      ~ ! [A3: fun_Pr272337284026993017e_term,B3: code_integer,C4: code_integer] : Y != produc7384524951051686153nteger(A3,produc1086072967326762835nteger(B3,C4)) ).

% prod_cases3
tff(fact_1119_prod__cases3,axiom,
    ! [Y: produc6108883039684432031et_nat] :
      ~ ! [A3: fun_Pr6001797050664311790t_bool,B3: heap_e7401611519738050253t_unit,C4: set_nat] : Y != produc8148480564485311129et_nat(A3,produc7507926704131184380et_nat(B3,C4)) ).

% prod_cases3
tff(fact_1120_prod__cases3,axiom,
    ! [Y: produc1542783094787327164et_nat] :
      ~ ! [A3: fun_Pr6001797050664311790t_bool,B3: fun_Pr6001797050664311790t_bool,C4: produc3658429121746597890et_nat] : Y != produc3083477295402903094et_nat(A3,produc8148480564485311129et_nat(B3,C4)) ).

% prod_cases3
tff(fact_1121_prod__cases3,axiom,
    ! [Y: produc4441460161354545860nt_int] :
      ~ ! [A3: fun_Pr9164807099015279978e_term,B3: int,C4: int] : Y != produc1560874099174258678nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C4)) ).

% prod_cases3
tff(fact_1122_prod__cases3,axiom,
    ! [Y: produc6768802789813714928nt_int] :
      ~ ! [A3: fun_in6429348457993719614e_term,B3: int,C4: int] : Y != produc4852315488734385442nt_int(A3,aa_int7462565939943104719nt_int(product_Pair_int_int(B3),C4)) ).

% prod_cases3
tff(fact_1123_Pair__inject,axiom,
    ! [A2: fun_Pr272337284026993017e_term,B2: produc8923325533196201883nteger,A5: fun_Pr272337284026993017e_term,B5: produc8923325533196201883nteger] :
      ( ( produc7384524951051686153nteger(A2,B2) = produc7384524951051686153nteger(A5,B5) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
tff(fact_1124_Pair__inject,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: produc3658429121746597890et_nat,A5: fun_Pr6001797050664311790t_bool,B5: produc3658429121746597890et_nat] :
      ( ( produc8148480564485311129et_nat(A2,B2) = produc8148480564485311129et_nat(A5,B5) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
tff(fact_1125_Pair__inject,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: produc6108883039684432031et_nat,A5: fun_Pr6001797050664311790t_bool,B5: produc6108883039684432031et_nat] :
      ( ( produc3083477295402903094et_nat(A2,B2) = produc3083477295402903094et_nat(A5,B5) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
tff(fact_1126_Pair__inject,axiom,
    ! [A2: fun_Pr9164807099015279978e_term,B2: product_prod_int_int,A5: fun_Pr9164807099015279978e_term,B5: product_prod_int_int] :
      ( ( produc1560874099174258678nt_int(A2,B2) = produc1560874099174258678nt_int(A5,B5) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
tff(fact_1127_Pair__inject,axiom,
    ! [A2: fun_in6429348457993719614e_term,B2: product_prod_int_int,A5: fun_in6429348457993719614e_term,B5: product_prod_int_int] :
      ( ( produc4852315488734385442nt_int(A2,B2) = produc4852315488734385442nt_int(A5,B5) )
     => ~ ( ( A2 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
tff(fact_1128_prod__cases,axiom,
    ! [P: fun_Pr5991498150630465241r_bool,P4: produc1616345218596728535nteger] :
      ( ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] : pp(aa_Pro3640759179027349698r_bool(P,produc7384524951051686153nteger(A3,B3)))
     => pp(aa_Pro3640759179027349698r_bool(P,P4)) ) ).

% prod_cases
tff(fact_1129_prod__cases,axiom,
    ! [P: fun_Pr3073222722562426257t_bool,P4: produc6108883039684432031et_nat] :
      ( ! [A3: fun_Pr6001797050664311790t_bool,B3: produc3658429121746597890et_nat] : pp(aa_Pro7454463142485379962t_bool(P,produc8148480564485311129et_nat(A3,B3)))
     => pp(aa_Pro7454463142485379962t_bool(P,P4)) ) ).

% prod_cases
tff(fact_1130_prod__cases,axiom,
    ! [P: fun_Pr229822445725053876t_bool,P4: produc1542783094787327164et_nat] :
      ( ! [A3: fun_Pr6001797050664311790t_bool,B3: produc6108883039684432031et_nat] : pp(aa_Pro6092667153363046813t_bool(P,produc3083477295402903094et_nat(A3,B3)))
     => pp(aa_Pro6092667153363046813t_bool(P,P4)) ) ).

% prod_cases
tff(fact_1131_prod__cases,axiom,
    ! [P: fun_Pr6019674992625747500t_bool,P4: produc4441460161354545860nt_int] :
      ( ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] : pp(aa_Pro5031108791230793749t_bool(P,produc1560874099174258678nt_int(A3,B3)))
     => pp(aa_Pro5031108791230793749t_bool(P,P4)) ) ).

% prod_cases
tff(fact_1132_prod__cases,axiom,
    ! [P: fun_Pr5504123824928244736t_bool,P4: produc6768802789813714928nt_int] :
      ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] : pp(aa_Pro7591400126937138665t_bool(P,produc4852315488734385442nt_int(A3,B3)))
     => pp(aa_Pro7591400126937138665t_bool(P,P4)) ) ).

% prod_cases
tff(fact_1133_surj__pair,axiom,
    ! [P4: produc1616345218596728535nteger] :
    ? [X2: fun_Pr272337284026993017e_term,Y2: produc8923325533196201883nteger] : P4 = produc7384524951051686153nteger(X2,Y2) ).

% surj_pair
tff(fact_1134_surj__pair,axiom,
    ! [P4: produc6108883039684432031et_nat] :
    ? [X2: fun_Pr6001797050664311790t_bool,Y2: produc3658429121746597890et_nat] : P4 = produc8148480564485311129et_nat(X2,Y2) ).

% surj_pair
tff(fact_1135_surj__pair,axiom,
    ! [P4: produc1542783094787327164et_nat] :
    ? [X2: fun_Pr6001797050664311790t_bool,Y2: produc6108883039684432031et_nat] : P4 = produc3083477295402903094et_nat(X2,Y2) ).

% surj_pair
tff(fact_1136_surj__pair,axiom,
    ! [P4: produc4441460161354545860nt_int] :
    ? [X2: fun_Pr9164807099015279978e_term,Y2: product_prod_int_int] : P4 = produc1560874099174258678nt_int(X2,Y2) ).

% surj_pair
tff(fact_1137_surj__pair,axiom,
    ! [P4: produc6768802789813714928nt_int] :
    ? [X2: fun_in6429348457993719614e_term,Y2: product_prod_int_int] : P4 = produc4852315488734385442nt_int(X2,Y2) ).

% surj_pair
tff(fact_1138_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1616345218596728535nteger] :
      ~ ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] : Y != produc7384524951051686153nteger(A3,B3) ).

% old.prod.exhaust
tff(fact_1139_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6108883039684432031et_nat] :
      ~ ! [A3: fun_Pr6001797050664311790t_bool,B3: produc3658429121746597890et_nat] : Y != produc8148480564485311129et_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_1140_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1542783094787327164et_nat] :
      ~ ! [A3: fun_Pr6001797050664311790t_bool,B3: produc6108883039684432031et_nat] : Y != produc3083477295402903094et_nat(A3,B3) ).

% old.prod.exhaust
tff(fact_1141_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4441460161354545860nt_int] :
      ~ ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] : Y != produc1560874099174258678nt_int(A3,B3) ).

% old.prod.exhaust
tff(fact_1142_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6768802789813714928nt_int] :
      ~ ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] : Y != produc4852315488734385442nt_int(A3,B3) ).

% old.prod.exhaust
tff(fact_1143_remove__code_I1_J,axiom,
    ! [X: int,Xs: list_int] : remove_int(X,set_int2(Xs)) = set_int2(removeAll_int(X,Xs)) ).

% remove_code(1)
tff(fact_1144_remove__code_I1_J,axiom,
    ! [X: nat,Xs: list_nat] : remove_nat(X,set_nat2(Xs)) = set_nat2(removeAll_nat(X,Xs)) ).

% remove_code(1)
tff(fact_1145_remove__code_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : remove6142905432236673558n_assn(X,set_Pr1139785259514867910n_assn(Xs)) = set_Pr1139785259514867910n_assn(remove1763081955827946941n_assn(X,Xs)) ).

% remove_code(1)
tff(fact_1146_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,one_one_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat))) ).

% mod_emp_simp
tff(fact_1147_length__removeAll__less,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s1588839187904960729at_nat(remove4908499291543849587at_nat(X,Xs))),size_s1588839187904960729at_nat(Xs))) ) ).

% length_removeAll_less
tff(fact_1148_length__removeAll__less,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s5460976970255530739at_nat(remove3673390508374433037at_nat(X,Xs))),size_s5460976970255530739at_nat(Xs))) ) ).

% length_removeAll_less
tff(fact_1149_length__removeAll__less,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s1066996506189871328t_bool(remove7324150401486087674t_bool(X,Xs))),size_s1066996506189871328t_bool(Xs))) ) ).

% length_removeAll_less
tff(fact_1150_length__removeAll__less,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6829681357464350627n_assn(remove1763081955827946941n_assn(X,Xs))),size_s6829681357464350627n_assn(Xs))) ) ).

% length_removeAll_less
tff(fact_1151_length__removeAll__less,axiom,
    ! [X: c,Xs: list_c] :
      ( member_c(X,set_c2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_c(removeAll_c(X,Xs))),size_size_list_c(Xs))) ) ).

% length_removeAll_less
tff(fact_1152_length__removeAll__less,axiom,
    ! [X: d,Xs: list_d] :
      ( member_d(X,set_d2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_d(removeAll_d(X,Xs))),size_size_list_d(Xs))) ) ).

% length_removeAll_less
tff(fact_1153_length__removeAll__less,axiom,
    ! [X: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(removeAll_nat(X,Xs))),size_size_list_nat(Xs))) ) ).

% length_removeAll_less
tff(fact_1154_length__removeAll__less,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(removeAll_int(X,Xs))),size_size_list_int(Xs))) ) ).

% length_removeAll_less
tff(fact_1155_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_rat_bool(ord_less_rat(one_one_rat),one_one_rat)) ).

% less_numeral_extra(4)
tff(fact_1156_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),one_one_Code_integer)) ).

% less_numeral_extra(4)
tff(fact_1157_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),one_one_nat)) ).

% less_numeral_extra(4)
tff(fact_1158_less__numeral__extra_I4_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),one_one_int)) ).

% less_numeral_extra(4)
tff(fact_1159_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: bool,Xa: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(pure_a825153325127701367it_nat(X),Xa))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H,As) )
           => ( ( As = bot_bot_set_nat )
              & pp(X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_1160_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: bool,Xa: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(pure_a825153325127701367it_nat(X),Xa))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H,As) )
           => ~ ( ( As = bot_bot_set_nat )
                & pp(X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_1161_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: bool,Xa: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(pure_a825153325127701367it_nat(X),Xa))
      <=> pp(Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H,As) )
           => ( pp(Y)
            <=> ~ ( ( As = bot_bot_set_nat )
                  & pp(X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_1162_pure__assn__raw_Osimps,axiom,
    ! [B2: bool,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(pure_a825153325127701367it_nat(B2),produc7507926704131184380et_nat(H2,As2)))
    <=> ( ( As2 = bot_bot_set_nat )
        & pp(B2) ) ) ).

% pure_assn_raw.simps
tff(fact_1163_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Ys)))
       => ( nth_Product_prod_c_c(zip_c_c(Xs,Ys),I) = aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,nth_c(Xs,I)),nth_c(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1164_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Ys)))
       => ( nth_Product_prod_c_d(zip_c_d(Xs,Ys),I) = aa_d_P6917246732237330211od_c_d(aa_c_f7904789294214393838od_c_d(product_Pair_c_d,nth_c(Xs,I)),nth_d(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1165_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ys)))
       => ( nth_Pr1708982275578427192_c_nat(zip_c_nat(Xs,Ys),I) = aa_nat2369501448413299853_c_nat(aa_c_f4242068757942246672_c_nat(product_Pair_c_nat,nth_c(Xs,I)),aa_nat_nat(nth_nat(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1166_nth__zip,axiom,
    ! [I: nat,Xs: list_c,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Ys)))
       => ( nth_Pr6754503292924006292_c_int(zip_c_int(Xs,Ys),I) = aa_int4739195317257975621_c_int(aa_c_f8299530501364392392_c_int(product_Pair_c_int,nth_c(Xs,I)),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1167_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Ys)))
       => ( nth_Product_prod_d_c(zip_d_c(Xs,Ys),I) = aa_c_P4769231446293241506od_d_c(aa_d_f479461486583208174od_d_c(product_Pair_d_c,nth_d(Xs,I)),nth_c(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1168_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Ys)))
       => ( nth_Product_prod_d_d(zip_d_d(Xs,Ys),I) = aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,nth_d(Xs,I)),nth_d(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1169_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ys)))
       => ( nth_Pr2944426604534821689_d_nat(zip_d_nat(Xs,Ys),I) = aa_nat3604945777369694350_d_nat(aa_d_f7443782029664826770_d_nat(product_Pair_d_nat,nth_d(Xs,I)),aa_nat_nat(nth_nat(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1170_nth__zip,axiom,
    ! [I: nat,Xs: list_d,Ys: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(Ys)))
       => ( nth_Pr7989947621880400789_d_int(zip_d_int(Xs,Ys),I) = aa_int5974639646214370118_d_int(aa_d_f2277871736232196682_d_int(product_Pair_d_int,nth_d(Xs,I)),aa_nat_int(nth_int(Ys),I)) ) ) ) ).

% nth_zip
tff(fact_1171_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_c(Ys)))
       => ( nth_Pr2476257089995772766_nat_c(zip_nat_c(Xs,Ys),I) = aa_c_P1942785857434746857_nat_c(aa_nat1477143957913088426_nat_c(product_Pair_nat_c,aa_nat_nat(nth_nat(Xs),I)),nth_c(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1172_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_d(Ys)))
       => ( nth_Pr2476257094299001567_nat_d(zip_nat_d(Xs,Ys),I) = aa_d_P9049729840602276459_nat_d(aa_nat8655121976823735212_nat_d(product_Pair_nat_d,aa_nat_nat(nth_nat(Xs),I)),nth_d(Ys,I)) ) ) ) ).

% nth_zip
tff(fact_1173_curryI,axiom,
    ! [F3: fun_Pr5991498150630465241r_bool,A2: fun_Pr272337284026993017e_term,B2: produc8923325533196201883nteger] :
      ( pp(aa_Pro3640759179027349698r_bool(F3,produc7384524951051686153nteger(A2,B2)))
     => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(produc8830213710918866432r_bool(F3),A2),B2)) ) ).

% curryI
tff(fact_1174_curryI,axiom,
    ! [F3: fun_Pr3073222722562426257t_bool,A2: fun_Pr6001797050664311790t_bool,B2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro7454463142485379962t_bool(F3,produc8148480564485311129et_nat(A2,B2)))
     => pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(produc2846001210079453936t_bool(F3),A2),B2)) ) ).

% curryI
tff(fact_1175_curryI,axiom,
    ! [F3: fun_Pr229822445725053876t_bool,A2: fun_Pr6001797050664311790t_bool,B2: produc6108883039684432031et_nat] :
      ( pp(aa_Pro6092667153363046813t_bool(F3,produc3083477295402903094et_nat(A2,B2)))
     => pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(produc3943886704551727507t_bool(F3),A2),B2)) ) ).

% curryI
tff(fact_1176_curryI,axiom,
    ! [F3: fun_Pr6019674992625747500t_bool,A2: fun_Pr9164807099015279978e_term,B2: product_prod_int_int] :
      ( pp(aa_Pro5031108791230793749t_bool(F3,produc1560874099174258678nt_int(A2,B2)))
     => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(produc7693670431277826643t_bool(F3),A2),B2)) ) ).

% curryI
tff(fact_1177_curryI,axiom,
    ! [F3: fun_Pr5504123824928244736t_bool,A2: fun_in6429348457993719614e_term,B2: product_prod_int_int] :
      ( pp(aa_Pro7591400126937138665t_bool(F3,produc4852315488734385442nt_int(A2,B2)))
     => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(produc6897183239252912167t_bool(F3),A2),B2)) ) ).

% curryI
tff(fact_1178_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_c,A5: list_c,B5: list_c] :
      ( ( size_size_list_c(A2) = size_size_list_c(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_c(B5) )
       => ( ( zip_c_c(A2,B2) = zip_c_c(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1179_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_d,A5: list_c,B5: list_d] :
      ( ( size_size_list_c(A2) = size_size_list_d(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_d(B5) )
       => ( ( zip_c_d(A2,B2) = zip_c_d(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1180_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_nat,A5: list_c,B5: list_nat] :
      ( ( size_size_list_c(A2) = size_size_list_nat(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_nat(B5) )
       => ( ( zip_c_nat(A2,B2) = zip_c_nat(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1181_zip__eq__zip__same__len,axiom,
    ! [A2: list_c,B2: list_int,A5: list_c,B5: list_int] :
      ( ( size_size_list_c(A2) = size_size_list_int(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_int(B5) )
       => ( ( zip_c_int(A2,B2) = zip_c_int(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1182_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_c,A5: list_d,B5: list_c] :
      ( ( size_size_list_d(A2) = size_size_list_c(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_c(B5) )
       => ( ( zip_d_c(A2,B2) = zip_d_c(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1183_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_d,A5: list_d,B5: list_d] :
      ( ( size_size_list_d(A2) = size_size_list_d(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_d(B5) )
       => ( ( zip_d_d(A2,B2) = zip_d_d(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1184_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_nat,A5: list_d,B5: list_nat] :
      ( ( size_size_list_d(A2) = size_size_list_nat(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_nat(B5) )
       => ( ( zip_d_nat(A2,B2) = zip_d_nat(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1185_zip__eq__zip__same__len,axiom,
    ! [A2: list_d,B2: list_int,A5: list_d,B5: list_int] :
      ( ( size_size_list_d(A2) = size_size_list_int(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_int(B5) )
       => ( ( zip_d_int(A2,B2) = zip_d_int(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1186_zip__eq__zip__same__len,axiom,
    ! [A2: list_nat,B2: list_c,A5: list_nat,B5: list_c] :
      ( ( size_size_list_nat(A2) = size_size_list_c(B2) )
     => ( ( size_size_list_nat(A5) = size_size_list_c(B5) )
       => ( ( zip_nat_c(A2,B2) = zip_nat_c(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1187_zip__eq__zip__same__len,axiom,
    ! [A2: list_nat,B2: list_d,A5: list_nat,B5: list_d] :
      ( ( size_size_list_nat(A2) = size_size_list_d(B2) )
     => ( ( size_size_list_nat(A5) = size_size_list_d(B5) )
       => ( ( zip_nat_d(A2,B2) = zip_nat_d(A5,B5) )
        <=> ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_1188_zip__inj,axiom,
    ! [A2: list_c,B2: list_c,A5: list_c,B5: list_c] :
      ( ( size_size_list_c(A2) = size_size_list_c(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_c(B5) )
       => ( ( zip_c_c(A2,B2) = zip_c_c(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1189_zip__inj,axiom,
    ! [A2: list_c,B2: list_d,A5: list_c,B5: list_d] :
      ( ( size_size_list_c(A2) = size_size_list_d(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_d(B5) )
       => ( ( zip_c_d(A2,B2) = zip_c_d(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1190_zip__inj,axiom,
    ! [A2: list_c,B2: list_nat,A5: list_c,B5: list_nat] :
      ( ( size_size_list_c(A2) = size_size_list_nat(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_nat(B5) )
       => ( ( zip_c_nat(A2,B2) = zip_c_nat(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1191_zip__inj,axiom,
    ! [A2: list_c,B2: list_int,A5: list_c,B5: list_int] :
      ( ( size_size_list_c(A2) = size_size_list_int(B2) )
     => ( ( size_size_list_c(A5) = size_size_list_int(B5) )
       => ( ( zip_c_int(A2,B2) = zip_c_int(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1192_zip__inj,axiom,
    ! [A2: list_d,B2: list_c,A5: list_d,B5: list_c] :
      ( ( size_size_list_d(A2) = size_size_list_c(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_c(B5) )
       => ( ( zip_d_c(A2,B2) = zip_d_c(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1193_zip__inj,axiom,
    ! [A2: list_d,B2: list_d,A5: list_d,B5: list_d] :
      ( ( size_size_list_d(A2) = size_size_list_d(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_d(B5) )
       => ( ( zip_d_d(A2,B2) = zip_d_d(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1194_zip__inj,axiom,
    ! [A2: list_d,B2: list_nat,A5: list_d,B5: list_nat] :
      ( ( size_size_list_d(A2) = size_size_list_nat(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_nat(B5) )
       => ( ( zip_d_nat(A2,B2) = zip_d_nat(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1195_zip__inj,axiom,
    ! [A2: list_d,B2: list_int,A5: list_d,B5: list_int] :
      ( ( size_size_list_d(A2) = size_size_list_int(B2) )
     => ( ( size_size_list_d(A5) = size_size_list_int(B5) )
       => ( ( zip_d_int(A2,B2) = zip_d_int(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1196_zip__inj,axiom,
    ! [A2: list_nat,B2: list_c,A5: list_nat,B5: list_c] :
      ( ( size_size_list_nat(A2) = size_size_list_c(B2) )
     => ( ( size_size_list_nat(A5) = size_size_list_c(B5) )
       => ( ( zip_nat_c(A2,B2) = zip_nat_c(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1197_zip__inj,axiom,
    ! [A2: list_nat,B2: list_d,A5: list_nat,B5: list_d] :
      ( ( size_size_list_nat(A2) = size_size_list_d(B2) )
     => ( ( size_size_list_nat(A5) = size_size_list_d(B5) )
       => ( ( zip_nat_d(A2,B2) = zip_nat_d(A5,B5) )
         => ( ( A2 = A5 )
            & ( B2 = B5 ) ) ) ) ) ).

% zip_inj
tff(fact_1198_pair__list__split,axiom,
    ! [L: list_P267710562049004973od_c_c] :
      ~ ! [L1: list_c,L22: list_c] :
          ( ( L = zip_c_c(L1,L22) )
         => ( ( size_size_list_c(L1) = size_size_list_c(L22) )
           => ( size_s2756448709718749209od_c_c(L) != size_size_list_c(L22) ) ) ) ).

% pair_list_split
tff(fact_1199_pair__list__split,axiom,
    ! [L: list_P338744602095350958od_c_d] :
      ~ ! [L1: list_c,L22: list_d] :
          ( ( L = zip_c_d(L1,L22) )
         => ( ( size_size_list_c(L1) = size_size_list_d(L22) )
           => ( size_s2827482749765095194od_c_d(L) != size_size_list_d(L22) ) ) ) ).

% pair_list_split
tff(fact_1200_pair__list__split,axiom,
    ! [L: list_P5561950507774946575_c_nat] :
      ~ ! [L1: list_c,L22: list_nat] :
          ( ( L = zip_c_nat(L1,L22) )
         => ( ( size_size_list_c(L1) = size_size_list_nat(L22) )
           => ( size_s2954062820725854115_c_nat(L) != size_size_list_nat(L22) ) ) ) ).

% pair_list_split
tff(fact_1201_pair__list__split,axiom,
    ! [L: list_P3071867362843219819_c_int] :
      ~ ! [L1: list_c,L22: list_int] :
          ( ( L = zip_c_int(L1,L22) )
         => ( ( size_size_list_c(L1) = size_size_list_int(L22) )
           => ( size_s463979675794127359_c_int(L) != size_size_list_int(L22) ) ) ) ).

% pair_list_split
tff(fact_1202_pair__list__split,axiom,
    ! [L: list_P8855433598298794092od_d_c] :
      ~ ! [L1: list_d,L22: list_c] :
          ( ( L = zip_d_c(L1,L22) )
         => ( ( size_size_list_d(L1) = size_size_list_c(L22) )
           => ( size_s2120799709113762520od_d_c(L) != size_size_list_c(L22) ) ) ) ).

% pair_list_split
tff(fact_1203_pair__list__split,axiom,
    ! [L: list_P8926467638345140077od_d_d] :
      ~ ! [L1: list_d,L22: list_d] :
          ( ( L = zip_d_d(L1,L22) )
         => ( ( size_size_list_d(L1) = size_size_list_d(L22) )
           => ( size_s2191833749160108505od_d_d(L) != size_size_list_d(L22) ) ) ) ).

% pair_list_split
tff(fact_1204_pair__list__split,axiom,
    ! [L: list_P1934797086108301456_d_nat] :
      ~ ! [L1: list_d,L22: list_nat] :
          ( ( L = zip_d_nat(L1,L22) )
         => ( ( size_size_list_d(L1) = size_size_list_nat(L22) )
           => ( size_s8550281435913984804_d_nat(L) != size_size_list_nat(L22) ) ) ) ).

% pair_list_split
tff(fact_1205_pair__list__split,axiom,
    ! [L: list_P8668085978031350508_d_int] :
      ~ ! [L1: list_d,L22: list_int] :
          ( ( L = zip_d_int(L1,L22) )
         => ( ( size_size_list_d(L1) = size_size_list_int(L22) )
           => ( size_s6060198290982258048_d_int(L) != size_size_list_int(L22) ) ) ) ).

% pair_list_split
tff(fact_1206_pair__list__split,axiom,
    ! [L: list_P2993859830824179253_nat_c] :
      ~ ! [L1: list_nat,L22: list_c] :
          ( ( L = zip_nat_c(L1,L22) )
         => ( ( size_size_list_nat(L1) = size_size_list_c(L22) )
           => ( size_s385972143775086793_nat_c(L) != size_size_list_c(L22) ) ) ) ).

% pair_list_split
tff(fact_1207_pair__list__split,axiom,
    ! [L: list_P3064893870870525238_nat_d] :
      ~ ! [L1: list_nat,L22: list_d] :
          ( ( L = zip_nat_d(L1,L22) )
         => ( ( size_size_list_nat(L1) = size_size_list_d(L22) )
           => ( size_s457006183821432778_nat_d(L) != size_size_list_d(L22) ) ) ) ).

% pair_list_split
tff(fact_1208_curryD,axiom,
    ! [F3: fun_Pr5991498150630465241r_bool,A2: fun_Pr272337284026993017e_term,B2: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(produc8830213710918866432r_bool(F3),A2),B2))
     => pp(aa_Pro3640759179027349698r_bool(F3,produc7384524951051686153nteger(A2,B2))) ) ).

% curryD
tff(fact_1209_curryD,axiom,
    ! [F3: fun_Pr3073222722562426257t_bool,A2: fun_Pr6001797050664311790t_bool,B2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(produc2846001210079453936t_bool(F3),A2),B2))
     => pp(aa_Pro7454463142485379962t_bool(F3,produc8148480564485311129et_nat(A2,B2))) ) ).

% curryD
tff(fact_1210_curryD,axiom,
    ! [F3: fun_Pr229822445725053876t_bool,A2: fun_Pr6001797050664311790t_bool,B2: produc6108883039684432031et_nat] :
      ( pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(produc3943886704551727507t_bool(F3),A2),B2))
     => pp(aa_Pro6092667153363046813t_bool(F3,produc3083477295402903094et_nat(A2,B2))) ) ).

% curryD
tff(fact_1211_curryD,axiom,
    ! [F3: fun_Pr6019674992625747500t_bool,A2: fun_Pr9164807099015279978e_term,B2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(produc7693670431277826643t_bool(F3),A2),B2))
     => pp(aa_Pro5031108791230793749t_bool(F3,produc1560874099174258678nt_int(A2,B2))) ) ).

% curryD
tff(fact_1212_curryD,axiom,
    ! [F3: fun_Pr5504123824928244736t_bool,A2: fun_in6429348457993719614e_term,B2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(produc6897183239252912167t_bool(F3),A2),B2))
     => pp(aa_Pro7591400126937138665t_bool(F3,produc4852315488734385442nt_int(A2,B2))) ) ).

% curryD
tff(fact_1213_curryE,axiom,
    ! [F3: fun_Pr5991498150630465241r_bool,A2: fun_Pr272337284026993017e_term,B2: produc8923325533196201883nteger] :
      ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(produc8830213710918866432r_bool(F3),A2),B2))
     => pp(aa_Pro3640759179027349698r_bool(F3,produc7384524951051686153nteger(A2,B2))) ) ).

% curryE
tff(fact_1214_curryE,axiom,
    ! [F3: fun_Pr3073222722562426257t_bool,A2: fun_Pr6001797050664311790t_bool,B2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(produc2846001210079453936t_bool(F3),A2),B2))
     => pp(aa_Pro7454463142485379962t_bool(F3,produc8148480564485311129et_nat(A2,B2))) ) ).

% curryE
tff(fact_1215_curryE,axiom,
    ! [F3: fun_Pr229822445725053876t_bool,A2: fun_Pr6001797050664311790t_bool,B2: produc6108883039684432031et_nat] :
      ( pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(produc3943886704551727507t_bool(F3),A2),B2))
     => pp(aa_Pro6092667153363046813t_bool(F3,produc3083477295402903094et_nat(A2,B2))) ) ).

% curryE
tff(fact_1216_curryE,axiom,
    ! [F3: fun_Pr6019674992625747500t_bool,A2: fun_Pr9164807099015279978e_term,B2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(produc7693670431277826643t_bool(F3),A2),B2))
     => pp(aa_Pro5031108791230793749t_bool(F3,produc1560874099174258678nt_int(A2,B2))) ) ).

% curryE
tff(fact_1217_curryE,axiom,
    ! [F3: fun_Pr5504123824928244736t_bool,A2: fun_in6429348457993719614e_term,B2: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(produc6897183239252912167t_bool(F3),A2),B2))
     => pp(aa_Pro7591400126937138665t_bool(F3,produc4852315488734385442nt_int(A2,B2))) ) ).

% curryE
tff(fact_1218_set__zip__rightD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),set_Pr3765526544606949372at_nat(zip_se5600341670672612855at_nat(Xs,Ys)))
     => member2643936169264416010at_nat(Y,set_se5049602875457034614at_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_1219_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys))))
     => member_nat(Y,set_nat2(Ys)) ) ).

% set_zip_rightD
tff(fact_1220_set__zip__rightD,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( member7957490590177025114n_assn(produc118845697133431529n_assn(X,Y),set_Pr1139785259514867910n_assn(zip_assn_assn(Xs,Ys)))
     => member_assn(Y,set_assn2(Ys)) ) ).

% set_zip_rightD
tff(fact_1221_set__zip__rightD,axiom,
    ! [X: fun_Pr272337284026993017e_term,Y: produc8923325533196201883nteger,Xs: list_f4295387047331035401e_term,Ys: list_P5578671422887162913nteger] :
      ( member6423679177046879214nteger(produc7384524951051686153nteger(X,Y),set_Pr8538088337767396482nteger(zip_fu197485079333405161nteger(Xs,Ys)))
     => member157494554546826820nteger(Y,set_Pr920681315882439344nteger(Ys)) ) ).

% set_zip_rightD
tff(fact_1222_set__zip__rightD,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Y: produc3658429121746597890et_nat,Xs: list_f5857023252961307252t_bool,Ys: list_P9062070895058802706et_nat] :
      ( member8615881266069945910et_nat(produc8148480564485311129et_nat(X,Y),set_Pr7877154892198005962et_nat(zip_fu8308979540302171001et_nat(Xs,Ys)))
     => member6260224972018164377et_nat(Y,set_Pr3864771053587467565et_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_1223_set__zip__rightD,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Y: produc6108883039684432031et_nat,Xs: list_f5857023252961307252t_bool,Ys: list_P5162959638400604975et_nat] :
      ( member2818932950976722259et_nat(produc3083477295402903094et_nat(X,Y),set_Pr6142456159824422887et_nat(zip_fu8753176646203213590et_nat(Xs,Ys)))
     => member8615881266069945910et_nat(Y,set_Pr7877154892198005962et_nat(Ys)) ) ).

% set_zip_rightD
tff(fact_1224_set__zip__rightD,axiom,
    ! [X: fun_Pr9164807099015279978e_term,Y: product_prod_int_int,Xs: list_f8504123576994025594e_term,Ys: list_P5707943133018811711nt_int] :
      ( member690604328116145115nt_int(produc1560874099174258678nt_int(X,Y),set_Pr7767296851990987887nt_int(zip_fu2408773928738089174nt_int(Xs,Ys)))
     => member5262025264175285858nt_int(Y,set_Pr2470121279949933262nt_int(Ys)) ) ).

% set_zip_rightD
tff(fact_1225_set__zip__rightD,axiom,
    ! [X: fun_in6429348457993719614e_term,Y: product_prod_int_int,Xs: list_f6829383385585334350e_term,Ys: list_P5707943133018811711nt_int] :
      ( member500491123954006279nt_int(produc4852315488734385442nt_int(X,Y),set_Pr2267922651014402459nt_int(zip_fu9102111156773635586nt_int(Xs,Ys)))
     => member5262025264175285858nt_int(Y,set_Pr2470121279949933262nt_int(Ys)) ) ).

% set_zip_rightD
tff(fact_1226_set__zip__leftD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),set_Pr3765526544606949372at_nat(zip_se5600341670672612855at_nat(Xs,Ys)))
     => member2643936169264416010at_nat(X,set_se5049602875457034614at_nat(Xs)) ) ).

% set_zip_leftD
tff(fact_1227_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys))))
     => member_nat(X,set_nat2(Xs)) ) ).

% set_zip_leftD
tff(fact_1228_set__zip__leftD,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( member7957490590177025114n_assn(produc118845697133431529n_assn(X,Y),set_Pr1139785259514867910n_assn(zip_assn_assn(Xs,Ys)))
     => member_assn(X,set_assn2(Xs)) ) ).

% set_zip_leftD
tff(fact_1229_set__zip__leftD,axiom,
    ! [X: fun_Pr272337284026993017e_term,Y: produc8923325533196201883nteger,Xs: list_f4295387047331035401e_term,Ys: list_P5578671422887162913nteger] :
      ( member6423679177046879214nteger(produc7384524951051686153nteger(X,Y),set_Pr8538088337767396482nteger(zip_fu197485079333405161nteger(Xs,Ys)))
     => member4565687819040644368e_term(X,set_fu53807654081409956e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1230_set__zip__leftD,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Y: produc3658429121746597890et_nat,Xs: list_f5857023252961307252t_bool,Ys: list_P9062070895058802706et_nat] :
      ( member8615881266069945910et_nat(produc8148480564485311129et_nat(X,Y),set_Pr7877154892198005962et_nat(zip_fu8308979540302171001et_nat(Xs,Ys)))
     => member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs)) ) ).

% set_zip_leftD
tff(fact_1231_set__zip__leftD,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Y: produc6108883039684432031et_nat,Xs: list_f5857023252961307252t_bool,Ys: list_P5162959638400604975et_nat] :
      ( member2818932950976722259et_nat(produc3083477295402903094et_nat(X,Y),set_Pr6142456159824422887et_nat(zip_fu8753176646203213590et_nat(Xs,Ys)))
     => member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs)) ) ).

% set_zip_leftD
tff(fact_1232_set__zip__leftD,axiom,
    ! [X: fun_Pr9164807099015279978e_term,Y: product_prod_int_int,Xs: list_f8504123576994025594e_term,Ys: list_P5707943133018811711nt_int] :
      ( member690604328116145115nt_int(produc1560874099174258678nt_int(X,Y),set_Pr7767296851990987887nt_int(zip_fu2408773928738089174nt_int(Xs,Ys)))
     => member2948136515907314945e_term(X,set_fu4765906003373919125e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1233_set__zip__leftD,axiom,
    ! [X: fun_in6429348457993719614e_term,Y: product_prod_int_int,Xs: list_f6829383385585334350e_term,Ys: list_P5707943133018811711nt_int] :
      ( member500491123954006279nt_int(produc4852315488734385442nt_int(X,Y),set_Pr2267922651014402459nt_int(zip_fu9102111156773635586nt_int(Xs,Ys)))
     => member7723273712176309973e_term(X,set_fu1494195493796392297e_term(Xs)) ) ).

% set_zip_leftD
tff(fact_1234_in__set__zipE,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Ys)))
     => ~ ( member_int(X,set_int2(Xs))
         => ~ member_int(Y,set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1235_in__set__zipE,axiom,
    ! [X: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( member216504246829706758nt_nat(product_Pair_int_nat(X,Y),set_Pr6647972299459129970nt_nat(zip_int_nat(Xs,Ys)))
     => ~ ( member_int(X,set_int2(Xs))
         => ~ member_nat(Y,set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1236_in__set__zipE,axiom,
    ! [X: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(X,Y),set_Pr1470767568048878706at_int(zip_nat_int(Xs,Ys)))
     => ~ ( member_nat(X,set_nat2(Xs))
         => ~ member_int(Y,set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1237_in__set__zipE,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Ys))))
     => ~ ( member_nat(X,set_nat2(Xs))
         => ~ member_nat(Y,set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1238_in__set__zipE,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( member7957490590177025114n_assn(produc118845697133431529n_assn(X,Y),set_Pr1139785259514867910n_assn(zip_assn_assn(Xs,Ys)))
     => ~ ( member_assn(X,set_assn2(Xs))
         => ~ member_assn(Y,set_assn2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1239_in__set__zipE,axiom,
    ! [X: product_prod_nat_nat,Y: int,Xs: list_P6011104703257516679at_nat,Ys: list_int] :
      ( member8394280151737582451at_int(produc6348220600061155286at_int(X,Y),set_Pr3658594827066574855at_int(zip_Pr6866960147343648950at_int(Xs,Ys)))
     => ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
         => ~ member_int(Y,set_int2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1240_in__set__zipE,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( member3348759134392003351at_nat(produc6350711070570205562at_nat(X,Y),set_Pr7836445846575771563at_nat(zip_Pr6869450617852699226at_nat(Xs,Ys)))
     => ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
         => ~ member_nat(Y,set_nat2(Ys)) ) ) ).

% in_set_zipE
tff(fact_1241_in__set__zipE,axiom,
    ! [X: int,Y: product_prod_nat_nat,Xs: list_int,Ys: list_P6011104703257516679at_nat] :
      ( member5261114234888836065at_nat(produc3525228511222278652at_nat(X,Y),set_Pr525428910217828469at_nat(zip_in4043968058504772316at_nat(Xs,Ys)))
     => ~ ( member_int(X,set_int2(Xs))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),set_Pr5648618587558075414at_nat(Ys))) ) ) ).

% in_set_zipE
tff(fact_1242_in__set__zipE,axiom,
    ! [X: int,Y: produc6575502325842934193n_assn,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( member241480614033477777n_assn(produc3313821056773121708n_assn(X,Y),set_Pr4510407026980144421n_assn(zip_in7098856637274830732n_assn(Xs,Ys)))
     => ~ ( member_int(X,set_int2(Xs))
         => ~ member7957490590177025114n_assn(Y,set_Pr1139785259514867910n_assn(Ys)) ) ) ).

% in_set_zipE
tff(fact_1243_in__set__zipE,axiom,
    ! [X: nat,Y: product_prod_nat_nat,Xs: list_nat,Ys: list_P6011104703257516679at_nat] :
      ( member2223272150424702269at_nat(produc487386426758144856at_nat(X,Y),set_Pr6710958862608470481at_nat(zip_na1006125974040638520at_nat(Xs,Ys)))
     => ~ ( member_nat(X,set_nat2(Xs))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),set_Pr5648618587558075414at_nat(Ys))) ) ) ).

% in_set_zipE
tff(fact_1244_zip__same,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member6009668390874684176at_nat(produc2335810615602371415at_nat(A2,B2),set_Pr4552580085251402108at_nat(zip_Pr2522280425463452279at_nat(Xs,Xs)))
    <=> ( member8757157785044589968at_nat(A2,set_Pr3765526544606949372at_nat(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1245_zip__same,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(A2,B2),set_Pr5518436109238095868at_nat(zip_Pr4664179122662387191at_nat(Xs,Xs)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),set_Pr5648618587558075414at_nat(Xs)))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1246_zip__same,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,B2: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member9182180520733025616t_bool(produc4314488481105410327t_bool(A2,B2),set_Pr2442920830988923836t_bool(zip_fu586808474262450231t_bool(Xs,Xs)))
    <=> ( member4125802036136091927t_bool(A2,set_fu1437410266168021891t_bool(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1247_zip__same,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),set_Pr3765526544606949372at_nat(zip_se5600341670672612855at_nat(Xs,Xs)))
    <=> ( member2643936169264416010at_nat(A2,set_se5049602875457034614at_nat(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1248_zip__same,axiom,
    ! [A2: int,B2: int,Xs: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),set_Pr2470121279949933262nt_int(zip_int_int(Xs,Xs)))
    <=> ( member_int(A2,set_int2(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1249_zip__same,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),set_Pr5648618587558075414at_nat(zip_nat_nat(Xs,Xs))))
    <=> ( member_nat(A2,set_nat2(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1250_zip__same,axiom,
    ! [A2: produc6575502325842934193n_assn,B2: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member4743710847862083984n_assn(produc6128706242671166167n_assn(A2,B2),set_Pr8975451644279219196n_assn(zip_Pr8806919808394484215n_assn(Xs,Xs)))
    <=> ( member7957490590177025114n_assn(A2,set_Pr1139785259514867910n_assn(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1251_zip__same,axiom,
    ! [A2: assn,B2: assn,Xs: list_assn] :
      ( member7957490590177025114n_assn(produc118845697133431529n_assn(A2,B2),set_Pr1139785259514867910n_assn(zip_assn_assn(Xs,Xs)))
    <=> ( member_assn(A2,set_assn2(Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_1252_zip__update,axiom,
    ! [Xs: list_c,I: nat,X: c,Ys: list_c,Y: c] : zip_c_c(list_update_c(Xs,I,X),list_update_c(Ys,I,Y)) = list_u8616212504517173525od_c_c(zip_c_c(Xs,Ys),I,aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,X),Y)) ).

% zip_update
tff(fact_1253_zip__update,axiom,
    ! [Xs: list_c,I: nat,X: c,Ys: list_d,Y: d] : zip_c_d(list_update_c(Xs,I,X),list_update_d(Ys,I,Y)) = list_u8616212508820402326od_c_d(zip_c_d(Xs,Ys),I,aa_d_P6917246732237330211od_c_d(aa_c_f7904789294214393838od_c_d(product_Pair_c_d,X),Y)) ).

% zip_update
tff(fact_1254_zip__update,axiom,
    ! [Xs: list_d,I: nat,X: d,Ys: list_c,Y: c] : zip_d_c(list_update_d(Xs,I,X),list_update_c(Ys,I,Y)) = list_u5828756922544497492od_d_c(zip_d_c(Xs,Ys),I,aa_c_P4769231446293241506od_d_c(aa_d_f479461486583208174od_d_c(product_Pair_d_c,X),Y)) ).

% zip_update
tff(fact_1255_zip__update,axiom,
    ! [Xs: list_d,I: nat,X: d,Ys: list_d,Y: d] : zip_d_d(list_update_d(Xs,I,X),list_update_d(Ys,I,Y)) = list_u5828756926847726293od_d_d(zip_d_d(Xs,Ys),I,aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,X),Y)) ).

% zip_update
tff(fact_1256_zip__update,axiom,
    ! [Xs: list_f4295387047331035401e_term,I: nat,X: fun_Pr272337284026993017e_term,Ys: list_P5578671422887162913nteger,Y: produc8923325533196201883nteger] : zip_fu197485079333405161nteger(list_u6547189781710427531e_term(Xs,I,X),list_u2254550707601501961nteger(Ys,I,Y)) = list_u5284324012848482281nteger(zip_fu197485079333405161nteger(Xs,Ys),I,produc7384524951051686153nteger(X,Y)) ).

% zip_update
tff(fact_1257_zip__update,axiom,
    ! [Xs: list_f5857023252961307252t_bool,I: nat,X: fun_Pr6001797050664311790t_bool,Ys: list_P9062070895058802706et_nat,Y: produc3658429121746597890et_nat] : zip_fu8308979540302171001et_nat(list_u922235751406726748t_bool(Xs,I,X),list_u9060326803697358356et_nat(Ys,I,Y)) = list_u5271228338798995633et_nat(zip_fu8308979540302171001et_nat(Xs,Ys),I,produc8148480564485311129et_nat(X,Y)) ).

% zip_update
tff(fact_1258_zip__update,axiom,
    ! [Xs: list_f5857023252961307252t_bool,I: nat,X: fun_Pr6001797050664311790t_bool,Ys: list_P5162959638400604975et_nat,Y: produc6108883039684432031et_nat] : zip_fu8753176646203213590et_nat(list_u922235751406726748t_bool(Xs,I,X),list_u5271228338798995633et_nat(Ys,I,Y)) = list_u8301521716676237006et_nat(zip_fu8753176646203213590et_nat(Xs,Ys),I,produc3083477295402903094et_nat(X,Y)) ).

% zip_update
tff(fact_1259_zip__update,axiom,
    ! [Xs: list_f8504123576994025594e_term,I: nat,X: fun_Pr9164807099015279978e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] : zip_fu2408773928738089174nt_int(list_u3754337643414342012e_term(Xs,I,X),list_u3002344382305578791nt_int(Ys,I,Y)) = list_u2433715936953936854nt_int(zip_fu2408773928738089174nt_int(Xs,Ys),I,produc1560874099174258678nt_int(X,Y)) ).

% zip_update
tff(fact_1260_zip__update,axiom,
    ! [Xs: list_f6829383385585334350e_term,I: nat,X: fun_in6429348457993719614e_term,Ys: list_P5707943133018811711nt_int,Y: product_prod_int_int] : zip_fu9102111156773635586nt_int(list_u8051947491296383824e_term(Xs,I,X),list_u3002344382305578791nt_int(Ys,I,Y)) = list_u6288444477985464578nt_int(zip_fu9102111156773635586nt_int(Xs,Ys),I,produc4852315488734385442nt_int(X,Y)) ).

% zip_update
tff(fact_1261_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_assn,Ys: list_assn,X: assn] :
      ( ( size_size_list_assn(Xs) = size_size_list_assn(Ys) )
     => ( member_assn(X,set_assn2(Xs))
       => ~ ! [Y2: assn] : ~ member7957490590177025114n_assn(produc118845697133431529n_assn(X,Y2),set_Pr1139785259514867910n_assn(zip_assn_assn(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1262_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_c,X: c] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( member_c(X,set_c2(Xs))
       => ~ ! [Y2: c] : ~ member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,X),Y2),set_Product_prod_c_c2(zip_c_c(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1263_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_d,X: c] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( member_c(X,set_c2(Xs))
       => ~ ! [Y2: d] : ~ member5074992363344545361od_c_d(aa_d_P6917246732237330211od_c_d(aa_c_f7904789294214393838od_c_d(product_Pair_c_d,X),Y2),set_Product_prod_c_d2(zip_c_d(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1264_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_nat,X: c] :
      ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
     => ( member_c(X,set_c2(Xs))
       => ~ ! [Y2: nat] : ~ member8195077246299207702_c_nat(aa_nat2369501448413299853_c_nat(aa_c_f4242068757942246672_c_nat(product_Pair_c_nat,X),Y2),set_Pr3395872032415823530_c_nat(zip_c_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1265_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_c,Ys: list_int,X: c] :
      ( ( size_size_list_c(Xs) = size_size_list_int(Ys) )
     => ( member_c(X,set_c2(Xs))
       => ~ ! [Y2: int] : ~ member4017226226790010994_c_int(aa_int4739195317257975621_c_int(aa_c_f8299530501364392392_c_int(product_Pair_c_int,X),Y2),set_Pr8441393049761402630_c_int(zip_c_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1266_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_c,X: d] :
      ( ( size_size_list_d(Xs) = size_size_list_c(Ys) )
     => ( member_d(X,set_d2(Xs))
       => ~ ! [Y2: c] : ~ member2287536777068640527od_d_c(aa_c_P4769231446293241506od_d_c(aa_d_f479461486583208174od_d_c(product_Pair_d_c,X),Y2),set_Product_prod_d_c2(zip_d_c(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1267_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_d,X: d] :
      ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
     => ( member_d(X,set_d2(Xs))
       => ~ ! [Y2: d] : ~ member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,X),Y2),set_Product_prod_d_d2(zip_d_d(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1268_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_nat,X: d] :
      ( ( size_size_list_d(Xs) = size_size_list_nat(Ys) )
     => ( member_d(X,set_d2(Xs))
       => ~ ! [Y2: nat] : ~ member207149538400826391_d_nat(aa_nat3604945777369694350_d_nat(aa_d_f7443782029664826770_d_nat(product_Pair_d_nat,X),Y2),set_Pr4631316361372218027_d_nat(zip_d_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1269_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_d,Ys: list_int,X: d] :
      ( ( size_size_list_d(Xs) = size_size_list_int(Ys) )
     => ( member_d(X,set_d2(Xs))
       => ~ ! [Y2: int] : ~ member5252670555746405491_d_int(aa_int5974639646214370118_d_int(aa_d_f2277871736232196682_d_int(product_Pair_d_int,X),Y2),set_Pr453465341863021319_d_int(zip_d_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1270_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_c,X: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_c(Ys) )
     => ( member_nat(X,set_nat2(Xs))
       => ~ ! [Y2: c] : ~ member8962352060716553276_nat_c(aa_c_P1942785857434746857_nat_c(aa_nat1477143957913088426_nat_c(product_Pair_nat_c,X),Y2),set_Pr4163146846833169104_nat_c(zip_nat_c(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_1271_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_assn,Ys: list_assn,Y: assn] :
      ( ( size_size_list_assn(Xs) = size_size_list_assn(Ys) )
     => ( member_assn(Y,set_assn2(Ys))
       => ~ ! [X2: assn] : ~ member7957490590177025114n_assn(produc118845697133431529n_assn(X2,Y),set_Pr1139785259514867910n_assn(zip_assn_assn(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1272_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_c,Y: c] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( member_c(Y,set_c2(Ys))
       => ~ ! [X2: c] : ~ member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,X2),Y),set_Product_prod_c_c2(zip_c_c(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1273_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_d,Y: d] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( member_d(Y,set_d2(Ys))
       => ~ ! [X2: c] : ~ member5074992363344545361od_c_d(aa_d_P6917246732237330211od_c_d(aa_c_f7904789294214393838od_c_d(product_Pair_c_d,X2),Y),set_Product_prod_c_d2(zip_c_d(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1274_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_nat,Y: nat] :
      ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
     => ( member_nat(Y,set_nat2(Ys))
       => ~ ! [X2: c] : ~ member8195077246299207702_c_nat(aa_nat2369501448413299853_c_nat(aa_c_f4242068757942246672_c_nat(product_Pair_c_nat,X2),Y),set_Pr3395872032415823530_c_nat(zip_c_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1275_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_c,Ys: list_int,Y: int] :
      ( ( size_size_list_c(Xs) = size_size_list_int(Ys) )
     => ( member_int(Y,set_int2(Ys))
       => ~ ! [X2: c] : ~ member4017226226790010994_c_int(aa_int4739195317257975621_c_int(aa_c_f8299530501364392392_c_int(product_Pair_c_int,X2),Y),set_Pr8441393049761402630_c_int(zip_c_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1276_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_c,Y: c] :
      ( ( size_size_list_d(Xs) = size_size_list_c(Ys) )
     => ( member_c(Y,set_c2(Ys))
       => ~ ! [X2: d] : ~ member2287536777068640527od_d_c(aa_c_P4769231446293241506od_d_c(aa_d_f479461486583208174od_d_c(product_Pair_d_c,X2),Y),set_Product_prod_d_c2(zip_d_c(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1277_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_d,Y: d] :
      ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
     => ( member_d(Y,set_d2(Ys))
       => ~ ! [X2: d] : ~ member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,X2),Y),set_Product_prod_d_d2(zip_d_d(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1278_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_nat,Y: nat] :
      ( ( size_size_list_d(Xs) = size_size_list_nat(Ys) )
     => ( member_nat(Y,set_nat2(Ys))
       => ~ ! [X2: d] : ~ member207149538400826391_d_nat(aa_nat3604945777369694350_d_nat(aa_d_f7443782029664826770_d_nat(product_Pair_d_nat,X2),Y),set_Pr4631316361372218027_d_nat(zip_d_nat(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1279_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_d,Ys: list_int,Y: int] :
      ( ( size_size_list_d(Xs) = size_size_list_int(Ys) )
     => ( member_int(Y,set_int2(Ys))
       => ~ ! [X2: d] : ~ member5252670555746405491_d_int(aa_int5974639646214370118_d_int(aa_d_f2277871736232196682_d_int(product_Pair_d_int,X2),Y),set_Pr453465341863021319_d_int(zip_d_int(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1280_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_c,Y: c] :
      ( ( size_size_list_nat(Xs) = size_size_list_c(Ys) )
     => ( member_c(Y,set_c2(Ys))
       => ~ ! [X2: nat] : ~ member8962352060716553276_nat_c(aa_c_P1942785857434746857_nat_c(aa_nat1477143957913088426_nat_c(product_Pair_nat_c,X2),Y),set_Pr4163146846833169104_nat_c(zip_nat_c(Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_1281_length__remove1,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
       => ( size_s1588839187904960729at_nat(remove8895909992688781315at_nat(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s1588839187904960729at_nat(Xs)),one_one_nat) ) )
      & ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
       => ( size_s1588839187904960729at_nat(remove8895909992688781315at_nat(X,Xs)) = size_s1588839187904960729at_nat(Xs) ) ) ) ).

% length_remove1
tff(fact_1282_length__remove1,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => ( size_s5460976970255530739at_nat(remove5766854097993331357at_nat(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s5460976970255530739at_nat(Xs)),one_one_nat) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => ( size_s5460976970255530739at_nat(remove5766854097993331357at_nat(X,Xs)) = size_s5460976970255530739at_nat(Xs) ) ) ) ).

% length_remove1
tff(fact_1283_length__remove1,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
       => ( size_s1066996506189871328t_bool(remove7684436675009651594t_bool(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s1066996506189871328t_bool(Xs)),one_one_nat) ) )
      & ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
       => ( size_s1066996506189871328t_bool(remove7684436675009651594t_bool(X,Xs)) = size_s1066996506189871328t_bool(Xs) ) ) ) ).

% length_remove1
tff(fact_1284_length__remove1,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
       => ( size_s6829681357464350627n_assn(remove1670527618125605709n_assn(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s6829681357464350627n_assn(Xs)),one_one_nat) ) )
      & ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
       => ( size_s6829681357464350627n_assn(remove1670527618125605709n_assn(X,Xs)) = size_s6829681357464350627n_assn(Xs) ) ) ) ).

% length_remove1
tff(fact_1285_length__remove1,axiom,
    ! [X: c,Xs: list_c] :
      ( ( member_c(X,set_c2(Xs))
       => ( size_size_list_c(remove1_c(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),one_one_nat) ) )
      & ( ~ member_c(X,set_c2(Xs))
       => ( size_size_list_c(remove1_c(X,Xs)) = size_size_list_c(Xs) ) ) ) ).

% length_remove1
tff(fact_1286_length__remove1,axiom,
    ! [X: d,Xs: list_d] :
      ( ( member_d(X,set_d2(Xs))
       => ( size_size_list_d(remove1_d(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),one_one_nat) ) )
      & ( ~ member_d(X,set_d2(Xs))
       => ( size_size_list_d(remove1_d(X,Xs)) = size_size_list_d(Xs) ) ) ) ).

% length_remove1
tff(fact_1287_length__remove1,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat(X,set_nat2(Xs))
       => ( size_size_list_nat(remove1_nat(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat) ) )
      & ( ~ member_nat(X,set_nat2(Xs))
       => ( size_size_list_nat(remove1_nat(X,Xs)) = size_size_list_nat(Xs) ) ) ) ).

% length_remove1
tff(fact_1288_length__remove1,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int(X,set_int2(Xs))
       => ( size_size_list_int(remove1_int(X,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat) ) )
      & ( ~ member_int(X,set_int2(Xs))
       => ( size_size_list_int(remove1_int(X,Xs)) = size_size_list_int(Xs) ) ) ) ).

% length_remove1
tff(fact_1289_insert__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : insert_nat2(X,coset_nat(Xs)) = coset_nat(removeAll_nat(X,Xs)) ).

% insert_code(2)
tff(fact_1290_insert__code_I2_J,axiom,
    ! [X: int,Xs: list_int] : insert_int2(X,coset_int(Xs)) = coset_int(removeAll_int(X,Xs)) ).

% insert_code(2)
tff(fact_1291_insert__code_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : insert8211810215607154385at_nat(X,coset_6809516720700967924at_nat(Xs)) = coset_6809516720700967924at_nat(remove3673390508374433037at_nat(X,Xs)) ).

% insert_code(2)
tff(fact_1292_insert__code_I2_J,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] : insert9069300056098147895at_nat(X,coset_5307619257043544794at_nat(Xs)) = coset_5307619257043544794at_nat(remove4908499291543849587at_nat(X,Xs)) ).

% insert_code(2)
tff(fact_1293_insert__code_I2_J,axiom,
    ! [X: bool,Xs: list_bool] : insert_bool2(X,coset_bool(Xs)) = coset_bool(removeAll_bool(X,Xs)) ).

% insert_code(2)
tff(fact_1294_zipf__zip,axiom,
    ! [L12: list_c,L23: list_c] :
      ( ( size_size_list_c(L12) = size_size_list_c(L23) )
     => ( zipf_c684006316856269033od_c_c(product_Pair_c_c,L12,L23) = zip_c_c(L12,L23) ) ) ).

% zipf_zip
tff(fact_1295_zipf__zip,axiom,
    ! [L12: list_c,L23: list_d] :
      ( ( size_size_list_c(L12) = size_size_list_d(L23) )
     => ( zipf_c44566020827681705od_c_d(product_Pair_c_d,L12,L23) = zip_c_d(L12,L23) ) ) ).

% zipf_zip
tff(fact_1296_zipf__zip,axiom,
    ! [L12: list_c,L23: list_nat] :
      ( ( size_size_list_c(L12) = size_size_list_nat(L23) )
     => ( zipf_c7084520048314076179_c_nat(product_Pair_c_nat,L12,L23) = zip_c_nat(L12,L23) ) ) ).

% zipf_zip
tff(fact_1297_zipf__zip,axiom,
    ! [L12: list_c,L23: list_int] :
      ( ( size_size_list_c(L12) = size_size_list_int(L23) )
     => ( zipf_c230841880303976139_c_int(product_Pair_c_int,L12,L23) = zip_c_int(L12,L23) ) ) ).

% zipf_zip
tff(fact_1298_zipf__zip,axiom,
    ! [L12: list_d,L23: list_c] :
      ( ( size_size_list_d(L12) = size_size_list_c(L23) )
     => ( zipf_d1836499321508429225od_d_c(product_Pair_d_c,L12,L23) = zip_d_c(L12,L23) ) ) ).

% zipf_zip
tff(fact_1299_zipf__zip,axiom,
    ! [L12: list_d,L23: list_d] :
      ( ( size_size_list_d(L12) = size_size_list_d(L23) )
     => ( zipf_d1197059025479841897od_d_d(product_Pair_d_d,L12,L23) = zip_d_d(L12,L23) ) ) ).

% zipf_zip
tff(fact_1300_zipf__zip,axiom,
    ! [L12: list_d,L23: list_nat] :
      ( ( size_size_list_d(L12) = size_size_list_nat(L23) )
     => ( zipf_d7851395264309270293_d_nat(product_Pair_d_nat,L12,L23) = zip_d_nat(L12,L23) ) ) ).

% zipf_zip
tff(fact_1301_zipf__zip,axiom,
    ! [L12: list_d,L23: list_int] :
      ( ( size_size_list_d(L12) = size_size_list_int(L23) )
     => ( zipf_d997717096299170253_d_int(product_Pair_d_int,L12,L23) = zip_d_int(L12,L23) ) ) ).

% zipf_zip
tff(fact_1302_zipf__zip,axiom,
    ! [L12: list_nat,L23: list_c] :
      ( ( size_size_list_nat(L12) = size_size_list_c(L23) )
     => ( zipf_n5773672615478942615_nat_c(product_Pair_nat_c,L12,L23) = zip_nat_c(L12,L23) ) ) ).

% zipf_zip
tff(fact_1303_zipf__zip,axiom,
    ! [L12: list_nat,L23: list_d] :
      ( ( size_size_list_nat(L12) = size_size_list_d(L23) )
     => ( zipf_n3657244561791696409_nat_d(product_Pair_nat_d,L12,L23) = zip_nat_d(L12,L23) ) ) ).

% zipf_zip
tff(fact_1304_one__assn__raw_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(one_assn_raw,produc7507926704131184380et_nat(H2,As2)))
    <=> ( As2 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_1305_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(one_assn_raw,X))
      <=> pp(Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( pp(Y)
            <=> ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_1306_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(one_assn_raw,X))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_1307_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(one_assn_raw,X))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_1308_psubset__insert__iff,axiom,
    ! [A: set_fu6080891106736489294t_bool,X: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le3815294503984575994t_bool(A,insert969342481230318014t_bool(X,B))
    <=> ( ( member4125802036136091927t_bool(X,B)
         => ord_le3815294503984575994t_bool(A,B) )
        & ( ~ member4125802036136091927t_bool(X,B)
         => ( ( member4125802036136091927t_bool(X,A)
             => ord_le3815294503984575994t_bool(minus_5708602904570203829t_bool(A,insert969342481230318014t_bool(X,bot_bo2714889297266208186t_bool)),B) )
            & ( ~ member4125802036136091927t_bool(X,A)
             => ord_le387542735852657390t_bool(A,B) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_1309_psubset__insert__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(A,insert9069300056098147895at_nat(X,B))
    <=> ( ( member8757157785044589968at_nat(X,B)
         => ord_le2604355607129572851at_nat(A,B) )
        & ( ~ member8757157785044589968at_nat(X,B)
         => ( ( member8757157785044589968at_nat(X,A)
             => ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)),B) )
            & ( ~ member8757157785044589968at_nat(X,A)
             => ord_le1268244103169919719at_nat(A,B) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_1310_psubset__insert__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A),insert8211810215607154385at_nat(X,B)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A),B)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
         => ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
             => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B)) )
            & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
             => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_1311_psubset__insert__iff,axiom,
    ! [A: set_bool,X: bool,B: set_bool] :
      ( ord_less_set_bool(A,insert_bool2(X,B))
    <=> ( ( member_bool(X,B)
         => ord_less_set_bool(A,B) )
        & ( ~ member_bool(X,B)
         => ( ( member_bool(X,A)
             => ord_less_set_bool(minus_minus_set_bool(A,insert_bool2(X,bot_bot_set_bool)),B) )
            & ( ~ member_bool(X,A)
             => ord_less_eq_set_bool(A,B) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_1312_psubset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ord_less_set_int(A,insert_int2(X,B))
    <=> ( ( member_int(X,B)
         => ord_less_set_int(A,B) )
        & ( ~ member_int(X,B)
         => ( ( member_int(X,A)
             => ord_less_set_int(minus_minus_set_int(A,insert_int2(X,bot_bot_set_int)),B) )
            & ( ~ member_int(X,A)
             => ord_less_eq_set_int(A,B) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_1313_psubset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ord_less_set_nat(A,insert_nat2(X,B))
    <=> ( ( member_nat(X,B)
         => ord_less_set_nat(A,B) )
        & ( ~ member_nat(X,B)
         => ( ( member_nat(X,A)
             => ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(X,bot_bot_set_nat)),B) )
            & ( ~ member_nat(X,A)
             => ord_less_eq_set_nat(A,B) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_1314_lexn__length,axiom,
    ! [Xs: list_c,Ys: list_c,R2: set_Product_prod_c_c,N: nat] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(Xs,Ys),lexn_c(R2,N))
     => ( ( size_size_list_c(Xs) = N )
        & ( size_size_list_c(Ys) = N ) ) ) ).

% lexn_length
tff(fact_1315_lexn__length,axiom,
    ! [Xs: list_d,Ys: list_d,R2: set_Product_prod_d_d,N: nat] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(Xs,Ys),lexn_d(R2,N))
     => ( ( size_size_list_d(Xs) = N )
        & ( size_size_list_d(Ys) = N ) ) ) ).

% lexn_length
tff(fact_1316_lexn__length,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,N: nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),lexn_nat(R2,N))
     => ( ( size_size_list_nat(Xs) = N )
        & ( size_size_list_nat(Ys) = N ) ) ) ).

% lexn_length
tff(fact_1317_lexn__length,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int,N: nat] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),lexn_int(R2,N))
     => ( ( size_size_list_int(Xs) = N )
        & ( size_size_list_int(Ys) = N ) ) ) ).

% lexn_length
tff(fact_1318_pairwise__alt,axiom,
    ! [R: fun_Pr124437560471516333t_bool,S: set_Pr4329608150637261639at_nat] :
      ( pairwi3222497012029234898at_nat(R,S)
    <=> ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,S)
         => ! [Xa2: produc3843707927480180839at_nat] :
              ( member8757157785044589968at_nat(Xa2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,S),insert9069300056098147895at_nat(X5,bot_bo228742789529271731at_nat)))
             => pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(R,X5),Xa2)) ) ) ) ).

% pairwise_alt
tff(fact_1319_pairwise__alt,axiom,
    ! [R: fun_Pr248683899324676269t_bool,S: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(R),S))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),S))
         => ! [Xa2: product_prod_nat_nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,S),insert8211810215607154385at_nat(X5,bot_bo2099793752762293965at_nat))))
             => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X5),Xa2)) ) ) ) ).

% pairwise_alt
tff(fact_1320_pairwise__alt,axiom,
    ! [R: fun_bo1841576699660343469l_bool,S: set_bool] :
      ( pairwise_bool(R,S)
    <=> ! [X5: bool] :
          ( member_bool(X5,S)
         => ! [Xa2: bool] :
              ( member_bool(Xa2,minus_minus_set_bool(S,insert_bool2(X5,bot_bot_set_bool)))
             => pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R,X5),Xa2)) ) ) ) ).

% pairwise_alt
tff(fact_1321_pairwise__alt,axiom,
    ! [R: fun_int_fun_int_bool,S: set_int] :
      ( pairwise_int(R,S)
    <=> ! [X5: int] :
          ( member_int(X5,S)
         => ! [Xa2: int] :
              ( member_int(Xa2,minus_minus_set_int(S,insert_int2(X5,bot_bot_set_int)))
             => pp(aa_int_bool(aa_int_fun_int_bool(R,X5),Xa2)) ) ) ) ).

% pairwise_alt
tff(fact_1322_pairwise__alt,axiom,
    ! [R: fun_nat_fun_nat_bool,S: set_nat] :
      ( pairwise_nat(R,S)
    <=> ! [X5: nat] :
          ( member_nat(X5,S)
         => ! [Xa2: nat] :
              ( member_nat(Xa2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,S),insert_nat2(X5,bot_bot_set_nat)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X5),Xa2)) ) ) ) ).

% pairwise_alt
tff(fact_1323_and__extract__pure__left__iff,axiom,
    ! [B2: bool,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,pure_assn(B2)),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),Q)),pure_assn(B2)) ).

% and_extract_pure_left_iff
tff(fact_1324_and__extract__pure__right__iff,axiom,
    ! [P: assn,B2: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),pure_assn(B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),P)),pure_assn(B2)) ).

% and_extract_pure_right_iff
tff(fact_1325_order__refl,axiom,
    ! [X: assn] : pp(aa_assn_bool(ord_less_eq_assn(X),X)) ).

% order_refl
tff(fact_1326_order__refl,axiom,
    ! [X: filter_nat] : ord_le2510731241096832064er_nat(X,X) ).

% order_refl
tff(fact_1327_order__refl,axiom,
    ! [X: set_nat] : ord_less_eq_set_nat(X,X) ).

% order_refl
tff(fact_1328_order__refl,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X)) ).

% order_refl
tff(fact_1329_order__refl,axiom,
    ! [X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X)) ).

% order_refl
tff(fact_1330_dual__order_Orefl,axiom,
    ! [A2: assn] : pp(aa_assn_bool(ord_less_eq_assn(A2),A2)) ).

% dual_order.refl
tff(fact_1331_dual__order_Orefl,axiom,
    ! [A2: filter_nat] : ord_le2510731241096832064er_nat(A2,A2) ).

% dual_order.refl
tff(fact_1332_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ord_less_eq_set_nat(A2,A2) ).

% dual_order.refl
tff(fact_1333_dual__order_Orefl,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A2)) ).

% dual_order.refl
tff(fact_1334_dual__order_Orefl,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),A2)) ).

% dual_order.refl
tff(fact_1335_subsetI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ! [X2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X2,A)
         => member8757157785044589968at_nat(X2,B) )
     => ord_le1268244103169919719at_nat(A,B) ) ).

% subsetI
tff(fact_1336_subsetI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),B)) )
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B)) ) ).

% subsetI
tff(fact_1337_subsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X2: int] :
          ( member_int(X2,A)
         => member_int(X2,B) )
     => ord_less_eq_set_int(A,B) ) ).

% subsetI
tff(fact_1338_subsetI,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X2,A)
         => member4125802036136091927t_bool(X2,B) )
     => ord_le387542735852657390t_bool(A,B) ) ).

% subsetI
tff(fact_1339_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( member_nat(X2,A)
         => member_nat(X2,B) )
     => ord_less_eq_set_nat(A,B) ) ).

% subsetI
tff(fact_1340_subset__antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( ord_less_eq_set_nat(B,A)
       => ( A = B ) ) ) ).

% subset_antisym
tff(fact_1341_inf_Oidem,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),A2) = A2 ).

% inf.idem
tff(fact_1342_inf_Oidem,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),A2) = A2 ).

% inf.idem
tff(fact_1343_inf_Oidem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),A2) = A2 ).

% inf.idem
tff(fact_1344_inf__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% inf_idem
tff(fact_1345_inf__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),X) = X ).

% inf_idem
tff(fact_1346_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),X) = X ).

% inf_idem
tff(fact_1347_inf_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) ).

% inf.left_idem
tff(fact_1348_inf_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% inf.left_idem
tff(fact_1349_inf_Oleft__idem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% inf.left_idem
tff(fact_1350_inf__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_left_idem
tff(fact_1351_inf__left__idem,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,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_left_idem
tff(fact_1352_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_left_idem
tff(fact_1353_inf_Oright__idem,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) ).

% inf.right_idem
tff(fact_1354_inf_Oright__idem,axiom,
    ! [A2: set_nat,B2: 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,A2),B2)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% inf.right_idem
tff(fact_1355_inf_Oright__idem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% inf.right_idem
tff(fact_1356_inf__right__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_right_idem
tff(fact_1357_inf__right__idem,axiom,
    ! [X: set_nat,Y: 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),Y)),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_right_idem
tff(fact_1358_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_right_idem
tff(fact_1359_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)))
    <=> ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Y))
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Z)) ) ) ).

% le_inf_iff
tff(fact_1360_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
        & pp(aa_assn_bool(ord_less_eq_assn(X),Z)) ) ) ).

% le_inf_iff
tff(fact_1361_le__inf__iff,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z))
    <=> ( ord_le2510731241096832064er_nat(X,Y)
        & ord_le2510731241096832064er_nat(X,Z) ) ) ).

% le_inf_iff
tff(fact_1362_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
    <=> ( ord_less_eq_set_nat(X,Y)
        & ord_less_eq_set_nat(X,Z) ) ) ).

% le_inf_iff
tff(fact_1363_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% le_inf_iff
tff(fact_1364_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).

% le_inf_iff
tff(fact_1365_inf_Obounded__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C)))
    <=> ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_1366_inf_Obounded__iff,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C)))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
        & pp(aa_assn_bool(ord_less_eq_assn(A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_1367_inf_Obounded__iff,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,B2),C))
    <=> ( ord_le2510731241096832064er_nat(A2,B2)
        & ord_le2510731241096832064er_nat(A2,C) ) ) ).

% inf.bounded_iff
tff(fact_1368_inf_Obounded__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C))
    <=> ( ord_less_eq_set_nat(A2,B2)
        & ord_less_eq_set_nat(A2,C) ) ) ).

% inf.bounded_iff
tff(fact_1369_inf_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_1370_inf_Obounded__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% inf.bounded_iff
tff(fact_1371_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_1372_inf__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),bot_bo228742789529271731at_nat) = bot_bo228742789529271731at_nat ).

% inf_bot_right
tff(fact_1373_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% inf_bot_right
tff(fact_1374_inf__bot__right,axiom,
    ! [X: set_bool] : inf_inf_set_bool(X,bot_bot_set_bool) = bot_bot_set_bool ).

% inf_bot_right
tff(fact_1375_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_1376_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_1377_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_1378_inf__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,bot_bo228742789529271731at_nat),X) = bot_bo228742789529271731at_nat ).

% inf_bot_left
tff(fact_1379_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% inf_bot_left
tff(fact_1380_inf__bot__left,axiom,
    ! [X: set_bool] : inf_inf_set_bool(bot_bot_set_bool,X) = bot_bot_set_bool ).

% inf_bot_left
tff(fact_1381_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_1382_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_1383_subset__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,bot_bo228742789529271731at_nat)
    <=> ( A = bot_bo228742789529271731at_nat ) ) ).

% subset_empty
tff(fact_1384_subset__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat))
    <=> ( A = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
tff(fact_1385_subset__empty,axiom,
    ! [A: set_bool] :
      ( ord_less_eq_set_bool(A,bot_bot_set_bool)
    <=> ( A = bot_bot_set_bool ) ) ).

% subset_empty
tff(fact_1386_subset__empty,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1387_subset__empty,axiom,
    ! [A: set_nat] :
      ( ord_less_eq_set_nat(A,bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1388_empty__subsetI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(bot_bo228742789529271731at_nat,A) ).

% empty_subsetI
tff(fact_1389_empty__subsetI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A)) ).

% empty_subsetI
tff(fact_1390_empty__subsetI,axiom,
    ! [A: set_bool] : ord_less_eq_set_bool(bot_bot_set_bool,A) ).

% empty_subsetI
tff(fact_1391_empty__subsetI,axiom,
    ! [A: set_int] : ord_less_eq_set_int(bot_bot_set_int,A) ).

% empty_subsetI
tff(fact_1392_empty__subsetI,axiom,
    ! [A: set_nat] : ord_less_eq_set_nat(bot_bot_set_nat,A) ).

% empty_subsetI
tff(fact_1393_insert__subset,axiom,
    ! [X: bool,A: set_bool,B: set_bool] :
      ( ord_less_eq_set_bool(insert_bool2(X,A),B)
    <=> ( member_bool(X,B)
        & ord_less_eq_set_bool(A,B) ) ) ).

% insert_subset
tff(fact_1394_insert__subset,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,A),B)
    <=> ( member8757157785044589968at_nat(X,B)
        & ord_le1268244103169919719at_nat(A,B) ) ) ).

% insert_subset
tff(fact_1395_insert__subset,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(X,A)),B))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B))
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B)) ) ) ).

% insert_subset
tff(fact_1396_insert__subset,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ord_less_eq_set_int(insert_int2(X,A),B)
    <=> ( member_int(X,B)
        & ord_less_eq_set_int(A,B) ) ) ).

% insert_subset
tff(fact_1397_insert__subset,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le387542735852657390t_bool(insert969342481230318014t_bool(X,A),B)
    <=> ( member4125802036136091927t_bool(X,B)
        & ord_le387542735852657390t_bool(A,B) ) ) ).

% insert_subset
tff(fact_1398_insert__subset,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(insert_nat2(X,A),B)
    <=> ( member_nat(X,B)
        & ord_less_eq_set_nat(A,B) ) ) ).

% insert_subset
tff(fact_1399_psubsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( ( A != B )
       => ord_less_set_nat(A,B) ) ) ).

% psubsetI
tff(fact_1400_merge__pure__and,axiom,
    ! [A2: bool,B2: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,pure_assn(A2)),pure_assn(B2)) = pure_assn(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,A2),B2)) ).

% merge_pure_and
tff(fact_1401_singleton__insert__inj__eq,axiom,
    ! [B2: produc3843707927480180839at_nat,A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat) = insert9069300056098147895at_nat(A2,A) )
    <=> ( ( A2 = B2 )
        & ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1402_singleton__insert__inj__eq,axiom,
    ! [B2: product_prod_nat_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat) = insert8211810215607154385at_nat(A2,A) )
    <=> ( ( A2 = B2 )
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq
tff(fact_1403_singleton__insert__inj__eq,axiom,
    ! [B2: bool,A2: bool,A: set_bool] :
      ( ( insert_bool2(B2,bot_bot_set_bool) = insert_bool2(A2,A) )
    <=> ( ( pp(A2)
        <=> pp(B2) )
        & ord_less_eq_set_bool(A,insert_bool2(B2,bot_bot_set_bool)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1404_singleton__insert__inj__eq,axiom,
    ! [B2: int,A2: int,A: set_int] :
      ( ( insert_int2(B2,bot_bot_set_int) = insert_int2(A2,A) )
    <=> ( ( A2 = B2 )
        & ord_less_eq_set_int(A,insert_int2(B2,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1405_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( insert_nat2(B2,bot_bot_set_nat) = insert_nat2(A2,A) )
    <=> ( ( A2 = B2 )
        & ord_less_eq_set_nat(A,insert_nat2(B2,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1406_singleton__insert__inj__eq_H,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( insert9069300056098147895at_nat(A2,A) = insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat) )
    <=> ( ( A2 = B2 )
        & ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1407_singleton__insert__inj__eq_H,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( insert8211810215607154385at_nat(A2,A) = insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat) )
    <=> ( ( A2 = B2 )
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1408_singleton__insert__inj__eq_H,axiom,
    ! [A2: bool,A: set_bool,B2: bool] :
      ( ( insert_bool2(A2,A) = insert_bool2(B2,bot_bot_set_bool) )
    <=> ( ( pp(A2)
        <=> pp(B2) )
        & ord_less_eq_set_bool(A,insert_bool2(B2,bot_bot_set_bool)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1409_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B2: int] :
      ( ( insert_int2(A2,A) = insert_int2(B2,bot_bot_set_int) )
    <=> ( ( A2 = B2 )
        & ord_less_eq_set_int(A,insert_int2(B2,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1410_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( insert_nat2(A2,A) = insert_nat2(B2,bot_bot_set_nat) )
    <=> ( ( A2 = B2 )
        & ord_less_eq_set_nat(A,insert_nat2(B2,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1411_Diff__eq__empty__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ord_le1268244103169919719at_nat(A,B) ) ).

% Diff_eq_empty_iff
tff(fact_1412_Diff__eq__empty__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B)) ) ).

% Diff_eq_empty_iff
tff(fact_1413_Diff__eq__empty__iff,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( minus_minus_set_bool(A,B) = bot_bot_set_bool )
    <=> ord_less_eq_set_bool(A,B) ) ).

% Diff_eq_empty_iff
tff(fact_1414_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( minus_minus_set_int(A,B) = bot_bot_set_int )
    <=> ord_less_eq_set_int(A,B) ) ).

% Diff_eq_empty_iff
tff(fact_1415_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) = bot_bot_set_nat )
    <=> ord_less_eq_set_nat(A,B) ) ).

% Diff_eq_empty_iff
tff(fact_1416_and__extract__pure__right__ctx__iff,axiom,
    ! [P: assn,Q: assn,B2: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),pure_assn(B2))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),pure_assn(B2)) ).

% and_extract_pure_right_ctx_iff
tff(fact_1417_and__extract__pure__left__ctx__iff,axiom,
    ! [P: assn,B2: bool,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),pure_assn(B2))),Q) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),pure_assn(B2)) ).

% and_extract_pure_left_ctx_iff
tff(fact_1418_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
        & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),produc7507926704131184380et_nat(H2,bot_bot_set_nat))) ) ) ).

% mod_h_bot_iff(6)
tff(fact_1419_in__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( member8757157785044589968at_nat(X,A)
       => member8757157785044589968at_nat(X,B) ) ) ).

% in_mono
tff(fact_1420_in__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),B)) ) ) ).

% in_mono
tff(fact_1421_in__mono,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ord_less_eq_set_int(A,B)
     => ( member_int(X,A)
       => member_int(X,B) ) ) ).

% in_mono
tff(fact_1422_in__mono,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool,X: fun_Pr6001797050664311790t_bool] :
      ( ord_le387542735852657390t_bool(A,B)
     => ( member4125802036136091927t_bool(X,A)
       => member4125802036136091927t_bool(X,B) ) ) ).

% in_mono
tff(fact_1423_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( member_nat(X,A)
       => member_nat(X,B) ) ) ).

% in_mono
tff(fact_1424_subsetD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( member8757157785044589968at_nat(C,A)
       => member8757157785044589968at_nat(C,B) ) ) ).

% subsetD
tff(fact_1425_subsetD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% subsetD
tff(fact_1426_subsetD,axiom,
    ! [A: set_int,B: set_int,C: int] :
      ( ord_less_eq_set_int(A,B)
     => ( member_int(C,A)
       => member_int(C,B) ) ) ).

% subsetD
tff(fact_1427_subsetD,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool,C: fun_Pr6001797050664311790t_bool] :
      ( ord_le387542735852657390t_bool(A,B)
     => ( member4125802036136091927t_bool(C,A)
       => member4125802036136091927t_bool(C,B) ) ) ).

% subsetD
tff(fact_1428_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( member_nat(C,A)
       => member_nat(C,B) ) ) ).

% subsetD
tff(fact_1429_equalityE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ~ ( ord_less_eq_set_nat(A,B)
         => ~ ord_less_eq_set_nat(B,A) ) ) ).

% equalityE
tff(fact_1430_pairwiseD,axiom,
    ! [R: fun_Pr124437560471516333t_bool,S: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( pairwi3222497012029234898at_nat(R,S)
     => ( member8757157785044589968at_nat(X,S)
       => ( member8757157785044589968at_nat(Y,S)
         => ( ( X != Y )
           => pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(R,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_1431_pairwiseD,axiom,
    ! [R: fun_Pr248683899324676269t_bool,S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(R),S))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),S))
       => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),S))
         => ( ( X != Y )
           => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_1432_pairwiseD,axiom,
    ! [R: fun_nat_fun_nat_bool,S: set_nat,X: nat,Y: nat] :
      ( pairwise_nat(R,S)
     => ( member_nat(X,S)
       => ( member_nat(Y,S)
         => ( ( X != Y )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_1433_pairwiseD,axiom,
    ! [R: fun_int_fun_int_bool,S: set_int,X: int,Y: int] :
      ( pairwise_int(R,S)
     => ( member_int(X,S)
       => ( member_int(Y,S)
         => ( ( X != Y )
           => pp(aa_int_bool(aa_int_fun_int_bool(R,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_1434_pairwiseD,axiom,
    ! [R: fun_fu7365307207611327597l_bool,S: set_fu6080891106736489294t_bool,X: fun_Pr6001797050664311790t_bool,Y: fun_Pr6001797050664311790t_bool] :
      ( pairwi2704567075835687513t_bool(R,S)
     => ( member4125802036136091927t_bool(X,S)
       => ( member4125802036136091927t_bool(Y,S)
         => ( ( X != Y )
           => pp(aa_fun5588795374382764133l_bool(aa_fun4580269573335628190l_bool(R,X),Y)) ) ) ) ) ).

% pairwiseD
tff(fact_1435_pairwiseI,axiom,
    ! [S: set_Pr4329608150637261639at_nat,R: fun_Pr124437560471516333t_bool] :
      ( ! [X2: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X2,S)
         => ( member8757157785044589968at_nat(Y2,S)
           => ( ( X2 != Y2 )
             => pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(R,X2),Y2)) ) ) )
     => pairwi3222497012029234898at_nat(R,S) ) ).

% pairwiseI
tff(fact_1436_pairwiseI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,R: fun_Pr248683899324676269t_bool] :
      ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),S))
         => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),S))
           => ( ( X2 != Y2 )
             => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R,X2),Y2)) ) ) )
     => pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(R),S)) ) ).

% pairwiseI
tff(fact_1437_pairwiseI,axiom,
    ! [S: set_nat,R: fun_nat_fun_nat_bool] :
      ( ! [X2: nat,Y2: nat] :
          ( member_nat(X2,S)
         => ( member_nat(Y2,S)
           => ( ( X2 != Y2 )
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X2),Y2)) ) ) )
     => pairwise_nat(R,S) ) ).

% pairwiseI
tff(fact_1438_pairwiseI,axiom,
    ! [S: set_int,R: fun_int_fun_int_bool] :
      ( ! [X2: int,Y2: int] :
          ( member_int(X2,S)
         => ( member_int(Y2,S)
           => ( ( X2 != Y2 )
             => pp(aa_int_bool(aa_int_fun_int_bool(R,X2),Y2)) ) ) )
     => pairwise_int(R,S) ) ).

% pairwiseI
tff(fact_1439_pairwiseI,axiom,
    ! [S: set_fu6080891106736489294t_bool,R: fun_fu7365307207611327597l_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool,Y2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X2,S)
         => ( member4125802036136091927t_bool(Y2,S)
           => ( ( X2 != Y2 )
             => pp(aa_fun5588795374382764133l_bool(aa_fun4580269573335628190l_bool(R,X2),Y2)) ) ) )
     => pairwi2704567075835687513t_bool(R,S) ) ).

% pairwiseI
tff(fact_1440_subset__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
    <=> ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,A)
         => member8757157785044589968at_nat(X5,B) ) ) ).

% subset_eq
tff(fact_1441_subset__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),B)) ) ) ).

% subset_eq
tff(fact_1442_subset__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
    <=> ! [X5: int] :
          ( member_int(X5,A)
         => member_int(X5,B) ) ) ).

% subset_eq
tff(fact_1443_subset__eq,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le387542735852657390t_bool(A,B)
    <=> ! [X5: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X5,A)
         => member4125802036136091927t_bool(X5,B) ) ) ).

% subset_eq
tff(fact_1444_subset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
    <=> ! [X5: nat] :
          ( member_nat(X5,A)
         => member_nat(X5,B) ) ) ).

% subset_eq
tff(fact_1445_equalityD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ord_less_eq_set_nat(A,B) ) ).

% equalityD1
tff(fact_1446_equalityD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ord_less_eq_set_nat(B,A) ) ).

% equalityD2
tff(fact_1447_subset__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
    <=> ! [T2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(T2,A)
         => member8757157785044589968at_nat(T2,B) ) ) ).

% subset_iff
tff(fact_1448_subset__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
    <=> ! [T2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(T2),A))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(T2),B)) ) ) ).

% subset_iff
tff(fact_1449_subset__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
    <=> ! [T2: int] :
          ( member_int(T2,A)
         => member_int(T2,B) ) ) ).

% subset_iff
tff(fact_1450_subset__iff,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le387542735852657390t_bool(A,B)
    <=> ! [T2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(T2,A)
         => member4125802036136091927t_bool(T2,B) ) ) ).

% subset_iff
tff(fact_1451_subset__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
    <=> ! [T2: nat] :
          ( member_nat(T2,A)
         => member_nat(T2,B) ) ) ).

% subset_iff
tff(fact_1452_subset__refl,axiom,
    ! [A: set_nat] : ord_less_eq_set_nat(A,A) ).

% subset_refl
tff(fact_1453_Collect__mono,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( ! [X2: product_prod_nat_nat] :
          ( pp(aa_Pro2232559814647311250t_bool(P,X2))
         => pp(aa_Pro2232559814647311250t_bool(Q,X2)) )
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q))) ) ).

% Collect_mono
tff(fact_1454_Collect__mono,axiom,
    ! [P: fun_fu9160419385450930796l_bool,Q: fun_fu9160419385450930796l_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool] :
          ( pp(aa_fun5588795374382764133l_bool(P,X2))
         => pp(aa_fun5588795374382764133l_bool(Q,X2)) )
     => ord_le387542735852657390t_bool(collec3007137136614167897t_bool(P),collec3007137136614167897t_bool(Q)) ) ).

% Collect_mono
tff(fact_1455_Collect__mono,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_nat_bool(P,X2))
         => pp(aa_nat_bool(Q,X2)) )
     => ord_less_eq_set_nat(collect_nat(P),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1456_subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( ord_less_eq_set_nat(B,C2)
       => ord_less_eq_set_nat(A,C2) ) ) ).

% subset_trans
tff(fact_1457_pairwise__mono,axiom,
    ! [P: fun_nat_fun_nat_bool,A: set_nat,Q: fun_nat_fun_nat_bool,B: set_nat] :
      ( pairwise_nat(P,A)
     => ( ! [X2: nat,Y2: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(Q,X2),Y2)) )
       => ( ord_less_eq_set_nat(B,A)
         => pairwise_nat(Q,B) ) ) ) ).

% pairwise_mono
tff(fact_1458_set__eq__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( ord_less_eq_set_nat(A,B)
        & ord_less_eq_set_nat(B,A) ) ) ).

% set_eq_subset
tff(fact_1459_pairwise__subset,axiom,
    ! [P: fun_nat_fun_nat_bool,S: set_nat,T3: set_nat] :
      ( pairwise_nat(P,S)
     => ( ord_less_eq_set_nat(T3,S)
       => pairwise_nat(P,T3) ) ) ).

% pairwise_subset
tff(fact_1460_Collect__mono__iff,axiom,
    ! [P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,collec3392354462482085612at_nat(P)),collec3392354462482085612at_nat(Q)))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_Pro2232559814647311250t_bool(P,X5))
         => pp(aa_Pro2232559814647311250t_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1461_Collect__mono__iff,axiom,
    ! [P: fun_fu9160419385450930796l_bool,Q: fun_fu9160419385450930796l_bool] :
      ( ord_le387542735852657390t_bool(collec3007137136614167897t_bool(P),collec3007137136614167897t_bool(Q))
    <=> ! [X5: fun_Pr6001797050664311790t_bool] :
          ( pp(aa_fun5588795374382764133l_bool(P,X5))
         => pp(aa_fun5588795374382764133l_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1462_Collect__mono__iff,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( ord_less_eq_set_nat(collect_nat(P),collect_nat(Q))
    <=> ! [X5: nat] :
          ( pp(aa_nat_bool(P,X5))
         => pp(aa_nat_bool(Q,X5)) ) ) ).

% Collect_mono_iff
tff(fact_1463_ord__eq__le__eq__trans,axiom,
    ! [A2: assn,B2: assn,C: assn,D2: assn] :
      ( ( A2 = B2 )
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ( C = D2 )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1464_ord__eq__le__eq__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat,D2: filter_nat] :
      ( ( A2 = B2 )
     => ( ord_le2510731241096832064er_nat(B2,C)
       => ( ( C = D2 )
         => ord_le2510731241096832064er_nat(A2,D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1465_ord__eq__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat,D2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat(B2,C)
       => ( ( C = D2 )
         => ord_less_eq_set_nat(A2,D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1466_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( A2 = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ( C = D2 )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1467_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( A2 = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ( C = D2 )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),D2)) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1468_subset__Collect__conv,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,S),collec3392354462482085612at_nat(P)))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),S))
         => pp(aa_Pro2232559814647311250t_bool(P,X5)) ) ) ).

% subset_Collect_conv
tff(fact_1469_subset__Collect__conv,axiom,
    ! [S: set_fu6080891106736489294t_bool,P: fun_fu9160419385450930796l_bool] :
      ( ord_le387542735852657390t_bool(S,collec3007137136614167897t_bool(P))
    <=> ! [X5: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X5,S)
         => pp(aa_fun5588795374382764133l_bool(P,X5)) ) ) ).

% subset_Collect_conv
tff(fact_1470_subset__Collect__conv,axiom,
    ! [S: set_nat,P: fun_nat_bool] :
      ( ord_less_eq_set_nat(S,collect_nat(P))
    <=> ! [X5: nat] :
          ( member_nat(X5,S)
         => pp(aa_nat_bool(P,X5)) ) ) ).

% subset_Collect_conv
tff(fact_1471_assn__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% assn_aci(4)
tff(fact_1472_assn__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% assn_aci(3)
tff(fact_1473_assn__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% assn_aci(1)
tff(fact_1474_norm__assertion__simps_I31_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% norm_assertion_simps(31)
tff(fact_1475_norm__assertion__simps_I14_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% norm_assertion_simps(14)
tff(fact_1476_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_1477_nle__le,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_1478_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) )
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y)) )
         => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),Y))
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) )
           => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
               => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X)) )
             => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
                 => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1479_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
       => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y)) )
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),Y))
             => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) )
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
               => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X)) )
             => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
                 => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ) ) ) ) ).

% le_cases3
tff(fact_1480_order__class_Oorder__eq__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
        & pp(aa_assn_bool(ord_less_eq_assn(Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_1481_order__class_Oorder__eq__iff,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
    <=> ( ord_le2510731241096832064er_nat(X,Y)
        & ord_le2510731241096832064er_nat(Y,X) ) ) ).

% order_class.order_eq_iff
tff(fact_1482_order__class_Oorder__eq__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
    <=> ( ord_less_eq_set_nat(X,Y)
        & ord_less_eq_set_nat(Y,X) ) ) ).

% order_class.order_eq_iff
tff(fact_1483_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_1484_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% order_class.order_eq_iff
tff(fact_1485_ord__eq__le__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( A2 = B2 )
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => pp(aa_assn_bool(ord_less_eq_assn(A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1486_ord__eq__le__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ( A2 = B2 )
     => ( ord_le2510731241096832064er_nat(B2,C)
       => ord_le2510731241096832064er_nat(A2,C) ) ) ).

% ord_eq_le_trans
tff(fact_1487_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat(B2,C)
       => ord_less_eq_set_nat(A2,C) ) ) ).

% ord_eq_le_trans
tff(fact_1488_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1489_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 = B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% ord_eq_le_trans
tff(fact_1490_ord__le__eq__trans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ( B2 = C )
       => pp(aa_assn_bool(ord_less_eq_assn(A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1491_ord__le__eq__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( ( B2 = C )
       => ord_le2510731241096832064er_nat(A2,C) ) ) ).

% ord_le_eq_trans
tff(fact_1492_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( ( B2 = C )
       => ord_less_eq_set_nat(A2,C) ) ) ).

% ord_le_eq_trans
tff(fact_1493_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( B2 = C )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1494_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( B2 = C )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% ord_le_eq_trans
tff(fact_1495_order__antisym,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1496_order__antisym,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( ord_le2510731241096832064er_nat(Y,X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1497_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( ord_less_eq_set_nat(Y,X)
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1498_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1499_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
       => ( X = Y ) ) ) ).

% order_antisym
tff(fact_1500_order_Otrans,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => pp(aa_assn_bool(ord_less_eq_assn(A2),C)) ) ) ).

% order.trans
tff(fact_1501_order_Otrans,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( ord_le2510731241096832064er_nat(B2,C)
       => ord_le2510731241096832064er_nat(A2,C) ) ) ).

% order.trans
tff(fact_1502_order_Otrans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( ord_less_eq_set_nat(B2,C)
       => ord_less_eq_set_nat(A2,C) ) ) ).

% order.trans
tff(fact_1503_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% order.trans
tff(fact_1504_order_Otrans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% order.trans
tff(fact_1505_order__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_eq_assn(Y),Z))
       => pp(aa_assn_bool(ord_less_eq_assn(X),Z)) ) ) ).

% order_trans
tff(fact_1506_order__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( ord_le2510731241096832064er_nat(Y,Z)
       => ord_le2510731241096832064er_nat(X,Z) ) ) ).

% order_trans
tff(fact_1507_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( ord_less_eq_set_nat(Y,Z)
       => ord_less_eq_set_nat(X,Z) ) ) ).

% order_trans
tff(fact_1508_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z)) ) ) ).

% order_trans
tff(fact_1509_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z)) ) ) ).

% order_trans
tff(fact_1510_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_bool,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A3),B3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
     => ( ! [A3: nat,B3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,B3),A3))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A3),B3)) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,A2),B2)) ) ) ).

% linorder_wlog
tff(fact_1511_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_bool,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A3),B3))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
     => ( ! [A3: int,B3: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(P,B3),A3))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,A3),B3)) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A2),B2)) ) ) ).

% linorder_wlog
tff(fact_1512_dual__order_Oeq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 = B2 )
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
        & pp(aa_assn_bool(ord_less_eq_assn(A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_1513_dual__order_Oeq__iff,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2 = B2 )
    <=> ( ord_le2510731241096832064er_nat(B2,A2)
        & ord_le2510731241096832064er_nat(A2,B2) ) ) ).

% dual_order.eq_iff
tff(fact_1514_dual__order_Oeq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( ord_less_eq_set_nat(B2,A2)
        & ord_less_eq_set_nat(A2,B2) ) ) ).

% dual_order.eq_iff
tff(fact_1515_dual__order_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_1516_dual__order_Oeq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% dual_order.eq_iff
tff(fact_1517_dual__order_Oantisym,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_1518_dual__order_Oantisym,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( ord_le2510731241096832064er_nat(A2,B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_1519_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( ord_less_eq_set_nat(A2,B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_1520_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_1521_dual__order_Oantisym,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_1522_dual__order_Otrans,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(C),B2))
       => pp(aa_assn_bool(ord_less_eq_assn(C),A2)) ) ) ).

% dual_order.trans
tff(fact_1523_dual__order_Otrans,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( ord_le2510731241096832064er_nat(C,B2)
       => ord_le2510731241096832064er_nat(C,A2) ) ) ).

% dual_order.trans
tff(fact_1524_dual__order_Otrans,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( ord_less_eq_set_nat(C,B2)
       => ord_less_eq_set_nat(C,A2) ) ) ).

% dual_order.trans
tff(fact_1525_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% dual_order.trans
tff(fact_1526_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% dual_order.trans
tff(fact_1527_antisym,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_1528_antisym,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( ord_le2510731241096832064er_nat(B2,A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_1529_antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( ord_less_eq_set_nat(B2,A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_1530_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_1531_antisym,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_1532_Orderings_Oorder__eq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 = B2 )
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
        & pp(aa_assn_bool(ord_less_eq_assn(B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1533_Orderings_Oorder__eq__iff,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2 = B2 )
    <=> ( ord_le2510731241096832064er_nat(A2,B2)
        & ord_le2510731241096832064er_nat(B2,A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1534_Orderings_Oorder__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( ord_less_eq_set_nat(A2,B2)
        & ord_less_eq_set_nat(B2,A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_1535_Orderings_Oorder__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1536_Orderings_Oorder__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ).

% Orderings.order_eq_iff
tff(fact_1537_order__subst1,axiom,
    ! [A2: assn,F3: fun_assn_assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1538_order__subst1,axiom,
    ! [A2: assn,F3: fun_nat_assn,B2: nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),aa_nat_assn(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_nat_assn(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1539_order__subst1,axiom,
    ! [A2: assn,F3: fun_int_assn,B2: int,C: int] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),aa_int_assn(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,X2)),aa_int_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_int_assn(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1540_order__subst1,axiom,
    ! [A2: nat,F3: fun_assn_nat,B2: assn,C: assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_assn_nat(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_assn_nat(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1541_order__subst1,axiom,
    ! [A2: nat,F3: fun_nat_nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,X2)),aa_nat_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1542_order__subst1,axiom,
    ! [A2: nat,F3: fun_int_nat,B2: int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_int_nat(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,X2)),aa_int_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_int_nat(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1543_order__subst1,axiom,
    ! [A2: int,F3: fun_assn_int,B2: assn,C: assn] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_assn_int(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_assn_int(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1544_order__subst1,axiom,
    ! [A2: int,F3: fun_nat_int,B2: nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,X2)),aa_nat_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1545_order__subst1,axiom,
    ! [A2: int,F3: fun_int_int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,X2)),aa_int_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1546_order__subst1,axiom,
    ! [A2: assn,F3: fun_filter_nat_assn,B2: filter_nat,C: filter_nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),aa_filter_nat_assn(F3,B2)))
     => ( ord_le2510731241096832064er_nat(B2,C)
       => ( ! [X2: filter_nat,Y2: filter_nat] :
              ( ord_le2510731241096832064er_nat(X2,Y2)
             => pp(aa_assn_bool(ord_less_eq_assn(aa_filter_nat_assn(F3,X2)),aa_filter_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_filter_nat_assn(F3,C))) ) ) ) ).

% order_subst1
tff(fact_1547_order__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1548_order__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1549_order__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_int,C: int] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1550_order__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_assn,C: assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1551_order__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,X2)),aa_nat_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1552_order__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,X2)),aa_nat_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1553_order__subst2,axiom,
    ! [A2: int,B2: int,F3: fun_int_assn,C: assn] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,B2)),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,X2)),aa_int_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1554_order__subst2,axiom,
    ! [A2: int,B2: int,F3: fun_int_nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,B2)),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,X2)),aa_int_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1555_order__subst2,axiom,
    ! [A2: int,B2: int,F3: fun_int_int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,B2)),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,X2)),aa_int_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,A2)),C)) ) ) ) ).

% order_subst2
tff(fact_1556_order__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_filter_nat,C: filter_nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ord_le2510731241096832064er_nat(aa_assn_filter_nat(F3,B2),C)
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => ord_le2510731241096832064er_nat(aa_assn_filter_nat(F3,X2),aa_assn_filter_nat(F3,Y2)) )
         => ord_le2510731241096832064er_nat(aa_assn_filter_nat(F3,A2),C) ) ) ) ).

% order_subst2
tff(fact_1557_order__eq__refl,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
     => pp(aa_assn_bool(ord_less_eq_assn(X),Y)) ) ).

% order_eq_refl
tff(fact_1558_order__eq__refl,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ( X = Y )
     => ord_le2510731241096832064er_nat(X,Y) ) ).

% order_eq_refl
tff(fact_1559_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ord_less_eq_set_nat(X,Y) ) ).

% order_eq_refl
tff(fact_1560_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_eq_refl
tff(fact_1561_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_eq_refl
tff(fact_1562_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_linear
tff(fact_1563_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_linear
tff(fact_1564_ord__eq__le__subst,axiom,
    ! [A2: assn,F3: fun_assn_assn,B2: assn,C: assn] :
      ( ( A2 = aa_assn_assn(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1565_ord__eq__le__subst,axiom,
    ! [A2: nat,F3: fun_assn_nat,B2: assn,C: assn] :
      ( ( A2 = aa_assn_nat(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_assn_nat(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1566_ord__eq__le__subst,axiom,
    ! [A2: int,F3: fun_assn_int,B2: assn,C: assn] :
      ( ( A2 = aa_assn_int(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_assn_int(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1567_ord__eq__le__subst,axiom,
    ! [A2: assn,F3: fun_nat_assn,B2: nat,C: nat] :
      ( ( A2 = aa_nat_assn(F3,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_nat_assn(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1568_ord__eq__le__subst,axiom,
    ! [A2: nat,F3: fun_nat_nat,B2: nat,C: nat] :
      ( ( A2 = aa_nat_nat(F3,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,X2)),aa_nat_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1569_ord__eq__le__subst,axiom,
    ! [A2: int,F3: fun_nat_int,B2: nat,C: nat] :
      ( ( A2 = aa_nat_int(F3,B2) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,X2)),aa_nat_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_nat_int(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1570_ord__eq__le__subst,axiom,
    ! [A2: assn,F3: fun_int_assn,B2: int,C: int] :
      ( ( A2 = aa_int_assn(F3,B2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,X2)),aa_int_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_int_assn(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1571_ord__eq__le__subst,axiom,
    ! [A2: nat,F3: fun_int_nat,B2: int,C: int] :
      ( ( A2 = aa_int_nat(F3,B2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,X2)),aa_int_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_int_nat(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1572_ord__eq__le__subst,axiom,
    ! [A2: int,F3: fun_int_int,B2: int,C: int] :
      ( ( A2 = aa_int_int(F3,B2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,X2)),aa_int_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(F3,C))) ) ) ) ).

% ord_eq_le_subst
tff(fact_1573_ord__eq__le__subst,axiom,
    ! [A2: filter_nat,F3: fun_assn_filter_nat,B2: assn,C: assn] :
      ( ( A2 = aa_assn_filter_nat(F3,B2) )
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => ord_le2510731241096832064er_nat(aa_assn_filter_nat(F3,X2),aa_assn_filter_nat(F3,Y2)) )
         => ord_le2510731241096832064er_nat(A2,aa_assn_filter_nat(F3,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_1574_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ( aa_assn_assn(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1575_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ( aa_assn_nat(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1576_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_int,C: int] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ( aa_assn_int(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1577_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_assn,C: assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( aa_nat_assn(F3,B2) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1578_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( aa_nat_nat(F3,B2) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,X2)),aa_nat_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1579_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( aa_nat_int(F3,B2) = C )
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,X2)),aa_nat_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1580_ord__le__eq__subst,axiom,
    ! [A2: int,B2: int,F3: fun_int_assn,C: assn] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( aa_int_assn(F3,B2) = C )
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,X2)),aa_int_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_eq_assn(aa_int_assn(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1581_ord__le__eq__subst,axiom,
    ! [A2: int,B2: int,F3: fun_int_nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( aa_int_nat(F3,B2) = C )
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,X2)),aa_int_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_int_nat(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1582_ord__le__eq__subst,axiom,
    ! [A2: int,B2: int,F3: fun_int_int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( aa_int_int(F3,B2) = C )
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,X2)),aa_int_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(F3,A2)),C)) ) ) ) ).

% ord_le_eq_subst
tff(fact_1583_ord__le__eq__subst,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_filter_nat,C: filter_nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ( aa_assn_filter_nat(F3,B2) = C )
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => ord_le2510731241096832064er_nat(aa_assn_filter_nat(F3,X2),aa_assn_filter_nat(F3,Y2)) )
         => ord_le2510731241096832064er_nat(aa_assn_filter_nat(F3,A2),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_1584_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_le_cases
tff(fact_1585_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_le_cases
tff(fact_1586_order__antisym__conv,axiom,
    ! [Y: assn,X: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
     => ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1587_order__antisym__conv,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,X)
     => ( ord_le2510731241096832064er_nat(X,Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1588_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ord_less_eq_set_nat(Y,X)
     => ( ord_less_eq_set_nat(X,Y)
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1589_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1590_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% order_antisym_conv
tff(fact_1591_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_sup_aci(4)
tff(fact_1592_inf__sup__aci_I4_J,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,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_1593_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_1594_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_aci(3)
tff(fact_1595_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: 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),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_1596_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_1597_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_1598_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: 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),Y)),Z) = 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),Z)) ).

% inf_sup_aci(2)
tff(fact_1599_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_1600_inf__sup__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_sup_aci(1)
tff(fact_1601_inf__sup__aci_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_1602_inf__sup__aci_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_1603_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_1604_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_1605_inf__sup__ord_I2_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y),Y) ).

% inf_sup_ord(2)
tff(fact_1606_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y),Y) ).

% inf_sup_ord(2)
tff(fact_1607_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_1608_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y)) ).

% inf_sup_ord(2)
tff(fact_1609_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_1610_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_1611_inf__sup__ord_I1_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y),X) ).

% inf_sup_ord(1)
tff(fact_1612_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y),X) ).

% inf_sup_ord(1)
tff(fact_1613_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_1614_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X)) ).

% inf_sup_ord(1)
tff(fact_1615_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X)) ).

% inf_le1
tff(fact_1616_inf__le1,axiom,
    ! [X: assn,Y: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X)) ).

% inf_le1
tff(fact_1617_inf__le1,axiom,
    ! [X: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y),X) ).

% inf_le1
tff(fact_1618_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y),X) ).

% inf_le1
tff(fact_1619_inf__le1,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X)) ).

% inf_le1
tff(fact_1620_inf__le1,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X)) ).

% inf_le1
tff(fact_1621_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_1622_inf__le2,axiom,
    ! [X: assn,Y: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y)) ).

% inf_le2
tff(fact_1623_inf__le2,axiom,
    ! [X: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y),Y) ).

% inf_le2
tff(fact_1624_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y),Y) ).

% inf_le2
tff(fact_1625_inf__le2,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y)) ).

% inf_le2
tff(fact_1626_inf__le2,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y)) ).

% inf_le2
tff(fact_1627_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)))
     => ~ ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),A2))
         => ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),B2)) ) ) ).

% le_infE
tff(fact_1628_le__infE,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)))
     => ~ ( pp(aa_assn_bool(ord_less_eq_assn(X),A2))
         => ~ pp(aa_assn_bool(ord_less_eq_assn(X),B2)) ) ) ).

% le_infE
tff(fact_1629_le__infE,axiom,
    ! [X: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2))
     => ~ ( ord_le2510731241096832064er_nat(X,A2)
         => ~ ord_le2510731241096832064er_nat(X,B2) ) ) ).

% le_infE
tff(fact_1630_le__infE,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => ~ ( ord_less_eq_set_nat(X,A2)
         => ~ ord_less_eq_set_nat(X,B2) ) ) ).

% le_infE
tff(fact_1631_le__infE,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B2)) ) ) ).

% le_infE
tff(fact_1632_le__infE,axiom,
    ! [X: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B2)) ) ) ).

% le_infE
tff(fact_1633_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),A2))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),B2))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))) ) ) ).

% le_infI
tff(fact_1634_le__infI,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),A2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(X),B2))
       => pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2))) ) ) ).

% le_infI
tff(fact_1635_le__infI,axiom,
    ! [X: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,A2)
     => ( ord_le2510731241096832064er_nat(X,B2)
       => ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2)) ) ) ).

% le_infI
tff(fact_1636_le__infI,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(X,A2)
     => ( ord_less_eq_set_nat(X,B2)
       => ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% le_infI
tff(fact_1637_le__infI,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2))) ) ) ).

% le_infI
tff(fact_1638_le__infI,axiom,
    ! [X: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2))) ) ) ).

% le_infI
tff(fact_1639_inf__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),C))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B2),D2))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),D2))) ) ) ).

% inf_mono
tff(fact_1640_inf__mono,axiom,
    ! [A2: assn,C: assn,B2: assn,D2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),C))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),D2))
       => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,C),D2))) ) ) ).

% inf_mono
tff(fact_1641_inf__mono,axiom,
    ! [A2: filter_nat,C: filter_nat,B2: filter_nat,D2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,C)
     => ( ord_le2510731241096832064er_nat(B2,D2)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_1642_inf__mono,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
      ( ord_less_eq_set_nat(A2,C)
     => ( ord_less_eq_set_nat(B2,D2)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_1643_inf__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D2))) ) ) ).

% inf_mono
tff(fact_1644_inf__mono,axiom,
    ! [A2: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D2))) ) ) ).

% inf_mono
tff(fact_1645_le__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),X))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),X)) ) ).

% le_infI1
tff(fact_1646_le__infI1,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),X))
     => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),X)) ) ).

% le_infI1
tff(fact_1647_le__infI1,axiom,
    ! [A2: filter_nat,X: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,X)
     => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),X) ) ).

% le_infI1
tff(fact_1648_le__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,X)
     => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),X) ) ).

% le_infI1
tff(fact_1649_le__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),X)) ) ).

% le_infI1
tff(fact_1650_le__infI1,axiom,
    ! [A2: int,X: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),X)) ) ).

% le_infI1
tff(fact_1651_le__infI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B2),X))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),X)) ) ).

% le_infI2
tff(fact_1652_le__infI2,axiom,
    ! [B2: assn,X: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),X))
     => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),X)) ) ).

% le_infI2
tff(fact_1653_le__infI2,axiom,
    ! [B2: filter_nat,X: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,X)
     => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),X) ) ).

% le_infI2
tff(fact_1654_le__infI2,axiom,
    ! [B2: set_nat,X: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,X)
     => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),X) ) ).

% le_infI2
tff(fact_1655_le__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),X)) ) ).

% le_infI2
tff(fact_1656_le__infI2,axiom,
    ! [B2: int,X: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),X)) ) ).

% le_infI2
tff(fact_1657_inf_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C)) ).

% inf.assoc
tff(fact_1658_inf_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C: 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,A2),B2)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C)) ).

% inf.assoc
tff(fact_1659_inf_Oassoc,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C)) ).

% inf.assoc
tff(fact_1660_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_assoc
tff(fact_1661_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: 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),Y)),Z) = 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),Z)) ).

% inf_assoc
tff(fact_1662_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ).

% inf_assoc
tff(fact_1663_inf_OorderE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
     => ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ).

% inf.orderE
tff(fact_1664_inf_OorderE,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) ) ) ).

% inf.orderE
tff(fact_1665_inf_OorderE,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) ) ) ).

% inf.orderE
tff(fact_1666_inf_OorderE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ).

% inf.orderE
tff(fact_1667_inf_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) ) ) ).

% inf.orderE
tff(fact_1668_inf_OorderE,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) ) ) ).

% inf.orderE
tff(fact_1669_inf_OorderI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) )
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2)) ) ).

% inf.orderI
tff(fact_1670_inf_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) )
     => pp(aa_assn_bool(ord_less_eq_assn(A2),B2)) ) ).

% inf.orderI
tff(fact_1671_inf_OorderI,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) )
     => ord_le2510731241096832064er_nat(A2,B2) ) ).

% inf.orderI
tff(fact_1672_inf_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) )
     => ord_less_eq_set_nat(A2,B2) ) ).

% inf.orderI
tff(fact_1673_inf_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% inf.orderI
tff(fact_1674_inf_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% inf.orderI
tff(fact_1675_inf__unique,axiom,
    ! [F3: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F3,X2),Y2)),X2))
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F3,X2),Y2)),Y2))
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X2),Y2))
             => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X2),Z3))
               => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F3,Y2),Z3))) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F3,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1676_inf__unique,axiom,
    ! [F3: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F3,X2),Y2)),X2))
     => ( ! [X2: assn,Y2: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F3,X2),Y2)),Y2))
       => ( ! [X2: assn,Y2: assn,Z3: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => ( pp(aa_assn_bool(ord_less_eq_assn(X2),Z3))
               => pp(aa_assn_bool(ord_less_eq_assn(X2),aa_assn_assn(aa_ass954072293579886259n_assn(F3,Y2),Z3))) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F3,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1677_inf__unique,axiom,
    ! [F3: fun_fi7557781610047541018er_nat,X: filter_nat,Y: filter_nat] :
      ( ! [X2: filter_nat,Y2: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,X2),Y2),X2)
     => ( ! [X2: filter_nat,Y2: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,X2),Y2),Y2)
       => ( ! [X2: filter_nat,Y2: filter_nat,Z3: filter_nat] :
              ( ord_le2510731241096832064er_nat(X2,Y2)
             => ( ord_le2510731241096832064er_nat(X2,Z3)
               => ord_le2510731241096832064er_nat(X2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,Y2),Z3)) ) )
         => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1678_inf__unique,axiom,
    ! [F3: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,X2),Y2),X2)
     => ( ! [X2: set_nat,Y2: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,X2),Y2),Y2)
       => ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
              ( ord_less_eq_set_nat(X2,Y2)
             => ( ord_less_eq_set_nat(X2,Z3)
               => ord_less_eq_set_nat(X2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,Y2),Z3)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1679_inf__unique,axiom,
    ! [F3: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F3,X2),Y2)),X2))
     => ( ! [X2: nat,Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F3,X2),Y2)),Y2))
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Z3))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F3,Y2),Z3))) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F3,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1680_inf__unique,axiom,
    ! [F3: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F3,X2),Y2)),X2))
     => ( ! [X2: int,Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F3,X2),Y2)),Y2))
       => ( ! [X2: int,Y2: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Y2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),Z3))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F3,Y2),Z3))) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F3,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1681_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Y))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1682_le__iff__inf,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1683_le__iff__inf,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1684_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1685_le__iff__inf,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1686_le__iff__inf,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1687_inf_Oabsorb1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = A2 ) ) ).

% inf.absorb1
tff(fact_1688_inf_Oabsorb1,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = A2 ) ) ).

% inf.absorb1
tff(fact_1689_inf_Oabsorb1,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) = A2 ) ) ).

% inf.absorb1
tff(fact_1690_inf_Oabsorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% inf.absorb1
tff(fact_1691_inf_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = A2 ) ) ).

% inf.absorb1
tff(fact_1692_inf_Oabsorb1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) = A2 ) ) ).

% inf.absorb1
tff(fact_1693_inf_Oabsorb2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B2),A2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_1694_inf_Oabsorb2,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_1695_inf_Oabsorb2,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_1696_inf_Oabsorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_1697_inf_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_1698_inf_Oabsorb2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) = B2 ) ) ).

% inf.absorb2
tff(fact_1699_inf_Ocommute,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),A2) ).

% inf.commute
tff(fact_1700_inf_Ocommute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) ).

% inf.commute
tff(fact_1701_inf_Ocommute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) ).

% inf.commute
tff(fact_1702_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Y))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1703_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1704_inf__absorb1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1705_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1706_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1707_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1708_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,Y),X))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1709_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1710_inf__absorb2,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,X)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1711_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ord_less_eq_set_nat(Y,X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1712_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1713_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1714_inf__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_commute
tff(fact_1715_inf__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).

% inf_commute
tff(fact_1716_inf__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ).

% inf_commute
tff(fact_1717_inf_OboundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C)))
     => ~ ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
         => ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),C)) ) ) ).

% inf.boundedE
tff(fact_1718_inf_OboundedE,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C)))
     => ~ ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
         => ~ pp(aa_assn_bool(ord_less_eq_assn(A2),C)) ) ) ).

% inf.boundedE
tff(fact_1719_inf_OboundedE,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,B2),C))
     => ~ ( ord_le2510731241096832064er_nat(A2,B2)
         => ~ ord_le2510731241096832064er_nat(A2,C) ) ) ).

% inf.boundedE
tff(fact_1720_inf_OboundedE,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C))
     => ~ ( ord_less_eq_set_nat(A2,B2)
         => ~ ord_less_eq_set_nat(A2,C) ) ) ).

% inf.boundedE
tff(fact_1721_inf_OboundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% inf.boundedE
tff(fact_1722_inf_OboundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% inf.boundedE
tff(fact_1723_inf_OboundedI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),C))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C))) ) ) ).

% inf.boundedI
tff(fact_1724_inf_OboundedI,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(A2),C))
       => pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C))) ) ) ).

% inf.boundedI
tff(fact_1725_inf_OboundedI,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( ord_le2510731241096832064er_nat(A2,C)
       => ord_le2510731241096832064er_nat(A2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,B2),C)) ) ) ).

% inf.boundedI
tff(fact_1726_inf_OboundedI,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( ord_less_eq_set_nat(A2,C)
       => ord_less_eq_set_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C)) ) ) ).

% inf.boundedI
tff(fact_1727_inf_OboundedI,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C))) ) ) ).

% inf.boundedI
tff(fact_1728_inf_OboundedI,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C))) ) ) ).

% inf.boundedI
tff(fact_1729_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Y))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Z))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))) ) ) ).

% inf_greatest
tff(fact_1730_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_eq_assn(X),Z))
       => pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))) ) ) ).

% inf_greatest
tff(fact_1731_inf__greatest,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( ord_le2510731241096832064er_nat(X,Z)
       => ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1732_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( ord_less_eq_set_nat(X,Z)
       => ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1733_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))) ) ) ).

% inf_greatest
tff(fact_1734_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))) ) ) ).

% inf_greatest
tff(fact_1735_inf_Oorder__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
    <=> ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ).

% inf.order_iff
tff(fact_1736_inf_Oorder__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
    <=> ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) ) ) ).

% inf.order_iff
tff(fact_1737_inf_Oorder__iff,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
    <=> ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) ) ) ).

% inf.order_iff
tff(fact_1738_inf_Oorder__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
    <=> ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ).

% inf.order_iff
tff(fact_1739_inf_Oorder__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) ) ) ).

% inf.order_iff
tff(fact_1740_inf_Oorder__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) ) ) ).

% inf.order_iff
tff(fact_1741_inf_Ocobounded1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),A2)) ).

% inf.cobounded1
tff(fact_1742_inf_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),A2)) ).

% inf.cobounded1
tff(fact_1743_inf_Ocobounded1,axiom,
    ! [A2: filter_nat,B2: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),A2) ).

% inf.cobounded1
tff(fact_1744_inf_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),A2) ).

% inf.cobounded1
tff(fact_1745_inf_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),A2)) ).

% inf.cobounded1
tff(fact_1746_inf_Ocobounded1,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),A2)) ).

% inf.cobounded1
tff(fact_1747_inf_Ocobounded2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),B2)) ).

% inf.cobounded2
tff(fact_1748_inf_Ocobounded2,axiom,
    ! [A2: assn,B2: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),B2)) ).

% inf.cobounded2
tff(fact_1749_inf_Ocobounded2,axiom,
    ! [A2: filter_nat,B2: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),B2) ).

% inf.cobounded2
tff(fact_1750_inf_Ocobounded2,axiom,
    ! [A2: set_nat,B2: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),B2) ).

% inf.cobounded2
tff(fact_1751_inf_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),B2)) ).

% inf.cobounded2
tff(fact_1752_inf_Ocobounded2,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),B2)) ).

% inf.cobounded2
tff(fact_1753_inf_Oabsorb__iff1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),B2))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_1754_inf_Oabsorb__iff1,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_1755_inf_Oabsorb__iff1,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_1756_inf_Oabsorb__iff1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_1757_inf_Oabsorb__iff1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_1758_inf_Oabsorb__iff1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) = A2 ) ) ).

% inf.absorb_iff1
tff(fact_1759_inf_Oabsorb__iff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B2),A2))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_1760_inf_Oabsorb__iff2,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_1761_inf_Oabsorb__iff2,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_1762_inf_Oabsorb__iff2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_1763_inf_Oabsorb__iff2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_1764_inf_Oabsorb__iff2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) = B2 ) ) ).

% inf.absorb_iff2
tff(fact_1765_inf_OcoboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),C))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C)) ) ).

% inf.coboundedI1
tff(fact_1766_inf_OcoboundedI1,axiom,
    ! [A2: assn,C: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),C))
     => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C)) ) ).

% inf.coboundedI1
tff(fact_1767_inf_OcoboundedI1,axiom,
    ! [A2: filter_nat,C: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,C)
     => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),C) ) ).

% inf.coboundedI1
tff(fact_1768_inf_OcoboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,C)
     => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),C) ) ).

% inf.coboundedI1
tff(fact_1769_inf_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),C)) ) ).

% inf.coboundedI1
tff(fact_1770_inf_OcoboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),C)) ) ).

% inf.coboundedI1
tff(fact_1771_inf_OcoboundedI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B2),C))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C)) ) ).

% inf.coboundedI2
tff(fact_1772_inf_OcoboundedI2,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
     => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C)) ) ).

% inf.coboundedI2
tff(fact_1773_inf_OcoboundedI2,axiom,
    ! [B2: filter_nat,C: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,C)
     => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,A2),B2),C) ) ).

% inf.coboundedI2
tff(fact_1774_inf_OcoboundedI2,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,C)
     => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),C) ) ).

% inf.coboundedI2
tff(fact_1775_inf_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),C)) ) ).

% inf.coboundedI2
tff(fact_1776_inf_OcoboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),C)) ) ).

% inf.coboundedI2
tff(fact_1777_inf_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C)) ).

% inf.left_commute
tff(fact_1778_inf_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C)) ).

% inf.left_commute
tff(fact_1779_inf_Oleft__commute,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C)) ).

% inf.left_commute
tff(fact_1780_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_left_commute
tff(fact_1781_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: 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),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_left_commute
tff(fact_1782_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_left_commute
tff(fact_1783_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,C)
     => ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),C) ) ).

% inf.strict_coboundedI2
tff(fact_1784_inf_Ostrict__coboundedI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,B2),C))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_1785_inf_Ostrict__coboundedI2,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),C))
     => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_1786_inf_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_1787_inf_Ostrict__coboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),C)) ) ).

% inf.strict_coboundedI2
tff(fact_1788_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,C)
     => ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),C) ) ).

% inf.strict_coboundedI1
tff(fact_1789_inf_Ostrict__coboundedI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),C))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_1790_inf_Ostrict__coboundedI1,axiom,
    ! [A2: assn,C: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),C))
     => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_1791_inf_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_1792_inf_Ostrict__coboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),C)) ) ).

% inf.strict_coboundedI1
tff(fact_1793_inf_Ostrict__order__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,B2)
    <=> ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_1794_inf_Ostrict__order__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),B2))
    <=> ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_1795_inf_Ostrict__order__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
    <=> ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_1796_inf_Ostrict__order__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_1797_inf_Ostrict__order__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% inf.strict_order_iff
tff(fact_1798_inf_Ostrict__boundedE,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_set_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C))
     => ~ ( ord_less_set_nat(A2,B2)
         => ~ ord_less_set_nat(A2,C) ) ) ).

% inf.strict_boundedE
tff(fact_1799_inf_Ostrict__boundedE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C)))
     => ~ ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),B2))
         => ~ pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_1800_inf_Ostrict__boundedE,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C)))
     => ~ ( pp(aa_assn_bool(ord_less_assn(A2),B2))
         => ~ pp(aa_assn_bool(ord_less_assn(A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_1801_inf_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B2),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_1802_inf_Ostrict__boundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(inf_inf_int,B2),C)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% inf.strict_boundedE
tff(fact_1803_inf_Oabsorb4,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_1804_inf_Oabsorb4,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,B2),A2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_1805_inf_Oabsorb4,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_1806_inf_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_1807_inf_Oabsorb4,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) = B2 ) ) ).

% inf.absorb4
tff(fact_1808_inf_Oabsorb3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% inf.absorb3
tff(fact_1809_inf_Oabsorb3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),B2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = A2 ) ) ).

% inf.absorb3
tff(fact_1810_inf_Oabsorb3,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) = A2 ) ) ).

% inf.absorb3
tff(fact_1811_inf_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2) = A2 ) ) ).

% inf.absorb3
tff(fact_1812_inf_Oabsorb3,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2) = A2 ) ) ).

% inf.absorb3
tff(fact_1813_less__infI2,axiom,
    ! [B2: set_nat,X: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,X)
     => ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),X) ) ).

% less_infI2
tff(fact_1814_less__infI2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,B2),X))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),X)) ) ).

% less_infI2
tff(fact_1815_less__infI2,axiom,
    ! [B2: assn,X: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),X))
     => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),X)) ) ).

% less_infI2
tff(fact_1816_less__infI2,axiom,
    ! [B2: nat,X: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),X)) ) ).

% less_infI2
tff(fact_1817_less__infI2,axiom,
    ! [B2: int,X: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),X)) ) ).

% less_infI2
tff(fact_1818_less__infI1,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,X)
     => ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2),X) ) ).

% less_infI1
tff(fact_1819_less__infI1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,A2),X))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),X)) ) ).

% less_infI1
tff(fact_1820_less__infI1,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),X))
     => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),X)) ) ).

% less_infI1
tff(fact_1821_less__infI1,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B2)),X)) ) ).

% less_infI1
tff(fact_1822_less__infI1,axiom,
    ! [A2: int,X: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A2),B2)),X)) ) ).

% less_infI1
tff(fact_1823_pinf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X6),T)) ) ).

% pinf(6)
tff(fact_1824_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X6),T)) ) ).

% pinf(6)
tff(fact_1825_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),T)) ) ).

% pinf(6)
tff(fact_1826_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T)) ) ).

% pinf(6)
tff(fact_1827_pinf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z3),X6))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,T),X6)) ) ).

% pinf(8)
tff(fact_1828_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z3),X6))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X6)) ) ).

% pinf(8)
tff(fact_1829_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z3),X6))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X6)) ) ).

% pinf(8)
tff(fact_1830_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z3),X6))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6)) ) ).

% pinf(8)
tff(fact_1831_minf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X6),T)) ) ).

% minf(6)
tff(fact_1832_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X6),T)) ) ).

% minf(6)
tff(fact_1833_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X6),T)) ) ).

% minf(6)
tff(fact_1834_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T)) ) ).

% minf(6)
tff(fact_1835_minf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X6),Z3))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,T),X6)) ) ).

% minf(8)
tff(fact_1836_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X6),Z3))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,T),X6)) ) ).

% minf(8)
tff(fact_1837_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Z3))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,T),X6)) ) ).

% minf(8)
tff(fact_1838_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),Z3))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6)) ) ).

% minf(8)
tff(fact_1839_order__le__imp__less__or__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1840_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1841_order__le__imp__less__or__eq,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_assn(X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1842_order__le__imp__less__or__eq,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( ord_less_filter_nat(X,Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1843_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( ord_less_set_nat(X,Y)
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1844_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1845_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_1846_linorder__le__less__linear,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1847_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1848_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1849_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_le_less_linear
tff(fact_1850_order__less__le__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Co8499238673658658103nteger,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,X2)),aa_Cod1783974652533396728nteger(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1851_order__less__le__subst2,axiom,
    ! [A2: code_integer,B2: code_integer,F3: fun_Code_integer_num,C: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_Code_integer_num(F3,B2)),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,X2)),aa_Code_integer_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1852_order__less__le__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_as9138348466139350018nteger,C: code_integer] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_assn_Code_integer(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1853_order__less__le__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_num,C: num] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_assn_num(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1854_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F3: fun_num_Code_integer,C: code_integer] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(F3,B2)),C))
       => ( ! [X2: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F3,X2)),aa_num_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1855_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F3: fun_num_num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_num_num(F3,B2)),C))
       => ( ! [X2: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F3,X2)),aa_num_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1856_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_Code_integer,C: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,X2)),aa_nat_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1857_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,X2)),aa_nat_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1858_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F3: fun_int_Code_integer,C: code_integer] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_int_Code_integer(F3,B2)),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F3,X2)),aa_int_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1859_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F3: fun_int_num,C: num] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_int_num(F3,B2)),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_int_num(F3,X2)),aa_int_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_int_num(F3,A2)),C)) ) ) ) ).

% order_less_le_subst2
tff(fact_1860_order__less__le__subst1,axiom,
    ! [A2: code_integer,F3: fun_as9138348466139350018nteger,B2: assn,C: assn] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1861_order__less__le__subst1,axiom,
    ! [A2: num,F3: fun_assn_num,B2: assn,C: assn] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_assn_num(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_assn_num(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1862_order__less__le__subst1,axiom,
    ! [A2: assn,F3: fun_assn_assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_assn_assn(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_assn_assn(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1863_order__less__le__subst1,axiom,
    ! [A2: nat,F3: fun_assn_nat,B2: assn,C: assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_assn_nat(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_assn_nat(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1864_order__less__le__subst1,axiom,
    ! [A2: int,F3: fun_assn_int,B2: assn,C: assn] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_assn_int(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_assn_int(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1865_order__less__le__subst1,axiom,
    ! [A2: code_integer,F3: fun_nat_Code_integer,B2: nat,C: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F3,X2)),aa_nat_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1866_order__less__le__subst1,axiom,
    ! [A2: num,F3: fun_nat_num,B2: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_nat_num(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F3,X2)),aa_nat_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_nat_num(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1867_order__less__le__subst1,axiom,
    ! [A2: assn,F3: fun_nat_assn,B2: nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_assn(A2),aa_nat_assn(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(A2),aa_nat_assn(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1868_order__less__le__subst1,axiom,
    ! [A2: nat,F3: fun_nat_nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,X2)),aa_nat_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1869_order__less__le__subst1,axiom,
    ! [A2: int,F3: fun_nat_int,B2: nat,C: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,X2)),aa_nat_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(F3,C))) ) ) ) ).

% order_less_le_subst1
tff(fact_1870_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_as9138348466139350018nteger,C: code_integer] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1871_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_num,C: num] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1872_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(F3,X2)),aa_assn_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(aa_assn_assn(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1873_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_nat,C: nat] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_assn_nat(F3,X2)),aa_assn_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_assn_nat(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1874_order__le__less__subst2,axiom,
    ! [A2: assn,B2: assn,F3: fun_assn_int,C: int] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,B2)),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_assn_int(F3,X2)),aa_assn_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_assn_int(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1875_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_Code_integer,C: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(F3,X2)),aa_nat_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1876_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_num,C: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(F3,X2)),aa_nat_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1877_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_assn,C: assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_assn_bool(ord_less_eq_assn(aa_nat_assn(F3,X2)),aa_nat_assn(F3,Y2))) )
         => pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1878_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,X2)),aa_nat_nat(F3,Y2))) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1879_order__le__less__subst2,axiom,
    ! [A2: nat,B2: nat,F3: fun_nat_int,C: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F3,B2)),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,X2)),aa_nat_int(F3,Y2))) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F3,A2)),C)) ) ) ) ).

% order_le_less_subst2
tff(fact_1880_order__le__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_Co8499238673658658103nteger,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(F3,B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(F3,X2)),aa_Cod1783974652533396728nteger(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1881_order__le__less__subst1,axiom,
    ! [A2: num,F3: fun_Code_integer_num,B2: code_integer,C: code_integer] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_Code_integer_num(F3,B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => ( ! [X2: code_integer,Y2: code_integer] :
              ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_Code_integer_num(F3,X2)),aa_Code_integer_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_Code_integer_num(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1882_order__le__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_as9138348466139350018nteger,B2: assn,C: assn] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_assn_Code_integer(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_assn_Code_integer(F3,X2)),aa_assn_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_assn_Code_integer(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1883_order__le__less__subst1,axiom,
    ! [A2: num,F3: fun_assn_num,B2: assn,C: assn] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_assn_num(F3,B2)))
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => ( ! [X2: assn,Y2: assn] :
              ( pp(aa_assn_bool(ord_less_assn(X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_assn_num(F3,X2)),aa_assn_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_assn_num(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1884_order__le__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_num_Code_integer,B2: num,C: num] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_num_Code_integer(F3,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => ( ! [X2: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_num_Code_integer(F3,X2)),aa_num_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_num_Code_integer(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1885_order__le__less__subst1,axiom,
    ! [A2: num,F3: fun_num_num,B2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_num_num(F3,B2)))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => ( ! [X2: num,Y2: num] :
              ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(F3,X2)),aa_num_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_num_num(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1886_order__le__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_nat_Code_integer,B2: nat,C: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_nat_Code_integer(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,X2)),aa_nat_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_nat_Code_integer(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1887_order__le__less__subst1,axiom,
    ! [A2: num,F3: fun_nat_num,B2: nat,C: nat] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_nat_num(F3,B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => ( ! [X2: nat,Y2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,X2)),aa_nat_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_nat_num(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1888_order__le__less__subst1,axiom,
    ! [A2: code_integer,F3: fun_int_Code_integer,B2: int,C: int] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_int_Code_integer(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_int_Code_integer(F3,X2)),aa_int_Code_integer(F3,Y2))) )
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_int_Code_integer(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1889_order__le__less__subst1,axiom,
    ! [A2: num,F3: fun_int_num,B2: int,C: int] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),aa_int_num(F3,B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => ( ! [X2: int,Y2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Y2))
             => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_int_num(F3,X2)),aa_int_num(F3,Y2))) )
         => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_int_num(F3,C))) ) ) ) ).

% order_le_less_subst1
tff(fact_1890_order__less__le__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),Z))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_1891_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_1892_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_eq_assn(Y),Z))
       => pp(aa_assn_bool(ord_less_assn(X),Z)) ) ) ).

% order_less_le_trans
tff(fact_1893_order__less__le__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_less_filter_nat(X,Y)
     => ( ord_le2510731241096832064er_nat(Y,Z)
       => ord_less_filter_nat(X,Z) ) ) ).

% order_less_le_trans
tff(fact_1894_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ord_less_set_nat(X,Y)
     => ( ord_less_eq_set_nat(Y,Z)
       => ord_less_set_nat(X,Z) ) ) ).

% order_less_le_trans
tff(fact_1895_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_1896_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_less_le_trans
tff(fact_1897_order__le__less__trans,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_1898_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_1899_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_assn(Y),Z))
       => pp(aa_assn_bool(ord_less_assn(X),Z)) ) ) ).

% order_le_less_trans
tff(fact_1900_order__le__less__trans,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( ord_less_filter_nat(Y,Z)
       => ord_less_filter_nat(X,Z) ) ) ).

% order_le_less_trans
tff(fact_1901_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( ord_less_set_nat(Y,Z)
       => ord_less_set_nat(X,Z) ) ) ).

% order_le_less_trans
tff(fact_1902_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_1903_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z)) ) ) ).

% order_le_less_trans
tff(fact_1904_order__neq__le__trans,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != B2 )
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_1905_order__neq__le__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 != B2 )
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_1906_order__neq__le__trans,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 != B2 )
     => ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
       => pp(aa_assn_bool(ord_less_assn(A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_1907_order__neq__le__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2 != B2 )
     => ( ord_le2510731241096832064er_nat(A2,B2)
       => ord_less_filter_nat(A2,B2) ) ) ).

% order_neq_le_trans
tff(fact_1908_order__neq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != B2 )
     => ( ord_less_eq_set_nat(A2,B2)
       => ord_less_set_nat(A2,B2) ) ) ).

% order_neq_le_trans
tff(fact_1909_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_1910_order__neq__le__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != B2 )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% order_neq_le_trans
tff(fact_1911_order__le__neq__trans,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_1912_order__le__neq__trans,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_1913_order__le__neq__trans,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_assn_bool(ord_less_assn(A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_1914_order__le__neq__trans,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( ( A2 != B2 )
       => ord_less_filter_nat(A2,B2) ) ) ).

% order_le_neq_trans
tff(fact_1915_order__le__neq__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( ( A2 != B2 )
       => ord_less_set_nat(A2,B2) ) ) ).

% order_le_neq_trans
tff(fact_1916_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_1917_order__le__neq__trans,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( ( A2 != B2 )
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% order_le_neq_trans
tff(fact_1918_order__less__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ).

% order_less_imp_le
tff(fact_1919_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y)) ) ).

% order_less_imp_le
tff(fact_1920_order__less__imp__le,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
     => pp(aa_assn_bool(ord_less_eq_assn(X),Y)) ) ).

% order_less_imp_le
tff(fact_1921_order__less__imp__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_less_filter_nat(X,Y)
     => ord_le2510731241096832064er_nat(X,Y) ) ).

% order_less_imp_le
tff(fact_1922_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_set_nat(X,Y)
     => ord_less_eq_set_nat(X,Y) ) ).

% order_less_imp_le
tff(fact_1923_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ).

% order_less_imp_le
tff(fact_1924_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ).

% order_less_imp_le
tff(fact_1925_linorder__not__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X)) ) ).

% linorder_not_less
tff(fact_1926_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% linorder_not_less
tff(fact_1927_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% linorder_not_less
tff(fact_1928_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% linorder_not_less
tff(fact_1929_linorder__not__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),X)) ) ).

% linorder_not_le
tff(fact_1930_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),X)) ) ).

% linorder_not_le
tff(fact_1931_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ).

% linorder_not_le
tff(fact_1932_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),X)) ) ).

% linorder_not_le
tff(fact_1933_order__less__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1934_order__less__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1935_order__less__le,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1936_order__less__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_less_filter_nat(X,Y)
    <=> ( ord_le2510731241096832064er_nat(X,Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1937_order__less__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_set_nat(X,Y)
    <=> ( ord_less_eq_set_nat(X,Y)
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1938_order__less__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1939_order__less__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ( X != Y ) ) ) ).

% order_less_le
tff(fact_1940_order__le__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1941_order__le__less,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1942_order__le__less,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
    <=> ( pp(aa_assn_bool(ord_less_assn(X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1943_order__le__less,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
    <=> ( ord_less_filter_nat(X,Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1944_order__le__less,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
    <=> ( ord_less_set_nat(X,Y)
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1945_order__le__less,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1946_order__le__less,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
        | ( X = Y ) ) ) ).

% order_le_less
tff(fact_1947_dual__order_Ostrict__implies__order,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_1948_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_1949_dual__order_Ostrict__implies__order,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => pp(aa_assn_bool(ord_less_eq_assn(B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_1950_dual__order_Ostrict__implies__order,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_less_filter_nat(B2,A2)
     => ord_le2510731241096832064er_nat(B2,A2) ) ).

% dual_order.strict_implies_order
tff(fact_1951_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,A2)
     => ord_less_eq_set_nat(B2,A2) ) ).

% dual_order.strict_implies_order
tff(fact_1952_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_1953_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% dual_order.strict_implies_order
tff(fact_1954_order_Ostrict__implies__order,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_1955_order_Ostrict__implies__order,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_1956_order_Ostrict__implies__order,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => pp(aa_assn_bool(ord_less_eq_assn(A2),B2)) ) ).

% order.strict_implies_order
tff(fact_1957_order_Ostrict__implies__order,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_less_filter_nat(A2,B2)
     => ord_le2510731241096832064er_nat(A2,B2) ) ).

% order.strict_implies_order
tff(fact_1958_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,B2)
     => ord_less_eq_set_nat(A2,B2) ) ).

% order.strict_implies_order
tff(fact_1959_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_1960_order_Ostrict__implies__order,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% order.strict_implies_order
tff(fact_1961_dual__order_Ostrict__iff__not,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
        & ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1962_dual__order_Ostrict__iff__not,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1963_dual__order_Ostrict__iff__not,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
        & ~ pp(aa_assn_bool(ord_less_eq_assn(A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1964_dual__order_Ostrict__iff__not,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_less_filter_nat(B2,A2)
    <=> ( ord_le2510731241096832064er_nat(B2,A2)
        & ~ ord_le2510731241096832064er_nat(A2,B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1965_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,A2)
    <=> ( ord_less_eq_set_nat(B2,A2)
        & ~ ord_less_eq_set_nat(A2,B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_1966_dual__order_Ostrict__iff__not,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1967_dual__order_Ostrict__iff__not,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% dual_order.strict_iff_not
tff(fact_1968_dual__order_Ostrict__trans2,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_1969_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,C),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_1970_dual__order_Ostrict__trans2,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(C),B2))
       => pp(aa_assn_bool(ord_less_assn(C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_1971_dual__order_Ostrict__trans2,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ord_less_filter_nat(B2,A2)
     => ( ord_le2510731241096832064er_nat(C,B2)
       => ord_less_filter_nat(C,A2) ) ) ).

% dual_order.strict_trans2
tff(fact_1972_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ord_less_set_nat(B2,A2)
     => ( ord_less_eq_set_nat(C,B2)
       => ord_less_set_nat(C,A2) ) ) ).

% dual_order.strict_trans2
tff(fact_1973_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_1974_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% dual_order.strict_trans2
tff(fact_1975_dual__order_Ostrict__trans1,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_1976_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),B2))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_1977_dual__order_Ostrict__trans1,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( pp(aa_assn_bool(ord_less_assn(C),B2))
       => pp(aa_assn_bool(ord_less_assn(C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_1978_dual__order_Ostrict__trans1,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( ord_less_filter_nat(C,B2)
       => ord_less_filter_nat(C,A2) ) ) ).

% dual_order.strict_trans1
tff(fact_1979_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( ord_less_set_nat(C,B2)
       => ord_less_set_nat(C,A2) ) ) ).

% dual_order.strict_trans1
tff(fact_1980_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_1981_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% dual_order.strict_trans1
tff(fact_1982_dual__order_Ostrict__iff__order,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1983_dual__order_Ostrict__iff__order,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1984_dual__order_Ostrict__iff__order,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1985_dual__order_Ostrict__iff__order,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_less_filter_nat(B2,A2)
    <=> ( ord_le2510731241096832064er_nat(B2,A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1986_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,A2)
    <=> ( ord_less_eq_set_nat(B2,A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1987_dual__order_Ostrict__iff__order,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1988_dual__order_Ostrict__iff__order,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_1989_dual__order_Oorder__iff__strict,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1990_dual__order_Oorder__iff__strict,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1991_dual__order_Oorder__iff__strict,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
    <=> ( pp(aa_assn_bool(ord_less_assn(B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1992_dual__order_Oorder__iff__strict,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
    <=> ( ord_less_filter_nat(B2,A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1993_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
    <=> ( ord_less_set_nat(B2,A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1994_dual__order_Oorder__iff__strict,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1995_dual__order_Oorder__iff__strict,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_1996_order_Ostrict__iff__not,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
        & ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_1997_order_Ostrict__iff__not,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_1998_order_Ostrict__iff__not,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
        & ~ pp(aa_assn_bool(ord_less_eq_assn(B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_1999_order_Ostrict__iff__not,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_less_filter_nat(A2,B2)
    <=> ( ord_le2510731241096832064er_nat(A2,B2)
        & ~ ord_le2510731241096832064er_nat(B2,A2) ) ) ).

% order.strict_iff_not
tff(fact_2000_order_Ostrict__iff__not,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,B2)
    <=> ( ord_less_eq_set_nat(A2,B2)
        & ~ ord_less_eq_set_nat(B2,A2) ) ) ).

% order.strict_iff_not
tff(fact_2001_order_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_2002_order_Ostrict__iff__not,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ).

% order.strict_iff_not
tff(fact_2003_order_Ostrict__trans2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_2004_order_Ostrict__trans2,axiom,
    ! [A2: num,B2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B2),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_2005_order_Ostrict__trans2,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),C))
       => pp(aa_assn_bool(ord_less_assn(A2),C)) ) ) ).

% order.strict_trans2
tff(fact_2006_order_Ostrict__trans2,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_less_filter_nat(A2,B2)
     => ( ord_le2510731241096832064er_nat(B2,C)
       => ord_less_filter_nat(A2,C) ) ) ).

% order.strict_trans2
tff(fact_2007_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_set_nat(A2,B2)
     => ( ord_less_eq_set_nat(B2,C)
       => ord_less_set_nat(A2,C) ) ) ).

% order.strict_trans2
tff(fact_2008_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_2009_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% order.strict_trans2
tff(fact_2010_order_Ostrict__trans1,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_2011_order_Ostrict__trans1,axiom,
    ! [A2: num,B2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_2012_order_Ostrict__trans1,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( pp(aa_assn_bool(ord_less_assn(B2),C))
       => pp(aa_assn_bool(ord_less_assn(A2),C)) ) ) ).

% order.strict_trans1
tff(fact_2013_order_Ostrict__trans1,axiom,
    ! [A2: filter_nat,B2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( ord_less_filter_nat(B2,C)
       => ord_less_filter_nat(A2,C) ) ) ).

% order.strict_trans1
tff(fact_2014_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( ord_less_set_nat(B2,C)
       => ord_less_set_nat(A2,C) ) ) ).

% order.strict_trans1
tff(fact_2015_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_2016_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% order.strict_trans1
tff(fact_2017_order_Ostrict__iff__order,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2018_order_Ostrict__iff__order,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2019_order_Ostrict__iff__order,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2020_order_Ostrict__iff__order,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_less_filter_nat(A2,B2)
    <=> ( ord_le2510731241096832064er_nat(A2,B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2021_order_Ostrict__iff__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,B2)
    <=> ( ord_less_eq_set_nat(A2,B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2022_order_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2023_order_Ostrict__iff__order,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_2024_order_Oorder__iff__strict,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2025_order_Oorder__iff__strict,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2026_order_Oorder__iff__strict,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
    <=> ( pp(aa_assn_bool(ord_less_assn(A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2027_order_Oorder__iff__strict,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
    <=> ( ord_less_filter_nat(A2,B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2028_order_Oorder__iff__strict,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
    <=> ( ord_less_set_nat(A2,B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2029_order_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2030_order_Oorder__iff__strict,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_2031_not__le__imp__less,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).

% not_le_imp_less
tff(fact_2032_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% not_le_imp_less
tff(fact_2033_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% not_le_imp_less
tff(fact_2034_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% not_le_imp_less
tff(fact_2035_less__le__not__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
        & ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2036_less__le__not__le,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
        & ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2037_less__le__not__le,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),Y))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
        & ~ pp(aa_assn_bool(ord_less_eq_assn(Y),X)) ) ) ).

% less_le_not_le
tff(fact_2038_less__le__not__le,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_less_filter_nat(X,Y)
    <=> ( ord_le2510731241096832064er_nat(X,Y)
        & ~ ord_le2510731241096832064er_nat(Y,X) ) ) ).

% less_le_not_le
tff(fact_2039_less__le__not__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_set_nat(X,Y)
    <=> ( ord_less_eq_set_nat(X,Y)
        & ~ ord_less_eq_set_nat(Y,X) ) ) ).

% less_le_not_le
tff(fact_2040_less__le__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2041_less__le__not__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ) ).

% less_le_not_le
tff(fact_2042_antisym__conv2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
     => ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2043_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
     => ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2044_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( ~ pp(aa_assn_bool(ord_less_assn(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2045_antisym__conv2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( ~ ord_less_filter_nat(X,Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2046_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( ~ ord_less_set_nat(X,Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2047_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2048_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv2
tff(fact_2049_antisym__conv1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2050_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2051_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ pp(aa_assn_bool(ord_less_assn(X),Y))
     => ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2052_antisym__conv1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ~ ord_less_filter_nat(X,Y)
     => ( ord_le2510731241096832064er_nat(X,Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2053_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ord_less_set_nat(X,Y)
     => ( ord_less_eq_set_nat(X,Y)
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2054_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2055_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
      <=> ( X = Y ) ) ) ).

% antisym_conv1
tff(fact_2056_nless__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
    <=> ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2057_nless__le,axiom,
    ! [A2: num,B2: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2058_nless__le,axiom,
    ! [A2: assn,B2: assn] :
      ( ~ pp(aa_assn_bool(ord_less_assn(A2),B2))
    <=> ( ~ pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2059_nless__le,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ~ ord_less_filter_nat(A2,B2)
    <=> ( ~ ord_le2510731241096832064er_nat(A2,B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2060_nless__le,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ~ ord_less_set_nat(A2,B2)
    <=> ( ~ ord_less_eq_set_nat(A2,B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2061_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2062_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_2063_leI,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X)) ) ).

% leI
tff(fact_2064_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X)) ) ).

% leI
tff(fact_2065_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X)) ) ).

% leI
tff(fact_2066_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X)) ) ).

% leI
tff(fact_2067_leD,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),X))
     => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ).

% leD
tff(fact_2068_leD,axiom,
    ! [Y: num,X: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,Y),X))
     => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Y)) ) ).

% leD
tff(fact_2069_leD,axiom,
    ! [Y: assn,X: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
     => ~ pp(aa_assn_bool(ord_less_assn(X),Y)) ) ).

% leD
tff(fact_2070_leD,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,X)
     => ~ ord_less_filter_nat(X,Y) ) ).

% leD
tff(fact_2071_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ord_less_eq_set_nat(Y,X)
     => ~ ord_less_set_nat(X,Y) ) ).

% leD
tff(fact_2072_leD,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% leD
tff(fact_2073_leD,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ).

% leD
tff(fact_2074_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,bot_bo228742789529271731at_nat)
     => ( A2 = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2075_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),bot_bo2099793752762293965at_nat))
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2076_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_bool] :
      ( ord_less_eq_set_bool(A2,bot_bot_set_bool)
     => ( A2 = bot_bot_set_bool ) ) ).

% bot.extremum_uniqueI
tff(fact_2077_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( ord_less_eq_set_int(A2,bot_bot_set_int)
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_2078_bot_Oextremum__uniqueI,axiom,
    ! [A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),bot_bot_assn))
     => ( A2 = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
tff(fact_2079_bot_Oextremum__uniqueI,axiom,
    ! [A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,bot_bot_filter_nat)
     => ( A2 = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2080_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ord_less_eq_set_nat(A2,bot_bot_set_nat)
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2081_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),bot_bot_nat))
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_2082_bot_Oextremum__unique,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,bot_bo228742789529271731at_nat)
    <=> ( A2 = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_unique
tff(fact_2083_bot_Oextremum__unique,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),bot_bo2099793752762293965at_nat))
    <=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
tff(fact_2084_bot_Oextremum__unique,axiom,
    ! [A2: set_bool] :
      ( ord_less_eq_set_bool(A2,bot_bot_set_bool)
    <=> ( A2 = bot_bot_set_bool ) ) ).

% bot.extremum_unique
tff(fact_2085_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( ord_less_eq_set_int(A2,bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_2086_bot_Oextremum__unique,axiom,
    ! [A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),bot_bot_assn))
    <=> ( A2 = bot_bot_assn ) ) ).

% bot.extremum_unique
tff(fact_2087_bot_Oextremum__unique,axiom,
    ! [A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,bot_bot_filter_nat)
    <=> ( A2 = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_2088_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ord_less_eq_set_nat(A2,bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_2089_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),bot_bot_nat))
    <=> ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_2090_bot_Oextremum,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(bot_bo228742789529271731at_nat,A2) ).

% bot.extremum
tff(fact_2091_bot_Oextremum,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,bot_bo2099793752762293965at_nat),A2)) ).

% bot.extremum
tff(fact_2092_bot_Oextremum,axiom,
    ! [A2: set_bool] : ord_less_eq_set_bool(bot_bot_set_bool,A2) ).

% bot.extremum
tff(fact_2093_bot_Oextremum,axiom,
    ! [A2: set_int] : ord_less_eq_set_int(bot_bot_set_int,A2) ).

% bot.extremum
tff(fact_2094_bot_Oextremum,axiom,
    ! [A2: assn] : pp(aa_assn_bool(ord_less_eq_assn(bot_bot_assn),A2)) ).

% bot.extremum
tff(fact_2095_bot_Oextremum,axiom,
    ! [A2: filter_nat] : ord_le2510731241096832064er_nat(bot_bot_filter_nat,A2) ).

% bot.extremum
tff(fact_2096_bot_Oextremum,axiom,
    ! [A2: set_nat] : ord_less_eq_set_nat(bot_bot_set_nat,A2) ).

% bot.extremum
tff(fact_2097_bot_Oextremum,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,bot_bot_nat),A2)) ).

% bot.extremum
tff(fact_2098_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
          ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y2),R2)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y2),S2) )
     => ord_le1268244103169919719at_nat(R2,S2) ) ).

% subrelI
tff(fact_2099_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),R2))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),S2)) )
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,R2),S2)) ) ).

% subrelI
tff(fact_2100_subrelI,axiom,
    ! [R2: set_Pr5663546166630115725nteger,S2: set_Pr5663546166630115725nteger] :
      ( ! [X2: fun_Pr272337284026993017e_term,Y2: produc8923325533196201883nteger] :
          ( member6423679177046879214nteger(produc7384524951051686153nteger(X2,Y2),R2)
         => member6423679177046879214nteger(produc7384524951051686153nteger(X2,Y2),S2) )
     => ord_le3337582088779512301nteger(R2,S2) ) ).

% subrelI
tff(fact_2101_subrelI,axiom,
    ! [R2: set_Pr1106803562331729365et_nat,S2: set_Pr1106803562331729365et_nat] :
      ( ! [X2: fun_Pr6001797050664311790t_bool,Y2: produc3658429121746597890et_nat] :
          ( member8615881266069945910et_nat(produc8148480564485311129et_nat(X2,Y2),R2)
         => member8615881266069945910et_nat(produc8148480564485311129et_nat(X2,Y2),S2) )
     => ord_le8271263703964716597et_nat(R2,S2) ) ).

% subrelI
tff(fact_2102_subrelI,axiom,
    ! [R2: set_Pr9222568342844664434et_nat,S2: set_Pr9222568342844664434et_nat] :
      ( ! [X2: fun_Pr6001797050664311790t_bool,Y2: produc6108883039684432031et_nat] :
          ( member2818932950976722259et_nat(produc3083477295402903094et_nat(X2,Y2),R2)
         => member2818932950976722259et_nat(produc3083477295402903094et_nat(X2,Y2),S2) )
     => ord_le4695037514429983442et_nat(R2,S2) ) ).

% subrelI
tff(fact_2103_subrelI,axiom,
    ! [R2: set_Pr233528076387193594nt_int,S2: set_Pr233528076387193594nt_int] :
      ( ! [X2: fun_Pr9164807099015279978e_term,Y2: product_prod_int_int] :
          ( member690604328116145115nt_int(produc1560874099174258678nt_int(X2,Y2),R2)
         => member690604328116145115nt_int(produc1560874099174258678nt_int(X2,Y2),S2) )
     => ord_le5172437211506631514nt_int(R2,S2) ) ).

% subrelI
tff(fact_2104_subrelI,axiom,
    ! [R2: set_Pr5033440245338114598nt_int,S2: set_Pr5033440245338114598nt_int] :
      ( ! [X2: fun_in6429348457993719614e_term,Y2: product_prod_int_int] :
          ( member500491123954006279nt_int(produc4852315488734385442nt_int(X2,Y2),R2)
         => member500491123954006279nt_int(produc4852315488734385442nt_int(X2,Y2),S2) )
     => ord_le172202941055982214nt_int(R2,S2) ) ).

% subrelI
tff(fact_2105_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( minus_minus_rat(A2,B2) = minus_minus_rat(C,D2) )
     => ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
      <=> pp(aa_rat_bool(ord_less_eq_rat(C),D2)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2106_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D2)) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2107_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => pp(aa_rat_bool(ord_less_eq_rat(minus_minus_rat(A2,C)),minus_minus_rat(B2,C))) ) ).

% diff_right_mono
tff(fact_2108_diff__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C))) ) ).

% diff_right_mono
tff(fact_2109_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(B2),A2))
     => pp(aa_rat_bool(ord_less_eq_rat(minus_minus_rat(C,A2)),minus_minus_rat(C,B2))) ) ).

% diff_left_mono
tff(fact_2110_diff__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))) ) ).

% diff_left_mono
tff(fact_2111_diff__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(D2),C))
       => pp(aa_rat_bool(ord_less_eq_rat(minus_minus_rat(A2,C)),minus_minus_rat(B2,D2))) ) ) ).

% diff_mono
tff(fact_2112_diff__mono,axiom,
    ! [A2: int,B2: int,D2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D2))) ) ) ).

% diff_mono
tff(fact_2113_mod__and__dist,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q)),H2))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
        & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),H2)) ) ) ).

% mod_and_dist
tff(fact_2114_ent__conjI,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),B))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),C2))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C2))) ) ) ).

% ent_conjI
tff(fact_2115_ent__conjE1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),C2))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2)) ) ).

% ent_conjE1
tff(fact_2116_ent__conjE2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),C2))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C2)) ) ).

% ent_conjE2
tff(fact_2117_subset__insertI2,axiom,
    ! [A: set_int,B: set_int,B2: int] :
      ( ord_less_eq_set_int(A,B)
     => ord_less_eq_set_int(A,insert_int2(B2,B)) ) ).

% subset_insertI2
tff(fact_2118_subset__insertI2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(B2,B))) ) ).

% subset_insertI2
tff(fact_2119_subset__insertI2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(B2,B)) ) ).

% subset_insertI2
tff(fact_2120_subset__insertI2,axiom,
    ! [A: set_bool,B: set_bool,B2: bool] :
      ( ord_less_eq_set_bool(A,B)
     => ord_less_eq_set_bool(A,insert_bool2(B2,B)) ) ).

% subset_insertI2
tff(fact_2121_subset__insertI2,axiom,
    ! [A: set_nat,B: set_nat,B2: nat] :
      ( ord_less_eq_set_nat(A,B)
     => ord_less_eq_set_nat(A,insert_nat2(B2,B)) ) ).

% subset_insertI2
tff(fact_2122_subset__insertI,axiom,
    ! [B: set_int,A2: int] : ord_less_eq_set_int(B,insert_int2(A2,B)) ).

% subset_insertI
tff(fact_2123_subset__insertI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B),insert8211810215607154385at_nat(A2,B))) ).

% subset_insertI
tff(fact_2124_subset__insertI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] : ord_le1268244103169919719at_nat(B,insert9069300056098147895at_nat(A2,B)) ).

% subset_insertI
tff(fact_2125_subset__insertI,axiom,
    ! [B: set_bool,A2: bool] : ord_less_eq_set_bool(B,insert_bool2(A2,B)) ).

% subset_insertI
tff(fact_2126_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : ord_less_eq_set_nat(B,insert_nat2(A2,B)) ).

% subset_insertI
tff(fact_2127_subset__insert,axiom,
    ! [X: bool,A: set_bool,B: set_bool] :
      ( ~ member_bool(X,A)
     => ( ord_less_eq_set_bool(A,insert_bool2(X,B))
      <=> ord_less_eq_set_bool(A,B) ) ) ).

% subset_insert
tff(fact_2128_subset__insert,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(X,A)
     => ( ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(X,B))
      <=> ord_le1268244103169919719at_nat(A,B) ) ) ).

% subset_insert
tff(fact_2129_subset__insert,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(X,B)))
      <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B)) ) ) ).

% subset_insert
tff(fact_2130_subset__insert,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ member_int(X,A)
     => ( ord_less_eq_set_int(A,insert_int2(X,B))
      <=> ord_less_eq_set_int(A,B) ) ) ).

% subset_insert
tff(fact_2131_subset__insert,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(X,A)
     => ( ord_le387542735852657390t_bool(A,insert969342481230318014t_bool(X,B))
      <=> ord_le387542735852657390t_bool(A,B) ) ) ).

% subset_insert
tff(fact_2132_subset__insert,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ member_nat(X,A)
     => ( ord_less_eq_set_nat(A,insert_nat2(X,B))
      <=> ord_less_eq_set_nat(A,B) ) ) ).

% subset_insert
tff(fact_2133_insert__mono,axiom,
    ! [C2: set_int,D: set_int,A2: int] :
      ( ord_less_eq_set_int(C2,D)
     => ord_less_eq_set_int(insert_int2(A2,C2),insert_int2(A2,D)) ) ).

% insert_mono
tff(fact_2134_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),D))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(A2,C2)),insert8211810215607154385at_nat(A2,D))) ) ).

% insert_mono
tff(fact_2135_insert__mono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(C2,D)
     => ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(A2,C2),insert9069300056098147895at_nat(A2,D)) ) ).

% insert_mono
tff(fact_2136_insert__mono,axiom,
    ! [C2: set_bool,D: set_bool,A2: bool] :
      ( ord_less_eq_set_bool(C2,D)
     => ord_less_eq_set_bool(insert_bool2(A2,C2),insert_bool2(A2,D)) ) ).

% insert_mono
tff(fact_2137_insert__mono,axiom,
    ! [C2: set_nat,D: set_nat,A2: nat] :
      ( ord_less_eq_set_nat(C2,D)
     => ord_less_eq_set_nat(insert_nat2(A2,C2),insert_nat2(A2,D)) ) ).

% insert_mono
tff(fact_2138_subset__code_I1_J,axiom,
    ! [Xs: list_P5464809261938338413at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(Xs),B)
    <=> ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,set_Pr3765526544606949372at_nat(Xs))
         => member8757157785044589968at_nat(X5,B) ) ) ).

% subset_code(1)
tff(fact_2139_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Xs)),B))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),set_Pr5648618587558075414at_nat(Xs)))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),B)) ) ) ).

% subset_code(1)
tff(fact_2140_subset__code_I1_J,axiom,
    ! [Xs: list_f5857023252961307252t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le387542735852657390t_bool(set_fu1437410266168021891t_bool(Xs),B)
    <=> ! [X5: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X5,set_fu1437410266168021891t_bool(Xs))
         => member4125802036136091927t_bool(X5,B) ) ) ).

% subset_code(1)
tff(fact_2141_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B)
    <=> ! [X5: int] :
          ( member_int(X5,set_int2(Xs))
         => member_int(X5,B) ) ) ).

% subset_code(1)
tff(fact_2142_subset__code_I1_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn,B: set_Pr5949110396991348497n_assn] :
      ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Xs),B)
    <=> ! [X5: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X5,set_Pr1139785259514867910n_assn(Xs))
         => member7957490590177025114n_assn(X5,B) ) ) ).

% subset_code(1)
tff(fact_2143_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( ord_less_eq_set_nat(set_nat2(Xs),B)
    <=> ! [X5: nat] :
          ( member_nat(X5,set_nat2(Xs))
         => member_nat(X5,B) ) ) ).

% subset_code(1)
tff(fact_2144_double__diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( ord_less_eq_set_nat(B,C2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),A)) = A ) ) ) ).

% double_diff
tff(fact_2145_Diff__subset,axiom,
    ! [A: set_nat,B: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B),A) ).

% Diff_subset
tff(fact_2146_Diff__mono,axiom,
    ! [A: set_nat,C2: set_nat,D: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,C2)
     => ( ord_less_eq_set_nat(D,B)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,C2),D)) ) ) ).

% Diff_mono
tff(fact_2147_subset__iff__psubset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
    <=> ( ord_less_set_nat(A,B)
        | ( A = B ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2148_subset__psubset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( ord_less_set_nat(B,C2)
       => ord_less_set_nat(A,C2) ) ) ).

% subset_psubset_trans
tff(fact_2149_subset__not__subset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_set_nat(A,B)
    <=> ( ord_less_eq_set_nat(A,B)
        & ~ ord_less_eq_set_nat(B,A) ) ) ).

% subset_not_subset_eq
tff(fact_2150_psubset__subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ord_less_set_nat(A,B)
     => ( ord_less_eq_set_nat(B,C2)
       => ord_less_set_nat(A,C2) ) ) ).

% psubset_subset_trans
tff(fact_2151_psubset__imp__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_set_nat(A,B)
     => ord_less_eq_set_nat(A,B) ) ).

% psubset_imp_subset
tff(fact_2152_psubset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_set_nat(A,B)
    <=> ( ord_less_eq_set_nat(A,B)
        & ( A != B ) ) ) ).

% psubset_eq
tff(fact_2153_psubsetE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_set_nat(A,B)
     => ~ ( ord_less_eq_set_nat(A,B)
         => ord_less_eq_set_nat(B,A) ) ) ).

% psubsetE
tff(fact_2154_norm__assertion__simps_I9_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% norm_assertion_simps(9)
tff(fact_2155_norm__assertion__simps_I10_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% norm_assertion_simps(10)
tff(fact_2156_pairwise__empty,axiom,
    ! [P: fun_Pr124437560471516333t_bool] : pairwi3222497012029234898at_nat(P,bot_bo228742789529271731at_nat) ).

% pairwise_empty
tff(fact_2157_pairwise__empty,axiom,
    ! [P: fun_Pr248683899324676269t_bool] : pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(P),bot_bo2099793752762293965at_nat)) ).

% pairwise_empty
tff(fact_2158_pairwise__empty,axiom,
    ! [P: fun_bo1841576699660343469l_bool] : pairwise_bool(P,bot_bot_set_bool) ).

% pairwise_empty
tff(fact_2159_pairwise__empty,axiom,
    ! [P: fun_nat_fun_nat_bool] : pairwise_nat(P,bot_bot_set_nat) ).

% pairwise_empty
tff(fact_2160_pairwise__empty,axiom,
    ! [P: fun_int_fun_int_bool] : pairwise_int(P,bot_bot_set_int) ).

% pairwise_empty
tff(fact_2161_pairwise__insert,axiom,
    ! [R2: fun_bo1841576699660343469l_bool,X: bool,S2: set_bool] :
      ( pairwise_bool(R2,insert_bool2(X,S2))
    <=> ( ! [Y4: bool] :
            ( ( member_bool(Y4,S2)
              & ~ ( pp(Y4)
                <=> pp(X) ) )
           => ( pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R2,X),Y4))
              & pp(aa_bool_bool(aa_boo8970175072276269406l_bool(R2,Y4),X)) ) )
        & pairwise_bool(R2,S2) ) ) ).

% pairwise_insert
tff(fact_2162_pairwise__insert,axiom,
    ! [R2: fun_Pr124437560471516333t_bool,X: produc3843707927480180839at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( pairwi3222497012029234898at_nat(R2,insert9069300056098147895at_nat(X,S2))
    <=> ( ! [Y4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat(Y4,S2)
              & ( Y4 != X ) )
           => ( pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(R2,X),Y4))
              & pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(R2,Y4),X)) ) )
        & pairwi3222497012029234898at_nat(R2,S2) ) ) ).

% pairwise_insert
tff(fact_2163_pairwise__insert,axiom,
    ! [R2: fun_Pr248683899324676269t_bool,X: product_prod_nat_nat,S2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(R2),insert8211810215607154385at_nat(X,S2)))
    <=> ( ! [Y4: product_prod_nat_nat] :
            ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y4),S2))
              & ( Y4 != X ) )
           => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R2,X),Y4))
              & pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(R2,Y4),X)) ) )
        & pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(R2),S2)) ) ) ).

% pairwise_insert
tff(fact_2164_pairwise__insert,axiom,
    ! [R2: fun_nat_fun_nat_bool,X: nat,S2: set_nat] :
      ( pairwise_nat(R2,insert_nat2(X,S2))
    <=> ( ! [Y4: nat] :
            ( ( member_nat(Y4,S2)
              & ( Y4 != X ) )
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,X),Y4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(R2,Y4),X)) ) )
        & pairwise_nat(R2,S2) ) ) ).

% pairwise_insert
tff(fact_2165_pairwise__insert,axiom,
    ! [R2: fun_int_fun_int_bool,X: int,S2: set_int] :
      ( pairwise_int(R2,insert_int2(X,S2))
    <=> ( ! [Y4: int] :
            ( ( member_int(Y4,S2)
              & ( Y4 != X ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(R2,X),Y4))
              & pp(aa_int_bool(aa_int_fun_int_bool(R2,Y4),X)) ) )
        & pairwise_int(R2,S2) ) ) ).

% pairwise_insert
tff(fact_2166_pairwise__insert,axiom,
    ! [R2: fun_fu7365307207611327597l_bool,X: fun_Pr6001797050664311790t_bool,S2: set_fu6080891106736489294t_bool] :
      ( pairwi2704567075835687513t_bool(R2,insert969342481230318014t_bool(X,S2))
    <=> ( ! [Y4: fun_Pr6001797050664311790t_bool] :
            ( ( member4125802036136091927t_bool(Y4,S2)
              & ( Y4 != X ) )
           => ( pp(aa_fun5588795374382764133l_bool(aa_fun4580269573335628190l_bool(R2,X),Y4))
              & pp(aa_fun5588795374382764133l_bool(aa_fun4580269573335628190l_bool(R2,Y4),X)) ) )
        & pairwi2704567075835687513t_bool(R2,S2) ) ) ).

% pairwise_insert
tff(fact_2167_subset__singletonD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat))
     => ( ( A = bot_bo228742789529271731at_nat )
        | ( A = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singletonD
tff(fact_2168_subset__singletonD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat)))
     => ( ( A = bot_bo2099793752762293965at_nat )
        | ( A = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singletonD
tff(fact_2169_subset__singletonD,axiom,
    ! [A: set_bool,X: bool] :
      ( ord_less_eq_set_bool(A,insert_bool2(X,bot_bot_set_bool))
     => ( ( A = bot_bot_set_bool )
        | ( A = insert_bool2(X,bot_bot_set_bool) ) ) ) ).

% subset_singletonD
tff(fact_2170_subset__singletonD,axiom,
    ! [A: set_int,X: int] :
      ( ord_less_eq_set_int(A,insert_int2(X,bot_bot_set_int))
     => ( ( A = bot_bot_set_int )
        | ( A = insert_int2(X,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_2171_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ord_less_eq_set_nat(A,insert_nat2(X,bot_bot_set_nat))
     => ( ( A = bot_bot_set_nat )
        | ( A = insert_nat2(X,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_2172_subset__singleton__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(X3,insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat))
    <=> ( ( X3 = bot_bo228742789529271731at_nat )
        | ( X3 = insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_2173_subset__singleton__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X3),insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)))
    <=> ( ( X3 = bot_bo2099793752762293965at_nat )
        | ( X3 = insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_2174_subset__singleton__iff,axiom,
    ! [X3: set_bool,A2: bool] :
      ( ord_less_eq_set_bool(X3,insert_bool2(A2,bot_bot_set_bool))
    <=> ( ( X3 = bot_bot_set_bool )
        | ( X3 = insert_bool2(A2,bot_bot_set_bool) ) ) ) ).

% subset_singleton_iff
tff(fact_2175_subset__singleton__iff,axiom,
    ! [X3: set_int,A2: int] :
      ( ord_less_eq_set_int(X3,insert_int2(A2,bot_bot_set_int))
    <=> ( ( X3 = bot_bot_set_int )
        | ( X3 = insert_int2(A2,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_2176_subset__singleton__iff,axiom,
    ! [X3: set_nat,A2: nat] :
      ( ord_less_eq_set_nat(X3,insert_nat2(A2,bot_bot_set_nat))
    <=> ( ( X3 = bot_bot_set_nat )
        | ( X3 = insert_nat2(A2,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_2177_subset__minus__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) = bot_bo228742789529271731at_nat ) ) ).

% subset_minus_empty
tff(fact_2178_subset__minus__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
tff(fact_2179_subset__minus__empty,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ord_less_eq_set_bool(A,B)
     => ( minus_minus_set_bool(A,B) = bot_bot_set_bool ) ) ).

% subset_minus_empty
tff(fact_2180_subset__minus__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( minus_minus_set_int(A,B) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_2181_subset__minus__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_2182_subset__Diff__insert,axiom,
    ! [A: set_bool,B: set_bool,X: bool,C2: set_bool] :
      ( ord_less_eq_set_bool(A,minus_minus_set_bool(B,insert_bool2(X,C2)))
    <=> ( ord_less_eq_set_bool(A,minus_minus_set_bool(B,C2))
        & ~ member_bool(X,A) ) ) ).

% subset_Diff_insert
tff(fact_2183_subset__Diff__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),insert9069300056098147895at_nat(X,C2)))
    <=> ( ord_le1268244103169919719at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),C2))
        & ~ member8757157785044589968at_nat(X,A) ) ) ).

% subset_Diff_insert
tff(fact_2184_subset__Diff__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),insert8211810215607154385at_nat(X,C2))))
    <=> ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),C2)))
        & ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A)) ) ) ).

% subset_Diff_insert
tff(fact_2185_subset__Diff__insert,axiom,
    ! [A: set_int,B: set_int,X: int,C2: set_int] :
      ( ord_less_eq_set_int(A,minus_minus_set_int(B,insert_int2(X,C2)))
    <=> ( ord_less_eq_set_int(A,minus_minus_set_int(B,C2))
        & ~ member_int(X,A) ) ) ).

% subset_Diff_insert
tff(fact_2186_subset__Diff__insert,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool,X: fun_Pr6001797050664311790t_bool,C2: set_fu6080891106736489294t_bool] :
      ( ord_le387542735852657390t_bool(A,minus_5708602904570203829t_bool(B,insert969342481230318014t_bool(X,C2)))
    <=> ( ord_le387542735852657390t_bool(A,minus_5708602904570203829t_bool(B,C2))
        & ~ member4125802036136091927t_bool(X,A) ) ) ).

% subset_Diff_insert
tff(fact_2187_subset__Diff__insert,axiom,
    ! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
      ( ord_less_eq_set_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),insert_nat2(X,C2)))
    <=> ( ord_less_eq_set_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),C2))
        & ~ member_nat(X,A) ) ) ).

% subset_Diff_insert
tff(fact_2188_set__update__subsetI,axiom,
    ! [Xs: list_P5464809261938338413at_nat,A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,I: nat] :
      ( ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(Xs),A)
     => ( member8757157785044589968at_nat(X,A)
       => ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2189_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(Xs)),A))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),A)) ) ) ).

% set_update_subsetI
tff(fact_2190_set__update__subsetI,axiom,
    ! [Xs: list_f5857023252961307252t_bool,A: set_fu6080891106736489294t_bool,X: fun_Pr6001797050664311790t_bool,I: nat] :
      ( ord_le387542735852657390t_bool(set_fu1437410266168021891t_bool(Xs),A)
     => ( member4125802036136091927t_bool(X,A)
       => ord_le387542735852657390t_bool(set_fu1437410266168021891t_bool(list_u922235751406726748t_bool(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2191_set__update__subsetI,axiom,
    ! [Xs: list_int,A: set_int,X: int,I: nat] :
      ( ord_less_eq_set_int(set_int2(Xs),A)
     => ( member_int(X,A)
       => ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2192_set__update__subsetI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,A: set_Pr5949110396991348497n_assn,X: produc6575502325842934193n_assn,I: nat] :
      ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Xs),A)
     => ( member7957490590177025114n_assn(X,A)
       => ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2193_set__update__subsetI,axiom,
    ! [Xs: list_c,A: set_c,X: c,I: nat] :
      ( ord_less_eq_set_c(set_c2(Xs),A)
     => ( member_c(X,A)
       => ord_less_eq_set_c(set_c2(list_update_c(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2194_set__update__subsetI,axiom,
    ! [Xs: list_d,A: set_d,X: d,I: nat] :
      ( ord_less_eq_set_d(set_d2(Xs),A)
     => ( member_d(X,A)
       => ord_less_eq_set_d(set_d2(list_update_d(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2195_set__update__subsetI,axiom,
    ! [Xs: list_nat,A: set_nat,X: nat,I: nat] :
      ( ord_less_eq_set_nat(set_nat2(Xs),A)
     => ( member_nat(X,A)
       => ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X)),A) ) ) ).

% set_update_subsetI
tff(fact_2196_set__remove1__subset,axiom,
    ! [X: int,Xs: list_int] : ord_less_eq_set_int(set_int2(remove1_int(X,Xs)),set_int2(Xs)) ).

% set_remove1_subset
tff(fact_2197_set__remove1__subset,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(remove1670527618125605709n_assn(X,Xs)),set_Pr1139785259514867910n_assn(Xs)) ).

% set_remove1_subset
tff(fact_2198_set__remove1__subset,axiom,
    ! [X: nat,Xs: list_nat] : ord_less_eq_set_nat(set_nat2(remove1_nat(X,Xs)),set_nat2(Xs)) ).

% set_remove1_subset
tff(fact_2199_subset__code_I2_J,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Ys: list_P5464809261938338413at_nat] :
      ( ord_le1268244103169919719at_nat(A,coset_5307619257043544794at_nat(Ys))
    <=> ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,set_Pr3765526544606949372at_nat(Ys))
         => ~ member8757157785044589968at_nat(X5,A) ) ) ).

% subset_code(2)
tff(fact_2200_subset__code_I2_J,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Ys: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),coset_6809516720700967924at_nat(Ys)))
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),set_Pr5648618587558075414at_nat(Ys)))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A)) ) ) ).

% subset_code(2)
tff(fact_2201_subset__code_I2_J,axiom,
    ! [A: set_fu6080891106736489294t_bool,Ys: list_f5857023252961307252t_bool] :
      ( ord_le387542735852657390t_bool(A,coset_1498173199935290977t_bool(Ys))
    <=> ! [X5: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X5,set_fu1437410266168021891t_bool(Ys))
         => ~ member4125802036136091927t_bool(X5,A) ) ) ).

% subset_code(2)
tff(fact_2202_subset__code_I2_J,axiom,
    ! [A: set_int,Ys: list_int] :
      ( ord_less_eq_set_int(A,coset_int(Ys))
    <=> ! [X5: int] :
          ( member_int(X5,set_int2(Ys))
         => ~ member_int(X5,A) ) ) ).

% subset_code(2)
tff(fact_2203_subset__code_I2_J,axiom,
    ! [A: set_Pr5949110396991348497n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ord_le171416862856029873n_assn(A,coset_703488708621576356n_assn(Ys))
    <=> ! [X5: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X5,set_Pr1139785259514867910n_assn(Ys))
         => ~ member7957490590177025114n_assn(X5,A) ) ) ).

% subset_code(2)
tff(fact_2204_subset__code_I2_J,axiom,
    ! [A: set_nat,Ys: list_nat] :
      ( ord_less_eq_set_nat(A,coset_nat(Ys))
    <=> ! [X5: nat] :
          ( member_nat(X5,set_nat2(Ys))
         => ~ member_nat(X5,A) ) ) ).

% subset_code(2)
tff(fact_2205_pairwise__singleton,axiom,
    ! [P: fun_Pr124437560471516333t_bool,A: produc3843707927480180839at_nat] : pairwi3222497012029234898at_nat(P,insert9069300056098147895at_nat(A,bot_bo228742789529271731at_nat)) ).

% pairwise_singleton
tff(fact_2206_pairwise__singleton,axiom,
    ! [P: fun_Pr248683899324676269t_bool,A: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(pairwi6357175311820755948at_nat(P),insert8211810215607154385at_nat(A,bot_bo2099793752762293965at_nat))) ).

% pairwise_singleton
tff(fact_2207_pairwise__singleton,axiom,
    ! [P: fun_bo1841576699660343469l_bool,A: bool] : pairwise_bool(P,insert_bool2(A,bot_bot_set_bool)) ).

% pairwise_singleton
tff(fact_2208_pairwise__singleton,axiom,
    ! [P: fun_nat_fun_nat_bool,A: nat] : pairwise_nat(P,insert_nat2(A,bot_bot_set_nat)) ).

% pairwise_singleton
tff(fact_2209_pairwise__singleton,axiom,
    ! [P: fun_int_fun_int_bool,A: int] : pairwise_int(P,insert_int2(A,bot_bot_set_int)) ).

% pairwise_singleton
tff(fact_2210_subset__insert__iff,axiom,
    ! [A: set_fu6080891106736489294t_bool,X: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool] :
      ( ord_le387542735852657390t_bool(A,insert969342481230318014t_bool(X,B))
    <=> ( ( member4125802036136091927t_bool(X,A)
         => ord_le387542735852657390t_bool(minus_5708602904570203829t_bool(A,insert969342481230318014t_bool(X,bot_bo2714889297266208186t_bool)),B) )
        & ( ~ member4125802036136091927t_bool(X,A)
         => ord_le387542735852657390t_bool(A,B) ) ) ) ).

% subset_insert_iff
tff(fact_2211_subset__insert__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(X,B))
    <=> ( ( member8757157785044589968at_nat(X,A)
         => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)),B) )
        & ( ~ member8757157785044589968at_nat(X,A)
         => ord_le1268244103169919719at_nat(A,B) ) ) ) ).

% subset_insert_iff
tff(fact_2212_subset__insert__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(X,B)))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B)) )
        & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B)) ) ) ) ).

% subset_insert_iff
tff(fact_2213_subset__insert__iff,axiom,
    ! [A: set_bool,X: bool,B: set_bool] :
      ( ord_less_eq_set_bool(A,insert_bool2(X,B))
    <=> ( ( member_bool(X,A)
         => ord_less_eq_set_bool(minus_minus_set_bool(A,insert_bool2(X,bot_bot_set_bool)),B) )
        & ( ~ member_bool(X,A)
         => ord_less_eq_set_bool(A,B) ) ) ) ).

% subset_insert_iff
tff(fact_2214_subset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ord_less_eq_set_int(A,insert_int2(X,B))
    <=> ( ( member_int(X,A)
         => ord_less_eq_set_int(minus_minus_set_int(A,insert_int2(X,bot_bot_set_int)),B) )
        & ( ~ member_int(X,A)
         => ord_less_eq_set_int(A,B) ) ) ) ).

% subset_insert_iff
tff(fact_2215_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,insert_nat2(X,B))
    <=> ( ( member_nat(X,A)
         => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(X,bot_bot_set_nat)),B) )
        & ( ~ member_nat(X,A)
         => ord_less_eq_set_nat(A,B) ) ) ) ).

% subset_insert_iff
tff(fact_2216_Diff__single__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat)),B)
     => ord_le1268244103169919719at_nat(A,insert9069300056098147895at_nat(X,B)) ) ).

% Diff_single_insert
tff(fact_2217_Diff__single__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat))),B))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),insert8211810215607154385at_nat(X,B))) ) ).

% Diff_single_insert
tff(fact_2218_Diff__single__insert,axiom,
    ! [A: set_bool,X: bool,B: set_bool] :
      ( ord_less_eq_set_bool(minus_minus_set_bool(A,insert_bool2(X,bot_bot_set_bool)),B)
     => ord_less_eq_set_bool(A,insert_bool2(X,B)) ) ).

% Diff_single_insert
tff(fact_2219_Diff__single__insert,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ord_less_eq_set_int(minus_minus_set_int(A,insert_int2(X,bot_bot_set_int)),B)
     => ord_less_eq_set_int(A,insert_int2(X,B)) ) ).

% Diff_single_insert
tff(fact_2220_Diff__single__insert,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),insert_nat2(X,bot_bot_set_nat)),B)
     => ord_less_eq_set_nat(A,insert_nat2(X,B)) ) ).

% Diff_single_insert
tff(fact_2221_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,X))),insert8211810215607154385at_nat(X,set_Pr5648618587558075414at_nat(Xs)))) ).

% set_update_subset_insert
tff(fact_2222_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I: nat,X: produc3843707927480180839at_nat] : ord_le1268244103169919719at_nat(set_Pr3765526544606949372at_nat(list_u4696772448584712917at_nat(Xs,I,X)),insert9069300056098147895at_nat(X,set_Pr3765526544606949372at_nat(Xs))) ).

% set_update_subset_insert
tff(fact_2223_set__update__subset__insert,axiom,
    ! [Xs: list_bool,I: nat,X: bool] : ord_less_eq_set_bool(set_bool2(list_update_bool(Xs,I,X)),insert_bool2(X,set_bool2(Xs))) ).

% set_update_subset_insert
tff(fact_2224_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ord_less_eq_set_int(set_int2(list_update_int(Xs,I,X)),insert_int2(X,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_2225_set__update__subset__insert,axiom,
    ! [Xs: list_P8527749157015355191n_assn,I: nat,X: produc6575502325842934193n_assn] : ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,I,X)),insert5290817439147925377n_assn(X,set_Pr1139785259514867910n_assn(Xs))) ).

% set_update_subset_insert
tff(fact_2226_set__update__subset__insert,axiom,
    ! [Xs: list_c,I: nat,X: c] : ord_less_eq_set_c(set_c2(list_update_c(Xs,I,X)),insert_c(X,set_c2(Xs))) ).

% set_update_subset_insert
tff(fact_2227_set__update__subset__insert,axiom,
    ! [Xs: list_d,I: nat,X: d] : ord_less_eq_set_d(set_d2(list_update_d(Xs,I,X)),insert_d(X,set_d2(Xs))) ).

% set_update_subset_insert
tff(fact_2228_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,X)),insert_nat2(X,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_2229_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_2230_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),bot_bo228742789529271731at_nat) = bot_bo228742789529271731at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_2231_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_2232_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_bool] : inf_inf_set_bool(X,bot_bot_set_bool) = bot_bot_set_bool ).

% boolean_algebra.conj_zero_right
tff(fact_2233_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_2234_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_2235_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_2236_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,bot_bo228742789529271731at_nat),X) = bot_bo228742789529271731at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_2237_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_2238_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_bool] : inf_inf_set_bool(bot_bot_set_bool,X) = bot_bot_set_bool ).

% boolean_algebra.conj_zero_left
tff(fact_2239_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_2240_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_2241_diff__shunt__var,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,X),Y) = bot_bo228742789529271731at_nat )
    <=> ord_le1268244103169919719at_nat(X,Y) ) ).

% diff_shunt_var
tff(fact_2242_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X),Y)) ) ).

% diff_shunt_var
tff(fact_2243_diff__shunt__var,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( minus_minus_set_bool(X,Y) = bot_bot_set_bool )
    <=> ord_less_eq_set_bool(X,Y) ) ).

% diff_shunt_var
tff(fact_2244_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( minus_minus_set_int(X,Y) = bot_bot_set_int )
    <=> ord_less_eq_set_int(X,Y) ) ).

% diff_shunt_var
tff(fact_2245_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = bot_bot_assn )
    <=> pp(aa_assn_bool(ord_less_eq_assn(X),Y)) ) ).

% diff_shunt_var
tff(fact_2246_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = bot_bot_set_nat )
    <=> ord_less_eq_set_nat(X,Y) ) ).

% diff_shunt_var
tff(fact_2247_length__product,axiom,
    ! [Xs: list_c,Ys: list_c] : size_s2756448709718749209od_c_c(product_c_c(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_c(Xs)),size_size_list_c(Ys)) ).

% length_product
tff(fact_2248_length__product,axiom,
    ! [Xs: list_c,Ys: list_d] : size_s2827482749765095194od_c_d(product_c_d(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_c(Xs)),size_size_list_d(Ys)) ).

% length_product
tff(fact_2249_length__product,axiom,
    ! [Xs: list_c,Ys: list_nat] : size_s2954062820725854115_c_nat(product_c_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_c(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2250_length__product,axiom,
    ! [Xs: list_c,Ys: list_int] : size_s463979675794127359_c_int(product_c_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_c(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2251_length__product,axiom,
    ! [Xs: list_d,Ys: list_c] : size_s2120799709113762520od_d_c(product_d_c(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_d(Xs)),size_size_list_c(Ys)) ).

% length_product
tff(fact_2252_length__product,axiom,
    ! [Xs: list_d,Ys: list_d] : size_s2191833749160108505od_d_d(product_d_d(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_d(Xs)),size_size_list_d(Ys)) ).

% length_product
tff(fact_2253_length__product,axiom,
    ! [Xs: list_d,Ys: list_nat] : size_s8550281435913984804_d_nat(product_d_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_d(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2254_length__product,axiom,
    ! [Xs: list_d,Ys: list_int] : size_s6060198290982258048_d_int(product_d_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_d(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2255_length__product,axiom,
    ! [Xs: list_nat,Ys: list_c] : size_s385972143775086793_nat_c(product_nat_c(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_c(Ys)) ).

% length_product
tff(fact_2256_length__product,axiom,
    ! [Xs: list_nat,Ys: list_d] : size_s457006183821432778_nat_d(product_nat_d(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_d(Ys)) ).

% length_product
tff(fact_2257_less__imp__diff__less,axiom,
    ! [J: nat,K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K3)) ) ).

% less_imp_diff_less
tff(fact_2258_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M2))) ) ) ).

% diff_less_mono2
tff(fact_2259_insert__subsetI,axiom,
    ! [X: bool,A: set_bool,X3: set_bool] :
      ( member_bool(X,A)
     => ( ord_less_eq_set_bool(X3,A)
       => ord_less_eq_set_bool(insert_bool2(X,X3),A) ) ) ).

% insert_subsetI
tff(fact_2260_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(X,A)
     => ( ord_le1268244103169919719at_nat(X3,A)
       => ord_le1268244103169919719at_nat(insert9069300056098147895at_nat(X,X3),A) ) ) ).

% insert_subsetI
tff(fact_2261_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),A))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,X3),A))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,insert8211810215607154385at_nat(X,X3)),A)) ) ) ).

% insert_subsetI
tff(fact_2262_insert__subsetI,axiom,
    ! [X: int,A: set_int,X3: set_int] :
      ( member_int(X,A)
     => ( ord_less_eq_set_int(X3,A)
       => ord_less_eq_set_int(insert_int2(X,X3),A) ) ) ).

% insert_subsetI
tff(fact_2263_insert__subsetI,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,X3: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(X,A)
     => ( ord_le387542735852657390t_bool(X3,A)
       => ord_le387542735852657390t_bool(insert969342481230318014t_bool(X,X3),A) ) ) ).

% insert_subsetI
tff(fact_2264_insert__subsetI,axiom,
    ! [X: nat,A: set_nat,X3: set_nat] :
      ( member_nat(X,A)
     => ( ord_less_eq_set_nat(X3,A)
       => ord_less_eq_set_nat(insert_nat2(X,X3),A) ) ) ).

% insert_subsetI
tff(fact_2265_subset__emptyI,axiom,
    ! [A: set_fu6080891106736489294t_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool] : ~ member4125802036136091927t_bool(X2,A)
     => ord_le387542735852657390t_bool(A,bot_bo2714889297266208186t_bool) ) ).

% subset_emptyI
tff(fact_2266_subset__emptyI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ! [X2: produc3843707927480180839at_nat] : ~ member8757157785044589968at_nat(X2,A)
     => ord_le1268244103169919719at_nat(A,bot_bo228742789529271731at_nat) ) ).

% subset_emptyI
tff(fact_2267_subset__emptyI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A))
     => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),bot_bo2099793752762293965at_nat)) ) ).

% subset_emptyI
tff(fact_2268_subset__emptyI,axiom,
    ! [A: set_bool] :
      ( ! [X2: bool] : ~ member_bool(X2,A)
     => ord_less_eq_set_bool(A,bot_bot_set_bool) ) ).

% subset_emptyI
tff(fact_2269_subset__emptyI,axiom,
    ! [A: set_int] :
      ( ! [X2: int] : ~ member_int(X2,A)
     => ord_less_eq_set_int(A,bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_2270_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X2: nat] : ~ member_nat(X2,A)
     => ord_less_eq_set_nat(A,bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_2271_complete__interval,axiom,
    ! [A2: nat,B2: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(P,A2))
       => ( ~ pp(aa_nat_bool(P,B2))
         => ? [C4: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C4))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C4),B2))
              & ! [X6: nat] :
                  ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X6))
                    & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),C4)) )
                 => pp(aa_nat_bool(P,X6)) )
              & ! [D4: nat] :
                  ( ! [X2: nat] :
                      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X2))
                        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),D4)) )
                     => pp(aa_nat_bool(P,X2)) )
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D4),C4)) ) ) ) ) ) ).

% complete_interval
tff(fact_2272_complete__interval,axiom,
    ! [A2: int,B2: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(P,A2))
       => ( ~ pp(aa_int_bool(P,B2))
         => ? [C4: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C4))
              & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C4),B2))
              & ! [X6: int] :
                  ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X6))
                    & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),C4)) )
                 => pp(aa_int_bool(P,X6)) )
              & ! [D4: int] :
                  ( ! [X2: int] :
                      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X2))
                        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),D4)) )
                     => pp(aa_int_bool(P,X2)) )
                 => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D4),C4)) ) ) ) ) ) ).

% complete_interval
tff(fact_2273_verit__comp__simplify1_I3_J,axiom,
    ! [B5: code_integer,A5: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B5),A5))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_2274_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,B5),A5))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_2275_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B5),A5))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_2276_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B5),A5))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A5),B5)) ) ).

% verit_comp_simplify1(3)
tff(fact_2277_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)))
    <=> ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),A))
        & pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),B)) ) ) ).

% Int_subset_iff
tff(fact_2278_Int__subset__iff,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
    <=> ( ord_less_eq_set_nat(C2,A)
        & ord_less_eq_set_nat(C2,B) ) ) ).

% Int_subset_iff
tff(fact_2279_Int__insert__left__if0,axiom,
    ! [A2: bool,C2: set_bool,B: set_bool] :
      ( ~ member_bool(A2,C2)
     => ( inf_inf_set_bool(insert_bool2(A2,B),C2) = inf_inf_set_bool(B,C2) ) ) ).

% Int_insert_left_if0
tff(fact_2280_Int__insert__left__if0,axiom,
    ! [A2: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(A2,C2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2) ) ) ).

% Int_insert_left_if0
tff(fact_2281_Int__insert__left__if0,axiom,
    ! [A2: int,C2: set_int,B: set_int] :
      ( ~ member_int(A2,C2)
     => ( inf_inf_set_int(insert_int2(A2,B),C2) = inf_inf_set_int(B,C2) ) ) ).

% Int_insert_left_if0
tff(fact_2282_Int__insert__left__if0,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,C2: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(A2,C2)
     => ( inf_in1689267372175091004t_bool(insert969342481230318014t_bool(A2,B),C2) = inf_in1689267372175091004t_bool(B,C2) ) ) ).

% Int_insert_left_if0
tff(fact_2283_Int__insert__left__if0,axiom,
    ! [A2: nat,C2: set_nat,B: set_nat] :
      ( ~ member_nat(A2,C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2) ) ) ).

% Int_insert_left_if0
tff(fact_2284_Int__insert__left__if0,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2) ) ) ).

% Int_insert_left_if0
tff(fact_2285_Int__insert__left__if1,axiom,
    ! [A2: bool,C2: set_bool,B: set_bool] :
      ( member_bool(A2,C2)
     => ( inf_inf_set_bool(insert_bool2(A2,B),C2) = insert_bool2(A2,inf_inf_set_bool(B,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_2286_Int__insert__left__if1,axiom,
    ! [A2: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,C2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,B)),C2) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_2287_Int__insert__left__if1,axiom,
    ! [A2: int,C2: set_int,B: set_int] :
      ( member_int(A2,C2)
     => ( inf_inf_set_int(insert_int2(A2,B),C2) = insert_int2(A2,inf_inf_set_int(B,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_2288_Int__insert__left__if1,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,C2: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,C2)
     => ( inf_in1689267372175091004t_bool(insert969342481230318014t_bool(A2,B),C2) = insert969342481230318014t_bool(A2,inf_in1689267372175091004t_bool(B,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_2289_Int__insert__left__if1,axiom,
    ! [A2: nat,C2: set_nat,B: set_nat] :
      ( member_nat(A2,C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,B)),C2) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_2290_Int__insert__left__if1,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B)),C2) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_2291_insert__inter__insert,axiom,
    ! [A2: int,A: set_int,B: set_int] : inf_inf_set_int(insert_int2(A2,A),insert_int2(A2,B)) = insert_int2(A2,inf_inf_set_int(A,B)) ).

% insert_inter_insert
tff(fact_2292_insert__inter__insert,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,A)),insert9069300056098147895at_nat(A2,B)) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)) ).

% insert_inter_insert
tff(fact_2293_insert__inter__insert,axiom,
    ! [A2: bool,A: set_bool,B: set_bool] : inf_inf_set_bool(insert_bool2(A2,A),insert_bool2(A2,B)) = insert_bool2(A2,inf_inf_set_bool(A,B)) ).

% insert_inter_insert
tff(fact_2294_insert__inter__insert,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,A)),insert_nat2(A2,B)) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ).

% insert_inter_insert
tff(fact_2295_insert__inter__insert,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,A)),insert8211810215607154385at_nat(A2,B)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ).

% insert_inter_insert
tff(fact_2296_Int__insert__right__if0,axiom,
    ! [A2: bool,A: set_bool,B: set_bool] :
      ( ~ member_bool(A2,A)
     => ( inf_inf_set_bool(A,insert_bool2(A2,B)) = inf_inf_set_bool(A,B) ) ) ).

% Int_insert_right_if0
tff(fact_2297_Int__insert__right__if0,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ~ member8757157785044589968at_nat(A2,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),insert9069300056098147895at_nat(A2,B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) ) ) ).

% Int_insert_right_if0
tff(fact_2298_Int__insert__right__if0,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ~ member_int(A2,A)
     => ( inf_inf_set_int(A,insert_int2(A2,B)) = inf_inf_set_int(A,B) ) ) ).

% Int_insert_right_if0
tff(fact_2299_Int__insert__right__if0,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ~ member4125802036136091927t_bool(A2,A)
     => ( inf_in1689267372175091004t_bool(A,insert969342481230318014t_bool(A2,B)) = inf_in1689267372175091004t_bool(A,B) ) ) ).

% Int_insert_right_if0
tff(fact_2300_Int__insert__right__if0,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ~ member_nat(A2,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),insert_nat2(A2,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% Int_insert_right_if0
tff(fact_2301_Int__insert__right__if0,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),insert8211810215607154385at_nat(A2,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% Int_insert_right_if0
tff(fact_2302_Int__insert__right__if1,axiom,
    ! [A2: bool,A: set_bool,B: set_bool] :
      ( member_bool(A2,A)
     => ( inf_inf_set_bool(A,insert_bool2(A2,B)) = insert_bool2(A2,inf_inf_set_bool(A,B)) ) ) ).

% Int_insert_right_if1
tff(fact_2303_Int__insert__right__if1,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(A2,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),insert9069300056098147895at_nat(A2,B)) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2304_Int__insert__right__if1,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( member_int(A2,A)
     => ( inf_inf_set_int(A,insert_int2(A2,B)) = insert_int2(A2,inf_inf_set_int(A,B)) ) ) ).

% Int_insert_right_if1
tff(fact_2305_Int__insert__right__if1,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(A2,A)
     => ( inf_in1689267372175091004t_bool(A,insert969342481230318014t_bool(A2,B)) = insert969342481230318014t_bool(A2,inf_in1689267372175091004t_bool(A,B)) ) ) ).

% Int_insert_right_if1
tff(fact_2306_Int__insert__right__if1,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( member_nat(A2,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),insert_nat2(A2,B)) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2307_Int__insert__right__if1,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),insert8211810215607154385at_nat(A2,B)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% Int_insert_right_if1
tff(fact_2308_insert__disjoint_I1_J,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ( inf_in1689267372175091004t_bool(insert969342481230318014t_bool(A2,A),B) = bot_bo2714889297266208186t_bool )
    <=> ( ~ member4125802036136091927t_bool(A2,B)
        & ( inf_in1689267372175091004t_bool(A,B) = bot_bo2714889297266208186t_bool ) ) ) ).

% insert_disjoint(1)
tff(fact_2309_insert__disjoint_I1_J,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,A)),B) = bot_bo228742789529271731at_nat )
    <=> ( ~ member8757157785044589968at_nat(A2,B)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_2310_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,A)),B) = bot_bo2099793752762293965at_nat )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B))
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_2311_insert__disjoint_I1_J,axiom,
    ! [A2: bool,A: set_bool,B: set_bool] :
      ( ( inf_inf_set_bool(insert_bool2(A2,A),B) = bot_bot_set_bool )
    <=> ( ~ member_bool(A2,B)
        & ( inf_inf_set_bool(A,B) = bot_bot_set_bool ) ) ) ).

% insert_disjoint(1)
tff(fact_2312_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,A)),B) = bot_bot_set_nat )
    <=> ( ~ member_nat(A2,B)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_2313_insert__disjoint_I1_J,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int(insert_int2(A2,A),B) = bot_bot_set_int )
    <=> ( ~ member_int(A2,B)
        & ( inf_inf_set_int(A,B) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_2314_insert__disjoint_I2_J,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ( bot_bo2714889297266208186t_bool = inf_in1689267372175091004t_bool(insert969342481230318014t_bool(A2,A),B) )
    <=> ( ~ member4125802036136091927t_bool(A2,B)
        & ( bot_bo2714889297266208186t_bool = inf_in1689267372175091004t_bool(A,B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2315_insert__disjoint_I2_J,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,A)),B) )
    <=> ( ~ member8757157785044589968at_nat(A2,B)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2316_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,A)),B) )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B))
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2317_insert__disjoint_I2_J,axiom,
    ! [A2: bool,A: set_bool,B: set_bool] :
      ( ( bot_bot_set_bool = inf_inf_set_bool(insert_bool2(A2,A),B) )
    <=> ( ~ member_bool(A2,B)
        & ( bot_bot_set_bool = inf_inf_set_bool(A,B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2318_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,A)),B) )
    <=> ( ~ member_nat(A2,B)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2319_insert__disjoint_I2_J,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(insert_int2(A2,A),B) )
    <=> ( ~ member_int(A2,B)
        & ( bot_bot_set_int = inf_inf_set_int(A,B) ) ) ) ).

% insert_disjoint(2)
tff(fact_2320_disjoint__insert_I1_J,axiom,
    ! [B: set_fu6080891106736489294t_bool,A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool] :
      ( ( inf_in1689267372175091004t_bool(B,insert969342481230318014t_bool(A2,A)) = bot_bo2714889297266208186t_bool )
    <=> ( ~ member4125802036136091927t_bool(A2,B)
        & ( inf_in1689267372175091004t_bool(B,A) = bot_bo2714889297266208186t_bool ) ) ) ).

% disjoint_insert(1)
tff(fact_2321_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),insert9069300056098147895at_nat(A2,A)) = bot_bo228742789529271731at_nat )
    <=> ( ~ member8757157785044589968at_nat(A2,B)
        & ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),A) = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_2322_disjoint__insert_I1_J,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),insert8211810215607154385at_nat(A2,A)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),B))
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_2323_disjoint__insert_I1_J,axiom,
    ! [B: set_bool,A2: bool,A: set_bool] :
      ( ( inf_inf_set_bool(B,insert_bool2(A2,A)) = bot_bot_set_bool )
    <=> ( ~ member_bool(A2,B)
        & ( inf_inf_set_bool(B,A) = bot_bot_set_bool ) ) ) ).

% disjoint_insert(1)
tff(fact_2324_disjoint__insert_I1_J,axiom,
    ! [B: set_nat,A2: nat,A: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),insert_nat2(A2,A)) = bot_bot_set_nat )
    <=> ( ~ member_nat(A2,B)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_2325_disjoint__insert_I1_J,axiom,
    ! [B: set_int,A2: int,A: set_int] :
      ( ( inf_inf_set_int(B,insert_int2(A2,A)) = bot_bot_set_int )
    <=> ( ~ member_int(A2,B)
        & ( inf_inf_set_int(B,A) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_2326_disjoint__insert_I2_J,axiom,
    ! [A: set_fu6080891106736489294t_bool,B2: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool] :
      ( ( bot_bo2714889297266208186t_bool = inf_in1689267372175091004t_bool(A,insert969342481230318014t_bool(B2,B)) )
    <=> ( ~ member4125802036136091927t_bool(B2,A)
        & ( bot_bo2714889297266208186t_bool = inf_in1689267372175091004t_bool(A,B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2327_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),insert9069300056098147895at_nat(B2,B)) )
    <=> ( ~ member8757157785044589968at_nat(B2,A)
        & ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2328_disjoint__insert_I2_J,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),insert8211810215607154385at_nat(B2,B)) )
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(B2),A))
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2329_disjoint__insert_I2_J,axiom,
    ! [A: set_bool,B2: bool,B: set_bool] :
      ( ( bot_bot_set_bool = inf_inf_set_bool(A,insert_bool2(B2,B)) )
    <=> ( ~ member_bool(B2,A)
        & ( bot_bot_set_bool = inf_inf_set_bool(A,B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2330_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B2: nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),insert_nat2(B2,B)) )
    <=> ( ~ member_nat(B2,A)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2331_disjoint__insert_I2_J,axiom,
    ! [A: set_int,B2: int,B: set_int] :
      ( ( bot_bot_set_int = inf_inf_set_int(A,insert_int2(B2,B)) )
    <=> ( ~ member_int(B2,A)
        & ( bot_bot_set_int = inf_inf_set_int(A,B) ) ) ) ).

% disjoint_insert(2)
tff(fact_2332_Diff__disjoint,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),A)) = bot_bo228742789529271731at_nat ).

% Diff_disjoint
tff(fact_2333_Diff__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),A)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_2334_Diff__disjoint,axiom,
    ! [A: set_bool,B: set_bool] : inf_inf_set_bool(A,minus_minus_set_bool(B,A)) = bot_bot_set_bool ).

% Diff_disjoint
tff(fact_2335_Diff__disjoint,axiom,
    ! [A: set_int,B: set_int] : inf_inf_set_int(A,minus_minus_set_int(B,A)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_2336_Diff__disjoint,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_2337_list__update__beyond,axiom,
    ! [Xs: list_c,I: nat,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_c(Xs)),I))
     => ( list_update_c(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_2338_list__update__beyond,axiom,
    ! [Xs: list_d,I: nat,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_d(Xs)),I))
     => ( list_update_d(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_2339_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),I))
     => ( list_update_nat(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_2340_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),I))
     => ( list_update_int(Xs,I,X) = Xs ) ) ).

% list_update_beyond
tff(fact_2341_nat__less__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
        & ( M2 != N ) ) ) ).

% nat_less_le
tff(fact_2342_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_imp_le_nat
tff(fact_2343_le__eq__less__or__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_2344_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_or_eq_imp_le
tff(fact_2345_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( ( M2 != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% le_neq_implies_less
tff(fact_2346_less__mono__imp__le__mono,axiom,
    ! [F3: fun_nat_nat,I: nat,J: nat] :
      ( ! [I5: nat,J3: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),J3))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,I5)),aa_nat_nat(F3,J3))) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(F3,I)),aa_nat_nat(F3,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_2347_disjoint__iff__not__equal,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,A)
         => ! [Xa2: produc3843707927480180839at_nat] :
              ( member8757157785044589968at_nat(Xa2,B)
             => ( X5 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_2348_disjoint__iff__not__equal,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A))
         => ! [Xa2: product_prod_nat_nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Xa2),B))
             => ( X5 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_2349_disjoint__iff__not__equal,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( inf_inf_set_bool(A,B) = bot_bot_set_bool )
    <=> ! [X5: bool] :
          ( member_bool(X5,A)
         => ! [Xa2: bool] :
              ( member_bool(Xa2,B)
             => ( pp(X5)
              <=> ~ pp(Xa2) ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_2350_disjoint__iff__not__equal,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat )
    <=> ! [X5: nat] :
          ( member_nat(X5,A)
         => ! [Xa2: nat] :
              ( member_nat(Xa2,B)
             => ( X5 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_2351_disjoint__iff__not__equal,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int(A,B) = bot_bot_set_int )
    <=> ! [X5: int] :
          ( member_int(X5,A)
         => ! [Xa2: int] :
              ( member_int(Xa2,B)
             => ( X5 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_2352_Int__empty__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),bot_bo228742789529271731at_nat) = bot_bo228742789529271731at_nat ).

% Int_empty_right
tff(fact_2353_Int__empty__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_2354_Int__empty__right,axiom,
    ! [A: set_bool] : inf_inf_set_bool(A,bot_bot_set_bool) = bot_bot_set_bool ).

% Int_empty_right
tff(fact_2355_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_2356_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_2357_Int__empty__left,axiom,
    ! [B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,bot_bo228742789529271731at_nat),B) = bot_bo228742789529271731at_nat ).

% Int_empty_left
tff(fact_2358_Int__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_2359_Int__empty__left,axiom,
    ! [B: set_bool] : inf_inf_set_bool(bot_bot_set_bool,B) = bot_bot_set_bool ).

% Int_empty_left
tff(fact_2360_Int__empty__left,axiom,
    ! [B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_2361_Int__empty__left,axiom,
    ! [B: set_int] : inf_inf_set_int(bot_bot_set_int,B) = bot_bot_set_int ).

% Int_empty_left
tff(fact_2362_disjoint__iff,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ( inf_in1689267372175091004t_bool(A,B) = bot_bo2714889297266208186t_bool )
    <=> ! [X5: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X5,A)
         => ~ member4125802036136091927t_bool(X5,B) ) ) ).

% disjoint_iff
tff(fact_2363_disjoint__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,A)
         => ~ member8757157785044589968at_nat(X5,B) ) ) ).

% disjoint_iff
tff(fact_2364_disjoint__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> ! [X5: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),A))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X5),B)) ) ) ).

% disjoint_iff
tff(fact_2365_disjoint__iff,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( inf_inf_set_bool(A,B) = bot_bot_set_bool )
    <=> ! [X5: bool] :
          ( member_bool(X5,A)
         => ~ member_bool(X5,B) ) ) ).

% disjoint_iff
tff(fact_2366_disjoint__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat )
    <=> ! [X5: nat] :
          ( member_nat(X5,A)
         => ~ member_nat(X5,B) ) ) ).

% disjoint_iff
tff(fact_2367_disjoint__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int(A,B) = bot_bot_set_int )
    <=> ! [X5: int] :
          ( member_int(X5,A)
         => ~ member_int(X5,B) ) ) ).

% disjoint_iff
tff(fact_2368_Int__emptyI,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X2,A)
         => ~ member4125802036136091927t_bool(X2,B) )
     => ( inf_in1689267372175091004t_bool(A,B) = bot_bo2714889297266208186t_bool ) ) ).

% Int_emptyI
tff(fact_2369_Int__emptyI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ! [X2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X2,A)
         => ~ member8757157785044589968at_nat(X2,B) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) = bot_bo228742789529271731at_nat ) ) ).

% Int_emptyI
tff(fact_2370_Int__emptyI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),B)) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_2371_Int__emptyI,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ! [X2: bool] :
          ( member_bool(X2,A)
         => ~ member_bool(X2,B) )
     => ( inf_inf_set_bool(A,B) = bot_bot_set_bool ) ) ).

% Int_emptyI
tff(fact_2372_Int__emptyI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( member_nat(X2,A)
         => ~ member_nat(X2,B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_2373_Int__emptyI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X2: int] :
          ( member_int(X2,A)
         => ~ member_int(X2,B) )
     => ( inf_inf_set_int(A,B) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_2374_disjointI,axiom,
    ! [A2: set_fu6080891106736489294t_bool,B2: set_fu6080891106736489294t_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X2,A2)
         => ~ member4125802036136091927t_bool(X2,B2) )
     => ( inf_in1689267372175091004t_bool(A2,B2) = bot_bo2714889297266208186t_bool ) ) ).

% disjointI
tff(fact_2375_disjointI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ! [X2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X2,A2)
         => ~ member8757157785044589968at_nat(X2,B2) )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2) = bot_bo228742789529271731at_nat ) ) ).

% disjointI
tff(fact_2376_disjointI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A2))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),B2)) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
tff(fact_2377_disjointI,axiom,
    ! [A2: set_bool,B2: set_bool] :
      ( ! [X2: bool] :
          ( member_bool(X2,A2)
         => ~ member_bool(X2,B2) )
     => ( inf_inf_set_bool(A2,B2) = bot_bot_set_bool ) ) ).

% disjointI
tff(fact_2378_disjointI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( member_nat(X2,A2)
         => ~ member_nat(X2,B2) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_2379_disjointI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X2: int] :
          ( member_int(X2,A2)
         => ~ member_int(X2,B2) )
     => ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_2380_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,S),S3))
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),S3) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),S3) )
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),S) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2381_inter__eq__subsetI,axiom,
    ! [S: set_nat,S3: set_nat,A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(S,S3)
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),S3) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),S3) )
       => ( 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,B),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_2382_Int__Collect__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,P: fun_Pr1607443439640623155t_bool,Q: fun_Pr1607443439640623155t_bool] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( ! [X2: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X2,A)
           => ( pp(aa_Pro1809015098595223084t_bool(P,X2))
             => pp(aa_Pro1809015098595223084t_bool(Q,X2)) ) )
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),collec6321179662152712658at_nat(P)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),collec6321179662152712658at_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2383_Int__Collect__mono,axiom,
    ! [A: set_int,B: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ord_less_eq_set_int(A,B)
     => ( ! [X2: int] :
            ( member_int(X2,A)
           => ( pp(aa_int_bool(P,X2))
             => pp(aa_int_bool(Q,X2)) ) )
       => ord_less_eq_set_int(inf_inf_set_int(A,collect_int(P)),inf_inf_set_int(B,collect_int(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2384_Int__Collect__mono,axiom,
    ! [A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool,P: fun_fu9160419385450930796l_bool,Q: fun_fu9160419385450930796l_bool] :
      ( ord_le387542735852657390t_bool(A,B)
     => ( ! [X2: fun_Pr6001797050664311790t_bool] :
            ( member4125802036136091927t_bool(X2,A)
           => ( pp(aa_fun5588795374382764133l_bool(P,X2))
             => pp(aa_fun5588795374382764133l_bool(Q,X2)) ) )
       => ord_le387542735852657390t_bool(inf_in1689267372175091004t_bool(A,collec3007137136614167897t_bool(P)),inf_in1689267372175091004t_bool(B,collec3007137136614167897t_bool(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2385_Int__Collect__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,P: fun_Pr7757163693017731481t_bool,Q: fun_Pr7757163693017731481t_bool] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
     => ( ! [X2: product_prod_nat_nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A))
           => ( pp(aa_Pro2232559814647311250t_bool(P,X2))
             => pp(aa_Pro2232559814647311250t_bool(Q,X2)) ) )
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),collec3392354462482085612at_nat(P))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),collec3392354462482085612at_nat(Q)))) ) ) ).

% Int_Collect_mono
tff(fact_2386_Int__Collect__mono,axiom,
    ! [A: set_nat,B: set_nat,P: fun_nat_bool,Q: fun_nat_bool] :
      ( ord_less_eq_set_nat(A,B)
     => ( ! [X2: nat] :
            ( member_nat(X2,A)
           => ( pp(aa_nat_bool(P,X2))
             => pp(aa_nat_bool(Q,X2)) ) )
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),collect_nat(P)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),collect_nat(Q))) ) ) ).

% Int_Collect_mono
tff(fact_2387_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),A))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),B))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))) ) ) ).

% Int_greatest
tff(fact_2388_Int__greatest,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(C2,A)
     => ( ord_less_eq_set_nat(C2,B)
       => ord_less_eq_set_nat(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% Int_greatest
tff(fact_2389_Int__absorb2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),B))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% Int_absorb2
tff(fact_2390_Int__absorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% Int_absorb2
tff(fact_2391_Int__absorb1,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B),A))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% Int_absorb1
tff(fact_2392_Int__absorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ord_less_eq_set_nat(B,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% Int_absorb1
tff(fact_2393_Int__lower2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B)) ).

% Int_lower2
tff(fact_2394_Int__lower2,axiom,
    ! [A: set_nat,B: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B),B) ).

% Int_lower2
tff(fact_2395_Int__lower1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),A)) ).

% Int_lower1
tff(fact_2396_Int__lower1,axiom,
    ! [A: set_nat,B: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B),A) ).

% Int_lower1
tff(fact_2397_Int__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A),C2))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B),D))
       => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D))) ) ) ).

% Int_mono
tff(fact_2398_Int__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ord_less_eq_set_nat(A,C2)
     => ( ord_less_eq_set_nat(B,D)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D)) ) ) ).

% Int_mono
tff(fact_2399_Int__insert__left,axiom,
    ! [A2: bool,C2: set_bool,B: set_bool] :
      ( ( member_bool(A2,C2)
       => ( inf_inf_set_bool(insert_bool2(A2,B),C2) = insert_bool2(A2,inf_inf_set_bool(B,C2)) ) )
      & ( ~ member_bool(A2,C2)
       => ( inf_inf_set_bool(insert_bool2(A2,B),C2) = inf_inf_set_bool(B,C2) ) ) ) ).

% Int_insert_left
tff(fact_2400_Int__insert__left,axiom,
    ! [A2: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat(A2,C2)
       => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,B)),C2) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2)) ) )
      & ( ~ member8757157785044589968at_nat(A2,C2)
       => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,insert9069300056098147895at_nat(A2,B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2) ) ) ) ).

% Int_insert_left
tff(fact_2401_Int__insert__left,axiom,
    ! [A2: int,C2: set_int,B: set_int] :
      ( ( member_int(A2,C2)
       => ( inf_inf_set_int(insert_int2(A2,B),C2) = insert_int2(A2,inf_inf_set_int(B,C2)) ) )
      & ( ~ member_int(A2,C2)
       => ( inf_inf_set_int(insert_int2(A2,B),C2) = inf_inf_set_int(B,C2) ) ) ) ).

% Int_insert_left
tff(fact_2402_Int__insert__left,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,C2: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ( member4125802036136091927t_bool(A2,C2)
       => ( inf_in1689267372175091004t_bool(insert969342481230318014t_bool(A2,B),C2) = insert969342481230318014t_bool(A2,inf_in1689267372175091004t_bool(B,C2)) ) )
      & ( ~ member4125802036136091927t_bool(A2,C2)
       => ( inf_in1689267372175091004t_bool(insert969342481230318014t_bool(A2,B),C2) = inf_in1689267372175091004t_bool(B,C2) ) ) ) ).

% Int_insert_left
tff(fact_2403_Int__insert__left,axiom,
    ! [A2: nat,C2: set_nat,B: set_nat] :
      ( ( member_nat(A2,C2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,B)),C2) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ) )
      & ( ~ member_nat(A2,C2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A2,B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2) ) ) ) ).

% Int_insert_left
tff(fact_2404_Int__insert__left,axiom,
    ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B)),C2) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),C2))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A2,B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2) ) ) ) ).

% Int_insert_left
tff(fact_2405_Int__insert__right,axiom,
    ! [A2: bool,A: set_bool,B: set_bool] :
      ( ( member_bool(A2,A)
       => ( inf_inf_set_bool(A,insert_bool2(A2,B)) = insert_bool2(A2,inf_inf_set_bool(A,B)) ) )
      & ( ~ member_bool(A2,A)
       => ( inf_inf_set_bool(A,insert_bool2(A2,B)) = inf_inf_set_bool(A,B) ) ) ) ).

% Int_insert_right
tff(fact_2406_Int__insert__right,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat(A2,A)
       => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),insert9069300056098147895at_nat(A2,B)) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)) ) )
      & ( ~ member8757157785044589968at_nat(A2,A)
       => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),insert9069300056098147895at_nat(A2,B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) ) ) ) ).

% Int_insert_right
tff(fact_2407_Int__insert__right,axiom,
    ! [A2: int,A: set_int,B: set_int] :
      ( ( member_int(A2,A)
       => ( inf_inf_set_int(A,insert_int2(A2,B)) = insert_int2(A2,inf_inf_set_int(A,B)) ) )
      & ( ~ member_int(A2,A)
       => ( inf_inf_set_int(A,insert_int2(A2,B)) = inf_inf_set_int(A,B) ) ) ) ).

% Int_insert_right
tff(fact_2408_Int__insert__right,axiom,
    ! [A2: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( ( member4125802036136091927t_bool(A2,A)
       => ( inf_in1689267372175091004t_bool(A,insert969342481230318014t_bool(A2,B)) = insert969342481230318014t_bool(A2,inf_in1689267372175091004t_bool(A,B)) ) )
      & ( ~ member4125802036136091927t_bool(A2,A)
       => ( inf_in1689267372175091004t_bool(A,insert969342481230318014t_bool(A2,B)) = inf_in1689267372175091004t_bool(A,B) ) ) ) ).

% Int_insert_right
tff(fact_2409_Int__insert__right,axiom,
    ! [A2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat(A2,A)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),insert_nat2(A2,B)) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) )
      & ( ~ member_nat(A2,A)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),insert_nat2(A2,B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ) ).

% Int_insert_right
tff(fact_2410_Int__insert__right,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),insert8211810215607154385at_nat(A2,B)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(A2),A))
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),insert8211810215607154385at_nat(A2,B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ) ).

% Int_insert_right
tff(fact_2411_Int__Diff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B),C2)) ).

% Int_Diff
tff(fact_2412_Int__Diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),C2)) ).

% Int_Diff
tff(fact_2413_Diff__Int2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)),B) ).

% Diff_Int2
tff(fact_2414_Diff__Int2,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C2)),B) ).

% Diff_Int2
tff(fact_2415_Diff__Diff__Int,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ).

% Diff_Diff_Int
tff(fact_2416_Diff__Diff__Int,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% Diff_Diff_Int
tff(fact_2417_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),B)) ).

% Diff_Int_distrib
tff(fact_2418_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B)) ).

% Diff_Int_distrib
tff(fact_2419_Diff__Int__distrib2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ).

% Diff_Int_distrib2
tff(fact_2420_Diff__Int__distrib2,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) ).

% Diff_Int_distrib2
tff(fact_2421_exists__leI,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ( ! [N3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N3),N))
           => ~ pp(aa_nat_bool(P,N3)) )
       => pp(aa_nat_bool(P,N)) )
     => ? [N4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N4),N))
          & pp(aa_nat_bool(P,N4)) ) ) ).

% exists_leI
tff(fact_2422_less__eq__assn__def,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
    <=> ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2) ) ) ).

% less_eq_assn_def
tff(fact_2423_less__diff__iff,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),K3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K3)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ) ).

% less_diff_iff
tff(fact_2424_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C))) ) ) ).

% diff_less_mono
tff(fact_2425_disjoint__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,A5)
     => ( ord_le1268244103169919719at_nat(B2,B5)
       => ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A5),B5) = bot_bo228742789529271731at_nat )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A2),B2) = bot_bo228742789529271731at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_2426_disjoint__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,A2),A5))
     => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,B2),B5))
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A5),B5) = bot_bo2099793752762293965at_nat )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_2427_disjoint__mono,axiom,
    ! [A2: set_bool,A5: set_bool,B2: set_bool,B5: set_bool] :
      ( ord_less_eq_set_bool(A2,A5)
     => ( ord_less_eq_set_bool(B2,B5)
       => ( ( inf_inf_set_bool(A5,B5) = bot_bot_set_bool )
         => ( inf_inf_set_bool(A2,B2) = bot_bot_set_bool ) ) ) ) ).

% disjoint_mono
tff(fact_2428_disjoint__mono,axiom,
    ! [A2: set_int,A5: set_int,B2: set_int,B5: set_int] :
      ( ord_less_eq_set_int(A2,A5)
     => ( ord_less_eq_set_int(B2,B5)
       => ( ( inf_inf_set_int(A5,B5) = bot_bot_set_int )
         => ( inf_inf_set_int(A2,B2) = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
tff(fact_2429_disjoint__mono,axiom,
    ! [A2: set_nat,A5: set_nat,B2: set_nat,B5: set_nat] :
      ( ord_less_eq_set_nat(A2,A5)
     => ( ord_less_eq_set_nat(B2,B5)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A5),B5) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_2430_disjoint__alt__simp2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) != A )
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) != bot_bo228742789529271731at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2431_disjoint__alt__simp2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) != A )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2432_disjoint__alt__simp2,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( minus_minus_set_bool(A,B) != A )
    <=> ( inf_inf_set_bool(A,B) != bot_bot_set_bool ) ) ).

% disjoint_alt_simp2
tff(fact_2433_disjoint__alt__simp2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( minus_minus_set_int(A,B) != A )
    <=> ( inf_inf_set_int(A,B) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_2434_disjoint__alt__simp2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) != A )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_2435_disjoint__alt__simp1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) = A )
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) = bot_bo228742789529271731at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2436_disjoint__alt__simp1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) = A )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2437_disjoint__alt__simp1,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( minus_minus_set_bool(A,B) = A )
    <=> ( inf_inf_set_bool(A,B) = bot_bot_set_bool ) ) ).

% disjoint_alt_simp1
tff(fact_2438_disjoint__alt__simp1,axiom,
    ! [A: set_int,B: set_int] :
      ( ( minus_minus_set_int(A,B) = A )
    <=> ( inf_inf_set_int(A,B) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_2439_disjoint__alt__simp1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) = A )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_2440_Int__Diff__disjoint,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)) = bot_bo228742789529271731at_nat ).

% Int_Diff_disjoint
tff(fact_2441_Int__Diff__disjoint,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_2442_Int__Diff__disjoint,axiom,
    ! [A: set_bool,B: set_bool] : inf_inf_set_bool(inf_inf_set_bool(A,B),minus_minus_set_bool(A,B)) = bot_bot_set_bool ).

% Int_Diff_disjoint
tff(fact_2443_Int__Diff__disjoint,axiom,
    ! [A: set_int,B: set_int] : inf_inf_set_int(inf_inf_set_int(A,B),minus_minus_set_int(A,B)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_2444_Int__Diff__disjoint,axiom,
    ! [A: 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,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_2445_Diff__triv,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) = bot_bo228742789529271731at_nat )
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B) = A ) ) ).

% Diff_triv
tff(fact_2446_Diff__triv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B) = A ) ) ).

% Diff_triv
tff(fact_2447_Diff__triv,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( inf_inf_set_bool(A,B) = bot_bot_set_bool )
     => ( minus_minus_set_bool(A,B) = A ) ) ).

% Diff_triv
tff(fact_2448_Diff__triv,axiom,
    ! [A: set_int,B: set_int] :
      ( ( inf_inf_set_int(A,B) = bot_bot_set_int )
     => ( minus_minus_set_int(A,B) = A ) ) ).

% Diff_triv
tff(fact_2449_Diff__triv,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B) = A ) ) ).

% Diff_triv
tff(fact_2450_foldl__absorb1,axiom,
    ! [X: code_integer,Zs: list_Code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),foldl_1542389171486570077nteger(times_3573771949741848930nteger,one_one_Code_integer,Zs)) = foldl_1542389171486570077nteger(times_3573771949741848930nteger,X,Zs) ).

% foldl_absorb1
tff(fact_2451_foldl__absorb1,axiom,
    ! [X: assn,Zs: list_assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),foldl_assn_assn(times_times_assn,one_one_assn,Zs)) = foldl_assn_assn(times_times_assn,X,Zs) ).

% foldl_absorb1
tff(fact_2452_foldl__absorb1,axiom,
    ! [X: rat,Zs: list_rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),foldl_rat_rat(times_times_rat,one_one_rat,Zs)) = foldl_rat_rat(times_times_rat,X,Zs) ).

% foldl_absorb1
tff(fact_2453_foldl__absorb1,axiom,
    ! [X: nat,Zs: list_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),foldl_nat_nat(times_times_nat,one_one_nat,Zs)) = foldl_nat_nat(times_times_nat,X,Zs) ).

% foldl_absorb1
tff(fact_2454_foldl__absorb1,axiom,
    ! [X: int,Zs: list_int] : aa_int_int(aa_int_fun_int_int(times_times_int,X),foldl_int_int(times_times_int,one_one_int,Zs)) = foldl_int_int(times_times_int,X,Zs) ).

% foldl_absorb1
tff(fact_2455_length__removeAll__less__eq,axiom,
    ! [X: c,Xs: list_c] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_c(removeAll_c(X,Xs))),size_size_list_c(Xs))) ).

% length_removeAll_less_eq
tff(fact_2456_length__removeAll__less__eq,axiom,
    ! [X: d,Xs: list_d] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_d(removeAll_d(X,Xs))),size_size_list_d(Xs))) ).

% length_removeAll_less_eq
tff(fact_2457_length__removeAll__less__eq,axiom,
    ! [X: nat,Xs: list_nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(removeAll_nat(X,Xs))),size_size_list_nat(Xs))) ).

% length_removeAll_less_eq
tff(fact_2458_length__removeAll__less__eq,axiom,
    ! [X: int,Xs: list_int] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(removeAll_int(X,Xs))),size_size_list_int(Xs))) ).

% length_removeAll_less_eq
tff(fact_2459_disjoint__alt__simp3,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B),A)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B) != bot_bo228742789529271731at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_2460_disjoint__alt__simp3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)),A))
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_2461_disjoint__alt__simp3,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ord_less_set_bool(minus_minus_set_bool(A,B),A)
    <=> ( inf_inf_set_bool(A,B) != bot_bot_set_bool ) ) ).

% disjoint_alt_simp3
tff(fact_2462_disjoint__alt__simp3,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(minus_minus_set_int(A,B),A)
    <=> ( inf_inf_set_int(A,B) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_2463_disjoint__alt__simp3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_2464_verit__comp__simplify1_I1_J,axiom,
    ! [A2: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_2465_verit__comp__simplify1_I1_J,axiom,
    ! [A2: assn] : ~ pp(aa_assn_bool(ord_less_assn(A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_2466_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] : ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_2467_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_2468_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),A2)) ).

% verit_comp_simplify1(1)
tff(fact_2469_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(R2,S2),R)
     => ( ( S4 = S2 )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_2470_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S2: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S2)),R))
     => ( ( S4 = S2 )
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(R2),S4)),R)) ) ) ).

% ssubst_Pair_rhs
tff(fact_2471_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr272337284026993017e_term,S2: produc8923325533196201883nteger,R: set_Pr5663546166630115725nteger,S4: produc8923325533196201883nteger] :
      ( member6423679177046879214nteger(produc7384524951051686153nteger(R2,S2),R)
     => ( ( S4 = S2 )
       => member6423679177046879214nteger(produc7384524951051686153nteger(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_2472_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr6001797050664311790t_bool,S2: produc3658429121746597890et_nat,R: set_Pr1106803562331729365et_nat,S4: produc3658429121746597890et_nat] :
      ( member8615881266069945910et_nat(produc8148480564485311129et_nat(R2,S2),R)
     => ( ( S4 = S2 )
       => member8615881266069945910et_nat(produc8148480564485311129et_nat(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_2473_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr6001797050664311790t_bool,S2: produc6108883039684432031et_nat,R: set_Pr9222568342844664434et_nat,S4: produc6108883039684432031et_nat] :
      ( member2818932950976722259et_nat(produc3083477295402903094et_nat(R2,S2),R)
     => ( ( S4 = S2 )
       => member2818932950976722259et_nat(produc3083477295402903094et_nat(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_2474_ssubst__Pair__rhs,axiom,
    ! [R2: fun_Pr9164807099015279978e_term,S2: product_prod_int_int,R: set_Pr233528076387193594nt_int,S4: product_prod_int_int] :
      ( member690604328116145115nt_int(produc1560874099174258678nt_int(R2,S2),R)
     => ( ( S4 = S2 )
       => member690604328116145115nt_int(produc1560874099174258678nt_int(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_2475_ssubst__Pair__rhs,axiom,
    ! [R2: fun_in6429348457993719614e_term,S2: product_prod_int_int,R: set_Pr5033440245338114598nt_int,S4: product_prod_int_int] :
      ( member500491123954006279nt_int(produc4852315488734385442nt_int(R2,S2),R)
     => ( ( S4 = S2 )
       => member500491123954006279nt_int(produc4852315488734385442nt_int(R2,S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_2476_distinct__length__le,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,Ys))
     => ( ( set_Pr1139785259514867910n_assn(Ys) = set_Pr1139785259514867910n_assn(Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6829681357464350627n_assn(Ys)),size_s6829681357464350627n_assn(Xs))) ) ) ).

% distinct_length_le
tff(fact_2477_distinct__length__le,axiom,
    ! [Ys: list_c,Xs: list_c] :
      ( pp(aa_list_c_bool(distinct_c,Ys))
     => ( ( set_c2(Ys) = set_c2(Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_c(Ys)),size_size_list_c(Xs))) ) ) ).

% distinct_length_le
tff(fact_2478_distinct__length__le,axiom,
    ! [Ys: list_d,Xs: list_d] :
      ( pp(aa_list_d_bool(distinct_d,Ys))
     => ( ( set_d2(Ys) = set_d2(Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_d(Ys)),size_size_list_d(Xs))) ) ) ).

% distinct_length_le
tff(fact_2479_distinct__length__le,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Ys))
     => ( ( set_nat2(Ys) = set_nat2(Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Ys)),size_size_list_nat(Xs))) ) ) ).

% distinct_length_le
tff(fact_2480_distinct__length__le,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Ys))
     => ( ( set_int2(Ys) = set_int2(Xs) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Ys)),size_size_list_int(Xs))) ) ) ).

% distinct_length_le
tff(fact_2481_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% nat_neq_iff
tff(fact_2482_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_not_refl
tff(fact_2483_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( M2 != N ) ) ).

% less_not_refl2
tff(fact_2484_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
     => ( S2 != T ) ) ).

% less_not_refl3
tff(fact_2485_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N)) ).

% less_irrefl_nat
tff(fact_2486_nat__less__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N5: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N5))
             => pp(aa_nat_bool(P,M3)) )
         => pp(aa_nat_bool(P,N5)) )
     => pp(aa_nat_bool(P,N)) ) ).

% nat_less_induct
tff(fact_2487_infinite__descent,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( ! [N5: nat] :
          ( ~ pp(aa_nat_bool(P,N5))
         => ? [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N5))
              & ~ pp(aa_nat_bool(P,M3)) ) )
     => pp(aa_nat_bool(P,N)) ) ).

% infinite_descent
tff(fact_2488_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_2489_size__neq__size__imp__neq,axiom,
    ! [X: list_c,Y: list_c] :
      ( ( size_size_list_c(X) != size_size_list_c(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2490_size__neq__size__imp__neq,axiom,
    ! [X: list_d,Y: list_d] :
      ( ( size_size_list_d(X) != size_size_list_d(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2491_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2492_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_2493_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_list_int_bool(distinct_int,Ys))
       => distin3744728255968310194nt_int(product_int_int(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_2494_distinct__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => ( pp(aa_list_nat_bool(distinct_nat,Ys))
       => distin7922579275477506902nt_nat(product_int_nat(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_2495_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_list_int_bool(distinct_int,Ys))
       => distin2745374544067255638at_int(product_nat_int(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_2496_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => ( pp(aa_list_nat_bool(distinct_nat,Ys))
       => distin6923225563576452346at_nat(product_nat_nat(Xs,Ys)) ) ) ).

% distinct_product
tff(fact_2497_distinct__sorted__mono,axiom,
    ! [L: list_Code_integer,I: nat,J: nat] :
      ( sorted710888440204495920nteger(ord_le3102999989581377725nteger,L)
     => ( distin1543349897113766820nteger(L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s3445333598471063425nteger(L)))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(nth_Code_integer(L),I)),aa_nat_Code_integer(nth_Code_integer(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_2498_distinct__sorted__mono,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( sorted_wrt_num(ord_less_eq_num,L)
     => ( distinct_num(L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_2499_distinct__sorted__mono,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
     => ( pp(aa_list_nat_bool(distinct_nat,L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(L)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_2500_distinct__sorted__mono,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
     => ( pp(aa_list_int_bool(distinct_int,L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(L)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J))) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_2501_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_Code_integer,I: nat,J: nat] :
      ( distin1543349897113766820nteger(L)
     => ( sorted710888440204495920nteger(ord_le3102999989581377725nteger,L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_s3445333598471063425nteger(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s3445333598471063425nteger(L)))
           => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(nth_Code_integer(L),I)),aa_nat_Code_integer(nth_Code_integer(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_2502_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_num,I: nat,J: nat] :
      ( distinct_num(L)
     => ( sorted_wrt_num(ord_less_eq_num,L)
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_num(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
           => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(nth_num(L),I)),aa_nat_num(nth_num(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_2503_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,L))
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(L)))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_2504_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(distinct_int,L))
     => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(L)))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_2505_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_c(Xs)))
       => ( size_size_list_c(slice_c(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_2506_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_d(Xs)))
       => ( size_size_list_d(slice_d(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_2507_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_nat(Xs)))
       => ( size_size_list_nat(slice_nat(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_2508_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_int(Xs)))
       => ( size_size_list_int(slice_int(From,To,Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From) ) ) ) ).

% slice_len
tff(fact_2509_rev__update,axiom,
    ! [K3: nat,Xs: list_c,Y: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),size_size_list_c(Xs)))
     => ( rev_c(list_update_c(Xs,K3,Y)) = list_update_c(rev_c(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),K3)),one_one_nat),Y) ) ) ).

% rev_update
tff(fact_2510_rev__update,axiom,
    ! [K3: nat,Xs: list_d,Y: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),size_size_list_d(Xs)))
     => ( rev_d(list_update_d(Xs,K3,Y)) = list_update_d(rev_d(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),K3)),one_one_nat),Y) ) ) ).

% rev_update
tff(fact_2511_rev__update,axiom,
    ! [K3: nat,Xs: list_nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),size_size_list_nat(Xs)))
     => ( rev_nat(list_update_nat(Xs,K3,Y)) = list_update_nat(rev_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),K3)),one_one_nat),Y) ) ) ).

% rev_update
tff(fact_2512_rev__update,axiom,
    ! [K3: nat,Xs: list_int,Y: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),size_size_list_int(Xs)))
     => ( rev_int(list_update_int(Xs,K3,Y)) = list_update_int(rev_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),K3)),one_one_nat),Y) ) ) ).

% rev_update
tff(fact_2513_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_c,X: c] :
      ( ( ( K3 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),one_one_nat) )
       => ( butlast_c(list_update_c(Xs,K3,X)) = butlast_c(Xs) ) )
      & ( ( K3 != aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),one_one_nat) )
       => ( butlast_c(list_update_c(Xs,K3,X)) = list_update_c(butlast_c(Xs),K3,X) ) ) ) ).

% butlast_list_update
tff(fact_2514_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_d,X: d] :
      ( ( ( K3 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),one_one_nat) )
       => ( butlast_d(list_update_d(Xs,K3,X)) = butlast_d(Xs) ) )
      & ( ( K3 != aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),one_one_nat) )
       => ( butlast_d(list_update_d(Xs,K3,X)) = list_update_d(butlast_d(Xs),K3,X) ) ) ) ).

% butlast_list_update
tff(fact_2515_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_nat,X: nat] :
      ( ( ( K3 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat) )
       => ( butlast_nat(list_update_nat(Xs,K3,X)) = butlast_nat(Xs) ) )
      & ( ( K3 != aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat) )
       => ( butlast_nat(list_update_nat(Xs,K3,X)) = list_update_nat(butlast_nat(Xs),K3,X) ) ) ) ).

% butlast_list_update
tff(fact_2516_butlast__list__update,axiom,
    ! [K3: nat,Xs: list_int,X: int] :
      ( ( ( K3 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat) )
       => ( butlast_int(list_update_int(Xs,K3,X)) = butlast_int(Xs) ) )
      & ( ( K3 != aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat) )
       => ( butlast_int(list_update_int(Xs,K3,X)) = list_update_int(butlast_int(Xs),K3,X) ) ) ) ).

% butlast_list_update
tff(fact_2517_distinct__sorted__mono__iff,axiom,
    ! [L: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(distinct_nat,L))
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(L)))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),aa_nat_nat(nth_nat(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_2518_distinct__sorted__mono__iff,axiom,
    ! [L: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(distinct_int,L))
     => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_int(L)))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(L)))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),I)),aa_nat_int(nth_int(L),J)))
            <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J)) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_2519_length__zip,axiom,
    ! [Xs: list_c,Ys: list_c] : size_s2756448709718749209od_c_c(zip_c_c(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_c(Xs)),size_size_list_c(Ys)) ).

% length_zip
tff(fact_2520_length__zip,axiom,
    ! [Xs: list_c,Ys: list_d] : size_s2827482749765095194od_c_d(zip_c_d(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_c(Xs)),size_size_list_d(Ys)) ).

% length_zip
tff(fact_2521_length__zip,axiom,
    ! [Xs: list_c,Ys: list_nat] : size_s2954062820725854115_c_nat(zip_c_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_c(Xs)),size_size_list_nat(Ys)) ).

% length_zip
tff(fact_2522_length__zip,axiom,
    ! [Xs: list_c,Ys: list_int] : size_s463979675794127359_c_int(zip_c_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_c(Xs)),size_size_list_int(Ys)) ).

% length_zip
tff(fact_2523_length__zip,axiom,
    ! [Xs: list_d,Ys: list_c] : size_s2120799709113762520od_d_c(zip_d_c(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_d(Xs)),size_size_list_c(Ys)) ).

% length_zip
tff(fact_2524_length__zip,axiom,
    ! [Xs: list_d,Ys: list_d] : size_s2191833749160108505od_d_d(zip_d_d(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_d(Xs)),size_size_list_d(Ys)) ).

% length_zip
tff(fact_2525_length__zip,axiom,
    ! [Xs: list_d,Ys: list_nat] : size_s8550281435913984804_d_nat(zip_d_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_d(Xs)),size_size_list_nat(Ys)) ).

% length_zip
tff(fact_2526_length__zip,axiom,
    ! [Xs: list_d,Ys: list_int] : size_s6060198290982258048_d_int(zip_d_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_d(Xs)),size_size_list_int(Ys)) ).

% length_zip
tff(fact_2527_length__zip,axiom,
    ! [Xs: list_nat,Ys: list_c] : size_s385972143775086793_nat_c(zip_nat_c(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_nat(Xs)),size_size_list_c(Ys)) ).

% length_zip
tff(fact_2528_length__zip,axiom,
    ! [Xs: list_nat,Ys: list_d] : size_s457006183821432778_nat_d(zip_nat_d(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,size_size_list_nat(Xs)),size_size_list_d(Ys)) ).

% length_zip
tff(fact_2529_star__assnI,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,As3: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H2,As2)))
     => ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),produc7507926704131184380et_nat(H2,As3)))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As3) = bot_bot_set_nat )
         => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)),produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3)))) ) ) ) ).

% star_assnI
tff(fact_2530_mod__star__conv,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H2))
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
          ( ( H2 = 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 )
          & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,A),produc7507926704131184380et_nat(Hr,As1)))
          & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,B),produc7507926704131184380et_nat(Hr,As22))) ) ) ).

% mod_star_conv
tff(fact_2531_rotate1__length01,axiom,
    ! [Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_c(Xs)),one_one_nat))
     => ( rotate1_c(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2532_rotate1__length01,axiom,
    ! [Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_d(Xs)),one_one_nat))
     => ( rotate1_d(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2533_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat))
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2534_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat))
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2535_sup_Oidem,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),A2) = A2 ).

% sup.idem
tff(fact_2536_sup_Oidem,axiom,
    ! [A2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),A2) = A2 ).

% sup.idem
tff(fact_2537_sup_Oidem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),A2) = A2 ).

% sup.idem
tff(fact_2538_sup__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),X) = X ).

% sup_idem
tff(fact_2539_sup__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).

% sup_idem
tff(fact_2540_sup__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),X) = X ).

% sup_idem
tff(fact_2541_sup_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ).

% sup.left_idem
tff(fact_2542_sup_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) ).

% sup.left_idem
tff(fact_2543_sup_Oleft__idem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) ).

% sup.left_idem
tff(fact_2544_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% sup_left_idem
tff(fact_2545_sup__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% sup_left_idem
tff(fact_2546_sup__left__idem,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ).

% sup_left_idem
tff(fact_2547_sup_Oright__idem,axiom,
    ! [A2: set_nat,B2: 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,A2),B2)),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ).

% sup.right_idem
tff(fact_2548_sup_Oright__idem,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) ).

% sup.right_idem
tff(fact_2549_sup_Oright__idem,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) ).

% sup.right_idem
tff(fact_2550_IntI,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,A)
     => ( member8757157785044589968at_nat(C,B)
       => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)) ) ) ).

% IntI
tff(fact_2551_IntI,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,A)
     => ( member_int(C,B)
       => member_int(C,inf_inf_set_int(A,B)) ) ) ).

% IntI
tff(fact_2552_IntI,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,A)
     => ( member4125802036136091927t_bool(C,B)
       => member4125802036136091927t_bool(C,inf_in1689267372175091004t_bool(A,B)) ) ) ).

% IntI
tff(fact_2553_IntI,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,A)
     => ( member_nat(C,B)
       => member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% IntI
tff(fact_2554_IntI,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))) ) ) ).

% IntI
tff(fact_2555_Int__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B))
    <=> ( member8757157785044589968at_nat(C,A)
        & member8757157785044589968at_nat(C,B) ) ) ).

% Int_iff
tff(fact_2556_Int__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,inf_inf_set_int(A,B))
    <=> ( member_int(C,A)
        & member_int(C,B) ) ) ).

% Int_iff
tff(fact_2557_Int__iff,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,inf_in1689267372175091004t_bool(A,B))
    <=> ( member4125802036136091927t_bool(C,A)
        & member4125802036136091927t_bool(C,B) ) ) ).

% Int_iff
tff(fact_2558_Int__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
    <=> ( member_nat(C,A)
        & member_nat(C,B) ) ) ).

% Int_iff
tff(fact_2559_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
        & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% Int_iff
tff(fact_2560_UnCI,axiom,
    ! [C: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A)) )
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B))) ) ).

% UnCI
tff(fact_2561_UnCI,axiom,
    ! [C: int,B: set_int,A: set_int] :
      ( ( ~ member_int(C,B)
       => member_int(C,A) )
     => member_int(C,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% UnCI
tff(fact_2562_UnCI,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool,A: set_fu6080891106736489294t_bool] :
      ( ( ~ member4125802036136091927t_bool(C,B)
       => member4125802036136091927t_bool(C,A) )
     => member4125802036136091927t_bool(C,sup_su7328640041279924514t_bool(A,B)) ) ).

% UnCI
tff(fact_2563_UnCI,axiom,
    ! [C: nat,B: set_nat,A: set_nat] :
      ( ( ~ member_nat(C,B)
       => member_nat(C,A) )
     => member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% UnCI
tff(fact_2564_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ~ member8757157785044589968at_nat(C,B)
       => member8757157785044589968at_nat(C,A) )
     => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% UnCI
tff(fact_2565_Un__iff,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B)))
    <=> ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% Un_iff
tff(fact_2566_Un__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))
    <=> ( member_int(C,A)
        | member_int(C,B) ) ) ).

% Un_iff
tff(fact_2567_Un__iff,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,sup_su7328640041279924514t_bool(A,B))
    <=> ( member4125802036136091927t_bool(C,A)
        | member4125802036136091927t_bool(C,B) ) ) ).

% Un_iff
tff(fact_2568_Un__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))
    <=> ( member_nat(C,A)
        | member_nat(C,B) ) ) ).

% Un_iff
tff(fact_2569_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B))
    <=> ( member8757157785044589968at_nat(C,A)
        | member8757157785044589968at_nat(C,B) ) ) ).

% Un_iff
tff(fact_2570_min_Oidem,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),A2) = A2 ).

% min.idem
tff(fact_2571_min_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) ).

% min.left_idem
tff(fact_2572_min_Oright__idem,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) ).

% min.right_idem
tff(fact_2573_le__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y),Z)
    <=> ( ord_le1268244103169919719at_nat(X,Z)
        & ord_le1268244103169919719at_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_2574_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(X),Z))
        & pp(aa_assn_bool(ord_less_eq_assn(Y),Z)) ) ) ).

% le_sup_iff
tff(fact_2575_le__sup__iff,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y),Z)
    <=> ( ord_le2510731241096832064er_nat(X,Z)
        & ord_le2510731241096832064er_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_2576_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y),Z)
    <=> ( ord_less_eq_set_nat(X,Z)
        & ord_less_eq_set_nat(Y,Z) ) ) ).

% le_sup_iff
tff(fact_2577_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_2578_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) ) ) ).

% le_sup_iff
tff(fact_2579_sup_Obounded__iff,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C),A2)
    <=> ( ord_le1268244103169919719at_nat(B2,A2)
        & ord_le1268244103169919719at_nat(C,A2) ) ) ).

% sup.bounded_iff
tff(fact_2580_sup_Obounded__iff,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C)),A2))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
        & pp(aa_assn_bool(ord_less_eq_assn(C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_2581_sup_Obounded__iff,axiom,
    ! [B2: filter_nat,C: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C),A2)
    <=> ( ord_le2510731241096832064er_nat(B2,A2)
        & ord_le2510731241096832064er_nat(C,A2) ) ) ).

% sup.bounded_iff
tff(fact_2582_sup_Obounded__iff,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C),A2)
    <=> ( ord_less_eq_set_nat(B2,A2)
        & ord_less_eq_set_nat(C,A2) ) ) ).

% sup.bounded_iff
tff(fact_2583_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C)),A2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_2584_sup_Obounded__iff,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C)),A2))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% sup.bounded_iff
tff(fact_2585_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_2586_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),bot_bo228742789529271731at_nat) = A2 ).

% sup_bot.right_neutral
tff(fact_2587_sup__bot_Oright__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),bot_bo2099793752762293965at_nat) = A2 ).

% sup_bot.right_neutral
tff(fact_2588_sup__bot_Oright__neutral,axiom,
    ! [A2: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),bot_bot_set_bool) = A2 ).

% sup_bot.right_neutral
tff(fact_2589_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_2590_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_2591_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) )
    <=> ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_2592_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) )
    <=> ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_2593_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2) )
    <=> ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_2594_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_bool,B2: set_bool] :
      ( ( bot_bot_set_bool = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),B2) )
    <=> ( ( A2 = bot_bot_set_bool )
        & ( B2 = bot_bot_set_bool ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_2595_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_2596_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_2597_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_2598_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_2599_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_2600_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,bot_bot_set_bool),A2) = A2 ).

% sup_bot.left_neutral
tff(fact_2601_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_2602_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_2603_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = bot_bot_assn )
    <=> ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_2604_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = bot_bo228742789529271731at_nat )
    <=> ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_2605_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_2606_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_bool,B2: set_bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A2),B2) = bot_bot_set_bool )
    <=> ( ( A2 = bot_bot_set_bool )
        & ( B2 = bot_bot_set_bool ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_2607_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_2608_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A2),B2) = bot_bot_set_int )
    <=> ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_2609_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_2610_sup__eq__bot__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = bot_bo228742789529271731at_nat )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_2611_sup__eq__bot__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_2612_sup__eq__bot__iff,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X),Y) = bot_bot_set_bool )
    <=> ( ( X = bot_bot_set_bool )
        & ( Y = bot_bot_set_bool ) ) ) ).

% sup_eq_bot_iff
tff(fact_2613_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_2614_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_2615_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_2616_bot__eq__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) )
    <=> ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_2617_bot__eq__sup__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) )
    <=> ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_2618_bot__eq__sup__iff,axiom,
    ! [X: set_bool,Y: set_bool] :
      ( ( bot_bot_set_bool = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X),Y) )
    <=> ( ( X = bot_bot_set_bool )
        & ( Y = bot_bot_set_bool ) ) ) ).

% bot_eq_sup_iff
tff(fact_2619_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_2620_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_2621_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_2622_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).

% sup_bot_right
tff(fact_2623_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ).

% sup_bot_right
tff(fact_2624_sup__bot__right,axiom,
    ! [X: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X),bot_bot_set_bool) = X ).

% sup_bot_right
tff(fact_2625_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_2626_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_2627_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_2628_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),X) = X ).

% sup_bot_left
tff(fact_2629_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),X) = X ).

% sup_bot_left
tff(fact_2630_sup__bot__left,axiom,
    ! [X: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,bot_bot_set_bool),X) = X ).

% sup_bot_left
tff(fact_2631_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_2632_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_2633_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2634_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2635_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2636_inf__sup__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_2637_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2638_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2639_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2640_sup__inf__absorb,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_2641_Un__empty,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = bot_bo228742789529271731at_nat )
    <=> ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
tff(fact_2642_Un__empty,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) = bot_bo2099793752762293965at_nat )
    <=> ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
tff(fact_2643_Un__empty,axiom,
    ! [A: set_bool,B: set_bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),B) = bot_bot_set_bool )
    <=> ( ( A = bot_bot_set_bool )
        & ( B = bot_bot_set_bool ) ) ) ).

% Un_empty
tff(fact_2644_Un__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
tff(fact_2645_Un__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% Un_empty
tff(fact_2646_min__arg__le_I2_J,axiom,
    ! [M2: assn,N: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(M2),aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N)))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2647_min__arg__le_I2_J,axiom,
    ! [M2: filter_nat,N: filter_nat] :
      ( ord_le2510731241096832064er_nat(M2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,M2),N))
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2648_min__arg__le_I2_J,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( ord_less_eq_set_nat(M2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2649_min__arg__le_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2650_min__arg__le_I2_J,axiom,
    ! [M2: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N)))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = M2 ) ) ).

% min_arg_le(2)
tff(fact_2651_min__arg__le_I1_J,axiom,
    ! [N: assn,M2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(N),aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N)))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2652_min__arg__le_I1_J,axiom,
    ! [N: filter_nat,M2: filter_nat] :
      ( ord_le2510731241096832064er_nat(N,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,M2),N))
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2653_min__arg__le_I1_J,axiom,
    ! [N: set_nat,M2: set_nat] :
      ( ord_less_eq_set_nat(N,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N))
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2654_min__arg__le_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2655_min__arg__le_I1_J,axiom,
    ! [N: int,M2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N)))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = N ) ) ).

% min_arg_le(1)
tff(fact_2656_min__eq__arg_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = N )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ).

% min_eq_arg(2)
tff(fact_2657_min__eq__arg_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = N )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),M2)) ) ).

% min_eq_arg(2)
tff(fact_2658_min__eq__arg_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = M2 )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% min_eq_arg(1)
tff(fact_2659_min__eq__arg_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = M2 )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,M2),N)) ) ).

% min_eq_arg(1)
tff(fact_2660_min_Oabsorb1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = A2 ) ) ).

% min.absorb1
tff(fact_2661_min_Oabsorb1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = A2 ) ) ).

% min.absorb1
tff(fact_2662_min_Oabsorb2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = B2 ) ) ).

% min.absorb2
tff(fact_2663_min_Oabsorb2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = B2 ) ) ).

% min.absorb2
tff(fact_2664_min_Obounded__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),C)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% min.bounded_iff
tff(fact_2665_min_Obounded__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(ord_min_int,B2),C)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% min.bounded_iff
tff(fact_2666_Un__subset__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B),C2)
    <=> ( ord_le1268244103169919719at_nat(A,C2)
        & ord_le1268244103169919719at_nat(B,C2) ) ) ).

% Un_subset_iff
tff(fact_2667_Un__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B),C2)
    <=> ( ord_less_eq_set_nat(A,C2)
        & ord_less_eq_set_nat(B,C2) ) ) ).

% Un_subset_iff
tff(fact_2668_Un__insert__right,axiom,
    ! [A: set_int,A2: int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),insert_int2(A2,B)) = insert_int2(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ).

% Un_insert_right
tff(fact_2669_Un__insert__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),insert8211810215607154385at_nat(A2,B)) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B)) ).

% Un_insert_right
tff(fact_2670_Un__insert__right,axiom,
    ! [A: set_bool,A2: bool,B: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),insert_bool2(A2,B)) = insert_bool2(A2,aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),B)) ).

% Un_insert_right
tff(fact_2671_Un__insert__right,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),insert_nat2(A2,B)) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% Un_insert_right
tff(fact_2672_Un__insert__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),insert9069300056098147895at_nat(A2,B)) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% Un_insert_right
tff(fact_2673_Un__insert__left,axiom,
    ! [A2: int,B: set_int,C2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int2(A2,B)),C2) = insert_int2(A2,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,B),C2)) ).

% Un_insert_left
tff(fact_2674_Un__insert__left,axiom,
    ! [A2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A2,B)),C2) = insert8211810215607154385at_nat(A2,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C2)) ).

% Un_insert_left
tff(fact_2675_Un__insert__left,axiom,
    ! [A2: bool,B: set_bool,C2: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,insert_bool2(A2,B)),C2) = insert_bool2(A2,aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,B),C2)) ).

% Un_insert_left
tff(fact_2676_Un__insert__left,axiom,
    ! [A2: nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat2(A2,B)),C2) = insert_nat2(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% Un_insert_left
tff(fact_2677_Un__insert__left,axiom,
    ! [A2: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A2,B)),C2) = insert9069300056098147895at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) ).

% Un_insert_left
tff(fact_2678_min__less__iff__conj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z),aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y)))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z),X))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Z),Y)) ) ) ).

% min_less_iff_conj
tff(fact_2679_min__less__iff__conj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),aa_num_num(ord_min_num(X),Y)))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),X))
        & pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Z),Y)) ) ) ).

% min_less_iff_conj
tff(fact_2680_min__less__iff__conj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),X))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Z),Y)) ) ) ).

% min_less_iff_conj
tff(fact_2681_min__less__iff__conj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),X))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),Y)) ) ) ).

% min_less_iff_conj
tff(fact_2682_min_Oabsorb4,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2) = B2 ) ) ).

% min.absorb4
tff(fact_2683_min_Oabsorb4,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( aa_num_num(ord_min_num(A2),B2) = B2 ) ) ).

% min.absorb4
tff(fact_2684_min_Oabsorb4,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = B2 ) ) ).

% min.absorb4
tff(fact_2685_min_Oabsorb4,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = B2 ) ) ).

% min.absorb4
tff(fact_2686_min_Oabsorb3,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2) = A2 ) ) ).

% min.absorb3
tff(fact_2687_min_Oabsorb3,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( aa_num_num(ord_min_num(A2),B2) = A2 ) ) ).

% min.absorb3
tff(fact_2688_min_Oabsorb3,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = A2 ) ) ).

% min.absorb3
tff(fact_2689_min_Oabsorb3,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = A2 ) ) ).

% min.absorb3
tff(fact_2690_min__arg__not__ge_I1_J,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(M2),N)),M2))
    <=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(M2),N) = M2 ) ) ).

% min_arg_not_ge(1)
tff(fact_2691_min__arg__not__ge_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(M2),N)),M2))
    <=> ( aa_num_num(ord_min_num(M2),N) = M2 ) ) ).

% min_arg_not_ge(1)
tff(fact_2692_min__arg__not__ge_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),M2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = M2 ) ) ).

% min_arg_not_ge(1)
tff(fact_2693_min__arg__not__ge_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N)),M2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = M2 ) ) ).

% min_arg_not_ge(1)
tff(fact_2694_min__arg__not__ge_I2_J,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(M2),N)),N))
    <=> ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(M2),N) = N ) ) ).

% min_arg_not_ge(2)
tff(fact_2695_min__arg__not__ge_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(M2),N)),N))
    <=> ( aa_num_num(ord_min_num(M2),N) = N ) ) ).

% min_arg_not_ge(2)
tff(fact_2696_min__arg__not__ge_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N)),N))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M2),N) = N ) ) ).

% min_arg_not_ge(2)
tff(fact_2697_min__arg__not__ge_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N)),N))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,M2),N) = N ) ) ).

% min_arg_not_ge(2)
tff(fact_2698_min__less__self__conv_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2)),A2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ).

% min_less_self_conv(1)
tff(fact_2699_min__less__self__conv_I1_J,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(A2),B2)),A2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2)) ) ).

% min_less_self_conv(1)
tff(fact_2700_min__less__self__conv_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),A2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2)) ) ).

% min_less_self_conv(1)
tff(fact_2701_min__less__self__conv_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),A2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% min_less_self_conv(1)
tff(fact_2702_min__less__self__conv_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2)),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% min_less_self_conv(2)
tff(fact_2703_min__less__self__conv_I2_J,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(A2),B2)),B2))
    <=> pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2)) ) ).

% min_less_self_conv(2)
tff(fact_2704_min__less__self__conv_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% min_less_self_conv(2)
tff(fact_2705_min__less__self__conv_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% min_less_self_conv(2)
tff(fact_2706_min__simps_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2) = A2 ) ) ).

% min_simps(1)
tff(fact_2707_min__simps_I1_J,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A2),B2) = A2 ) ) ).

% min_simps(1)
tff(fact_2708_min__simps_I1_J,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
     => ( aa_num_num(ord_min_num(A2),B2) = A2 ) ) ).

% min_simps(1)
tff(fact_2709_min__simps_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = A2 ) ) ).

% min_simps(1)
tff(fact_2710_min__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = A2 ) ) ).

% min_simps(1)
tff(fact_2711_min__simps_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2) = B2 ) ) ).

% min_simps(2)
tff(fact_2712_min__simps_I2_J,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A2),B2) = B2 ) ) ).

% min_simps(2)
tff(fact_2713_min__simps_I2_J,axiom,
    ! [B2: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),A2))
     => ( aa_num_num(ord_min_num(A2),B2) = B2 ) ) ).

% min_simps(2)
tff(fact_2714_min__simps_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = B2 ) ) ).

% min_simps(2)
tff(fact_2715_min__simps_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = B2 ) ) ).

% min_simps(2)
tff(fact_2716_min__bot2,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(ord_mi6693596345299712430at_nat,X),bot_bo228742789529271731at_nat) = bot_bo228742789529271731at_nat ).

% min_bot2
tff(fact_2717_min__bot2,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(ord_mi4194197829855655752at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% min_bot2
tff(fact_2718_min__bot2,axiom,
    ! [X: set_bool] : ord_min_set_bool(X,bot_bot_set_bool) = bot_bot_set_bool ).

% min_bot2
tff(fact_2719_min__bot2,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% min_bot2
tff(fact_2720_min__bot2,axiom,
    ! [X: set_int] : ord_min_set_int(X,bot_bot_set_int) = bot_bot_set_int ).

% min_bot2
tff(fact_2721_min__bot2,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),bot_bot_nat) = bot_bot_nat ).

% min_bot2
tff(fact_2722_min__bot,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(ord_mi6693596345299712430at_nat,bot_bo228742789529271731at_nat),X) = bot_bo228742789529271731at_nat ).

% min_bot
tff(fact_2723_min__bot,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(ord_mi4194197829855655752at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% min_bot
tff(fact_2724_min__bot,axiom,
    ! [X: set_bool] : ord_min_set_bool(bot_bot_set_bool,X) = bot_bot_set_bool ).

% min_bot
tff(fact_2725_min__bot,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% min_bot
tff(fact_2726_min__bot,axiom,
    ! [X: set_int] : ord_min_set_int(bot_bot_set_int,X) = bot_bot_set_int ).

% min_bot
tff(fact_2727_min__bot,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,bot_bot_nat),X) = bot_bot_nat ).

% min_bot
tff(fact_2728_Int__Un__eq_I4_J,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_2729_Int__Un__eq_I4_J,axiom,
    ! [T3: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_2730_Int__Un__eq_I4_J,axiom,
    ! [T3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,T3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_2731_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T3)) = S ).

% Int_Un_eq(3)
tff(fact_2732_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T3: 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),T3)) = S ).

% Int_Un_eq(3)
tff(fact_2733_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T3)) = S ).

% Int_Un_eq(3)
tff(fact_2734_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_2735_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T3: 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),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_2736_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_2737_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T3)),S) = S ).

% Int_Un_eq(1)
tff(fact_2738_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T3: 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),T3)),S) = S ).

% Int_Un_eq(1)
tff(fact_2739_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),T3)),S) = S ).

% Int_Un_eq(1)
tff(fact_2740_Un__Int__eq_I4_J,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T3),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_2741_Un__Int__eq_I4_J,axiom,
    ! [T3: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T3),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_2742_Un__Int__eq_I4_J,axiom,
    ! [T3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,T3),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_2743_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T3)) = S ).

% Un_Int_eq(3)
tff(fact_2744_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T3: 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),T3)) = S ).

% Un_Int_eq(3)
tff(fact_2745_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,S),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T3)) = S ).

% Un_Int_eq(3)
tff(fact_2746_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_2747_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T3: 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),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_2748_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_2749_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T3)),S) = S ).

% Un_Int_eq(1)
tff(fact_2750_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T3: 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),T3)),S) = S ).

% Un_Int_eq(1)
tff(fact_2751_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,S),T3)),S) = S ).

% Un_Int_eq(1)
tff(fact_2752_Un__Diff__cancel2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),A)),A) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),A) ).

% Un_Diff_cancel2
tff(fact_2753_Un__Diff__cancel2,axiom,
    ! [B: 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(minus_minus_set_nat,B),A)),A) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).

% Un_Diff_cancel2
tff(fact_2754_Un__Diff__cancel,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),A)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ).

% Un_Diff_cancel
tff(fact_2755_Un__Diff__cancel,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% Un_Diff_cancel
tff(fact_2756_set__rev,axiom,
    ! [Xs: list_int] : set_int2(rev_int(Xs)) = set_int2(Xs) ).

% set_rev
tff(fact_2757_set__rev,axiom,
    ! [Xs: list_nat] : set_nat2(rev_nat(Xs)) = set_nat2(Xs) ).

% set_rev
tff(fact_2758_set__rev,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(rev_Pr4855572775806611735n_assn(Xs)) = set_Pr1139785259514867910n_assn(Xs) ).

% set_rev
tff(fact_2759_length__rev,axiom,
    ! [Xs: list_c] : size_size_list_c(rev_c(Xs)) = size_size_list_c(Xs) ).

% length_rev
tff(fact_2760_length__rev,axiom,
    ! [Xs: list_d] : size_size_list_d(rev_d(Xs)) = size_size_list_d(Xs) ).

% length_rev
tff(fact_2761_length__rev,axiom,
    ! [Xs: list_nat] : size_size_list_nat(rev_nat(Xs)) = size_size_list_nat(Xs) ).

% length_rev
tff(fact_2762_length__rev,axiom,
    ! [Xs: list_int] : size_size_list_int(rev_int(Xs)) = size_size_list_int(Xs) ).

% length_rev
tff(fact_2763_distinct__rev,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,rev_int(Xs)))
    <=> pp(aa_list_int_bool(distinct_int,Xs)) ) ).

% distinct_rev
tff(fact_2764_distinct__rev,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,rev_nat(Xs)))
    <=> pp(aa_list_nat_bool(distinct_nat,Xs)) ) ).

% distinct_rev
tff(fact_2765_set__rotate1,axiom,
    ! [Xs: list_int] : set_int2(rotate1_int(Xs)) = set_int2(Xs) ).

% set_rotate1
tff(fact_2766_set__rotate1,axiom,
    ! [Xs: list_nat] : set_nat2(rotate1_nat(Xs)) = set_nat2(Xs) ).

% set_rotate1
tff(fact_2767_set__rotate1,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(rotate328796349445179396n_assn(Xs)) = set_Pr1139785259514867910n_assn(Xs) ).

% set_rotate1
tff(fact_2768_length__rotate1,axiom,
    ! [Xs: list_c] : size_size_list_c(rotate1_c(Xs)) = size_size_list_c(Xs) ).

% length_rotate1
tff(fact_2769_length__rotate1,axiom,
    ! [Xs: list_d] : size_size_list_d(rotate1_d(Xs)) = size_size_list_d(Xs) ).

% length_rotate1
tff(fact_2770_length__rotate1,axiom,
    ! [Xs: list_nat] : size_size_list_nat(rotate1_nat(Xs)) = size_size_list_nat(Xs) ).

% length_rotate1
tff(fact_2771_length__rotate1,axiom,
    ! [Xs: list_int] : size_size_list_int(rotate1_int(Xs)) = size_size_list_int(Xs) ).

% length_rotate1
tff(fact_2772_distinct1__rotate,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,rotate1_int(Xs)))
    <=> pp(aa_list_int_bool(distinct_int,Xs)) ) ).

% distinct1_rotate
tff(fact_2773_distinct1__rotate,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,rotate1_nat(Xs)))
    <=> pp(aa_list_nat_bool(distinct_nat,Xs)) ) ).

% distinct1_rotate
tff(fact_2774_list__ex__rev,axiom,
    ! [P: fun_int_bool,Xs: list_int] :
      ( pp(aa_list_int_bool(list_ex_int(P),rev_int(Xs)))
    <=> pp(aa_list_int_bool(list_ex_int(P),Xs)) ) ).

% list_ex_rev
tff(fact_2775_list__ex__rev,axiom,
    ! [P: fun_nat_bool,Xs: list_nat] :
      ( pp(aa_list_nat_bool(list_ex_nat(P),rev_nat(Xs)))
    <=> pp(aa_list_nat_bool(list_ex_nat(P),Xs)) ) ).

% list_ex_rev
tff(fact_2776_set__union,axiom,
    ! [Xs: list_int,Ys: list_int] : set_int2(union_int(Xs,Ys)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_int2(Xs)),set_int2(Ys)) ).

% set_union
tff(fact_2777_set__union,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(union_944050417204428971n_assn(Xs,Ys)) = sup_su1735938885662425829n_assn(set_Pr1139785259514867910n_assn(Xs),set_Pr1139785259514867910n_assn(Ys)) ).

% set_union
tff(fact_2778_set__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] : set_nat2(union_nat(Xs,Ys)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_nat2(Xs)),set_nat2(Ys)) ).

% set_union
tff(fact_2779_set__union,axiom,
    ! [Xs: list_P5464809261938338413at_nat,Ys: list_P5464809261938338413at_nat] : set_Pr3765526544606949372at_nat(union_4462254032241401953at_nat(Xs,Ys)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,set_Pr3765526544606949372at_nat(Xs)),set_Pr3765526544606949372at_nat(Ys)) ).

% set_union
tff(fact_2780_length__butlast,axiom,
    ! [Xs: list_c] : size_size_list_c(butlast_c(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),one_one_nat) ).

% length_butlast
tff(fact_2781_length__butlast,axiom,
    ! [Xs: list_d] : size_size_list_d(butlast_d(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),one_one_nat) ).

% length_butlast
tff(fact_2782_length__butlast,axiom,
    ! [Xs: list_nat] : size_size_list_nat(butlast_nat(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat) ).

% length_butlast
tff(fact_2783_length__butlast,axiom,
    ! [Xs: list_int] : size_size_list_int(butlast_int(Xs)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat) ).

% length_butlast
tff(fact_2784_IntE,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B))
     => ~ ( member8757157785044589968at_nat(C,A)
         => ~ member8757157785044589968at_nat(C,B) ) ) ).

% IntE
tff(fact_2785_IntE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,inf_inf_set_int(A,B))
     => ~ ( member_int(C,A)
         => ~ member_int(C,B) ) ) ).

% IntE
tff(fact_2786_IntE,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,inf_in1689267372175091004t_bool(A,B))
     => ~ ( member4125802036136091927t_bool(C,A)
         => ~ member4125802036136091927t_bool(C,B) ) ) ).

% IntE
tff(fact_2787_IntE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( member_nat(C,A)
         => ~ member_nat(C,B) ) ) ).

% IntE
tff(fact_2788_IntE,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)))
     => ~ ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
         => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% IntE
tff(fact_2789_IntD1,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B))
     => member8757157785044589968at_nat(C,A) ) ).

% IntD1
tff(fact_2790_IntD1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,inf_inf_set_int(A,B))
     => member_int(C,A) ) ).

% IntD1
tff(fact_2791_IntD1,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,inf_in1689267372175091004t_bool(A,B))
     => member4125802036136091927t_bool(C,A) ) ).

% IntD1
tff(fact_2792_IntD1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => member_nat(C,A) ) ).

% IntD1
tff(fact_2793_IntD1,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A)) ) ).

% IntD1
tff(fact_2794_IntD2,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B))
     => member8757157785044589968at_nat(C,B) ) ).

% IntD2
tff(fact_2795_IntD2,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,inf_inf_set_int(A,B))
     => member_int(C,B) ) ).

% IntD2
tff(fact_2796_IntD2,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,inf_in1689267372175091004t_bool(A,B))
     => member4125802036136091927t_bool(C,B) ) ).

% IntD2
tff(fact_2797_IntD2,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => member_nat(C,B) ) ).

% IntD2
tff(fact_2798_IntD2,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ).

% IntD2
tff(fact_2799_Int__assoc,axiom,
    ! [A: set_nat,B: set_nat,C2: 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),B)),C2) = 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,B),C2)) ).

% Int_assoc
tff(fact_2800_Int__assoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) ).

% Int_assoc
tff(fact_2801_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_2802_Int__absorb,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),A) = A ).

% Int_absorb
tff(fact_2803_Int__commute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) ).

% Int_commute
tff(fact_2804_Int__commute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A) ).

% Int_commute
tff(fact_2805_Un__Int__crazy,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A)) ).

% Un_Int_crazy
tff(fact_2806_Un__Int__crazy,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),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),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A)) ).

% Un_Int_crazy
tff(fact_2807_Un__Int__crazy,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2))),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A)) ).

% Un_Int_crazy
tff(fact_2808_Int__Un__distrib,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)) ).

% Int_Un_distrib
tff(fact_2809_Int__Un__distrib,axiom,
    ! [A: set_nat,B: set_nat,C2: 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,B),C2)) = 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),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C2)) ).

% Int_Un_distrib
tff(fact_2810_Int__Un__distrib,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),C2)) ).

% Int_Un_distrib
tff(fact_2811_Un__Int__distrib,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),C2)) ).

% Un_Int_distrib
tff(fact_2812_Un__Int__distrib,axiom,
    ! [A: set_nat,B: set_nat,C2: 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,B),C2)) = 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),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C2)) ).

% Un_Int_distrib
tff(fact_2813_Un__Int__distrib,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),C2)) ).

% Un_Int_distrib
tff(fact_2814_Int__Un__distrib2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C2)),A) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A)) ).

% Int_Un_distrib2
tff(fact_2815_Int__Un__distrib2,axiom,
    ! [B: set_nat,C2: 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,B),C2)),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,B),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A)) ).

% Int_Un_distrib2
tff(fact_2816_Int__Un__distrib2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)),A) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),A)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,C2),A)) ).

% Int_Un_distrib2
tff(fact_2817_Int__left__absorb,axiom,
    ! [A: set_nat,B: 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),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% Int_left_absorb
tff(fact_2818_Int__left__absorb,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ).

% Int_left_absorb
tff(fact_2819_Un__Int__distrib2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)),A) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A)) ).

% Un_Int_distrib2
tff(fact_2820_Un__Int__distrib2,axiom,
    ! [B: set_nat,C2: 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,B),C2)),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,B),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A)) ).

% Un_Int_distrib2
tff(fact_2821_Un__Int__distrib2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2)),A) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),A)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),A)) ).

% Un_Int_distrib2
tff(fact_2822_Int__left__commute,axiom,
    ! [A: set_nat,B: set_nat,C2: 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,B),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C2)) ).

% Int_left_commute
tff(fact_2823_Int__left__commute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C2)) ).

% Int_left_commute
tff(fact_2824_less__assn__def,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
    <=> ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
        & ( A2 != B2 ) ) ) ).

% less_assn_def
tff(fact_2825_UnE,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B)))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B)) ) ) ).

% UnE
tff(fact_2826_UnE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B))
     => ( ~ member_int(C,A)
       => member_int(C,B) ) ) ).

% UnE
tff(fact_2827_UnE,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,sup_su7328640041279924514t_bool(A,B))
     => ( ~ member4125802036136091927t_bool(C,A)
       => member4125802036136091927t_bool(C,B) ) ) ).

% UnE
tff(fact_2828_UnE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))
     => ( ~ member_nat(C,A)
       => member_nat(C,B) ) ) ).

% UnE
tff(fact_2829_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B))
     => ( ~ member8757157785044589968at_nat(C,A)
       => member8757157785044589968at_nat(C,B) ) ) ).

% UnE
tff(fact_2830_UnI1,axiom,
    ! [C: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),A))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B))) ) ).

% UnI1
tff(fact_2831_UnI1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( member_int(C,A)
     => member_int(C,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% UnI1
tff(fact_2832_UnI1,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,A: set_fu6080891106736489294t_bool,B: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,A)
     => member4125802036136091927t_bool(C,sup_su7328640041279924514t_bool(A,B)) ) ).

% UnI1
tff(fact_2833_UnI1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( member_nat(C,A)
     => member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% UnI1
tff(fact_2834_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,A)
     => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% UnI1
tff(fact_2835_UnI2,axiom,
    ! [C: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),B))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B))) ) ).

% UnI2
tff(fact_2836_UnI2,axiom,
    ! [C: int,B: set_int,A: set_int] :
      ( member_int(C,B)
     => member_int(C,aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B)) ) ).

% UnI2
tff(fact_2837_UnI2,axiom,
    ! [C: fun_Pr6001797050664311790t_bool,B: set_fu6080891106736489294t_bool,A: set_fu6080891106736489294t_bool] :
      ( member4125802036136091927t_bool(C,B)
     => member4125802036136091927t_bool(C,sup_su7328640041279924514t_bool(A,B)) ) ).

% UnI2
tff(fact_2838_UnI2,axiom,
    ! [C: nat,B: set_nat,A: set_nat] :
      ( member_nat(C,B)
     => member_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% UnI2
tff(fact_2839_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(C,B)
     => member8757157785044589968at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ) ).

% UnI2
tff(fact_2840_bex__Un,axiom,
    ! [A: set_nat,B: set_nat,P: fun_nat_bool] :
      ( ? [X5: nat] :
          ( member_nat(X5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))
          & pp(aa_nat_bool(P,X5)) )
    <=> ( ? [X5: nat] :
            ( member_nat(X5,A)
            & pp(aa_nat_bool(P,X5)) )
        | ? [X5: nat] :
            ( member_nat(X5,B)
            & pp(aa_nat_bool(P,X5)) ) ) ) ).

% bex_Un
tff(fact_2841_bex__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,P: fun_Pr1607443439640623155t_bool] :
      ( ? [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B))
          & pp(aa_Pro1809015098595223084t_bool(P,X5)) )
    <=> ( ? [X5: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X5,A)
            & pp(aa_Pro1809015098595223084t_bool(P,X5)) )
        | ? [X5: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X5,B)
            & pp(aa_Pro1809015098595223084t_bool(P,X5)) ) ) ) ).

% bex_Un
tff(fact_2842_ball__Un,axiom,
    ! [A: set_nat,B: set_nat,P: fun_nat_bool] :
      ( ! [X5: nat] :
          ( member_nat(X5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))
         => pp(aa_nat_bool(P,X5)) )
    <=> ( ! [X5: nat] :
            ( member_nat(X5,A)
           => pp(aa_nat_bool(P,X5)) )
        & ! [X5: nat] :
            ( member_nat(X5,B)
           => pp(aa_nat_bool(P,X5)) ) ) ) ).

% ball_Un
tff(fact_2843_ball__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,P: fun_Pr1607443439640623155t_bool] :
      ( ! [X5: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X5,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B))
         => pp(aa_Pro1809015098595223084t_bool(P,X5)) )
    <=> ( ! [X5: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X5,A)
           => pp(aa_Pro1809015098595223084t_bool(P,X5)) )
        & ! [X5: produc3843707927480180839at_nat] :
            ( member8757157785044589968at_nat(X5,B)
           => pp(aa_Pro1809015098595223084t_bool(P,X5)) ) ) ) ).

% ball_Un
tff(fact_2844_Un__assoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% Un_assoc
tff(fact_2845_Un__assoc,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) ).

% Un_assoc
tff(fact_2846_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_2847_Un__absorb,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),A) = A ).

% Un_absorb
tff(fact_2848_Un__commute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).

% Un_commute
tff(fact_2849_Un__commute,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),A) ).

% Un_commute
tff(fact_2850_Un__left__absorb,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% Un_left_absorb
tff(fact_2851_Un__left__absorb,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) ).

% Un_left_absorb
tff(fact_2852_Un__left__commute,axiom,
    ! [A: set_nat,B: set_nat,C2: 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,B),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C2)) ).

% Un_left_commute
tff(fact_2853_Un__left__commute,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),C2)) ).

% Un_left_commute
tff(fact_2854_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_2855_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% inf_sup_aci(8)
tff(fact_2856_inf__sup__aci_I8_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_2857_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% inf_sup_aci(7)
tff(fact_2858_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% inf_sup_aci(7)
tff(fact_2859_inf__sup__aci_I7_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).

% inf_sup_aci(7)
tff(fact_2860_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_2861_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_2862_inf__sup__aci_I6_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_2863_inf__sup__aci_I5_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_2864_inf__sup__aci_I5_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% inf_sup_aci(5)
tff(fact_2865_inf__sup__aci_I5_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_2866_min_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),C)) ).

% min.assoc
tff(fact_2867_min_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),A2) ).

% min.commute
tff(fact_2868_min_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),C)) ).

% min.left_commute
tff(fact_2869_sup_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C: 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,A2),B2)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C)) ).

% sup.assoc
tff(fact_2870_sup_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C)) ).

% sup.assoc
tff(fact_2871_sup_Oassoc,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)),C) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C)) ).

% sup.assoc
tff(fact_2872_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% sup_assoc
tff(fact_2873_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% sup_assoc
tff(fact_2874_sup__assoc,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),Z) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) ).

% sup_assoc
tff(fact_2875_sup_Ocommute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2) ).

% sup.commute
tff(fact_2876_sup_Ocommute,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),A2) ).

% sup.commute
tff(fact_2877_sup_Ocommute,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),A2) ).

% sup.commute
tff(fact_2878_sup__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% sup_commute
tff(fact_2879_sup__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% sup_commute
tff(fact_2880_sup__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X) ).

% sup_commute
tff(fact_2881_sup_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C)) ).

% sup.left_commute
tff(fact_2882_sup_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C)) ).

% sup.left_commute
tff(fact_2883_sup_Oleft__commute,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),C)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C)) ).

% sup.left_commute
tff(fact_2884_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% sup_left_commute
tff(fact_2885_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_left_commute
tff(fact_2886_sup__left__commute,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).

% sup_left_commute
tff(fact_2887_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_c] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( zip_c_c(rev_c(Xs),rev_c(Ys)) = rev_Product_prod_c_c(zip_c_c(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2888_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_d] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( zip_c_d(rev_c(Xs),rev_d(Ys)) = rev_Product_prod_c_d(zip_c_d(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2889_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_nat] :
      ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
     => ( zip_c_nat(rev_c(Xs),rev_nat(Ys)) = rev_Pr3799340238495523993_c_nat(zip_c_nat(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2890_zip__rev,axiom,
    ! [Xs: list_c,Ys: list_int] :
      ( ( size_size_list_c(Xs) = size_size_list_int(Ys) )
     => ( zip_c_int(rev_c(Xs),rev_int(Ys)) = rev_Pr8844861255841103093_c_int(zip_c_int(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2891_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_c] :
      ( ( size_size_list_d(Xs) = size_size_list_c(Ys) )
     => ( zip_d_c(rev_d(Xs),rev_c(Ys)) = rev_Product_prod_d_c(zip_d_c(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2892_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_d] :
      ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
     => ( zip_d_d(rev_d(Xs),rev_d(Ys)) = rev_Product_prod_d_d(zip_d_d(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2893_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_nat] :
      ( ( size_size_list_d(Xs) = size_size_list_nat(Ys) )
     => ( zip_d_nat(rev_d(Xs),rev_nat(Ys)) = rev_Pr5034784567451918490_d_nat(zip_d_nat(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2894_zip__rev,axiom,
    ! [Xs: list_d,Ys: list_int] :
      ( ( size_size_list_d(Xs) = size_size_list_int(Ys) )
     => ( zip_d_int(rev_d(Xs),rev_int(Ys)) = rev_Pr856933547942721782_d_int(zip_d_int(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2895_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_c] :
      ( ( size_size_list_nat(Xs) = size_size_list_c(Ys) )
     => ( zip_nat_c(rev_nat(Xs),rev_c(Ys)) = rev_Pr4566615052912869567_nat_c(zip_nat_c(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2896_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_d] :
      ( ( size_size_list_nat(Xs) = size_size_list_d(Ys) )
     => ( zip_nat_d(rev_nat(Xs),rev_d(Ys)) = rev_Pr4566615057216098368_nat_d(zip_nat_d(Xs,Ys)) ) ) ).

% zip_rev
tff(fact_2897_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ns))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),size_size_list_nat(Ns)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I))) ) ) ).

% sorted_wrt_less_idx
tff(fact_2898_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P5464809261938338413at_nat,P: fun_Pr124437560471516333t_bool,Q: fun_Pr124437560471516333t_bool] :
      ( ! [X2: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X2,set_Pr3765526544606949372at_nat(Xs))
         => ( member8757157785044589968at_nat(Y2,set_Pr3765526544606949372at_nat(Xs))
           => ( pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(P,X2),Y2))
             => pp(aa_Pro1809015098595223084t_bool(aa_Pro2859564254541897054t_bool(Q,X2),Y2)) ) ) )
     => ( sorted8831241538929206612at_nat(P,Xs)
       => sorted8831241538929206612at_nat(Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_2899_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr248683899324676269t_bool,Q: fun_Pr248683899324676269t_bool] :
      ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),set_Pr5648618587558075414at_nat(Xs)))
         => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),set_Pr5648618587558075414at_nat(Xs)))
           => ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(P,X2),Y2))
             => pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(Q,X2),Y2)) ) ) )
     => ( sorted5214655850825725294at_nat(P,Xs)
       => sorted5214655850825725294at_nat(Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_2900_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_f5857023252961307252t_bool,P: fun_fu7365307207611327597l_bool,Q: fun_fu7365307207611327597l_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool,Y2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X2,set_fu1437410266168021891t_bool(Xs))
         => ( member4125802036136091927t_bool(Y2,set_fu1437410266168021891t_bool(Xs))
           => ( pp(aa_fun5588795374382764133l_bool(aa_fun4580269573335628190l_bool(P,X2),Y2))
             => pp(aa_fun5588795374382764133l_bool(aa_fun4580269573335628190l_bool(Q,X2),Y2)) ) ) )
     => ( sorted4636103753318484187t_bool(P,Xs)
       => sorted4636103753318484187t_bool(Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_2901_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr6563606323009366189n_bool,Q: fun_Pr6563606323009366189n_bool] :
      ( ! [X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => ( member7957490590177025114n_assn(Y2,set_Pr1139785259514867910n_assn(Xs))
           => ( pp(aa_Pro3608213857613363938n_bool(aa_Pro3155263873353247326n_bool(P,X2),Y2))
             => pp(aa_Pro3608213857613363938n_bool(aa_Pro3155263873353247326n_bool(Q,X2),Y2)) ) ) )
     => ( pp(aa_lis5212129917091495004n_bool(sorted3986126766855596574n_assn(P),Xs))
       => pp(aa_lis5212129917091495004n_bool(sorted3986126766855596574n_assn(Q),Xs)) ) ) ).

% sorted_wrt_mono_rel
tff(fact_2902_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_nat,P: fun_nat_fun_nat_bool,Q: fun_nat_fun_nat_bool] :
      ( ! [X2: nat,Y2: nat] :
          ( member_nat(X2,set_nat2(Xs))
         => ( member_nat(Y2,set_nat2(Xs))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y2))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(Q,X2),Y2)) ) ) )
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs))
       => pp(aa_list_nat_bool(sorted_wrt_nat(Q),Xs)) ) ) ).

% sorted_wrt_mono_rel
tff(fact_2903_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_int,P: fun_int_fun_int_bool,Q: fun_int_fun_int_bool] :
      ( ! [X2: int,Y2: int] :
          ( member_int(X2,set_int2(Xs))
         => ( member_int(Y2,set_int2(Xs))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(P,X2),Y2))
             => pp(aa_int_bool(aa_int_fun_int_bool(Q,X2),Y2)) ) ) )
     => ( pp(aa_list_int_bool(sorted_wrt_int(P),Xs))
       => pp(aa_list_int_bool(sorted_wrt_int(Q),Xs)) ) ) ).

% sorted_wrt_mono_rel
tff(fact_2904_min__def,axiom,
    ! [A2: assn,B2: assn] :
      ( ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A2),B2) = A2 ) )
      & ( ~ pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,A2),B2) = B2 ) ) ) ).

% min_def
tff(fact_2905_min__def,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( ord_le2510731241096832064er_nat(A2,B2)
       => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,A2),B2) = A2 ) )
      & ( ~ ord_le2510731241096832064er_nat(A2,B2)
       => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,A2),B2) = B2 ) ) ) ).

% min_def
tff(fact_2906_min__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat(A2,B2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,A2),B2) = A2 ) )
      & ( ~ ord_less_eq_set_nat(A2,B2)
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,A2),B2) = B2 ) ) ) ).

% min_def
tff(fact_2907_min__def,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = A2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = B2 ) ) ) ).

% min_def
tff(fact_2908_min__def,axiom,
    ! [A2: int,B2: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = A2 ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = B2 ) ) ) ).

% min_def
tff(fact_2909_min__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),Y) = X ) ) ).

% min_absorb1
tff(fact_2910_min__absorb1,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,X),Y) = X ) ) ).

% min_absorb1
tff(fact_2911_min__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),Y) = X ) ) ).

% min_absorb1
tff(fact_2912_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y) = X ) ) ).

% min_absorb1
tff(fact_2913_min__absorb1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y) = X ) ) ).

% min_absorb1
tff(fact_2914_min__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(ord_min_assn,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_2915_min__absorb2,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,X)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(ord_min_filter_nat,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_2916_min__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ord_less_eq_set_nat(Y,X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(ord_min_set_nat,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_2917_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_2918_min__absorb2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y) = Y ) ) ).

% min_absorb2
tff(fact_2919_min_Omono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,C),D2))) ) ) ).

% min.mono
tff(fact_2920_min_Omono,axiom,
    ! [A2: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),aa_int_int(aa_int_fun_int_int(ord_min_int,C),D2))) ) ) ).

% min.mono
tff(fact_2921_min_OorderE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) ) ) ).

% min.orderE
tff(fact_2922_min_OorderE,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( A2 = aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) ) ) ).

% min.orderE
tff(fact_2923_min_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% min.orderI
tff(fact_2924_min_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% min.orderI
tff(fact_2925_min_OboundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C)) ) ) ).

% min.boundedE
tff(fact_2926_min_OboundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(ord_min_int,B2),C)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C)) ) ) ).

% min.boundedE
tff(fact_2927_min_OboundedI,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),C))) ) ) ).

% min.boundedI
tff(fact_2928_min_OboundedI,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(ord_min_int,B2),C))) ) ) ).

% min.boundedI
tff(fact_2929_min_Oorder__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) ) ) ).

% min.order_iff
tff(fact_2930_min_Oorder__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) ) ) ).

% min.order_iff
tff(fact_2931_min_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),A2)) ).

% min.cobounded1
tff(fact_2932_min_Ocobounded1,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),A2)) ).

% min.cobounded1
tff(fact_2933_min_Ocobounded2,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),B2)) ).

% min.cobounded2
tff(fact_2934_min_Ocobounded2,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),B2)) ).

% min.cobounded2
tff(fact_2935_min_Oabsorb__iff1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = A2 ) ) ).

% min.absorb_iff1
tff(fact_2936_min_Oabsorb__iff1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = A2 ) ) ).

% min.absorb_iff1
tff(fact_2937_min_Oabsorb__iff2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) = B2 ) ) ).

% min.absorb_iff2
tff(fact_2938_min_Oabsorb__iff2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) = B2 ) ) ).

% min.absorb_iff2
tff(fact_2939_min_OcoboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),C)) ) ).

% min.coboundedI1
tff(fact_2940_min_OcoboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),C)) ) ).

% min.coboundedI1
tff(fact_2941_min_OcoboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),C)) ) ).

% min.coboundedI2
tff(fact_2942_min_OcoboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),C)) ) ).

% min.coboundedI2
tff(fact_2943_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Z))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),Z)) ) ) ).

% min_le_iff_disj
tff(fact_2944_min__le__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Z))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),Z)) ) ) ).

% min_le_iff_disj
tff(fact_2945_min_Ostrict__coboundedI2,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2)),C)) ) ).

% min.strict_coboundedI2
tff(fact_2946_min_Ostrict__coboundedI2,axiom,
    ! [B2: num,C: num,A2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,B2),C))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(A2),B2)),C)) ) ).

% min.strict_coboundedI2
tff(fact_2947_min_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),C)) ) ).

% min.strict_coboundedI2
tff(fact_2948_min_Ostrict__coboundedI2,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),C)) ) ).

% min.strict_coboundedI2
tff(fact_2949_min_Ostrict__coboundedI1,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2)),C)) ) ).

% min.strict_coboundedI1
tff(fact_2950_min_Ostrict__coboundedI1,axiom,
    ! [A2: num,C: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C))
     => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(A2),B2)),C)) ) ).

% min.strict_coboundedI1
tff(fact_2951_min_Ostrict__coboundedI1,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),C)) ) ).

% min.strict_coboundedI1
tff(fact_2952_min_Ostrict__coboundedI1,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2)),C)) ) ).

% min.strict_coboundedI1
tff(fact_2953_min_Ostrict__order__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
    <=> ( ( A2 = aa_Cod1783974652533396728nteger(ord_min_Code_integer(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% min.strict_order_iff
tff(fact_2954_min_Ostrict__order__iff,axiom,
    ! [A2: num,B2: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
    <=> ( ( A2 = aa_num_num(ord_min_num(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% min.strict_order_iff
tff(fact_2955_min_Ostrict__order__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% min.strict_order_iff
tff(fact_2956_min_Ostrict__order__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_min_int,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% min.strict_order_iff
tff(fact_2957_min_Ostrict__boundedE,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(ord_min_Code_integer(B2),C)))
     => ~ ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
         => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),C)) ) ) ).

% min.strict_boundedE
tff(fact_2958_min_Ostrict__boundedE,axiom,
    ! [A2: num,B2: num,C: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),aa_num_num(ord_min_num(B2),C)))
     => ~ ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),B2))
         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,A2),C)) ) ) ).

% min.strict_boundedE
tff(fact_2959_min_Ostrict__boundedE,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),C)))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),C)) ) ) ).

% min.strict_boundedE
tff(fact_2960_min_Ostrict__boundedE,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(ord_min_int,B2),C)))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),C)) ) ) ).

% min.strict_boundedE
tff(fact_2961_min__less__iff__disj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_min_Code_integer(X),Y)),Z))
    <=> ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Z))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),Z)) ) ) ).

% min_less_iff_disj
tff(fact_2962_min__less__iff__disj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_num_num(ord_min_num(X),Y)),Z))
    <=> ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),Z))
        | pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,Y),Z)) ) ) ).

% min_less_iff_disj
tff(fact_2963_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Z))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ) ).

% min_less_iff_disj
tff(fact_2964_min__less__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Z))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),Z)) ) ) ).

% min_less_iff_disj
tff(fact_2965_min__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : minus_minus_rat(ord_min_rat(X,Y),Z) = ord_min_rat(minus_minus_rat(X,Z),minus_minus_rat(Y,Z)) ).

% min_diff_distrib_left
tff(fact_2966_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% min_diff_distrib_left
tff(fact_2967_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2968_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : pp(aa_assn_bool(ord_less_eq_assn(Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y))) ).

% inf_sup_ord(4)
tff(fact_2969_inf__sup__ord_I4_J,axiom,
    ! [Y: filter_nat,X: filter_nat] : ord_le2510731241096832064er_nat(Y,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2970_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : ord_less_eq_set_nat(Y,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_2971_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% inf_sup_ord(4)
tff(fact_2972_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% inf_sup_ord(4)
tff(fact_2973_inf__sup__ord_I3_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2974_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y))) ).

% inf_sup_ord(3)
tff(fact_2975_inf__sup__ord_I3_J,axiom,
    ! [X: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2976_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_2977_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% inf_sup_ord(3)
tff(fact_2978_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% inf_sup_ord(3)
tff(fact_2979_le__supE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2),X)
     => ~ ( ord_le1268244103169919719at_nat(A2,X)
         => ~ ord_le1268244103169919719at_nat(B2,X) ) ) ).

% le_supE
tff(fact_2980_le__supE,axiom,
    ! [A2: assn,B2: assn,X: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),X))
     => ~ ( pp(aa_assn_bool(ord_less_eq_assn(A2),X))
         => ~ pp(aa_assn_bool(ord_less_eq_assn(B2),X)) ) ) ).

% le_supE
tff(fact_2981_le__supE,axiom,
    ! [A2: filter_nat,B2: filter_nat,X: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2),X)
     => ~ ( ord_le2510731241096832064er_nat(A2,X)
         => ~ ord_le2510731241096832064er_nat(B2,X) ) ) ).

% le_supE
tff(fact_2982_le__supE,axiom,
    ! [A2: set_nat,B2: set_nat,X: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2),X)
     => ~ ( ord_less_eq_set_nat(A2,X)
         => ~ ord_less_eq_set_nat(B2,X) ) ) ).

% le_supE
tff(fact_2983_le__supE,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2)),X))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),X)) ) ) ).

% le_supE
tff(fact_2984_le__supE,axiom,
    ! [A2: int,B2: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2)),X))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),X)) ) ) ).

% le_supE
tff(fact_2985_le__supI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,X)
     => ( ord_le1268244103169919719at_nat(B2,X)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2),X) ) ) ).

% le_supI
tff(fact_2986_le__supI,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),X))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),X))
       => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),X)) ) ) ).

% le_supI
tff(fact_2987_le__supI,axiom,
    ! [A2: filter_nat,X: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,X)
     => ( ord_le2510731241096832064er_nat(B2,X)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2),X) ) ) ).

% le_supI
tff(fact_2988_le__supI,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,X)
     => ( ord_less_eq_set_nat(B2,X)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2),X) ) ) ).

% le_supI
tff(fact_2989_le__supI,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2)),X)) ) ) ).

% le_supI
tff(fact_2990_le__supI,axiom,
    ! [A2: int,X: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2)),X)) ) ) ).

% le_supI
tff(fact_2991_sup__ge1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge1
tff(fact_2992_sup__ge1,axiom,
    ! [X: assn,Y: assn] : pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y))) ).

% sup_ge1
tff(fact_2993_sup__ge1,axiom,
    ! [X: filter_nat,Y: filter_nat] : ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% sup_ge1
tff(fact_2994_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge1
tff(fact_2995_sup__ge1,axiom,
    ! [X: nat,Y: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% sup_ge1
tff(fact_2996_sup__ge1,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% sup_ge1
tff(fact_2997_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)) ).

% sup_ge2
tff(fact_2998_sup__ge2,axiom,
    ! [Y: assn,X: assn] : pp(aa_assn_bool(ord_less_eq_assn(Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y))) ).

% sup_ge2
tff(fact_2999_sup__ge2,axiom,
    ! [Y: filter_nat,X: filter_nat] : ord_le2510731241096832064er_nat(Y,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)) ).

% sup_ge2
tff(fact_3000_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : ord_less_eq_set_nat(Y,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge2
tff(fact_3001_sup__ge2,axiom,
    ! [Y: nat,X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y))) ).

% sup_ge2
tff(fact_3002_sup__ge2,axiom,
    ! [Y: int,X: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y))) ).

% sup_ge2
tff(fact_3003_le__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,A2)
     => ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% le_supI1
tff(fact_3004_le__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),A2))
     => pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% le_supI1
tff(fact_3005_le__supI1,axiom,
    ! [X: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,A2)
     => ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ) ).

% le_supI1
tff(fact_3006_le__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(X,A2)
     => ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% le_supI1
tff(fact_3007_le__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% le_supI1
tff(fact_3008_le__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% le_supI1
tff(fact_3009_le__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,B2)
     => ord_le1268244103169919719at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% le_supI2
tff(fact_3010_le__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),B2))
     => pp(aa_assn_bool(ord_less_eq_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% le_supI2
tff(fact_3011_le__supI2,axiom,
    ! [X: filter_nat,B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,B2)
     => ord_le2510731241096832064er_nat(X,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ) ).

% le_supI2
tff(fact_3012_le__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(X,B2)
     => ord_less_eq_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% le_supI2
tff(fact_3013_le__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% le_supI2
tff(fact_3014_le__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% le_supI2
tff(fact_3015_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C,A2)
     => ( ord_le1268244103169919719at_nat(D2,B2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ) ).

% sup.mono
tff(fact_3016_sup_Omono,axiom,
    ! [C: assn,A2: assn,D2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(C),A2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(D2),B2))
       => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ) ).

% sup.mono
tff(fact_3017_sup_Omono,axiom,
    ! [C: filter_nat,A2: filter_nat,D2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(C,A2)
     => ( ord_le2510731241096832064er_nat(D2,B2)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,C),D2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ) ) ).

% sup.mono
tff(fact_3018_sup_Omono,axiom,
    ! [C: set_nat,A2: set_nat,D2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(C,A2)
     => ( ord_less_eq_set_nat(D2,B2)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ) ).

% sup.mono
tff(fact_3019_sup_Omono,axiom,
    ! [C: nat,A2: nat,D2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,D2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ) ).

% sup.mono
tff(fact_3020_sup_Omono,axiom,
    ! [C: int,A2: int,D2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,D2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ) ).

% sup.mono
tff(fact_3021_sup__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,C)
     => ( ord_le1268244103169919719at_nat(B2,D2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3022_sup__mono,axiom,
    ! [A2: assn,C: assn,B2: assn,D2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),C))
     => ( pp(aa_assn_bool(ord_less_eq_assn(B2),D2))
       => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D2))) ) ) ).

% sup_mono
tff(fact_3023_sup__mono,axiom,
    ! [A2: filter_nat,C: filter_nat,B2: filter_nat,D2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,C)
     => ( ord_le2510731241096832064er_nat(B2,D2)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3024_sup__mono,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
      ( ord_less_eq_set_nat(A2,C)
     => ( ord_less_eq_set_nat(B2,D2)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_3025_sup__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2))) ) ) ).

% sup_mono
tff(fact_3026_sup__mono,axiom,
    ! [A2: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2))) ) ) ).

% sup_mono
tff(fact_3027_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Y,X)
     => ( ord_le1268244103169919719at_nat(Z,X)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_3028_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
     => ( pp(aa_assn_bool(ord_less_eq_assn(Z),X))
       => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_3029_sup__least,axiom,
    ! [Y: filter_nat,X: filter_nat,Z: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,X)
     => ( ord_le2510731241096832064er_nat(Z,X)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_3030_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( ord_less_eq_set_nat(Y,X)
     => ( ord_less_eq_set_nat(Z,X)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z),X) ) ) ).

% sup_least
tff(fact_3031_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z),X))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_3032_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z),X))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X)) ) ) ).

% sup_least
tff(fact_3033_le__iff__sup,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,Y)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3034_le__iff__sup,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3035_le__iff__sup,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3036_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3037_le__iff__sup,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3038_le__iff__sup,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_3039_sup_OorderE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B2,A2)
     => ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) ) ) ).

% sup.orderE
tff(fact_3040_sup_OorderE,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) ) ) ).

% sup.orderE
tff(fact_3041_sup_OorderE,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) ) ) ).

% sup.orderE
tff(fact_3042_sup_OorderE,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ) ) ).

% sup.orderE
tff(fact_3043_sup_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) ) ) ).

% sup.orderE
tff(fact_3044_sup_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) ) ) ).

% sup.orderE
tff(fact_3045_sup_OorderI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) )
     => ord_le1268244103169919719at_nat(B2,A2) ) ).

% sup.orderI
tff(fact_3046_sup_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) )
     => pp(aa_assn_bool(ord_less_eq_assn(B2),A2)) ) ).

% sup.orderI
tff(fact_3047_sup_OorderI,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) )
     => ord_le2510731241096832064er_nat(B2,A2) ) ).

% sup.orderI
tff(fact_3048_sup_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) )
     => ord_less_eq_set_nat(B2,A2) ) ).

% sup.orderI
tff(fact_3049_sup_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2)) ) ).

% sup.orderI
tff(fact_3050_sup_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% sup.orderI
tff(fact_3051_sup__unique,axiom,
    ! [F3: fun_se177437144654672861at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(X2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F3,X2),Y2))
     => ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(Y2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F3,X2),Y2))
       => ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
              ( ord_le1268244103169919719at_nat(Y2,X2)
             => ( ord_le1268244103169919719at_nat(Z3,X2)
               => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F3,Y2),Z3),X2) ) )
         => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(F3,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3052_sup__unique,axiom,
    ! [F3: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : pp(aa_assn_bool(ord_less_eq_assn(X2),aa_assn_assn(aa_ass954072293579886259n_assn(F3,X2),Y2)))
     => ( ! [X2: assn,Y2: assn] : pp(aa_assn_bool(ord_less_eq_assn(Y2),aa_assn_assn(aa_ass954072293579886259n_assn(F3,X2),Y2)))
       => ( ! [X2: assn,Y2: assn,Z3: assn] :
              ( pp(aa_assn_bool(ord_less_eq_assn(Y2),X2))
             => ( pp(aa_assn_bool(ord_less_eq_assn(Z3),X2))
               => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F3,Y2),Z3)),X2)) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F3,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3053_sup__unique,axiom,
    ! [F3: fun_fi7557781610047541018er_nat,X: filter_nat,Y: filter_nat] :
      ( ! [X2: filter_nat,Y2: filter_nat] : ord_le2510731241096832064er_nat(X2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,X2),Y2))
     => ( ! [X2: filter_nat,Y2: filter_nat] : ord_le2510731241096832064er_nat(Y2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,X2),Y2))
       => ( ! [X2: filter_nat,Y2: filter_nat,Z3: filter_nat] :
              ( ord_le2510731241096832064er_nat(Y2,X2)
             => ( ord_le2510731241096832064er_nat(Z3,X2)
               => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,Y2),Z3),X2) ) )
         => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(F3,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3054_sup__unique,axiom,
    ! [F3: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : ord_less_eq_set_nat(X2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,X2),Y2))
     => ( ! [X2: set_nat,Y2: set_nat] : ord_less_eq_set_nat(Y2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,X2),Y2))
       => ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
              ( ord_less_eq_set_nat(Y2,X2)
             => ( ord_less_eq_set_nat(Z3,X2)
               => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,Y2),Z3),X2) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F3,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3055_sup__unique,axiom,
    ! [F3: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F3,X2),Y2)))
     => ( ! [X2: nat,Y2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(F3,X2),Y2)))
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y2),X2))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Z3),X2))
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F3,Y2),Z3)),X2)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F3,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3056_sup__unique,axiom,
    ! [F3: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F3,X2),Y2)))
     => ( ! [X2: int,Y2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y2),aa_int_int(aa_int_fun_int_int(F3,X2),Y2)))
       => ( ! [X2: int,Y2: int,Z3: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y2),X2))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Z3),X2))
               => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F3,Y2),Z3)),X2)) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F3,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_3057_sup_Oabsorb1,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B2,A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = A2 ) ) ).

% sup.absorb1
tff(fact_3058_sup_Oabsorb1,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = A2 ) ) ).

% sup.absorb1
tff(fact_3059_sup_Oabsorb1,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) = A2 ) ) ).

% sup.absorb1
tff(fact_3060_sup_Oabsorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = A2 ) ) ).

% sup.absorb1
tff(fact_3061_sup_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = A2 ) ) ).

% sup.absorb1
tff(fact_3062_sup_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) = A2 ) ) ).

% sup.absorb1
tff(fact_3063_sup_Oabsorb2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,B2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_3064_sup_Oabsorb2,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_3065_sup_Oabsorb2,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_3066_sup_Oabsorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_3067_sup_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_3068_sup_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) = B2 ) ) ).

% sup.absorb2
tff(fact_3069_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(Y,X)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3070_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(Y),X))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3071_sup__absorb1,axiom,
    ! [Y: filter_nat,X: filter_nat] :
      ( ord_le2510731241096832064er_nat(Y,X)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3072_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ord_less_eq_set_nat(Y,X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3073_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3074_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),X))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_3075_sup__absorb2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(X,Y)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3076_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(X),Y))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3077_sup__absorb2,axiom,
    ! [X: filter_nat,Y: filter_nat] :
      ( ord_le2510731241096832064er_nat(X,Y)
     => ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3078_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(X,Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3079_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3080_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_3081_sup_OboundedE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C),A2)
     => ~ ( ord_le1268244103169919719at_nat(B2,A2)
         => ~ ord_le1268244103169919719at_nat(C,A2) ) ) ).

% sup.boundedE
tff(fact_3082_sup_OboundedE,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C)),A2))
     => ~ ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
         => ~ pp(aa_assn_bool(ord_less_eq_assn(C),A2)) ) ) ).

% sup.boundedE
tff(fact_3083_sup_OboundedE,axiom,
    ! [B2: filter_nat,C: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C),A2)
     => ~ ( ord_le2510731241096832064er_nat(B2,A2)
         => ~ ord_le2510731241096832064er_nat(C,A2) ) ) ).

% sup.boundedE
tff(fact_3084_sup_OboundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C),A2)
     => ~ ( ord_less_eq_set_nat(B2,A2)
         => ~ ord_less_eq_set_nat(C,A2) ) ) ).

% sup.boundedE
tff(fact_3085_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2)) ) ) ).

% sup.boundedE
tff(fact_3086_sup_OboundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2)) ) ) ).

% sup.boundedE
tff(fact_3087_sup_OboundedI,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B2,A2)
     => ( ord_le1268244103169919719at_nat(C,A2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C),A2) ) ) ).

% sup.boundedI
tff(fact_3088_sup_OboundedI,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
     => ( pp(aa_assn_bool(ord_less_eq_assn(C),A2))
       => pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_3089_sup_OboundedI,axiom,
    ! [B2: filter_nat,A2: filter_nat,C: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
     => ( ord_le2510731241096832064er_nat(C,A2)
       => ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,B2),C),A2) ) ) ).

% sup.boundedI
tff(fact_3090_sup_OboundedI,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
     => ( ord_less_eq_set_nat(C,A2)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C),A2) ) ) ).

% sup.boundedI
tff(fact_3091_sup_OboundedI,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_3092_sup_OboundedI,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C)),A2)) ) ) ).

% sup.boundedI
tff(fact_3093_sup_Oorder__iff,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B2,A2)
    <=> ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) ) ) ).

% sup.order_iff
tff(fact_3094_sup_Oorder__iff,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
    <=> ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) ) ) ).

% sup.order_iff
tff(fact_3095_sup_Oorder__iff,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
    <=> ( A2 = aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) ) ) ).

% sup.order_iff
tff(fact_3096_sup_Oorder__iff,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
    <=> ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) ) ) ).

% sup.order_iff
tff(fact_3097_sup_Oorder__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) ) ) ).

% sup.order_iff
tff(fact_3098_sup_Oorder__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) ) ) ).

% sup.order_iff
tff(fact_3099_sup_Ocobounded1,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(A2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ).

% sup.cobounded1
tff(fact_3100_sup_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : pp(aa_assn_bool(ord_less_eq_assn(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ).

% sup.cobounded1
tff(fact_3101_sup_Ocobounded1,axiom,
    ! [A2: filter_nat,B2: filter_nat] : ord_le2510731241096832064er_nat(A2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ).

% sup.cobounded1
tff(fact_3102_sup_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ord_less_eq_set_nat(A2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% sup.cobounded1
tff(fact_3103_sup_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ).

% sup.cobounded1
tff(fact_3104_sup_Ocobounded1,axiom,
    ! [A2: int,B2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ).

% sup.cobounded1
tff(fact_3105_sup_Ocobounded2,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(B2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ).

% sup.cobounded2
tff(fact_3106_sup_Ocobounded2,axiom,
    ! [B2: assn,A2: assn] : pp(aa_assn_bool(ord_less_eq_assn(B2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ).

% sup.cobounded2
tff(fact_3107_sup_Ocobounded2,axiom,
    ! [B2: filter_nat,A2: filter_nat] : ord_le2510731241096832064er_nat(B2,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ).

% sup.cobounded2
tff(fact_3108_sup_Ocobounded2,axiom,
    ! [B2: set_nat,A2: set_nat] : ord_less_eq_set_nat(B2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ).

% sup.cobounded2
tff(fact_3109_sup_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ).

% sup.cobounded2
tff(fact_3110_sup_Ocobounded2,axiom,
    ! [B2: int,A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ).

% sup.cobounded2
tff(fact_3111_sup_Oabsorb__iff1,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B2,A2)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_3112_sup_Oabsorb__iff1,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(B2),A2))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_3113_sup_Oabsorb__iff1,axiom,
    ! [B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(B2,A2)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_3114_sup_Oabsorb__iff1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(B2,A2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_3115_sup_Oabsorb__iff1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_3116_sup_Oabsorb__iff1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) = A2 ) ) ).

% sup.absorb_iff1
tff(fact_3117_sup_Oabsorb__iff2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A2,B2)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_3118_sup_Oabsorb__iff2,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(A2),B2))
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_3119_sup_Oabsorb__iff2,axiom,
    ! [A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(A2,B2)
    <=> ( aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_3120_sup_Oabsorb__iff2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A2,B2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_3121_sup_Oabsorb__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_3122_sup_Oabsorb__iff2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) = B2 ) ) ).

% sup.absorb_iff2
tff(fact_3123_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C,A2)
     => ord_le1268244103169919719at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% sup.coboundedI1
tff(fact_3124_sup_OcoboundedI1,axiom,
    ! [C: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(C),A2))
     => pp(aa_assn_bool(ord_less_eq_assn(C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% sup.coboundedI1
tff(fact_3125_sup_OcoboundedI1,axiom,
    ! [C: filter_nat,A2: filter_nat,B2: filter_nat] :
      ( ord_le2510731241096832064er_nat(C,A2)
     => ord_le2510731241096832064er_nat(C,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ) ).

% sup.coboundedI1
tff(fact_3126_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(C,A2)
     => ord_less_eq_set_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% sup.coboundedI1
tff(fact_3127_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% sup.coboundedI1
tff(fact_3128_sup_OcoboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% sup.coboundedI1
tff(fact_3129_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C,B2)
     => ord_le1268244103169919719at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% sup.coboundedI2
tff(fact_3130_sup_OcoboundedI2,axiom,
    ! [C: assn,B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_eq_assn(C),B2))
     => pp(aa_assn_bool(ord_less_eq_assn(C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% sup.coboundedI2
tff(fact_3131_sup_OcoboundedI2,axiom,
    ! [C: filter_nat,B2: filter_nat,A2: filter_nat] :
      ( ord_le2510731241096832064er_nat(C,B2)
     => ord_le2510731241096832064er_nat(C,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,A2),B2)) ) ).

% sup.coboundedI2
tff(fact_3132_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ord_less_eq_set_nat(C,B2)
     => ord_less_eq_set_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% sup.coboundedI2
tff(fact_3133_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% sup.coboundedI2
tff(fact_3134_sup_OcoboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% sup.coboundedI2
tff(fact_3135_inf__min,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_min
tff(fact_3136_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(C,B2)
     => ord_less_set_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_3137_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(C,B2)
     => ord_le2604355607129572851at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% sup.strict_coboundedI2
tff(fact_3138_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(C),B2))
     => pp(aa_assn_bool(ord_less_assn(C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% sup.strict_coboundedI2
tff(fact_3139_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% sup.strict_coboundedI2
tff(fact_3140_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% sup.strict_coboundedI2
tff(fact_3141_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(C,A2)
     => ord_less_set_nat(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_3142_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(C,A2)
     => ord_le2604355607129572851at_nat(C,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% sup.strict_coboundedI1
tff(fact_3143_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(C),A2))
     => pp(aa_assn_bool(ord_less_assn(C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% sup.strict_coboundedI1
tff(fact_3144_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% sup.strict_coboundedI1
tff(fact_3145_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% sup.strict_coboundedI1
tff(fact_3146_sup_Ostrict__order__iff,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,A2)
    <=> ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_3147_sup_Ostrict__order__iff,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(B2,A2)
    <=> ( ( A2 = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_3148_sup_Ostrict__order__iff,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
    <=> ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_3149_sup_Ostrict__order__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_3150_sup_Ostrict__order__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% sup.strict_order_iff
tff(fact_3151_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ord_less_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C),A2)
     => ~ ( ord_less_set_nat(B2,A2)
         => ~ ord_less_set_nat(C,A2) ) ) ).

% sup.strict_boundedE
tff(fact_3152_sup_Ostrict__boundedE,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B2),C),A2)
     => ~ ( ord_le2604355607129572851at_nat(B2,A2)
         => ~ ord_le2604355607129572851at_nat(C,A2) ) ) ).

% sup.strict_boundedE
tff(fact_3153_sup_Ostrict__boundedE,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C)),A2))
     => ~ ( pp(aa_assn_bool(ord_less_assn(B2),A2))
         => ~ pp(aa_assn_bool(ord_less_assn(C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_3154_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B2),C)),A2))
     => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_3155_sup_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B2),C)),A2))
     => ~ ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),A2)) ) ) ).

% sup.strict_boundedE
tff(fact_3156_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A2,B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_3157_sup_Oabsorb4,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(A2,B2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_3158_sup_Oabsorb4,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(A2),B2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_3159_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_3160_sup_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) = B2 ) ) ).

% sup.absorb4
tff(fact_3161_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(B2,A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2) = A2 ) ) ).

% sup.absorb3
tff(fact_3162_sup_Oabsorb3,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(B2,A2)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2) = A2 ) ) ).

% sup.absorb3
tff(fact_3163_sup_Oabsorb3,axiom,
    ! [B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(B2),A2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2) = A2 ) ) ).

% sup.absorb3
tff(fact_3164_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2) = A2 ) ) ).

% sup.absorb3
tff(fact_3165_sup_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2) = A2 ) ) ).

% sup.absorb3
tff(fact_3166_less__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ord_less_set_nat(X,B2)
     => ord_less_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% less_supI2
tff(fact_3167_less__supI2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(X,B2)
     => ord_le2604355607129572851at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% less_supI2
tff(fact_3168_less__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),B2))
     => pp(aa_assn_bool(ord_less_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% less_supI2
tff(fact_3169_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% less_supI2
tff(fact_3170_less__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% less_supI2
tff(fact_3171_less__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ord_less_set_nat(X,A2)
     => ord_less_set_nat(X,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) ) ).

% less_supI1
tff(fact_3172_less__supI1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ord_le2604355607129572851at_nat(X,A2)
     => ord_le2604355607129572851at_nat(X,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A2),B2)) ) ).

% less_supI1
tff(fact_3173_less__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(ord_less_assn(X),A2))
     => pp(aa_assn_bool(ord_less_assn(X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2))) ) ).

% less_supI1
tff(fact_3174_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A2),B2))) ) ).

% less_supI1
tff(fact_3175_less__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A2),B2))) ) ).

% less_supI1
tff(fact_3176_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_3177_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),bot_bo228742789529271731at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_3178_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),bot_bo2099793752762293965at_nat) = X ).

% boolean_algebra.disj_zero_right
tff(fact_3179_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,X),bot_bot_set_bool) = X ).

% boolean_algebra.disj_zero_right
tff(fact_3180_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_3181_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_3182_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y2),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Z3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_3183_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y2),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Z3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_3184_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y2),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Z3))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ) ) ).

% distrib_imp1
tff(fact_3185_distrib__imp1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y2),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),Y2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X2),Z3))
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_3186_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y2),Z3)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Z3))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_3187_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y2),Z3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Z3))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_3188_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z3: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y2),Z3)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Z3))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ) ) ).

% distrib_imp2
tff(fact_3189_distrib__imp2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y2),Z3)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Y2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X2),Z3))
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_3190_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_3191_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_3192_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_distrib1
tff(fact_3193_inf__sup__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_3194_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_3195_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_3196_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% inf_sup_distrib2
tff(fact_3197_inf__sup__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_3198_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_3199_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_3200_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_inf_distrib1
tff(fact_3201_sup__inf__distrib1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_3202_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_3203_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_3204_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% sup_inf_distrib2
tff(fact_3205_sup__inf__distrib2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),X) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),X)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_3206_Un__empty__left,axiom,
    ! [B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat),B) = B ).

% Un_empty_left
tff(fact_3207_Un__empty__left,axiom,
    ! [B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat),B) = B ).

% Un_empty_left
tff(fact_3208_Un__empty__left,axiom,
    ! [B: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,bot_bot_set_bool),B) = B ).

% Un_empty_left
tff(fact_3209_Un__empty__left,axiom,
    ! [B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),B) = B ).

% Un_empty_left
tff(fact_3210_Un__empty__left,axiom,
    ! [B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),B) = B ).

% Un_empty_left
tff(fact_3211_Un__empty__right,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),bot_bo228742789529271731at_nat) = A ).

% Un_empty_right
tff(fact_3212_Un__empty__right,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),bot_bo2099793752762293965at_nat) = A ).

% Un_empty_right
tff(fact_3213_Un__empty__right,axiom,
    ! [A: set_bool] : aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),bot_bot_set_bool) = A ).

% Un_empty_right
tff(fact_3214_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_3215_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_3216_foldl__un__empty__eq,axiom,
    ! [I: set_Pr4329608150637261639at_nat,Ww: list_s4242580248156217677at_nat] : foldl_4541198706405875041at_nat(sup_su5525570899277871387at_nat,I,Ww) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,I),foldl_4541198706405875041at_nat(sup_su5525570899277871387at_nat,bot_bo228742789529271731at_nat,Ww)) ).

% foldl_un_empty_eq
tff(fact_3217_foldl__un__empty__eq,axiom,
    ! [I: set_Pr1261947904930325089at_nat,Ww: list_s1210847774152347623at_nat] : foldl_5307762784932486881at_nat(sup_su6327502436637775413at_nat,I,Ww) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,I),foldl_5307762784932486881at_nat(sup_su6327502436637775413at_nat,bot_bo2099793752762293965at_nat,Ww)) ).

% foldl_un_empty_eq
tff(fact_3218_foldl__un__empty__eq,axiom,
    ! [I: set_bool,Ww: list_set_bool] : foldl_3150261563131881825t_bool(sup_sup_set_bool,I,Ww) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,I),foldl_3150261563131881825t_bool(sup_sup_set_bool,bot_bot_set_bool,Ww)) ).

% foldl_un_empty_eq
tff(fact_3219_foldl__un__empty__eq,axiom,
    ! [I: set_nat,Ww: list_set_nat] : foldl_4988731653086973103et_nat(sup_sup_set_nat,I,Ww) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,I),foldl_4988731653086973103et_nat(sup_sup_set_nat,bot_bot_set_nat,Ww)) ).

% foldl_un_empty_eq
tff(fact_3220_foldl__un__empty__eq,axiom,
    ! [I: set_int,Ww: list_set_int] : foldl_6819690284573351271et_int(sup_sup_set_int,I,Ww) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,I),foldl_6819690284573351271et_int(sup_sup_set_int,bot_bot_set_int,Ww)) ).

% foldl_un_empty_eq
tff(fact_3221_Un__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,C2)
     => ( ord_le1268244103169919719at_nat(B,D)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,C2),D)) ) ) ).

% Un_mono
tff(fact_3222_Un__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ord_less_eq_set_nat(A,C2)
     => ( ord_less_eq_set_nat(B,D)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),D)) ) ) ).

% Un_mono
tff(fact_3223_Un__least,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,C2)
     => ( ord_le1268244103169919719at_nat(B,C2)
       => ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B),C2) ) ) ).

% Un_least
tff(fact_3224_Un__least,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,C2)
     => ( ord_less_eq_set_nat(B,C2)
       => ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B),C2) ) ) ).

% Un_least
tff(fact_3225_Un__upper1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% Un_upper1
tff(fact_3226_Un__upper1,axiom,
    ! [A: set_nat,B: set_nat] : ord_less_eq_set_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% Un_upper1
tff(fact_3227_Un__upper2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(B,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)) ).

% Un_upper2
tff(fact_3228_Un__upper2,axiom,
    ! [B: set_nat,A: set_nat] : ord_less_eq_set_nat(B,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% Un_upper2
tff(fact_3229_Un__absorb1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% Un_absorb1
tff(fact_3230_Un__absorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% Un_absorb1
tff(fact_3231_Un__absorb2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(B,A)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = A ) ) ).

% Un_absorb2
tff(fact_3232_Un__absorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ord_less_eq_set_nat(B,A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% Un_absorb2
tff(fact_3233_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(C2,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B))
     => ~ ! [A6: set_Pr4329608150637261639at_nat] :
            ( ord_le1268244103169919719at_nat(A6,A)
           => ! [B6: set_Pr4329608150637261639at_nat] :
                ( ord_le1268244103169919719at_nat(B6,B)
               => ( C2 != aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A6),B6) ) ) ) ) ).

% subset_UnE
tff(fact_3234_subset__UnE,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(C2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B))
     => ~ ! [A6: set_nat] :
            ( ord_less_eq_set_nat(A6,A)
           => ! [B6: set_nat] :
                ( ord_less_eq_set_nat(B6,B)
               => ( C2 != aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A6),B6) ) ) ) ) ).

% subset_UnE
tff(fact_3235_subset__Un__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
    <=> ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = B ) ) ).

% subset_Un_eq
tff(fact_3236_subset__Un__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% subset_Un_eq
tff(fact_3237_Un__Int__assoc__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C2)) )
    <=> pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,C2),A)) ) ).

% Un_Int_assoc_eq
tff(fact_3238_Un__Int__assoc__eq,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) )
    <=> ord_le1268244103169919719at_nat(C2,A) ) ).

% Un_Int_assoc_eq
tff(fact_3239_Un__Int__assoc__eq,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C2) = 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,B),C2)) )
    <=> ord_less_eq_set_nat(C2,A) ) ).

% Un_Int_assoc_eq
tff(fact_3240_set__diff__diff__left,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) ).

% set_diff_diff_left
tff(fact_3241_set__diff__diff__left,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ).

% set_diff_diff_left
tff(fact_3242_Un__Diff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B)),C2) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),C2)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),C2)) ).

% Un_Diff
tff(fact_3243_Un__Diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),C2)) ).

% Un_Diff
tff(fact_3244_Diff__Un,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),C2)) ).

% Diff_Un
tff(fact_3245_Diff__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),C2)) ).

% Diff_Un
tff(fact_3246_Diff__Un,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),C2)) ).

% Diff_Un
tff(fact_3247_Diff__Int,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),C2)) ).

% Diff_Int
tff(fact_3248_Diff__Int,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,B),C2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),C2)) ).

% Diff_Int
tff(fact_3249_Diff__Int,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),C2)) ).

% Diff_Int
tff(fact_3250_Int__Diff__Un,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)) = A ).

% Int_Diff_Un
tff(fact_3251_Int__Diff__Un,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)) = A ).

% Int_Diff_Un
tff(fact_3252_Int__Diff__Un,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)) = A ).

% Int_Diff_Un
tff(fact_3253_Un__Diff__Int,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) = A ).

% Un_Diff_Int
tff(fact_3254_Un__Diff__Int,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,A),B)) = A ).

% Un_Diff_Int
tff(fact_3255_Un__Diff__Int,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) = A ).

% Un_Diff_Int
tff(fact_3256_in__set__butlastD,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(butlas2561766218336428473at_nat(Xs)))
     => member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs)) ) ).

% in_set_butlastD
tff(fact_3257_in__set__butlastD,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(butlas5569151904373679443at_nat(Xs))))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs))) ) ).

% in_set_butlastD
tff(fact_3258_in__set__butlastD,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(butlas4412598605973335360t_bool(Xs)))
     => member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs)) ) ).

% in_set_butlastD
tff(fact_3259_in__set__butlastD,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(butlast_int(Xs)))
     => member_int(X,set_int2(Xs)) ) ).

% in_set_butlastD
tff(fact_3260_in__set__butlastD,axiom,
    ! [X: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(butlast_nat(Xs)))
     => member_nat(X,set_nat2(Xs)) ) ).

% in_set_butlastD
tff(fact_3261_in__set__butlastD,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(butlas3012047794866324995n_assn(Xs)))
     => member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs)) ) ).

% in_set_butlastD
tff(fact_3262_distinct__butlast,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,Xs))
     => pp(aa_list_int_bool(distinct_int,butlast_int(Xs))) ) ).

% distinct_butlast
tff(fact_3263_distinct__butlast,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,Xs))
     => pp(aa_list_nat_bool(distinct_nat,butlast_nat(Xs))) ) ).

% distinct_butlast
tff(fact_3264_butlast__update_H,axiom,
    ! [L: list_nat,I: nat,X: nat] : list_update_nat(butlast_nat(L),I,X) = butlast_nat(list_update_nat(L,I,X)) ).

% butlast_update'
tff(fact_3265_butlast__update_H,axiom,
    ! [L: list_c,I: nat,X: c] : list_update_c(butlast_c(L),I,X) = butlast_c(list_update_c(L,I,X)) ).

% butlast_update'
tff(fact_3266_butlast__update_H,axiom,
    ! [L: list_d,I: nat,X: d] : list_update_d(butlast_d(L),I,X) = butlast_d(list_update_d(L,I,X)) ).

% butlast_update'
tff(fact_3267_sorted__rev__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),rev_nat(Xs)))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_nat(Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),J2)),aa_nat_nat(nth_nat(Xs),I4))) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_3268_sorted__rev__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),rev_int(Xs)))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),J2)),aa_nat_int(nth_int(Xs),I4))) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_3269_sorted__rev__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),rev_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),J)),aa_nat_nat(nth_nat(Xs),I))) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_3270_sorted__rev__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),rev_int(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),J)),aa_nat_int(nth_int(Xs),I))) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_3271_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_Code_integer] :
      ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,Xs)
     => sorted710888440204495920nteger(ord_le3102999989581377725nteger,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_3272_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_num] :
      ( sorted_wrt_num(ord_less_num,Xs)
     => sorted_wrt_num(ord_less_eq_num,Xs) ) ).

% strict_sorted_imp_sorted
tff(fact_3273_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Xs))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs)) ) ).

% strict_sorted_imp_sorted
tff(fact_3274_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Xs))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs)) ) ).

% strict_sorted_imp_sorted
tff(fact_3275_strict__sorted__equal,axiom,
    ! [Xs: list_Code_integer,Ys: list_Code_integer] :
      ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,Xs)
     => ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,Ys)
       => ( ( set_Code_integer2(Ys) = set_Code_integer2(Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_3276_strict__sorted__equal,axiom,
    ! [Xs: list_num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_num,Xs)
     => ( sorted_wrt_num(ord_less_num,Ys)
       => ( ( set_num2(Ys) = set_num2(Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_3277_strict__sorted__equal,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Xs))
     => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ys))
       => ( ( set_nat2(Ys) = set_nat2(Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_3278_strict__sorted__equal,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Xs))
     => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Ys))
       => ( ( set_int2(Ys) = set_int2(Xs) )
         => ( Ys = Xs ) ) ) ) ).

% strict_sorted_equal
tff(fact_3279_sorted__wrt01,axiom,
    ! [Xs: list_c,P: fun_c_fun_c_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_c(Xs)),one_one_nat))
     => pp(aa_list_c_bool(sorted_wrt_c(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3280_sorted__wrt01,axiom,
    ! [Xs: list_d,P: fun_d_fun_d_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_d(Xs)),one_one_nat))
     => pp(aa_list_d_bool(sorted_wrt_d(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3281_sorted__wrt01,axiom,
    ! [Xs: list_nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat))
     => pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3282_sorted__wrt01,axiom,
    ! [Xs: list_int,P: fun_int_fun_int_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat))
     => pp(aa_list_int_bool(sorted_wrt_int(P),Xs)) ) ).

% sorted_wrt01
tff(fact_3283_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)))) ).

% distrib_inf_le
tff(fact_3284_distrib__inf__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),Z)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3285_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)))) ).

% distrib_inf_le
tff(fact_3286_distrib__inf__le,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Y)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),Z)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3287_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_3288_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)))) ).

% distrib_inf_le
tff(fact_3289_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)))) ).

% distrib_inf_le
tff(fact_3290_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)))) ).

% distrib_sup_le
tff(fact_3291_distrib__sup__le,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,Y),Z)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(inf_in7913087082777306421at_nat,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Y)),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3292_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : pp(aa_assn_bool(ord_less_eq_assn(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)))) ).

% distrib_sup_le
tff(fact_3293_distrib__sup__le,axiom,
    ! [X: filter_nat,Y: filter_nat,Z: filter_nat] : ord_le2510731241096832064er_nat(aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,Y),Z)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(inf_inf_filter_nat,aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Y)),aa_fil4124504356706564990er_nat(aa_fil1282661720248192731er_nat(sup_sup_filter_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3294_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z))) ).

% distrib_sup_le
tff(fact_3295_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)))) ).

% distrib_sup_le
tff(fact_3296_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z)))) ).

% distrib_sup_le
tff(fact_3297_sorted__remove1,axiom,
    ! [Xs: list_nat,A2: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),remove1_nat(A2,Xs))) ) ).

% sorted_remove1
tff(fact_3298_sorted__remove1,axiom,
    ! [Xs: list_int,A2: int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),remove1_int(A2,Xs))) ) ).

% sorted_remove1
tff(fact_3299_insert__is__Un,axiom,
    ! [A2: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat] : insert9069300056098147895at_nat(A2,A) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A2,bot_bo228742789529271731at_nat)),A) ).

% insert_is_Un
tff(fact_3300_insert__is__Un,axiom,
    ! [A2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat] : insert8211810215607154385at_nat(A2,A) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A2,bot_bo2099793752762293965at_nat)),A) ).

% insert_is_Un
tff(fact_3301_insert__is__Un,axiom,
    ! [A2: bool,A: set_bool] : insert_bool2(A2,A) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,insert_bool2(A2,bot_bot_set_bool)),A) ).

% insert_is_Un
tff(fact_3302_insert__is__Un,axiom,
    ! [A2: nat,A: set_nat] : insert_nat2(A2,A) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat2(A2,bot_bot_set_nat)),A) ).

% insert_is_Un
tff(fact_3303_insert__is__Un,axiom,
    ! [A2: int,A: set_int] : insert_int2(A2,A) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int2(A2,bot_bot_set_int)),A) ).

% insert_is_Un
tff(fact_3304_Un__singleton__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
    <=> ( ( ( A = bot_bo228742789529271731at_nat )
          & ( B = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B = bot_bo228742789529271731at_nat ) )
        | ( ( A = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_3305_Un__singleton__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
    <=> ( ( ( A = bot_bo2099793752762293965at_nat )
          & ( B = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B = bot_bo2099793752762293965at_nat ) )
        | ( ( A = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_3306_Un__singleton__iff,axiom,
    ! [A: set_bool,B: set_bool,X: bool] :
      ( ( aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),B) = insert_bool2(X,bot_bot_set_bool) )
    <=> ( ( ( A = bot_bot_set_bool )
          & ( B = insert_bool2(X,bot_bot_set_bool) ) )
        | ( ( A = insert_bool2(X,bot_bot_set_bool) )
          & ( B = bot_bot_set_bool ) )
        | ( ( A = insert_bool2(X,bot_bot_set_bool) )
          & ( B = insert_bool2(X,bot_bot_set_bool) ) ) ) ) ).

% Un_singleton_iff
tff(fact_3307_Un__singleton__iff,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = insert_nat2(X,bot_bot_set_nat) )
    <=> ( ( ( A = bot_bot_set_nat )
          & ( B = insert_nat2(X,bot_bot_set_nat) ) )
        | ( ( A = insert_nat2(X,bot_bot_set_nat) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A = insert_nat2(X,bot_bot_set_nat) )
          & ( B = insert_nat2(X,bot_bot_set_nat) ) ) ) ) ).

% Un_singleton_iff
tff(fact_3308_Un__singleton__iff,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = insert_int2(X,bot_bot_set_int) )
    <=> ( ( ( A = bot_bot_set_int )
          & ( B = insert_int2(X,bot_bot_set_int) ) )
        | ( ( A = insert_int2(X,bot_bot_set_int) )
          & ( B = bot_bot_set_int ) )
        | ( ( A = insert_int2(X,bot_bot_set_int) )
          & ( B = insert_int2(X,bot_bot_set_int) ) ) ) ) ).

% Un_singleton_iff
tff(fact_3309_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),B) )
    <=> ( ( ( A = bot_bo228742789529271731at_nat )
          & ( B = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) )
        | ( ( A = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B = bot_bo228742789529271731at_nat ) )
        | ( ( A = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) )
          & ( B = insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_3310_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),B) )
    <=> ( ( ( A = bot_bo2099793752762293965at_nat )
          & ( B = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) )
        | ( ( A = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B = bot_bo2099793752762293965at_nat ) )
        | ( ( A = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) )
          & ( B = insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_3311_singleton__Un__iff,axiom,
    ! [X: bool,A: set_bool,B: set_bool] :
      ( ( insert_bool2(X,bot_bot_set_bool) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,A),B) )
    <=> ( ( ( A = bot_bot_set_bool )
          & ( B = insert_bool2(X,bot_bot_set_bool) ) )
        | ( ( A = insert_bool2(X,bot_bot_set_bool) )
          & ( B = bot_bot_set_bool ) )
        | ( ( A = insert_bool2(X,bot_bot_set_bool) )
          & ( B = insert_bool2(X,bot_bot_set_bool) ) ) ) ) ).

% singleton_Un_iff
tff(fact_3312_singleton__Un__iff,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( insert_nat2(X,bot_bot_set_nat) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( ( A = bot_bot_set_nat )
          & ( B = insert_nat2(X,bot_bot_set_nat) ) )
        | ( ( A = insert_nat2(X,bot_bot_set_nat) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A = insert_nat2(X,bot_bot_set_nat) )
          & ( B = insert_nat2(X,bot_bot_set_nat) ) ) ) ) ).

% singleton_Un_iff
tff(fact_3313_singleton__Un__iff,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ( insert_int2(X,bot_bot_set_int) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
    <=> ( ( ( A = bot_bot_set_int )
          & ( B = insert_int2(X,bot_bot_set_int) ) )
        | ( ( A = insert_int2(X,bot_bot_set_int) )
          & ( B = bot_bot_set_int ) )
        | ( ( A = insert_int2(X,bot_bot_set_int) )
          & ( B = insert_int2(X,bot_bot_set_int) ) ) ) ) ).

% singleton_Un_iff
tff(fact_3314_Diff__partition,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(A,B)
     => ( aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,A),aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,B),A)) = B ) ) ).

% Diff_partition
tff(fact_3315_Diff__partition,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ord_less_eq_set_nat(A,B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B),A)) = B ) ) ).

% Diff_partition
tff(fact_3316_Diff__subset__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ord_le1268244103169919719at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(minus_3314409938677909166at_nat,A),B),C2)
    <=> ord_le1268244103169919719at_nat(A,aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,B),C2)) ) ).

% Diff_subset_conv
tff(fact_3317_Diff__subset__conv,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ord_less_eq_set_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A),B),C2)
    <=> ord_less_eq_set_nat(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C2)) ) ).

% Diff_subset_conv
tff(fact_3318_strict__sorted__iff,axiom,
    ! [L: list_Code_integer] :
      ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,L)
    <=> ( sorted710888440204495920nteger(ord_le3102999989581377725nteger,L)
        & distin1543349897113766820nteger(L) ) ) ).

% strict_sorted_iff
tff(fact_3319_strict__sorted__iff,axiom,
    ! [L: list_num] :
      ( sorted_wrt_num(ord_less_num,L)
    <=> ( sorted_wrt_num(ord_less_eq_num,L)
        & distinct_num(L) ) ) ).

% strict_sorted_iff
tff(fact_3320_strict__sorted__iff,axiom,
    ! [L: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),L))
    <=> ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
        & pp(aa_list_nat_bool(distinct_nat,L)) ) ) ).

% strict_sorted_iff
tff(fact_3321_strict__sorted__iff,axiom,
    ! [L: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),L))
    <=> ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
        & pp(aa_list_int_bool(distinct_int,L)) ) ) ).

% strict_sorted_iff
tff(fact_3322_sorted01,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat))
     => pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs)) ) ).

% sorted01
tff(fact_3323_sorted01,axiom,
    ! [Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat))
     => pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs)) ) ).

% sorted01
tff(fact_3324_sorted__distinct__set__unique,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => ( pp(aa_list_nat_bool(distinct_nat,Xs))
       => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Ys))
         => ( pp(aa_list_nat_bool(distinct_nat,Ys))
           => ( ( set_nat2(Xs) = set_nat2(Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_3325_sorted__distinct__set__unique,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => ( pp(aa_list_int_bool(distinct_int,Xs))
       => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Ys))
         => ( pp(aa_list_int_bool(distinct_int,Ys))
           => ( ( set_int2(Xs) = set_int2(Ys) )
             => ( Xs = Ys ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_3326_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_c_fun_c_bool,Xs: list_c] :
      ( pp(aa_list_c_bool(sorted_wrt_c(P),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_c(Xs)))
           => pp(aa_c_bool(aa_c_fun_c_bool(P,nth_c(Xs,I4)),nth_c(Xs,J2))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3327_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_d_fun_d_bool,Xs: list_d] :
      ( pp(aa_list_d_bool(sorted_wrt_d(P),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_d(Xs)))
           => pp(aa_d_bool(aa_d_fun_d_bool(P,nth_d(Xs,I4)),nth_d(Xs,J2))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3328_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_nat_fun_nat_bool,Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_nat(Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),J2))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3329_sorted__wrt__iff__nth__less,axiom,
    ! [P: fun_int_fun_int_bool,Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(P),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),J2))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_3330_sorted__wrt__nth__less,axiom,
    ! [P: fun_c_fun_c_bool,Xs: list_c,I: nat,J: nat] :
      ( pp(aa_list_c_bool(sorted_wrt_c(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_c(Xs)))
         => pp(aa_c_bool(aa_c_fun_c_bool(P,nth_c(Xs,I)),nth_c(Xs,J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3331_sorted__wrt__nth__less,axiom,
    ! [P: fun_d_fun_d_bool,Xs: list_d,I: nat,J: nat] :
      ( pp(aa_list_d_bool(sorted_wrt_d(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_d(Xs)))
         => pp(aa_d_bool(aa_d_fun_d_bool(P,nth_d(Xs,I)),nth_d(Xs,J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3332_sorted__wrt__nth__less,axiom,
    ! [P: fun_nat_fun_nat_bool,Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3333_sorted__wrt__nth__less,axiom,
    ! [P: fun_int_fun_int_bool,Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(P),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_3334_nth__butlast,axiom,
    ! [N: nat,Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(butlast_c(Xs))))
     => ( nth_c(butlast_c(Xs),N) = nth_c(Xs,N) ) ) ).

% nth_butlast
tff(fact_3335_nth__butlast,axiom,
    ! [N: nat,Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(butlast_d(Xs))))
     => ( nth_d(butlast_d(Xs),N) = nth_d(Xs,N) ) ) ).

% nth_butlast
tff(fact_3336_nth__butlast,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(butlast_nat(Xs))))
     => ( aa_nat_nat(nth_nat(butlast_nat(Xs)),N) = aa_nat_nat(nth_nat(Xs),N) ) ) ).

% nth_butlast
tff(fact_3337_nth__butlast,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(butlast_int(Xs))))
     => ( aa_nat_int(nth_int(butlast_int(Xs)),N) = aa_nat_int(nth_int(Xs),N) ) ) ).

% nth_butlast
tff(fact_3338_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_nat(Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),J2))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3339_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),J2))) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_3340_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I)),aa_nat_nat(nth_nat(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3341_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),I)),aa_nat_int(nth_int(Xs),J))) ) ) ) ).

% sorted_nth_mono
tff(fact_3342_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_nat(Xs)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),J2))) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_3343_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
    <=> ! [I4: nat,J2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),J2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),size_size_list_int(Xs)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),J2))) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_3344_mergesort__remdups__correct,axiom,
    ! [L: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,merges1656613366846331073ps_nat(L)))
      & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),merges1656613366846331073ps_nat(L)))
      & ( set_nat2(merges1656613366846331073ps_nat(L)) = set_nat2(L) ) ) ).

% mergesort_remdups_correct
tff(fact_3345_mergesort__remdups__correct,axiom,
    ! [L: list_int] :
      ( pp(aa_list_int_bool(distinct_int,merges1654122896337280797ps_int(L)))
      & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),merges1654122896337280797ps_int(L)))
      & ( set_int2(merges1654122896337280797ps_int(L)) = set_int2(L) ) ) ).

% mergesort_remdups_correct
tff(fact_3346_times__assn__raw_Osimps,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Q: fun_Pr6001797050664311790t_bool,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(P),Q),produc7507926704131184380et_nat(H2,As2)))
    <=> ? [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 )
          & pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H2,As1)))
          & pp(aa_Pro2991721470636411351t_bool(Q,produc7507926704131184380et_nat(H2,As22))) ) ) ).

% times_assn_raw.simps
tff(fact_3347_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(X),Xa),Xb))
      <=> pp(Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( pp(Y)
            <=> ~ ? [As1: set_nat,As22: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                    & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H,As1)))
                    & pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H,As22))) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_3348_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(X),Xa),Xb))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ~ ? [As12: set_nat,As23: set_nat] :
                  ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                  & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                  & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H,As12)))
                  & pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H,As23))) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_3349_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(X),Xa),Xb))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ? [As13: set_nat,As24: set_nat] :
                ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H,As13)))
                & pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H,As24))) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_3350_nat__descend__induct,axiom,
    ! [N: nat,P: fun_nat_bool,M2: nat] :
      ( ! [K: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K))
         => pp(aa_nat_bool(P,K)) )
     => ( ! [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
           => ( ! [I6: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),I6))
                 => pp(aa_nat_bool(P,I6)) )
             => pp(aa_nat_bool(P,K)) ) )
       => pp(aa_nat_bool(P,M2)) ) ) ).

% nat_descend_induct
tff(fact_3351_merge__correct,axiom,
    ! [L12: list_nat,L23: list_nat] :
      ( ( pp(aa_list_nat_bool(distinct_nat,L12))
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L12)) )
     => ( ( pp(aa_list_nat_bool(distinct_nat,L23))
          & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L23)) )
       => ( pp(aa_list_nat_bool(distinct_nat,merge_nat(L12,L23)))
          & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),merge_nat(L12,L23)))
          & ( set_nat2(merge_nat(L12,L23)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,set_nat2(L12)),set_nat2(L23)) ) ) ) ) ).

% merge_correct
tff(fact_3352_merge__correct,axiom,
    ! [L12: list_int,L23: list_int] :
      ( ( pp(aa_list_int_bool(distinct_int,L12))
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L12)) )
     => ( ( pp(aa_list_int_bool(distinct_int,L23))
          & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L23)) )
       => ( pp(aa_list_int_bool(distinct_int,merge_int(L12,L23)))
          & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),merge_int(L12,L23)))
          & ( set_int2(merge_int(L12,L23)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,set_int2(L12)),set_int2(L23)) ) ) ) ) ).

% merge_correct
tff(fact_3353_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_c,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_c(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
         => ( nth_c(slice_c(From,To,Xs),I) = nth_c(Xs,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3354_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_d,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_d(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
         => ( nth_d(slice_d(From,To,Xs),I) = nth_d(Xs,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3355_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_nat(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
         => ( aa_nat_nat(nth_nat(slice_nat(From,To,Xs)),I) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3356_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,From),To))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,To),size_size_list_int(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,To),From)))
         => ( aa_nat_int(nth_int(slice_int(From,To,Xs)),I) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_3357_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),rev_nat(Xs)))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I4))),aa_nat_nat(nth_nat(Xs),I4))) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_3358_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),rev_int(Xs)))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_int(Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I4))),aa_nat_int(nth_int(Xs),I4))) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_3359_add__right__cancel,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(B2,A2) = plus_p7104986032573967614at_nat(C,A2) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_3360_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_3361_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_3362_add__right__cancel,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_3363_add__left__cancel,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(A2,B2) = plus_p7104986032573967614at_nat(A2,C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_3364_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_3365_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_3366_add__left__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_3367_mod__or__dist,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),H2))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
        | pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),H2)) ) ) ).

% mod_or_dist
tff(fact_3368_merge__pure__or,axiom,
    ! [A2: bool,B2: bool] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,pure_assn(A2)),pure_assn(B2)) = pure_assn(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,A2),B2)) ).

% merge_pure_or
tff(fact_3369_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ).

% add_le_cancel_right
tff(fact_3370_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_cancel_right
tff(fact_3371_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_cancel_right
tff(fact_3372_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ).

% add_le_cancel_left
tff(fact_3373_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_cancel_left
tff(fact_3374_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_cancel_left
tff(fact_3375_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
    <=> pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ).

% add_less_cancel_left
tff(fact_3376_add__less__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),A2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_3377_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_3378_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_cancel_left
tff(fact_3379_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
    <=> pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ).

% add_less_cancel_right
tff(fact_3380_add__less__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_3381_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_3382_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_cancel_right
tff(fact_3383_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2),B2) = A2 ).

% add_diff_cancel
tff(fact_3384_add__diff__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_3385_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A2,B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_3386_diff__add__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_3387_add__diff__cancel__left,axiom,
    ! [C: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(plus_p7104986032573967614at_nat(C,A2),plus_p7104986032573967614at_nat(C,B2)) = minus_4286766774447292334at_nat(A2,B2) ).

% add_diff_cancel_left
tff(fact_3388_add__diff__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: 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),B2)) = minus_minus_rat(A2,B2) ).

% add_diff_cancel_left
tff(fact_3389_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ).

% add_diff_cancel_left
tff(fact_3390_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) ).

% add_diff_cancel_left
tff(fact_3391_add__diff__cancel__left_H,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(plus_p7104986032573967614at_nat(A2,B2),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3392_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B2: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3393_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3394_add__diff__cancel__left_H,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_3395_add__diff__cancel__right,axiom,
    ! [A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(plus_p7104986032573967614at_nat(A2,C),plus_p7104986032573967614at_nat(B2,C)) = minus_4286766774447292334at_nat(A2,B2) ).

% add_diff_cancel_right
tff(fact_3396_add__diff__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: 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,B2),C)) = minus_minus_rat(A2,B2) ).

% add_diff_cancel_right
tff(fact_3397_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ).

% add_diff_cancel_right
tff(fact_3398_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) ).

% add_diff_cancel_right
tff(fact_3399_add__diff__cancel__right_H,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(plus_p7104986032573967614at_nat(A2,B2),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3400_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B2: rat] : minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3401_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3402_add__diff__cancel__right_H,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_3403_lessI,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,N))) ).

% lessI
tff(fact_3404_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) ) ).

% Suc_mono
tff(fact_3405_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_less_eq
tff(fact_3406_nat__add__left__cancel__less,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% nat_add_left_cancel_less
tff(fact_3407_le__add__diff__inverse,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(B2),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),minus_minus_rat(A2,B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3408_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3409_le__add__diff__inverse,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_3410_le__add__diff__inverse2,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(B2),A2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A2,B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3411_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3412_le__add__diff__inverse2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_3413_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A2)),B2) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_3414_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),aa_nat_nat(suc,A2)) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_3415_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H2,bot_bot_set_nat)))
        | pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),produc7507926704131184380et_nat(H2,bot_bot_set_nat))) ) ) ).

% mod_h_bot_iff(7)
tff(fact_3416_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),M2))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_3417_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ~ ! [Q4: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),Q4)) ) ).

% less_natE
tff(fact_3418_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)))) ).

% less_add_Suc1
tff(fact_3419_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),I)))) ).

% less_add_Suc2
tff(fact_3420_less__iff__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> ? [K4: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K4)) ) ).

% less_iff_Suc_add
tff(fact_3421_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ? [K: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K)) ) ).

% less_imp_Suc_add
tff(fact_3422_add__right__imp__eq,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(B2,A2) = plus_p7104986032573967614at_nat(C,A2) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_3423_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_3424_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_3425_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_3426_add__left__imp__eq,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(A2,B2) = plus_p7104986032573967614at_nat(A2,C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_3427_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_3428_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_3429_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_3430_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(B2,plus_p7104986032573967614at_nat(A2,C)) = plus_p7104986032573967614at_nat(A2,plus_p7104986032573967614at_nat(B2,C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3431_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),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,B2),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3432_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3433_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_3434_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(A2,B2) = plus_p7104986032573967614at_nat(B2,A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3435_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3436_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3437_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_3438_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A2) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_3439_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_3440_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_3441_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_3442_add_Oassoc,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(plus_p7104986032573967614at_nat(A2,B2),C) = plus_p7104986032573967614at_nat(A2,plus_p7104986032573967614at_nat(B2,C)) ).

% add.assoc
tff(fact_3443_add_Oassoc,axiom,
    ! [A2: rat,B2: 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),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% add.assoc
tff(fact_3444_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% add.assoc
tff(fact_3445_add_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% add.assoc
tff(fact_3446_group__cancel_Oadd2,axiom,
    ! [B: multis2468970476368604999at_nat,K3: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( B = plus_p7104986032573967614at_nat(K3,B2) )
     => ( plus_p7104986032573967614at_nat(A2,B) = plus_p7104986032573967614at_nat(K3,plus_p7104986032573967614at_nat(A2,B2)) ) ) ).

% group_cancel.add2
tff(fact_3447_group__cancel_Oadd2,axiom,
    ! [B: rat,K3: rat,B2: rat,A2: rat] :
      ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),B2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),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),B2)) ) ) ).

% group_cancel.add2
tff(fact_3448_group__cancel_Oadd2,axiom,
    ! [B: nat,K3: nat,B2: nat,A2: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_3449_group__cancel_Oadd2,axiom,
    ! [B: int,K3: int,B2: int,A2: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,K3),B2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_3450_group__cancel_Oadd1,axiom,
    ! [A: multis2468970476368604999at_nat,K3: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A = plus_p7104986032573967614at_nat(K3,A2) )
     => ( plus_p7104986032573967614at_nat(A,B2) = plus_p7104986032573967614at_nat(K3,plus_p7104986032573967614at_nat(A2,B2)) ) ) ).

% group_cancel.add1
tff(fact_3451_group__cancel_Oadd1,axiom,
    ! [A: rat,K3: rat,A2: rat,B2: 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),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3452_group__cancel_Oadd1,axiom,
    ! [A: nat,K3: nat,A2: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),A2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3453_group__cancel_Oadd1,axiom,
    ! [A: int,K3: int,A2: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K3),A2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K3),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_3454_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I = J )
        & ( K3 = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3455_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I = J )
        & ( K3 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3456_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( I = J )
        & ( K3 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_3457_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
     => pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3458_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3459_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_imp_le_right
tff(fact_3460_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
     => pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3461_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3462_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ).

% add_le_imp_le_left
tff(fact_3463_le__iff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
    <=> ? [C5: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C5) ) ).

% le_iff_add
tff(fact_3464_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => pp(aa_rat_bool(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,B2),C))) ) ).

% add_right_mono
tff(fact_3465_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))) ) ).

% add_right_mono
tff(fact_3466_add__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))) ) ).

% add_right_mono
tff(fact_3467_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ~ ! [C4: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C4) ) ).

% less_eqE
tff(fact_3468_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => pp(aa_rat_bool(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),B2))) ) ).

% add_left_mono
tff(fact_3469_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))) ) ).

% add_left_mono
tff(fact_3470_add__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% add_left_mono
tff(fact_3471_add__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),D2))
       => pp(aa_rat_bool(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,B2),D2))) ) ) ).

% add_mono
tff(fact_3472_add__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_mono
tff(fact_3473_add__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_mono
tff(fact_3474_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( pp(aa_rat_bool(ord_less_eq_rat(I),J))
        & pp(aa_rat_bool(ord_less_eq_rat(K3),L)) )
     => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3475_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3476_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K3),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_3477_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(ord_less_eq_rat(K3),L)) )
     => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3478_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3479_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K3),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_3480_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( pp(aa_rat_bool(ord_less_eq_rat(I),J))
        & ( K3 = L ) )
     => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3481_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & ( K3 = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3482_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & ( K3 = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_3483_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( pp(aa_rat_bool(ord_less_rat(I),J))
        & pp(aa_rat_bool(ord_less_rat(K3),L)) )
     => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3484_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K3),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I),K3)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3485_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3486_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K3),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_3487_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( ( I = J )
        & pp(aa_rat_bool(ord_less_rat(K3),L)) )
     => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3488_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( ( I = J )
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K3),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I),K3)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3489_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( ( I = J )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3490_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( ( I = J )
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K3),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_3491_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( pp(aa_rat_bool(ord_less_rat(I),J))
        & ( K3 = L ) )
     => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3492_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J))
        & ( K3 = L ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I),K3)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3493_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( K3 = L ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3494_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & ( K3 = L ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_3495_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(C),D2))
       => pp(aa_rat_bool(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,B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3496_add__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3497_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3498_add__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_strict_mono
tff(fact_3499_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => pp(aa_rat_bool(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),B2))) ) ).

% add_strict_left_mono
tff(fact_3500_add__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),A2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),B2))) ) ).

% add_strict_left_mono
tff(fact_3501_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))) ) ).

% add_strict_left_mono
tff(fact_3502_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% add_strict_left_mono
tff(fact_3503_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => pp(aa_rat_bool(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,B2),C))) ) ).

% add_strict_right_mono
tff(fact_3504_add__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C))) ) ).

% add_strict_right_mono
tff(fact_3505_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))) ) ).

% add_strict_right_mono
tff(fact_3506_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))) ) ).

% add_strict_right_mono
tff(fact_3507_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
     => pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3508_add__less__imp__less__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),A2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),B2)))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3509_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3510_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_imp_less_left
tff(fact_3511_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
     => pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3512_add__less__imp__less__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C)))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3513_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3514_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ).

% add_less_imp_less_right
tff(fact_3515_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_3516_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: 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),B2)),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,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_3517_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_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,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_3518_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_3519_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: 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,B2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_3520_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_3521_comm__semiring__class_Odistrib,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_3522_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: 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),B2)),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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_3523_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_3524_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_3525_distrib__left,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)) ).

% distrib_left
tff(fact_3526_distrib__left,axiom,
    ! [A2: rat,B2: 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,B2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)) ).

% distrib_left
tff(fact_3527_distrib__left,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) ).

% distrib_left
tff(fact_3528_distrib__left,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% distrib_left
tff(fact_3529_distrib__right,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ).

% distrib_right
tff(fact_3530_distrib__right,axiom,
    ! [A2: rat,B2: 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),B2)),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,B2),C)) ).

% distrib_right
tff(fact_3531_distrib__right,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_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,B2),C)) ).

% distrib_right
tff(fact_3532_distrib__right,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_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,B2),C)) ).

% distrib_right
tff(fact_3533_combine__common__factor,axiom,
    ! [A2: code_integer,E: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),C)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),E)),C) ).

% combine_common_factor
tff(fact_3534_combine__common__factor,axiom,
    ! [A2: rat,E: rat,B2: 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),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),E)),C) ).

% combine_common_factor
tff(fact_3535_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),E)),C) ).

% combine_common_factor
tff(fact_3536_combine__common__factor,axiom,
    ! [A2: int,E: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),E)),C) ).

% combine_common_factor
tff(fact_3537_group__cancel_Osub1,axiom,
    ! [A: rat,K3: rat,A2: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),A2) )
     => ( minus_minus_rat(A,B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K3),minus_minus_rat(A2,B2)) ) ) ).

% group_cancel.sub1
tff(fact_3538_group__cancel_Osub1,axiom,
    ! [A: int,K3: int,A2: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,K3),A2) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K3),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_3539_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat(A2,B2) = C )
    <=> ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2) ) ) ).

% diff_eq_eq
tff(fact_3540_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = C )
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2) ) ) ).

% diff_eq_eq
tff(fact_3541_eq__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( A2 = minus_minus_rat(C,B2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = C ) ) ).

% eq_diff_eq
tff(fact_3542_eq__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = C ) ) ).

% eq_diff_eq
tff(fact_3543_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2),C) ).

% add_diff_eq
tff(fact_3544_add__diff__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ).

% add_diff_eq
tff(fact_3545_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C: rat] : minus_minus_rat(A2,minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C),B2) ).

% diff_diff_eq2
tff(fact_3546_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B2) ).

% diff_diff_eq2
tff(fact_3547_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,minus_minus_rat(A2,B2)),C) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C),B2) ).

% diff_add_eq
tff(fact_3548_diff__add__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B2) ).

% diff_add_eq
tff(fact_3549_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C: rat] : minus_minus_rat(A2,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) = minus_minus_rat(minus_minus_rat(A2,C),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3550_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_3551_add__implies__diff,axiom,
    ! [C: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(C,B2) = A2 )
     => ( C = minus_4286766774447292334at_nat(A2,B2) ) ) ).

% add_implies_diff
tff(fact_3552_add__implies__diff,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2) = A2 )
     => ( C = minus_minus_rat(A2,B2) ) ) ).

% add_implies_diff
tff(fact_3553_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2) = A2 )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ) ) ).

% add_implies_diff
tff(fact_3554_add__implies__diff,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2) = A2 )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) ) ) ).

% add_implies_diff
tff(fact_3555_diff__diff__eq,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(minus_4286766774447292334at_nat(A2,B2),C) = minus_4286766774447292334at_nat(A2,plus_p7104986032573967614at_nat(B2,C)) ).

% diff_diff_eq
tff(fact_3556_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] : minus_minus_rat(minus_minus_rat(A2,B2),C) = minus_minus_rat(A2,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% diff_diff_eq
tff(fact_3557_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% diff_diff_eq
tff(fact_3558_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% diff_diff_eq
tff(fact_3559_star__or__dist1,axiom,
    ! [A: assn,B: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C2)) ).

% star_or_dist1
tff(fact_3560_star__or__dist2,axiom,
    ! [C2: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),A)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),B)) ).

% star_or_dist2
tff(fact_3561_Nat_OlessE,axiom,
    ! [I: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K3))
     => ( ( K3 != aa_nat_nat(suc,I) )
       => ~ ! [J3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J3))
             => ( K3 != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_3562_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_lessD
tff(fact_3563_Suc__lessE,axiom,
    ! [I: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K3))
     => ~ ! [J3: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J3))
           => ( K3 != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_3564_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( ( aa_nat_nat(suc,M2) != N )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),N)) ) ) ).

% Suc_lessI
tff(fact_3565_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( M2 = N ) ) ) ).

% less_SucE
tff(fact_3566_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N))) ) ).

% less_SucI
tff(fact_3567_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_3568_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( M2 = N ) ) ) ).

% less_Suc_eq
tff(fact_3569_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2))) ) ).

% not_less_eq
tff(fact_3570_All__less__Suc,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,N))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_3571_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),M2))
    <=> ? [M4: nat] :
          ( ( M2 = aa_nat_nat(suc,M4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M4)) ) ) ).

% Suc_less_eq2
tff(fact_3572_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
       => ( M2 = N ) ) ) ).

% less_antisym
tff(fact_3573_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_less_SucD
tff(fact_3574_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),K3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),K3)) ) ) ).

% less_trans_Suc
tff(fact_3575_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I5),aa_nat_nat(suc,I5)))
       => ( ! [I5: nat,J3: nat,K: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),J3))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J3),K))
               => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I5),J3))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,J3),K))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I5),K)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_3576_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( ! [I5: nat] :
            ( ( J = aa_nat_nat(suc,I5) )
           => pp(aa_nat_bool(P,I5)) )
       => ( ! [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),J))
             => ( pp(aa_nat_bool(P,aa_nat_nat(suc,I5)))
               => pp(aa_nat_bool(P,I5)) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_3577_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
      <=> ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
tff(fact_3578_add__lessD1,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K3))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),K3)) ) ).

% add_lessD1
tff(fact_3579_add__less__mono,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),L))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).

% add_less_mono
tff(fact_3580_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).

% not_add_less1
tff(fact_3581_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).

% not_add_less2
tff(fact_3582_add__less__mono1,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K3))) ) ).

% add_less_mono1
tff(fact_3583_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M2))) ) ).

% trans_less_add1
tff(fact_3584_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),J))) ) ).

% trans_less_add2
tff(fact_3585_less__add__eq__less,axiom,
    ! [K3: nat,L: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),L))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K3),N) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% less_add_eq_less
tff(fact_3586_ent__disjI2__direct,axiom,
    ! [B: assn,A: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B))) ).

% ent_disjI2_direct
tff(fact_3587_ent__disjI1__direct,axiom,
    ! [A: assn,B: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B))) ).

% ent_disjI1_direct
tff(fact_3588_ent__disjI2_H,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),C2))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2))) ) ).

% ent_disjI2'
tff(fact_3589_ent__disjI1_H,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),B))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2))) ) ).

% ent_disjI1'
tff(fact_3590_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),R))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,Q),R)) ) ).

% ent_disjI2
tff(fact_3591_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q)),R))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),R)) ) ).

% ent_disjI1
tff(fact_3592_ent__disjE,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),C2))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),C2))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C2)) ) ) ).

% ent_disjE
tff(fact_3593_min__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),ord_min_rat(Y,Z)) = ord_min_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)) ).

% min_add_distrib_right
tff(fact_3594_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_min_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% min_add_distrib_right
tff(fact_3595_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% min_add_distrib_right
tff(fact_3596_min__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,ord_min_rat(X,Y)),Z) = ord_min_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)) ).

% min_add_distrib_left
tff(fact_3597_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_min_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_min_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% min_add_distrib_left
tff(fact_3598_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% min_add_distrib_left
tff(fact_3599_entt__disjE,axiom,
    ! [A: assn,M5: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(A),M5))
     => ( pp(aa_assn_bool(entailst(B),M5))
       => pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),M5)) ) ) ).

% entt_disjE
tff(fact_3600_entt__disjD1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C2))
     => pp(aa_assn_bool(entailst(A),C2)) ) ).

% entt_disjD1
tff(fact_3601_entt__disjD2,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C2))
     => pp(aa_assn_bool(entailst(B),C2)) ) ).

% entt_disjD2
tff(fact_3602_entt__disjI1_H,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( pp(aa_assn_bool(entailst(A),B))
     => pp(aa_assn_bool(entailst(A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2))) ) ).

% entt_disjI1'
tff(fact_3603_entt__disjI2_H,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(A),C2))
     => pp(aa_assn_bool(entailst(A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C2))) ) ).

% entt_disjI2'
tff(fact_3604_entt__disjI1__direct,axiom,
    ! [A: assn,B: assn] : pp(aa_assn_bool(entailst(A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B))) ).

% entt_disjI1_direct
tff(fact_3605_entt__disjI2__direct,axiom,
    ! [B: assn,A: assn] : pp(aa_assn_bool(entailst(B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B))) ).

% entt_disjI2_direct
tff(fact_3606_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% norm_assertion_simps(6)
tff(fact_3607_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).

% norm_assertion_simps(5)
tff(fact_3608_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( pp(aa_rat_bool(ord_less_eq_rat(I),J))
        & pp(aa_rat_bool(ord_less_rat(K3),L)) )
     => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3609_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,I),J))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K3),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I),K3)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3610_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3611_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K3),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_3612_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K3: rat,L: rat] :
      ( ( pp(aa_rat_bool(ord_less_rat(I),J))
        & pp(aa_rat_bool(ord_less_eq_rat(K3),L)) )
     => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3613_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J: code_integer,K3: code_integer,L: code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,I),J))
        & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,K3),L)) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(I),K3)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3614_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K3: nat,L: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),L)) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3615_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K3: int,L: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K3),L)) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L))) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_3616_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(C),D2))
       => pp(aa_rat_bool(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,B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3617_add__le__less__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3618_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3619_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_le_less_mono
tff(fact_3620_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),D2))
       => pp(aa_rat_bool(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,B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3621_add__less__le__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3622_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3623_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D2))) ) ) ).

% add_less_le_mono
tff(fact_3624_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(minus_minus_rat(A2,B2)),C))
    <=> pp(aa_rat_bool(ord_less_eq_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2))) ) ).

% diff_le_eq
tff(fact_3625_diff__le__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% diff_le_eq
tff(fact_3626_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),minus_minus_rat(C,B2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),C)) ) ).

% le_diff_eq
tff(fact_3627_le__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C)) ) ).

% le_diff_eq
tff(fact_3628_add__le__imp__le__diff,axiom,
    ! [I: rat,K3: rat,N: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),N))
     => pp(aa_rat_bool(ord_less_eq_rat(I),minus_minus_rat(N,K3))) ) ).

% add_le_imp_le_diff
tff(fact_3629_add__le__imp__le__diff,axiom,
    ! [I: nat,K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K3))) ) ).

% add_le_imp_le_diff
tff(fact_3630_add__le__imp__le__diff,axiom,
    ! [I: int,K3: int,N: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),N))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K3))) ) ).

% add_le_imp_le_diff
tff(fact_3631_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),A2) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_3632_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K3: rat,N: rat,J: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),N))
     => ( pp(aa_rat_bool(ord_less_eq_rat(N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K3)))
       => ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K3)),N))
         => ( pp(aa_rat_bool(ord_less_eq_rat(N),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K3)))
           => pp(aa_rat_bool(ord_less_eq_rat(minus_minus_rat(N,K3)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3633_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K3: nat,N: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K3)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K3)))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K3)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3634_add__le__add__imp__diff__le,axiom,
    ! [I: int,K3: int,N: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),N))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K3)))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K3)),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K3)))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K3)),J)) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3635_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A2))) ) ).

% le_add_diff
tff(fact_3636_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),B2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3637_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3638_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3639_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3640_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3641_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3642_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3643_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2) = C )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3644_less__add__one,axiom,
    ! [A2: rat] : pp(aa_rat_bool(ord_less_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),one_one_rat))) ).

% less_add_one
tff(fact_3645_less__add__one,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),one_one_Code_integer))) ).

% less_add_one
tff(fact_3646_less__add__one,axiom,
    ! [A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat))) ).

% less_add_one
tff(fact_3647_less__add__one,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int))) ).

% less_add_one
tff(fact_3648_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => pp(aa_rat_bool(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,B2),one_one_rat))) ) ).

% add_mono1
tff(fact_3649_add__mono1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),one_one_Code_integer)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),one_one_Code_integer))) ) ).

% add_mono1
tff(fact_3650_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),one_one_nat))) ) ).

% add_mono1
tff(fact_3651_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int))) ) ).

% add_mono1
tff(fact_3652_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(minus_minus_rat(A2,B2)),C))
    <=> pp(aa_rat_bool(ord_less_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2))) ) ).

% diff_less_eq
tff(fact_3653_diff__less__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),B2))) ) ).

% diff_less_eq
tff(fact_3654_diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))) ) ).

% diff_less_eq
tff(fact_3655_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),minus_minus_rat(C,B2)))
    <=> pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),C)) ) ).

% less_diff_eq
tff(fact_3656_less__diff__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C)) ) ).

% less_diff_eq
tff(fact_3657_less__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C)) ) ).

% less_diff_eq
tff(fact_3658_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),minus_minus_rat(A2,B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3659_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3660_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3661_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B2: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3662_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(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(plus_p5714425477246183910nteger(X),Y)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),Y)) ).

% square_diff_square_factored
tff(fact_3663_square__diff__square__factored,axiom,
    ! [X: rat,Y: 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)),minus_minus_rat(X,Y)) ).

% square_diff_square_factored
tff(fact_3664_square__diff__square__factored,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_3665_eq__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),D2) )
    <=> ( C = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_3666_eq__add__iff2,axiom,
    ! [A2: rat,E: rat,C: rat,B2: 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),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D2) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B2,A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_3667_eq__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_3668_eq__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),D2) )
    <=> ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),E)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_3669_eq__add__iff1,axiom,
    ! [A2: rat,E: rat,C: rat,B2: 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),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A2,B2)),E)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_3670_eq__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_3671_lift__Suc__mono__less,axiom,
    ! [F3: fun_nat_Code_integer,N: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,N5)),aa_nat_Code_integer(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,N)),aa_nat_Code_integer(F3,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_3672_lift__Suc__mono__less,axiom,
    ! [F3: fun_nat_assn,N: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,N5)),aa_nat_assn(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,N)),aa_nat_assn(F3,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_3673_lift__Suc__mono__less,axiom,
    ! [F3: fun_nat_num,N: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,N5)),aa_nat_num(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,N)),aa_nat_num(F3,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_3674_lift__Suc__mono__less,axiom,
    ! [F3: fun_nat_nat,N: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,N5)),aa_nat_nat(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,N)),aa_nat_nat(F3,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_3675_lift__Suc__mono__less,axiom,
    ! [F3: fun_nat_int,N: nat,N6: nat] :
      ( ! [N5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F3,N5)),aa_nat_int(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),N6))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F3,N)),aa_nat_int(F3,N6))) ) ) ).

% lift_Suc_mono_less
tff(fact_3676_lift__Suc__mono__less__iff,axiom,
    ! [F3: fun_nat_Code_integer,N: nat,M2: nat] :
      ( ! [N5: nat] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,N5)),aa_nat_Code_integer(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(F3,N)),aa_nat_Code_integer(F3,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3677_lift__Suc__mono__less__iff,axiom,
    ! [F3: fun_nat_assn,N: nat,M2: nat] :
      ( ! [N5: nat] : pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,N5)),aa_nat_assn(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_assn_bool(ord_less_assn(aa_nat_assn(F3,N)),aa_nat_assn(F3,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3678_lift__Suc__mono__less__iff,axiom,
    ! [F3: fun_nat_num,N: nat,M2: nat] :
      ( ! [N5: nat] : pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,N5)),aa_nat_num(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,aa_nat_num(F3,N)),aa_nat_num(F3,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3679_lift__Suc__mono__less__iff,axiom,
    ! [F3: fun_nat_nat,N: nat,M2: nat] :
      ( ! [N5: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,N5)),aa_nat_nat(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,N)),aa_nat_nat(F3,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3680_lift__Suc__mono__less__iff,axiom,
    ! [F3: fun_nat_int,N: nat,M2: nat] :
      ( ! [N5: nat] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F3,N5)),aa_nat_int(F3,aa_nat_nat(suc,N5))))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(F3,N)),aa_nat_int(F3,M2)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_3681_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)) ) ).

% Suc_leI
tff(fact_3682_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_le_eq
tff(fact_3683_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,I))
       => ( ! [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N5))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),J))
               => ( pp(aa_nat_bool(P,N5))
                 => pp(aa_nat_bool(P,aa_nat_nat(suc,N5))) ) ) )
         => pp(aa_nat_bool(P,J)) ) ) ) ).

% dec_induct
tff(fact_3684_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(P,J))
       => ( ! [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),N5))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),J))
               => ( pp(aa_nat_bool(P,aa_nat_nat(suc,N5)))
                 => pp(aa_nat_bool(P,N5)) ) ) )
         => pp(aa_nat_bool(P,I)) ) ) ) ).

% inc_induct
tff(fact_3685_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,M2)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_le_lessD
tff(fact_3686_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,M2)))
      <=> ( N = M2 ) ) ) ).

% le_less_Suc_eq
tff(fact_3687_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% less_Suc_eq_le
tff(fact_3688_less__eq__Suc__le,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),M2)) ) ).

% less_eq_Suc_le
tff(fact_3689_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N))) ) ).

% le_imp_less_Suc
tff(fact_3690_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(suc,A2))) )
    <=> ( B2 = aa_nat_nat(suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_3691_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(suc,A2))) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_3692_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N) ) ) ).

% Suc_diff_Suc
tff(fact_3693_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_nat_nat(suc,M2))) ).

% diff_less_Suc
tff(fact_3694_mono__nat__linear__lb,axiom,
    ! [F3: fun_nat_nat,M2: nat,K3: nat] :
      ( ! [M6: nat,N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N5))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,M6)),aa_nat_nat(F3,N5))) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F3,M2)),K3)),aa_nat_nat(F3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),K3)))) ) ).

% mono_nat_linear_lb
tff(fact_3695_Suc__mult__less__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K3)),N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_3696_less__diff__conv,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),J)) ) ).

% less_diff_conv
tff(fact_3697_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)) = M2 ) ) ).

% add_diff_inverse_nat
tff(fact_3698_mlex__bound,axiom,
    ! [A2: nat,A: nat,B2: nat,N7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),N7))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N7))) ) ) ).

% mlex_bound
tff(fact_3699_mlex__fst__decrI,axiom,
    ! [A2: nat,A5: nat,B2: nat,N7: nat,B5: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),A5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),N7))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B5),N7))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N7)),B5))) ) ) ) ).

% mlex_fst_decrI
tff(fact_3700_mlex__snd__decrI,axiom,
    ! [A2: nat,A5: nat,B2: nat,B5: nat,N7: nat] :
      ( ( A2 = A5 )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N7)),B5))) ) ) ).

% mlex_snd_decrI
tff(fact_3701_mlex__leI,axiom,
    ! [A2: nat,A5: nat,B2: nat,B5: nat,N7: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),A5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),B5))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N7)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),N7)),B5))) ) ) ).

% mlex_leI
tff(fact_3702_le__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),D2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),A2)),E)),D2))) ) ).

% le_add_iff2
tff(fact_3703_le__add__iff2,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(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),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B2,A2)),E)),D2))) ) ).

% le_add_iff2
tff(fact_3704_le__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E)),D2))) ) ).

% le_add_iff2
tff(fact_3705_le__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),D2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),E)),C)),D2)) ) ).

% le_add_iff1
tff(fact_3706_le__add__iff1,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(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),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A2,B2)),E)),C)),D2)) ) ).

% le_add_iff1
tff(fact_3707_le__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E)),C)),D2)) ) ).

% le_add_iff1
tff(fact_3708_less__add__iff2,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(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),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D2)))
    <=> pp(aa_rat_bool(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(B2,A2)),E)),D2))) ) ).

% less_add_iff2
tff(fact_3709_less__add__iff2,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),D2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),A2)),E)),D2))) ) ).

% less_add_iff2
tff(fact_3710_less__add__iff2,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E)),D2))) ) ).

% less_add_iff2
tff(fact_3711_less__add__iff1,axiom,
    ! [A2: rat,E: rat,C: rat,B2: rat,D2: rat] :
      ( pp(aa_rat_bool(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),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D2)))
    <=> pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,minus_minus_rat(A2,B2)),E)),C)),D2)) ) ).

% less_add_iff1
tff(fact_3712_less__add__iff1,axiom,
    ! [A2: code_integer,E: code_integer,C: code_integer,B2: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),E)),C)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),E)),D2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),E)),C)),D2)) ) ).

% less_add_iff1
tff(fact_3713_less__add__iff1,axiom,
    ! [A2: int,E: int,C: int,B2: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E)),C)),D2)) ) ).

% less_add_iff1
tff(fact_3714_square__diff__one__factored,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),one_one_Code_integer)),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(X),one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_3715_square__diff__one__factored,axiom,
    ! [X: 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)),minus_minus_rat(X,one_one_rat)) ).

% square_diff_one_factored
tff(fact_3716_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_3717_less__diff__conv2,axiom,
    ! [K3: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K3)),I))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3))) ) ) ).

% less_diff_conv2
tff(fact_3718_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),M2))
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_3719_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Xs))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_nat(Xs)))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(Xs),I4)),aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,I4)))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_3720_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Xs))
    <=> ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I4)),size_size_list_int(Xs)))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(Xs),I4)),aa_nat_int(nth_int(Xs),aa_nat_nat(suc,I4)))) ) ) ).

% sorted_iff_nth_Suc
tff(fact_3721_rev__nth,axiom,
    ! [N: nat,Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(Xs)))
     => ( nth_c(rev_c(Xs),N) = nth_c(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),aa_nat_nat(suc,N))) ) ) ).

% rev_nth
tff(fact_3722_rev__nth,axiom,
    ! [N: nat,Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(Xs)))
     => ( nth_d(rev_d(Xs),N) = nth_d(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),aa_nat_nat(suc,N))) ) ) ).

% rev_nth
tff(fact_3723_rev__nth,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
     => ( aa_nat_nat(nth_nat(rev_nat(Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),aa_nat_nat(suc,N))) ) ) ).

% rev_nth
tff(fact_3724_rev__nth,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
     => ( aa_nat_int(nth_int(rev_int(Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),aa_nat_nat(suc,N))) ) ) ).

% rev_nth
tff(fact_3725_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M2)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_3726_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_3727_discrete,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),one_one_Code_integer)),B2)) ) ).

% discrete
tff(fact_3728_discrete,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B2)) ) ).

% discrete
tff(fact_3729_discrete,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),B2)) ) ).

% discrete
tff(fact_3730_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_Code_integer,X: code_integer] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_s3445333598471063425nteger(L)))
         => ( sorted710888440204495920nteger(ord_le3102999989581377725nteger,L)
           => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(nth_Code_integer(L),I)),X))
             => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),aa_nat_Code_integer(nth_Code_integer(L),J)))
               => ~ ! [K: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),J))
                       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(nth_Code_integer(L),K)),X))
                         => ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),aa_nat_Code_integer(nth_Code_integer(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_3731_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_num,X: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_num(L)))
         => ( sorted_wrt_num(ord_less_eq_num,L)
           => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),I)),X))
             => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),aa_nat_num(nth_num(L),J)))
               => ~ ! [K: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),J))
                       => ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_eq_num,aa_nat_num(nth_num(L),K)),X))
                         => ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),aa_nat_num(nth_num(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_3732_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_nat(L)))
         => ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),L))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),I)),X))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(nth_nat(L),J)))
               => ~ ! [K: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),J))
                       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(nth_nat(L),K)),X))
                         => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),aa_nat_nat(nth_nat(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_3733_sorted__in__between,axiom,
    ! [I: nat,J: nat,L: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J),size_size_list_int(L)))
         => ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),L))
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),I)),X))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(nth_int(L),J)))
               => ~ ! [K: nat] :
                      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K))
                     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),J))
                       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(nth_int(L),K)),X))
                         => ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),aa_nat_int(nth_int(L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_3734_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(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_3735_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_3736_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3737_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_3738_crossproduct__noteq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
    <=> ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2)) != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),D2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_3739_crossproduct__noteq,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( ( A2 != B2 )
        & ( 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,B2),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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_3740_crossproduct__noteq,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
    <=> ( aa_nat_nat(aa_nat_fun_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,B2),D2)) != aa_nat_nat(aa_nat_fun_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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_3741_crossproduct__noteq,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( ( ( A2 != B2 )
        & ( C != D2 ) )
    <=> ( aa_int_int(aa_int_fun_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,B2),D2)) != aa_int_int(aa_int_fun_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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_3742_nth__step__trancl,axiom,
    ! [Xs: list_s1210847774152347623at_nat,R: set_Pr4329608150637261639at_nat,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_s8736152011456118867at_nat(Xs)),one_one_nat)))
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(nth_se5679702624988424552at_nat(Xs,aa_nat_nat(suc,N5)),nth_se5679702624988424552at_nat(Xs,N5)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_s8736152011456118867at_nat(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(nth_se5679702624988424552at_nat(Xs,N),nth_se5679702624988424552at_nat(Xs,M2)),transi2703068831062848130at_nat(R)) ) ) ) ).

% nth_step_trancl
tff(fact_3743_nth__step__trancl,axiom,
    ! [Xs: list_c,R: set_Product_prod_c_c,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_c(Xs)),one_one_nat)))
         => member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,nth_c(Xs,aa_nat_nat(suc,N5))),nth_c(Xs,N5)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_c(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
         => member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,nth_c(Xs,N)),nth_c(Xs,M2)),transitive_trancl_c(R)) ) ) ) ).

% nth_step_trancl
tff(fact_3744_nth__step__trancl,axiom,
    ! [Xs: list_d,R: set_Product_prod_d_d,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_d(Xs)),one_one_nat)))
         => member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,nth_d(Xs,aa_nat_nat(suc,N5))),nth_d(Xs,N5)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_d(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
         => member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,nth_d(Xs,N)),nth_d(Xs,M2)),transitive_trancl_d(R)) ) ) ) ).

% nth_step_trancl
tff(fact_3745_nth__step__trancl,axiom,
    ! [Xs: list_int,R: set_Pr958786334691620121nt_int,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_int(Xs)),one_one_nat)))
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(suc,N5))),aa_nat_int(nth_int(Xs),N5)),R) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_int(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),N)),aa_nat_int(nth_int(Xs),M2)),transi6261509568448316235cl_int(R)) ) ) ) ).

% nth_step_trancl
tff(fact_3746_nth__step__trancl,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,N: nat,M2: nat] :
      ( ! [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,size_size_list_nat(Xs)),one_one_nat)))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),aa_nat_nat(suc,N5))),aa_nat_nat(nth_nat(Xs),N5))),R)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),size_size_list_nat(Xs)))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
         => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),N)),aa_nat_nat(nth_nat(Xs),M2))),transi6264000038957366511cl_nat(R))) ) ) ) ).

% nth_step_trancl
tff(fact_3747_dbl__dec__def,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(neg_nu7757733837767384882nteger,X) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),X)),one_one_Code_integer) ).

% dbl_dec_def
tff(fact_3748_dbl__dec__def,axiom,
    ! [X: rat] : neg_nu3179335615603231917ec_rat(X) = minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X),one_one_rat) ).

% dbl_dec_def
tff(fact_3749_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_3750_le__zero__eq,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_3751_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_3752_add_Oright__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),zero_z3403309356797280102nteger) = A2 ).

% add.right_neutral
tff(fact_3753_add_Oright__neutral,axiom,
    ! [A2: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(A2,zero_z1048942125864253310at_nat) = A2 ).

% add.right_neutral
tff(fact_3754_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_3755_add_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.right_neutral
tff(fact_3756_add_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.right_neutral
tff(fact_3757_double__zero__sym,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
tff(fact_3758_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_3759_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_3760_add__cancel__left__left,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2) = A2 )
    <=> ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
tff(fact_3761_add__cancel__left__left,axiom,
    ! [B2: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(B2,A2) = A2 )
    <=> ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
tff(fact_3762_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_3763_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_3764_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_3765_add__cancel__left__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2) = A2 )
    <=> ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
tff(fact_3766_add__cancel__left__right,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(A2,B2) = A2 )
    <=> ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
tff(fact_3767_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_3768_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_3769_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_3770_add__cancel__right__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2) )
    <=> ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
tff(fact_3771_add__cancel__right__left,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A2 = plus_p7104986032573967614at_nat(B2,A2) )
    <=> ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
tff(fact_3772_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_3773_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_3774_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_3775_add__cancel__right__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2) )
    <=> ( B2 = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
tff(fact_3776_add__cancel__right__right,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] :
      ( ( A2 = plus_p7104986032573967614at_nat(A2,B2) )
    <=> ( B2 = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
tff(fact_3777_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_3778_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_3779_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_3780_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_3781_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_3782_add__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),A2) = A2 ).

% add_0
tff(fact_3783_add__0,axiom,
    ! [A2: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(zero_z1048942125864253310at_nat,A2) = A2 ).

% add_0
tff(fact_3784_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_3785_add__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% add_0
tff(fact_3786_add__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add_0
tff(fact_3787_mult__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_3788_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: 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,B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_3789_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: 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,B2),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_3790_mult__cancel__right,axiom,
    ! [A2: int,C: int,B2: 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,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_3791_mult__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_3792_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: 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),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_3793_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: 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),B2) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_3794_mult__cancel__left,axiom,
    ! [C: int,A2: int,B2: 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),B2) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_3795_mult__eq__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) = zero_z3403309356797280102nteger )
    <=> ( ( A2 = zero_z3403309356797280102nteger )
        | ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_3796_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_3797_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_3798_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_3799_mult__zero__right,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_3800_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_3801_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_3802_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_3803_mult__zero__left,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A2) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_3804_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_3805_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_3806_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_3807_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),A2) = zero_z3403309356797280102nteger ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3808_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A2,A2) = zero_z1048942125864253310at_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3809_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] : minus_minus_rat(A2,A2) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3810_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),A2) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3811_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3812_diff__zero,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),zero_z3403309356797280102nteger) = A2 ).

% diff_zero
tff(fact_3813_diff__zero,axiom,
    ! [A2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A2,zero_z1048942125864253310at_nat) = A2 ).

% diff_zero
tff(fact_3814_diff__zero,axiom,
    ! [A2: rat] : minus_minus_rat(A2,zero_zero_rat) = A2 ).

% diff_zero
tff(fact_3815_diff__zero,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),zero_zero_nat) = A2 ).

% diff_zero
tff(fact_3816_diff__zero,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_zero
tff(fact_3817_zero__diff,axiom,
    ! [A2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(zero_z1048942125864253310at_nat,A2) = zero_z1048942125864253310at_nat ).

% zero_diff
tff(fact_3818_zero__diff,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A2) = zero_zero_nat ).

% zero_diff
tff(fact_3819_diff__0__right,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),zero_z3403309356797280102nteger) = A2 ).

% diff_0_right
tff(fact_3820_diff__0__right,axiom,
    ! [A2: rat] : minus_minus_rat(A2,zero_zero_rat) = A2 ).

% diff_0_right
tff(fact_3821_diff__0__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_0_right
tff(fact_3822_diff__self,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),A2) = zero_z3403309356797280102nteger ).

% diff_self
tff(fact_3823_diff__self,axiom,
    ! [A2: rat] : minus_minus_rat(A2,A2) = zero_zero_rat ).

% diff_self
tff(fact_3824_diff__self,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% diff_self
tff(fact_3825_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_nat_zero_code
tff(fact_3826_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% neq0_conv
tff(fact_3827_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_3828_trancl__single,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : transi2703068831062848130at_nat(insert9069300056098147895at_nat(produc2922128104949294807at_nat(A2,B2),bot_bo228742789529271731at_nat)) = insert9069300056098147895at_nat(produc2922128104949294807at_nat(A2,B2),bot_bo228742789529271731at_nat) ).

% trancl_single
tff(fact_3829_trancl__single,axiom,
    ! [A2: nat,B2: nat] : transi6264000038957366511cl_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),bot_bo2099793752762293965at_nat)) = insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),bot_bo2099793752762293965at_nat) ).

% trancl_single
tff(fact_3830_add__le__same__cancel1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2)),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% add_le_same_cancel1
tff(fact_3831_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2)),B2))
    <=> pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat)) ) ).

% add_le_same_cancel1
tff(fact_3832_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat)) ) ).

% add_le_same_cancel1
tff(fact_3833_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% add_le_same_cancel1
tff(fact_3834_add__le__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% add_le_same_cancel2
tff(fact_3835_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),B2))
    <=> pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat)) ) ).

% add_le_same_cancel2
tff(fact_3836_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat)) ) ).

% add_le_same_cancel2
tff(fact_3837_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% add_le_same_cancel2
tff(fact_3838_le__add__same__cancel1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)) ) ).

% le_add_same_cancel1
tff(fact_3839_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2)) ) ).

% le_add_same_cancel1
tff(fact_3840_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) ).

% le_add_same_cancel1
tff(fact_3841_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ).

% le_add_same_cancel1
tff(fact_3842_le__add__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)) ) ).

% le_add_same_cancel2
tff(fact_3843_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2)) ) ).

% le_add_same_cancel2
tff(fact_3844_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) ).

% le_add_same_cancel2
tff(fact_3845_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ).

% le_add_same_cancel2
tff(fact_3846_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3847_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3848_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int)) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_3849_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3850_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3851_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2)) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_3852_diff__ge__0__iff__ge,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_3853_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),minus_minus_rat(A2,B2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(B2),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_3854_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% diff_ge_0_iff_ge
tff(fact_3855_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3856_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),minus_minus_rat(A2,B2)))
    <=> pp(aa_rat_bool(ord_less_eq_rat(B2),A2)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3857_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ).

% zero_comp_diff_simps(1)
tff(fact_3858_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)))
    <=> pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3859_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3860_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2)) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_3861_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),A2)),zero_zero_rat))
    <=> pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3862_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),A2)),zero_z3403309356797280102nteger))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3863_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_3864_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2)))
    <=> pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2)) ) ).

% less_add_same_cancel2
tff(fact_3865_less__add__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)) ) ).

% less_add_same_cancel2
tff(fact_3866_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ).

% less_add_same_cancel2
tff(fact_3867_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ).

% less_add_same_cancel2
tff(fact_3868_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)))
    <=> pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2)) ) ).

% less_add_same_cancel1
tff(fact_3869_less__add__same__cancel1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)) ) ).

% less_add_same_cancel1
tff(fact_3870_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ).

% less_add_same_cancel1
tff(fact_3871_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ).

% less_add_same_cancel1
tff(fact_3872_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),B2))
    <=> pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat)) ) ).

% add_less_same_cancel2
tff(fact_3873_add__less__same__cancel2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% add_less_same_cancel2
tff(fact_3874_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ) ).

% add_less_same_cancel2
tff(fact_3875_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% add_less_same_cancel2
tff(fact_3876_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A2)),B2))
    <=> pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat)) ) ).

% add_less_same_cancel1
tff(fact_3877_add__less__same__cancel1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2)),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)) ) ).

% add_less_same_cancel1
tff(fact_3878_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ) ).

% add_less_same_cancel1
tff(fact_3879_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int)) ) ).

% add_less_same_cancel1
tff(fact_3880_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),minus_minus_rat(A2,B2)))
    <=> pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_3881_diff__gt__0__iff__gt,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_3882_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% diff_gt_0_iff_gt
tff(fact_3883_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),minus_minus_rat(A2,B2)))
    <=> pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3884_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3885_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ).

% zero_comp_diff_simps(2)
tff(fact_3886_mult__cancel__left1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B2 = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_3887_mult__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_3888_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_3889_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_3890_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_3891_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_3892_mult__cancel__right1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B2 = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_3893_mult__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_3894_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_3895_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_3896_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_3897_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_3898_diff__add__zero,axiom,
    ! [A2: multis2468970476368604999at_nat,B2: multis2468970476368604999at_nat] : minus_4286766774447292334at_nat(A2,plus_p7104986032573967614at_nat(A2,B2)) = zero_z1048942125864253310at_nat ).

% diff_add_zero
tff(fact_3899_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3900_diff__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_3901_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_3902_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_3903_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat)))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_3904_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N))) ).

% zero_less_Suc
tff(fact_3905_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% add_gr_0
tff(fact_3906_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% zero_less_diff
tff(fact_3907_nat__mult__less__cancel__disj,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_3908_mult__less__cancel2,axiom,
    ! [M2: nat,K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K3)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% mult_less_cancel2
tff(fact_3909_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_3910_less__one,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_3911_slice__complete,axiom,
    ! [Xs: list_c] : slice_c(zero_zero_nat,size_size_list_c(Xs),Xs) = Xs ).

% slice_complete
tff(fact_3912_slice__complete,axiom,
    ! [Xs: list_d] : slice_d(zero_zero_nat,size_size_list_d(Xs),Xs) = Xs ).

% slice_complete
tff(fact_3913_slice__complete,axiom,
    ! [Xs: list_nat] : slice_nat(zero_zero_nat,size_size_list_nat(Xs),Xs) = Xs ).

% slice_complete
tff(fact_3914_slice__complete,axiom,
    ! [Xs: list_int] : slice_int(zero_zero_nat,size_size_list_int(Xs),Xs) = Xs ).

% slice_complete
tff(fact_3915_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_3916_nat__mult__le__cancel__disj,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_3917_mult__le__cancel2,axiom,
    ! [M2: nat,K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K3)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% mult_le_cancel2
tff(fact_3918_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_3919_assn__aci_I8_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% assn_aci(8)
tff(fact_3920_assn__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% assn_aci(7)
tff(fact_3921_assn__aci_I5_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% assn_aci(5)
tff(fact_3922_norm__assertion__simps_I32_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).

% norm_assertion_simps(32)
tff(fact_3923_norm__assertion__simps_I15_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% norm_assertion_simps(15)
tff(fact_3924_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
    <=> ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
tff(fact_3925_zero__reorient,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = X )
    <=> ( X = zero_z1048942125864253310at_nat ) ) ).

% zero_reorient
tff(fact_3926_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_3927_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_3928_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_3929_trancl__mono__mp,axiom,
    ! [U2: set_Pr4329608150637261639at_nat,V: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ord_le1268244103169919719at_nat(U2,V)
     => ( member8757157785044589968at_nat(X,transi2703068831062848130at_nat(U2))
       => member8757157785044589968at_nat(X,transi2703068831062848130at_nat(V)) ) ) ).

% trancl_mono_mp
tff(fact_3930_trancl__mono__mp,axiom,
    ! [U2: set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,U2),V))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),transi6264000038957366511cl_nat(U2)))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),transi6264000038957366511cl_nat(V))) ) ) ).

% trancl_mono_mp
tff(fact_3931_trancl__sub,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,R),transi6264000038957366511cl_nat(R))) ).

% trancl_sub
tff(fact_3932_trancl__sub__insert__trancl,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] : ord_le1268244103169919719at_nat(transi2703068831062848130at_nat(R),transi2703068831062848130at_nat(insert9069300056098147895at_nat(X,R))) ).

% trancl_sub_insert_trancl
tff(fact_3933_trancl__sub__insert__trancl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,transi6264000038957366511cl_nat(R)),transi6264000038957366511cl_nat(insert8211810215607154385at_nat(X,R)))) ).

% trancl_sub_insert_trancl
tff(fact_3934_zero__le,axiom,
    ! [X: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X)) ).

% zero_le
tff(fact_3935_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_3936_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_3937_not__less__zero,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less_zero
tff(fact_3938_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr_zeroI
tff(fact_3939_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_rat_bool(ord_less_rat(zero_zero_rat),zero_zero_rat)) ).

% less_numeral_extra(3)
tff(fact_3940_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger)) ).

% less_numeral_extra(3)
tff(fact_3941_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),zero_zero_nat)) ).

% less_numeral_extra(3)
tff(fact_3942_less__numeral__extra_I3_J,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),zero_zero_int)) ).

% less_numeral_extra(3)
tff(fact_3943_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3944_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(zero_z1048942125864253310at_nat,A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3945_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_3946_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3947_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_3948_add_Ocomm__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),zero_z3403309356797280102nteger) = A2 ).

% add.comm_neutral
tff(fact_3949_add_Ocomm__neutral,axiom,
    ! [A2: multis2468970476368604999at_nat] : plus_p7104986032573967614at_nat(A2,zero_z1048942125864253310at_nat) = A2 ).

% add.comm_neutral
tff(fact_3950_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_3951_add_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.comm_neutral
tff(fact_3952_add_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.comm_neutral
tff(fact_3953_add_Ogroup__left__neutral,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),A2) = A2 ).

% add.group_left_neutral
tff(fact_3954_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_3955_add_Ogroup__left__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add.group_left_neutral
tff(fact_3956_mult__right__cancel,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_3957_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B2: 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,B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_3958_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B2: 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,B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_3959_mult__right__cancel,axiom,
    ! [C: int,A2: int,B2: 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,B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_3960_mult__left__cancel,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_3961_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B2: 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),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_3962_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B2: 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),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_3963_mult__left__cancel,axiom,
    ! [C: int,A2: int,B2: 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),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_3964_no__zero__divisors,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( B2 != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_3965_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_3966_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_3967_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_3968_divisors__zero,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) = zero_z3403309356797280102nteger )
     => ( ( A2 = zero_z3403309356797280102nteger )
        | ( B2 = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_3969_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2) = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_3970_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_3971_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_3972_mult__not__zero,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) != zero_z3403309356797280102nteger )
     => ( ( A2 != zero_z3403309356797280102nteger )
        & ( B2 != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_3973_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2) != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_3974_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_3975_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_3976_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_3977_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_3978_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_3979_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_3980_eq__iff__diff__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = B2 )
    <=> ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2) = zero_z3403309356797280102nteger ) ) ).

% eq_iff_diff_eq_0
tff(fact_3981_eq__iff__diff__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( minus_minus_rat(A2,B2) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3982_eq__iff__diff__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3983_infinite__descent0,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
           => ( ~ pp(aa_nat_bool(P,N5))
             => ? [M3: nat] :
                  ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M3),N5))
                  & ~ pp(aa_nat_bool(P,M3)) ) ) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% infinite_descent0
tff(fact_3984_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_3985_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% less_zeroE
tff(fact_3986_not__less0,axiom,
    ! [N: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),zero_zero_nat)) ).

% not_less0
tff(fact_3987_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_3988_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% gr0I
tff(fact_3989_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat)) ).

% bot_nat_0.extremum_strict
tff(fact_3990_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),C)) != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3991_add__scale__eq__noteq,axiom,
    ! [R2: rat,A2: rat,B2: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A2 = B2 )
          & ( 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,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3992_add__scale__eq__noteq,axiom,
    ! [R2: nat,A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3993_add__scale__eq__noteq,axiom,
    ! [R2: int,A2: int,B2: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A2 = B2 )
          & ( C != D2 ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3994_nat__mult__eq__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N) )
      <=> ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_3995_nat__mult__less__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_3996_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),zero_z3403309356797280102nteger))
       => ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3997_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(X),zero_zero_rat))
     => ( pp(aa_rat_bool(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_3998_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),zero_zero_nat))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_3999_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),zero_zero_int))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_4000_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4001_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),X))
     => ( pp(aa_rat_bool(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_4002_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),X))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),Y))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4003_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_4004_add__nonpos__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% add_nonpos_nonpos
tff(fact_4005_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),zero_zero_rat)) ) ) ).

% add_nonpos_nonpos
tff(fact_4006_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_nonpos_nonpos
tff(fact_4007_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_nonpos_nonpos
tff(fact_4008_add__nonneg__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_4009_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_4010_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_4011_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_nonneg_nonneg
tff(fact_4012_add__increasing2,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C))) ) ) ).

% add_increasing2
tff(fact_4013_add__increasing2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),A2))
       => pp(aa_rat_bool(ord_less_eq_rat(B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_increasing2
tff(fact_4014_add__increasing2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_increasing2
tff(fact_4015_add__increasing2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_increasing2
tff(fact_4016_add__decreasing2,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_4017_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_4018_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_4019_add__decreasing2,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B2)) ) ) ).

% add_decreasing2
tff(fact_4020_add__increasing,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C))) ) ) ).

% add_increasing
tff(fact_4021_add__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),C))
       => pp(aa_rat_bool(ord_less_eq_rat(B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_increasing
tff(fact_4022_add__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_increasing
tff(fact_4023_add__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_increasing
tff(fact_4024_add__decreasing,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C)),B2)) ) ) ).

% add_decreasing
tff(fact_4025_add__decreasing,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),B2))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C)),B2)) ) ) ).

% add_decreasing
tff(fact_4026_add__decreasing,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),B2)) ) ) ).

% add_decreasing
tff(fact_4027_add__decreasing,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B2)) ) ) ).

% add_decreasing
tff(fact_4028_mult__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_4029_mult__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),D2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
           => pp(aa_rat_bool(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,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_4030_mult__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_4031_mult__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_mono
tff(fact_4032_mult__mono_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_4033_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),D2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
           => pp(aa_rat_bool(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,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_4034_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_4035_mult__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_mono'
tff(fact_4036_zero__le__square,axiom,
    ! [A2: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2))) ).

% zero_le_square
tff(fact_4037_zero__le__square,axiom,
    ! [A2: rat] : pp(aa_rat_bool(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_4038_zero__le__square,axiom,
    ! [A2: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2))) ).

% zero_le_square
tff(fact_4039_split__mult__pos__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2))) ) ).

% split_mult_pos_le
tff(fact_4040_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
          & pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat)) ) )
     => pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2))) ) ).

% split_mult_pos_le
tff(fact_4041_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ).

% split_mult_pos_le
tff(fact_4042_mult__left__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_4043_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(B2),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
       => pp(aa_rat_bool(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),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_4044_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_left_mono_neg
tff(fact_4045_mult__nonpos__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_4046_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_4047_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_nonpos_nonpos
tff(fact_4048_mult__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))) ) ) ).

% mult_left_mono
tff(fact_4049_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(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),B2))) ) ) ).

% mult_left_mono
tff(fact_4050_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% mult_left_mono
tff(fact_4051_mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_left_mono
tff(fact_4052_mult__right__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))) ) ) ).

% mult_right_mono_neg
tff(fact_4053_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(B2),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
       => pp(aa_rat_bool(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,B2),C))) ) ) ).

% mult_right_mono_neg
tff(fact_4054_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_right_mono_neg
tff(fact_4055_mult__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))) ) ) ).

% mult_right_mono
tff(fact_4056_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(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,B2),C))) ) ) ).

% mult_right_mono
tff(fact_4057_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))) ) ) ).

% mult_right_mono
tff(fact_4058_mult__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_right_mono
tff(fact_4059_mult__le__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_4060_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
          & pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_4061_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) ) ) ).

% mult_le_0_iff
tff(fact_4062_split__mult__neg__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)) ) )
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger)) ) ).

% split_mult_neg_le
tff(fact_4063_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
          & pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2)) ) )
     => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat)) ) ).

% split_mult_neg_le
tff(fact_4064_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2)) ) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ).

% split_mult_neg_le
tff(fact_4065_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) ) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ).

% split_mult_neg_le
tff(fact_4066_mult__nonneg__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4067_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4068_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4069_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_nonneg_nonneg
tff(fact_4070_mult__nonneg__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4071_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4072_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4073_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos
tff(fact_4074_mult__nonpos__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4075_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4076_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4077_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_nonpos_nonneg
tff(fact_4078_mult__nonneg__nonpos2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)),zero_z3403309356797280102nteger)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4079_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A2)),zero_zero_rat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4080_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4081_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),zero_zero_int)) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4082_zero__le__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger)) ) ) ) ).

% zero_le_mult_iff
tff(fact_4083_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
          & pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat)) ) ) ) ).

% zero_le_mult_iff
tff(fact_4084_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int)) ) ) ) ).

% zero_le_mult_iff
tff(fact_4085_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4086_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(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),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4087_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4088_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4089_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% zero_less_one_class.zero_le_one
tff(fact_4090_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),one_one_rat)) ).

% zero_less_one_class.zero_le_one
tff(fact_4091_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one_class.zero_le_one
tff(fact_4092_zero__less__one__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% zero_less_one_class.zero_le_one
tff(fact_4093_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_4094_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),one_one_rat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_4095_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_4096_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_4097_not__one__le__zero,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).

% not_one_le_zero
tff(fact_4098_not__one__le__zero,axiom,
    ~ pp(aa_rat_bool(ord_less_eq_rat(one_one_rat),zero_zero_rat)) ).

% not_one_le_zero
tff(fact_4099_not__one__le__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).

% not_one_le_zero
tff(fact_4100_not__one__le__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).

% not_one_le_zero
tff(fact_4101_le__iff__diff__le__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),zero_z3403309356797280102nteger)) ) ).

% le_iff_diff_le_0
tff(fact_4102_le__iff__diff__le__0,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
    <=> pp(aa_rat_bool(ord_less_eq_rat(minus_minus_rat(A2,B2)),zero_zero_rat)) ) ).

% le_iff_diff_le_0
tff(fact_4103_le__iff__diff__le__0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),zero_zero_int)) ) ).

% le_iff_diff_le_0
tff(fact_4104_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_rat(X),zero_zero_rat))
        | pp(aa_rat_bool(ord_less_rat(Y),zero_zero_rat)) ) ) ).

% add_less_zeroD
tff(fact_4105_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),Y)),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger))
        | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),zero_z3403309356797280102nteger)) ) ) ).

% add_less_zeroD
tff(fact_4106_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),zero_zero_int))
        | pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),zero_zero_int)) ) ) ).

% add_less_zeroD
tff(fact_4107_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),zero_zero_rat)) ) ) ).

% add_neg_neg
tff(fact_4108_add__neg__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% add_neg_neg
tff(fact_4109_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_neg_neg
tff(fact_4110_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_neg_neg
tff(fact_4111_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2))) ) ) ).

% add_pos_pos
tff(fact_4112_add__pos__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))) ) ) ).

% add_pos_pos
tff(fact_4113_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_pos_pos
tff(fact_4114_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_pos_pos
tff(fact_4115_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ~ ! [C4: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C4) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_4116_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(B2),C))
       => pp(aa_rat_bool(ord_less_rat(B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% pos_add_strict
tff(fact_4117_pos__add__strict,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C))) ) ) ).

% pos_add_strict
tff(fact_4118_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% pos_add_strict
tff(fact_4119_pos__add__strict,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% pos_add_strict
tff(fact_4120_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2))) ) ) ).

% mult_neg_neg
tff(fact_4121_mult__neg__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2))) ) ) ).

% mult_neg_neg
tff(fact_4122_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_neg_neg
tff(fact_4123_not__square__less__zero,axiom,
    ! [A2: rat] : ~ pp(aa_rat_bool(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_4124_not__square__less__zero,axiom,
    ! [A2: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2)),zero_z3403309356797280102nteger)) ).

% not_square_less_zero
tff(fact_4125_not__square__less__zero,axiom,
    ! [A2: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(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_4126_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
          & pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat)) )
        | ( pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_4127_mult__less__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_4128_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ) ).

% mult_less_0_iff
tff(fact_4129_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat)) ) ) ).

% mult_neg_pos
tff(fact_4130_mult__neg__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% mult_neg_pos
tff(fact_4131_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_neg_pos
tff(fact_4132_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_neg_pos
tff(fact_4133_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),zero_zero_rat)) ) ) ).

% mult_pos_neg
tff(fact_4134_mult__pos__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% mult_pos_neg
tff(fact_4135_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat)) ) ) ).

% mult_pos_neg
tff(fact_4136_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)) ) ) ).

% mult_pos_neg
tff(fact_4137_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_4138_mult__pos__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_4139_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_4140_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))) ) ) ).

% mult_pos_pos
tff(fact_4141_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A2)),zero_zero_rat)) ) ) ).

% mult_pos_neg2
tff(fact_4142_mult__pos__neg2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)),zero_z3403309356797280102nteger)) ) ) ).

% mult_pos_neg2
tff(fact_4143_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat)) ) ) ).

% mult_pos_neg2
tff(fact_4144_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),zero_zero_int)) ) ) ).

% mult_pos_neg2
tff(fact_4145_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
          & pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2)) )
        | ( pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat)) ) ) ) ).

% zero_less_mult_iff
tff(fact_4146_zero__less__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger)) ) ) ) ).

% zero_less_mult_iff
tff(fact_4147_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int)) ) ) ) ).

% zero_less_mult_iff
tff(fact_4148_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_4149_zero__less__mult__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_4150_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_4151_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ).

% zero_less_mult_pos
tff(fact_4152_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A2)))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_4153_zero__less__mult__pos2,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_4154_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_4155_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2)) ) ) ).

% zero_less_mult_pos2
tff(fact_4156_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
     => ( pp(aa_rat_bool(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),B2)))
      <=> pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4157_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4158_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4159_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
     => ( pp(aa_rat_bool(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),B2)))
      <=> pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4160_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4161_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4162_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(B2),A2))
     => ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
       => pp(aa_rat_bool(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),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4163_mult__strict__left__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4164_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4165_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(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),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_4166_mult__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_4167_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_4168_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% mult_strict_left_mono
tff(fact_4169_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
          & pp(aa_rat_bool(ord_less_rat(A2),B2)) )
        | ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4170_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4171_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4172_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(B2),A2))
     => ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
       => pp(aa_rat_bool(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,B2),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4173_mult__strict__right__mono__neg,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4174_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4175_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(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,B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4176_mult__strict__right__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4177_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4178_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))) ) ) ).

% mult_strict_right_mono
tff(fact_4179_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
          & pp(aa_rat_bool(ord_less_rat(A2),B2)) )
        | ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
          & pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4180_mult__less__cancel__right__disj,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) )
        | ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
          & pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4181_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        | ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
          & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4182_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(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),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4183_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4184_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4185_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4186_not__one__less__zero,axiom,
    ~ pp(aa_rat_bool(ord_less_rat(one_one_rat),zero_zero_rat)) ).

% not_one_less_zero
tff(fact_4187_not__one__less__zero,axiom,
    ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),zero_z3403309356797280102nteger)) ).

% not_one_less_zero
tff(fact_4188_not__one__less__zero,axiom,
    ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).

% not_one_less_zero
tff(fact_4189_not__one__less__zero,axiom,
    ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),zero_zero_int)) ).

% not_one_less_zero
tff(fact_4190_zero__less__one,axiom,
    pp(aa_rat_bool(ord_less_rat(zero_zero_rat),one_one_rat)) ).

% zero_less_one
tff(fact_4191_zero__less__one,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% zero_less_one
tff(fact_4192_zero__less__one,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% zero_less_one
tff(fact_4193_zero__less__one,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% zero_less_one
tff(fact_4194_less__numeral__extra_I1_J,axiom,
    pp(aa_rat_bool(ord_less_rat(zero_zero_rat),one_one_rat)) ).

% less_numeral_extra(1)
tff(fact_4195_less__numeral__extra_I1_J,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),one_one_Code_integer)) ).

% less_numeral_extra(1)
tff(fact_4196_less__numeral__extra_I1_J,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).

% less_numeral_extra(1)
tff(fact_4197_less__numeral__extra_I1_J,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),one_one_int)) ).

% less_numeral_extra(1)
tff(fact_4198_less__iff__diff__less__0,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
    <=> pp(aa_rat_bool(ord_less_rat(minus_minus_rat(A2,B2)),zero_zero_rat)) ) ).

% less_iff_diff_less_0
tff(fact_4199_less__iff__diff__less__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
    <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),zero_z3403309356797280102nteger)) ) ).

% less_iff_diff_less_0
tff(fact_4200_less__iff__diff__less__0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),zero_zero_int)) ) ).

% less_iff_diff_less_0
tff(fact_4201_nat__mult__le__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_4202_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
          & pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        | ? [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
            & pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_4203_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
    <=> ? [M: nat] : N = aa_nat_nat(suc,M) ) ).

% gr0_conv_Suc
tff(fact_4204_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [I4: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),aa_nat_nat(suc,N)))
         => pp(aa_nat_bool(P,I4)) )
    <=> ( pp(aa_nat_bool(P,zero_zero_nat))
        & ! [I4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I4),N))
           => pp(aa_nat_bool(P,aa_nat_nat(suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_4205_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ? [M6: nat] : N = aa_nat_nat(suc,M6) ) ).

% gr0_implies_Suc
tff(fact_4206_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(suc,N)))
    <=> ( ( M2 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M2 = aa_nat_nat(suc,J2) )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_4207_nat__compl__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N5))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N5))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct
tff(fact_4208_nat__compl__induct_H,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N5: nat] :
            ( ! [Nn: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Nn),N5))
               => pp(aa_nat_bool(P,Nn)) )
           => pp(aa_nat_bool(P,aa_nat_nat(suc,N5))) )
       => pp(aa_nat_bool(P,N)) ) ) ).

% nat_compl_induct'
tff(fact_4209_ex__least__nat__le,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K),N))
            & ! [I6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I6),K))
               => ~ pp(aa_nat_bool(P,I6)) )
            & pp(aa_nat_bool(P,K)) ) ) ) ).

% ex_least_nat_le
tff(fact_4210_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ? [K: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K))
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_4211_Suc__to__right,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_nat(suc,N) = M2 )
     => ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_4212_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),M2)) ) ) ).

% diff_less
tff(fact_4213_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K3))) ) ) ).

% mult_less_mono1
tff(fact_4214_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),J))) ) ) ).

% mult_less_mono2
tff(fact_4215_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),X))
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_4216_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),zero_zero_rat)) ) ) ).

% add_neg_nonpos
tff(fact_4217_add__neg__nonpos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% add_neg_nonpos
tff(fact_4218_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_neg_nonpos
tff(fact_4219_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_neg_nonpos
tff(fact_4220_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_4221_add__nonneg__pos,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_4222_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_4223_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_nonneg_pos
tff(fact_4224_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),zero_zero_rat))
     => ( pp(aa_rat_bool(ord_less_rat(B2),zero_zero_rat))
       => pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2)),zero_zero_rat)) ) ) ).

% add_nonpos_neg
tff(fact_4225_add__nonpos__neg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),zero_z3403309356797280102nteger))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),zero_z3403309356797280102nteger)) ) ) ).

% add_nonpos_neg
tff(fact_4226_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),zero_zero_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),zero_zero_nat))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat)) ) ) ).

% add_nonpos_neg
tff(fact_4227_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),zero_zero_int))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int)) ) ) ).

% add_nonpos_neg
tff(fact_4228_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
       => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_4229_add__pos__nonneg,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_4230_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_4231_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))) ) ) ).

% add_pos_nonneg
tff(fact_4232_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(B2),C))
       => pp(aa_rat_bool(ord_less_rat(B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_4233_add__strict__increasing,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C))) ) ) ).

% add_strict_increasing
tff(fact_4234_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_4235_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_strict_increasing
tff(fact_4236_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_rat(B2),C))
       => pp(aa_rat_bool(ord_less_rat(B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_4237_add__strict__increasing2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_4238_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_4239_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C))) ) ) ).

% add_strict_increasing2
tff(fact_4240_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) )
        & ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(B2),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_4241_mult__le__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_4242_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ) ).

% mult_le_cancel_left
tff(fact_4243_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) )
        & ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(B2),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_4244_mult__le__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_4245_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ) ).

% mult_le_cancel_right
tff(fact_4246_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_4247_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_4248_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_4249_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% mult_left_less_imp_less
tff(fact_4250_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(C),D2))
       => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
           => pp(aa_rat_bool(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,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4251_mult__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4252_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4253_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_strict_mono
tff(fact_4254_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_rat(A2),B2)) )
        & ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_4255_mult__less__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_4256_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_left
tff(fact_4257_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(ord_less_rat(A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_4258_mult__right__less__imp__less,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_4259_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_4260_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) ) ) ).

% mult_right_less_imp_less
tff(fact_4261_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(C),D2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
           => pp(aa_rat_bool(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,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4262_mult__strict__mono_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4263_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4264_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4265_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_rat(A2),B2)) )
        & ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_rat(B2),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_4266_mult__less__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_4267_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),A2)) ) ) ) ).

% mult_less_cancel_right
tff(fact_4268_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
     => ( pp(aa_rat_bool(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),B2)))
      <=> pp(aa_rat_bool(ord_less_eq_rat(B2),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4269_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4270_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),A2)) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4271_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
     => ( pp(aa_rat_bool(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),B2)))
      <=> pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4272_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4273_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4274_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( pp(aa_rat_bool(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),B2)))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_4275_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_4276_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_4277_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% mult_left_le_imp_le
tff(fact_4278_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( pp(aa_rat_bool(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,B2),C)))
     => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
       => pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_4279_mult__right__le__imp__le,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_4280_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_4281_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ).

% mult_right_le_imp_le
tff(fact_4282_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_rat(C),D2))
       => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
           => pp(aa_rat_bool(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,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4283_mult__le__less__imp__less,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),D2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4284_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4285_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4286_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(A2),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(C),D2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
         => ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
           => pp(aa_rat_bool(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,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4287_mult__less__le__imp__less,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),D2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4288_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),D2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),C))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4289_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),D2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D2))) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4290_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_4291_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : pp(aa_rat_bool(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_4292_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))) ).

% sum_squares_ge_zero
tff(fact_4293_mult__left__le,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
       => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4294_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(C),one_one_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
       => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4295_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4296_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),A2)) ) ) ).

% mult_left_le
tff(fact_4297_mult__le__one,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),one_one_Code_integer))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),one_one_Code_integer)) ) ) ) ).

% mult_le_one
tff(fact_4298_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(A2),one_one_rat))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(B2),one_one_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2)),one_one_rat)) ) ) ) ).

% mult_le_one
tff(fact_4299_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),one_one_nat))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),one_one_nat))
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat)) ) ) ) ).

% mult_le_one
tff(fact_4300_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int)) ) ) ) ).

% mult_le_one
tff(fact_4301_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_4302_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),X))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),Y))
       => ( pp(aa_rat_bool(ord_less_eq_rat(Y),one_one_rat))
         => pp(aa_rat_bool(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_4303_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),X)) ) ) ) ).

% mult_right_le_one_le
tff(fact_4304_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),one_one_Code_integer))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_4305_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),X))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),Y))
       => ( pp(aa_rat_bool(ord_less_eq_rat(Y),one_one_rat))
         => pp(aa_rat_bool(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_4306_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),Y))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),one_one_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)),X)) ) ) ) ).

% mult_left_le_one_le
tff(fact_4307_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ pp(aa_rat_bool(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_4308_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(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_4309_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] : ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)) ).

% not_sum_squares_lt_zero
tff(fact_4310_zero__less__two,axiom,
    pp(aa_rat_bool(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_4311_zero__less__two,axiom,
    pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),one_one_Code_integer))) ).

% zero_less_two
tff(fact_4312_zero__less__two,axiom,
    pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% zero_less_two
tff(fact_4313_zero__less__two,axiom,
    pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int))) ).

% zero_less_two
tff(fact_4314_ex__least__nat__less,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( ~ pp(aa_nat_bool(P,zero_zero_nat))
       => ? [K: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K),N))
            & ! [I6: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I6),K))
               => ~ pp(aa_nat_bool(P,I6)) )
            & pp(aa_nat_bool(P,aa_nat_nat(suc,K))) ) ) ) ).

% ex_least_nat_less
tff(fact_4315_length__pos__if__in__set,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s1588839187904960729at_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4316_length__pos__if__in__set,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4317_length__pos__if__in__set,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s1066996506189871328t_bool(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4318_length__pos__if__in__set,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6829681357464350627n_assn(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4319_length__pos__if__in__set,axiom,
    ! [X: c,Xs: list_c] :
      ( member_c(X,set_c2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_c(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4320_length__pos__if__in__set,axiom,
    ! [X: d,Xs: list_d] :
      ( member_d(X,set_d2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_d(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4321_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( member_nat(X,set_nat2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4322_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))) ) ).

% length_pos_if_in_set
tff(fact_4323_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_4324_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2))) ) ) ).

% n_less_n_mult_m
tff(fact_4325_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% n_less_m_mult_n
tff(fact_4326_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))) ) ) ).

% one_less_mult
tff(fact_4327_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(P,one_one_nat))
       => ( ! [N5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
             => ( pp(aa_nat_bool(P,N5))
               => pp(aa_nat_bool(P,aa_nat_nat(suc,N5))) ) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_4328_nat__diff__split,axiom,
    ! [P: fun_nat_bool,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ! [D5: nat] :
            ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D5) )
           => pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_4329_nat__diff__split__asm,axiom,
    ! [P: fun_nat_bool,A2: nat,B2: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)))
    <=> ~ ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
            & ~ pp(aa_nat_bool(P,zero_zero_nat)) )
          | ? [D5: nat] :
              ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D5) )
              & ~ pp(aa_nat_bool(P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_4330_mult__le__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_eq_rat(one_one_rat),B2)) )
        & ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(B2),one_one_rat)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4331_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),one_one_Code_integer)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4332_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int)) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4333_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),C))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_eq_rat(A2),one_one_rat)) )
        & ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(one_one_rat),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4334_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4335_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2)) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4336_mult__le__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_eq_rat(one_one_rat),B2)) )
        & ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(B2),one_one_rat)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4337_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,B2),one_one_Code_integer)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4338_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),one_one_int)) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4339_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),C))
    <=> ( ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_eq_rat(A2),one_one_rat)) )
        & ( pp(aa_rat_bool(ord_less_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_eq_rat(one_one_rat),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4340_mult__le__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,one_one_Code_integer),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4341_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,one_one_int),A2)) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4342_mult__less__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_rat(one_one_rat),B2)) )
        & ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_rat(B2),one_one_rat)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4343_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),one_one_Code_integer)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4344_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int)) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4345_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A2)),C))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_rat(A2),one_one_rat)) )
        & ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_rat(one_one_rat),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4346_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4347_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2)) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4348_mult__less__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_rat(one_one_rat),B2)) )
        & ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_rat(B2),one_one_rat)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4349_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B2)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),one_one_Code_integer)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4350_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int)) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4351_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( pp(aa_rat_bool(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),C)),C))
    <=> ( ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),C))
         => pp(aa_rat_bool(ord_less_rat(A2),one_one_rat)) )
        & ( pp(aa_rat_bool(ord_less_eq_rat(C),zero_zero_rat))
         => pp(aa_rat_bool(ord_less_rat(one_one_rat),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4352_mult__less__cancel__right2,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),C))
    <=> ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,A2),one_one_Code_integer)) )
        & ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,C),zero_z3403309356797280102nteger))
         => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4353_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C))
    <=> ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),C))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),one_one_int)) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,C),zero_zero_int))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2)) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4354_convex__bound__le,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,Y),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V2))
           => ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(U),V2) = one_one_Code_integer )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V2),Y))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4355_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V2: rat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(X),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(Y),A2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),U))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),V2))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = one_one_rat )
             => pp(aa_rat_bool(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,V2),Y))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4356_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,Y),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V2))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A2)) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4357_nz__le__conv__less,axiom,
    ! [K3: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K3),aa_nat_nat(suc,zero_zero_nat))),M2)) ) ) ).

% nz_le_conv_less
tff(fact_4358_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4359_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4360_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V2: rat] :
      ( pp(aa_rat_bool(ord_less_rat(X),A2))
     => ( pp(aa_rat_bool(ord_less_rat(Y),A2))
       => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),U))
         => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),V2))
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = one_one_rat )
             => pp(aa_rat_bool(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,V2),Y))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4361_convex__bound__lt,axiom,
    ! [X: code_integer,A2: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,Y),A2))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),V2))
           => ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(U),V2) = one_one_Code_integer )
             => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V2),Y))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4362_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Y),A2))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),U))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),V2))
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = one_one_int )
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A2)) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4363_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),Z3))
         => ( pp(aa_rat_bool(ord_less_rat(Z3),one_one_rat))
           => pp(aa_rat_bool(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z3),X)),Y)) ) )
     => pp(aa_rat_bool(ord_less_eq_rat(X),Y)) ) ).

% field_le_mult_one_interval
tff(fact_4364_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(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_4365_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(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_4366_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4367_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(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_4368_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa_rat_bool(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_4369_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int))
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4370_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),Z))
     => ( pp(aa_rat_bool(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)))
      <=> pp(aa_rat_bool(ord_less_eq_rat(X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4371_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4372_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff1
tff(fact_4373_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),Z))
     => ( pp(aa_rat_bool(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)))
      <=> pp(aa_rat_bool(ord_less_eq_rat(X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4374_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4375_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y)) ) ) ).

% mult_le_cancel_iff2
tff(fact_4376_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),E2))
         => pp(aa_rat_bool(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2))) )
     => pp(aa_rat_bool(ord_less_eq_rat(X),Y)) ) ).

% field_le_epsilon
tff(fact_4377_divides__aux__eq,axiom,
    ! [Q5: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q5,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_4378_divides__aux__eq,axiom,
    ! [Q5: nat,R2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q5),R2)))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_4379_divides__aux__eq,axiom,
    ! [Q5: int,R2: int] :
      ( unique6319869463603278526ux_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_4380_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(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_4381_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_4382_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4383_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),Z))
     => ( pp(aa_rat_bool(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)))
      <=> pp(aa_rat_bool(ord_less_rat(X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4384_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Z))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z)))
      <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4385_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(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)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Y)) ) ) ).

% mult_less_iff1
tff(fact_4386_converse__trancl__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P: fun_se8074476271188067385t_bool] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),transi2703068831062848130at_nat(R2))
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y2,B2),R2)
           => pp(aa_set3135850010578026034t_bool(P,Y2)) )
       => ( ! [Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y2,Z3),R2)
             => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Z3,B2),transi2703068831062848130at_nat(R2))
               => ( pp(aa_set3135850010578026034t_bool(P,Z3))
                 => pp(aa_set3135850010578026034t_bool(P,Y2)) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,A2)) ) ) ) ).

% converse_trancl_induct
tff(fact_4387_converse__trancl__induct,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),transi6264000038957366511cl_nat(R2)))
     => ( ! [Y2: nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),B2)),R2))
           => pp(aa_nat_bool(P,Y2)) )
       => ( ! [Y2: nat,Z3: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3)),R2))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Z3),B2)),transi6264000038957366511cl_nat(R2)))
               => ( pp(aa_nat_bool(P,Z3))
                 => pp(aa_nat_bool(P,Y2)) ) ) )
         => pp(aa_nat_bool(P,A2)) ) ) ) ).

% converse_trancl_induct
tff(fact_4388_trancl__trans__induct,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P: fun_se1534075524646452397t_bool] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),transi2703068831062848130at_nat(R2))
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
            ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y2),R2)
           => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,X2),Y2)) )
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y2),transi2703068831062848130at_nat(R2))
             => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,X2),Y2))
               => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y2,Z3),transi2703068831062848130at_nat(R2))
                 => ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,Y2),Z3))
                   => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,X2),Z3)) ) ) ) )
         => pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(P,X),Y)) ) ) ) ).

% trancl_trans_induct
tff(fact_4389_trancl__trans__induct,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),transi6264000038957366511cl_nat(R2)))
     => ( ! [X2: nat,Y2: nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),R2))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y2)) )
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),transi6264000038957366511cl_nat(R2)))
             => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Y2))
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3)),transi6264000038957366511cl_nat(R2)))
                 => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,Y2),Z3))
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X2),Z3)) ) ) ) )
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,X),Y)) ) ) ) ).

% trancl_trans_induct
tff(fact_4390_trancl__into__trancl2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),R2)
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(B2,C),transi2703068831062848130at_nat(R2))
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,C),transi2703068831062848130at_nat(R2)) ) ) ).

% trancl_into_trancl2
tff(fact_4391_trancl__into__trancl2,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),R2))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),C)),transi6264000038957366511cl_nat(R2)))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C)),transi6264000038957366511cl_nat(R2))) ) ) ).

% trancl_into_trancl2
tff(fact_4392_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),transi2703068831062848130at_nat(R2))
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(B2,C),R2)
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,C),transi2703068831062848130at_nat(R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_4393_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,C: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),transi6264000038957366511cl_nat(R2)))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),C)),R2))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C)),transi6264000038957366511cl_nat(R2))) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_4394_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat] : ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,X2),transi2703068831062848130at_nat(R2))
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R2)
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_4395_irrefl__trancl__rD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ! [X2: nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),X2)),transi6264000038957366511cl_nat(R2)))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),R2))
       => ( X != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_4396_converse__tranclE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Z),transi2703068831062848130at_nat(R2))
     => ( ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Z),R2)
       => ~ ! [Y2: set_Pr1261947904930325089at_nat] :
              ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y2),R2)
             => ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(Y2,Z),transi2703068831062848130at_nat(R2)) ) ) ) ).

% converse_tranclE
tff(fact_4397_converse__tranclE,axiom,
    ! [X: nat,Z: nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),transi6264000038957366511cl_nat(R2)))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),R2))
       => ~ ! [Y2: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y2)),R2))
             => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z)),transi6264000038957366511cl_nat(R2))) ) ) ) ).

% converse_tranclE
tff(fact_4398_r__r__into__trancl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,C: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),R)
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(B2,C),R)
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,C),transi2703068831062848130at_nat(R)) ) ) ).

% r_r_into_trancl
tff(fact_4399_r__r__into__trancl,axiom,
    ! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,C: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),R))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),C)),R))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C)),transi6264000038957366511cl_nat(R))) ) ) ).

% r_r_into_trancl
tff(fact_4400_trancl__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P: fun_se8074476271188067385t_bool] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),transi2703068831062848130at_nat(R2))
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,Y2),R2)
           => pp(aa_set3135850010578026034t_bool(P,Y2)) )
       => ( ! [Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,Y2),transi2703068831062848130at_nat(R2))
             => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y2,Z3),R2)
               => ( pp(aa_set3135850010578026034t_bool(P,Y2))
                 => pp(aa_set3135850010578026034t_bool(P,Z3)) ) ) )
         => pp(aa_set3135850010578026034t_bool(P,B2)) ) ) ) ).

% trancl_induct
tff(fact_4401_trancl__induct,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat,P: fun_nat_bool] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),transi6264000038957366511cl_nat(R2)))
     => ( ! [Y2: nat] :
            ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y2)),R2))
           => pp(aa_nat_bool(P,Y2)) )
       => ( ! [Y2: nat,Z3: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),Y2)),transi6264000038957366511cl_nat(R2)))
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),Z3)),R2))
               => ( pp(aa_nat_bool(P,Y2))
                 => pp(aa_nat_bool(P,Z3)) ) ) )
         => pp(aa_nat_bool(P,B2)) ) ) ) ).

% trancl_induct
tff(fact_4402_trancl__induct2,axiom,
    ! [Ax: fun_Pr272337284026993017e_term,Ay: produc8923325533196201883nteger,Bx: fun_Pr272337284026993017e_term,By: produc8923325533196201883nteger,R2: set_Pr617690850500594159nteger,P: fun_fu1674507805476766837r_bool] :
      ( member5037371425510568504nteger(produc6779745771149221959nteger(produc7384524951051686153nteger(Ax,Ay),produc7384524951051686153nteger(Bx,By)),transi1277664463620755574nteger(R2))
     => ( ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger] :
            ( member5037371425510568504nteger(produc6779745771149221959nteger(produc7384524951051686153nteger(Ax,Ay),produc7384524951051686153nteger(A3,B3)),R2)
           => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(P,A3),B3)) )
       => ( ! [A3: fun_Pr272337284026993017e_term,B3: produc8923325533196201883nteger,Aa: fun_Pr272337284026993017e_term,Ba: produc8923325533196201883nteger] :
              ( member5037371425510568504nteger(produc6779745771149221959nteger(produc7384524951051686153nteger(Ax,Ay),produc7384524951051686153nteger(A3,B3)),transi1277664463620755574nteger(R2))
             => ( member5037371425510568504nteger(produc6779745771149221959nteger(produc7384524951051686153nteger(A3,B3),produc7384524951051686153nteger(Aa,Ba)),R2)
               => ( pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(P,A3),B3))
                 => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(P,Aa),Ba)) ) ) )
         => pp(aa_Pro5843519176478241656r_bool(aa_fun5449300841655810358r_bool(P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_4403_trancl__induct2,axiom,
    ! [Ax: fun_Pr6001797050664311790t_bool,Ay: produc3658429121746597890et_nat,Bx: fun_Pr6001797050664311790t_bool,By: produc3658429121746597890et_nat,R2: set_Pr4255865656320958847et_nat,P: fun_fu7398481913972163267t_bool] :
      ( member2599368984600772040et_nat(produc6525134947082930135et_nat(produc8148480564485311129et_nat(Ax,Ay),produc8148480564485311129et_nat(Bx,By)),transi1932502723865673406et_nat(R2))
     => ( ! [A3: fun_Pr6001797050664311790t_bool,B3: produc3658429121746597890et_nat] :
            ( member2599368984600772040et_nat(produc6525134947082930135et_nat(produc8148480564485311129et_nat(Ax,Ay),produc8148480564485311129et_nat(A3,B3)),R2)
           => pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(P,A3),B3)) )
       => ( ! [A3: fun_Pr6001797050664311790t_bool,B3: produc3658429121746597890et_nat,Aa: fun_Pr6001797050664311790t_bool,Ba: produc3658429121746597890et_nat] :
              ( member2599368984600772040et_nat(produc6525134947082930135et_nat(produc8148480564485311129et_nat(Ax,Ay),produc8148480564485311129et_nat(A3,B3)),transi1932502723865673406et_nat(R2))
             => ( member2599368984600772040et_nat(produc6525134947082930135et_nat(produc8148480564485311129et_nat(A3,B3),produc8148480564485311129et_nat(Aa,Ba)),R2)
               => ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(P,A3),B3))
                 => pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(P,Aa),Ba)) ) ) )
         => pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_4404_trancl__induct2,axiom,
    ! [Ax: fun_Pr6001797050664311790t_bool,Ay: produc6108883039684432031et_nat,Bx: fun_Pr6001797050664311790t_bool,By: produc6108883039684432031et_nat,R2: set_Pr6041871929502663225et_nat,P: fun_fu5758690788630904038t_bool] :
      ( member3796744270619190146et_nat(produc2636233771625430033et_nat(produc3083477295402903094et_nat(Ax,Ay),produc3083477295402903094et_nat(Bx,By)),transi873757729886363099et_nat(R2))
     => ( ! [A3: fun_Pr6001797050664311790t_bool,B3: produc6108883039684432031et_nat] :
            ( member3796744270619190146et_nat(produc2636233771625430033et_nat(produc3083477295402903094et_nat(Ax,Ay),produc3083477295402903094et_nat(A3,B3)),R2)
           => pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(P,A3),B3)) )
       => ( ! [A3: fun_Pr6001797050664311790t_bool,B3: produc6108883039684432031et_nat,Aa: fun_Pr6001797050664311790t_bool,Ba: produc6108883039684432031et_nat] :
              ( member3796744270619190146et_nat(produc2636233771625430033et_nat(produc3083477295402903094et_nat(Ax,Ay),produc3083477295402903094et_nat(A3,B3)),transi873757729886363099et_nat(R2))
             => ( member3796744270619190146et_nat(produc2636233771625430033et_nat(produc3083477295402903094et_nat(A3,B3),produc3083477295402903094et_nat(Aa,Ba)),R2)
               => ( pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(P,A3),B3))
                 => pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(P,Aa),Ba)) ) ) )
         => pp(aa_Pro7454463142485379962t_bool(aa_fun1003322038264691679t_bool(P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_4405_trancl__induct2,axiom,
    ! [Ax: fun_Pr9164807099015279978e_term,Ay: product_prod_int_int,Bx: fun_Pr9164807099015279978e_term,By: product_prod_int_int,R2: set_Pr445502709953361865nt_int,P: fun_fu4196931294019812808t_bool] :
      ( member5015713543875923218nt_int(produc5372120124174279073nt_int(produc1560874099174258678nt_int(Ax,Ay),produc1560874099174258678nt_int(Bx,By)),transi425565986406895715nt_int(R2))
     => ( ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int] :
            ( member5015713543875923218nt_int(produc5372120124174279073nt_int(produc1560874099174258678nt_int(Ax,Ay),produc1560874099174258678nt_int(A3,B3)),R2)
           => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(P,A3),B3)) )
       => ( ! [A3: fun_Pr9164807099015279978e_term,B3: product_prod_int_int,Aa: fun_Pr9164807099015279978e_term,Ba: product_prod_int_int] :
              ( member5015713543875923218nt_int(produc5372120124174279073nt_int(produc1560874099174258678nt_int(Ax,Ay),produc1560874099174258678nt_int(A3,B3)),transi425565986406895715nt_int(R2))
             => ( member5015713543875923218nt_int(produc5372120124174279073nt_int(produc1560874099174258678nt_int(A3,B3),produc1560874099174258678nt_int(Aa,Ba)),R2)
               => ( pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(P,A3),B3))
                 => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(P,Aa),Ba)) ) ) )
         => pp(aa_Pro4501881432599481050t_bool(aa_fun5300525129335604105t_bool(P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_4406_trancl__induct2,axiom,
    ! [Ax: fun_in6429348457993719614e_term,Ay: product_prod_int_int,Bx: fun_in6429348457993719614e_term,By: product_prod_int_int,R2: set_Pr2100580169568797217nt_int,P: fun_fu5117548769634986908t_bool] :
      ( member7649561347739665258nt_int(produc5207114200681655801nt_int(produc4852315488734385442nt_int(Ax,Ay),produc4852315488734385442nt_int(Bx,By)),transi1908933091705503119nt_int(R2))
     => ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int] :
            ( member7649561347739665258nt_int(produc5207114200681655801nt_int(produc4852315488734385442nt_int(Ax,Ay),produc4852315488734385442nt_int(A3,B3)),R2)
           => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(P,A3),B3)) )
       => ( ! [A3: fun_in6429348457993719614e_term,B3: product_prod_int_int,Aa: fun_in6429348457993719614e_term,Ba: product_prod_int_int] :
              ( member7649561347739665258nt_int(produc5207114200681655801nt_int(produc4852315488734385442nt_int(Ax,Ay),produc4852315488734385442nt_int(A3,B3)),transi1908933091705503119nt_int(R2))
             => ( member7649561347739665258nt_int(produc5207114200681655801nt_int(produc4852315488734385442nt_int(A3,B3),produc4852315488734385442nt_int(Aa,Ba)),R2)
               => ( pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(P,A3),B3))
                 => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(P,Aa),Ba)) ) ) )
         => pp(aa_Pro4501881432599481050t_bool(aa_fun4446166836006121821t_bool(P,Bx),By)) ) ) ) ).

% trancl_induct2
tff(fact_4407_trancl_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A1,A22),transi2703068831062848130at_nat(R2))
     => ( ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(A1,A22),R2)
       => ~ ! [B3: set_Pr1261947904930325089at_nat] :
              ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A1,B3),transi2703068831062848130at_nat(R2))
             => ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(B3,A22),R2) ) ) ) ).

% trancl.cases
tff(fact_4408_trancl_Ocases,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22)),transi6264000038957366511cl_nat(R2)))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22)),R2))
       => ~ ! [B3: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),B3)),transi6264000038957366511cl_nat(R2)))
             => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),A22)),R2)) ) ) ) ).

% trancl.cases
tff(fact_4409_trancl_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A1,A22),transi2703068831062848130at_nat(R2))
    <=> ( ? [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A7 )
            & ( A22 = B7 )
            & member8757157785044589968at_nat(produc2922128104949294807at_nat(A7,B7),R2) )
        | ? [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat,C5: set_Pr1261947904930325089at_nat] :
            ( ( A1 = A7 )
            & ( A22 = C5 )
            & member8757157785044589968at_nat(produc2922128104949294807at_nat(A7,B7),transi2703068831062848130at_nat(R2))
            & member8757157785044589968at_nat(produc2922128104949294807at_nat(B7,C5),R2) ) ) ) ).

% trancl.simps
tff(fact_4410_trancl_Osimps,axiom,
    ! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A1),A22)),transi6264000038957366511cl_nat(R2)))
    <=> ( ? [A7: nat,B7: nat] :
            ( ( A1 = A7 )
            & ( A22 = B7 )
            & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A7),B7)),R2)) )
        | ? [A7: nat,B7: nat,C5: nat] :
            ( ( A1 = A7 )
            & ( A22 = C5 )
            & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A7),B7)),transi6264000038957366511cl_nat(R2)))
            & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B7),C5)),R2)) ) ) ) ).

% trancl.simps
tff(fact_4411_trancl_Or__into__trancl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),R2)
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),transi2703068831062848130at_nat(R2)) ) ).

% trancl.r_into_trancl
tff(fact_4412_trancl_Or__into__trancl,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),R2))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),transi6264000038957366511cl_nat(R2))) ) ).

% trancl.r_into_trancl
tff(fact_4413_tranclE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),transi2703068831062848130at_nat(R2))
     => ( ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,B2),R2)
       => ~ ! [C4: set_Pr1261947904930325089at_nat] :
              ( member8757157785044589968at_nat(produc2922128104949294807at_nat(A2,C4),transi2703068831062848130at_nat(R2))
             => ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(C4,B2),R2) ) ) ) ).

% tranclE
tff(fact_4414_tranclE,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),transi6264000038957366511cl_nat(R2)))
     => ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)),R2))
       => ~ ! [C4: nat] :
              ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),C4)),transi6264000038957366511cl_nat(R2)))
             => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(C4),B2)),R2)) ) ) ) ).

% tranclE
tff(fact_4415_trancl__trans,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),transi2703068831062848130at_nat(R2))
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y,Z),transi2703068831062848130at_nat(R2))
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Z),transi2703068831062848130at_nat(R2)) ) ) ).

% trancl_trans
tff(fact_4416_trancl__trans,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),transi6264000038957366511cl_nat(R2)))
     => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Z)),transi6264000038957366511cl_nat(R2)))
       => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),transi6264000038957366511cl_nat(R2))) ) ) ).

% trancl_trans
tff(fact_4417_nonempty__has__size,axiom,
    ! [S: multis2468970476368604999at_nat] :
      ( ( S != zero_z1048942125864253310at_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s8510653225128441779at_nat(S))) ) ).

% nonempty_has_size
tff(fact_4418_bijective__Empty,axiom,
    biject3388518990857851981at_nat(bot_bo228742789529271731at_nat) ).

% bijective_Empty
tff(fact_4419_bijective__Empty,axiom,
    pp(aa_set3135850010578026034t_bool(bijective_nat_nat,bot_bo2099793752762293965at_nat)) ).

% bijective_Empty
tff(fact_4420_total__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : total_7212417256332265445at_nat(insert9200635055090092081at_nat(X,bot_bo3083307316010499117at_nat),insert9069300056098147895at_nat(produc2922128104949294807at_nat(X,X),bot_bo228742789529271731at_nat)) ).

% total_on_singleton
tff(fact_4421_total__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : total_1513685709984341995at_nat(insert9069300056098147895at_nat(X,bot_bo228742789529271731at_nat),insert398606708087465655at_nat(produc2335810615602371415at_nat(X,X),bot_bo7193250159860175667at_nat)) ).

% total_on_singleton
tff(fact_4422_total__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : total_3592101749530773125at_nat(insert8211810215607154385at_nat(X,bot_bo2099793752762293965at_nat),insert5050368324300391991at_nat(produc6161850002892822231at_nat(X,X),bot_bo5327735625951526323at_nat)) ).

% total_on_singleton
tff(fact_4423_total__on__singleton,axiom,
    ! [X: bool] : total_on_bool(insert_bool2(X,bot_bot_set_bool),insert3967033758906133175l_bool(produc8297744024737662807l_bool(X,X),bot_bo3663083580490196787l_bool)) ).

% total_on_singleton
tff(fact_4424_total__on__singleton,axiom,
    ! [X: nat] : pp(aa_set3135850010578026034t_bool(total_on_nat(insert_nat2(X,bot_bot_set_nat)),insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),X),bot_bo2099793752762293965at_nat))) ).

% total_on_singleton
tff(fact_4425_total__on__singleton,axiom,
    ! [X: int] : total_on_int(insert_int2(X,bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),X),bot_bo1796632182523588997nt_int)) ).

% total_on_singleton
tff(fact_4426_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4427_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4428_nth__Cons__pos,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),N) = nth_Pr1769885009046257848n_assn(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_4429_Field__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] : field_3145993188331922222at_nat(insert9069300056098147895at_nat(produc2922128104949294807at_nat(A2,B2),R2)) = sup_su3642409539654194069at_nat(insert9200635055090092081at_nat(A2,insert9200635055090092081at_nat(B2,bot_bo3083307316010499117at_nat)),field_3145993188331922222at_nat(R2)) ).

% Field_insert
tff(fact_4430_Field__insert,axiom,
    ! [A2: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] : field_2862844103579723188at_nat(insert398606708087465655at_nat(produc2335810615602371415at_nat(A2,B2),R2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,insert9069300056098147895at_nat(A2,insert9069300056098147895at_nat(B2,bot_bo228742789529271731at_nat))),field_2862844103579723188at_nat(R2)) ).

% Field_insert
tff(fact_4431_Field__insert,axiom,
    ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] : field_4706345496283612622at_nat(insert5050368324300391991at_nat(produc6161850002892822231at_nat(A2,B2),R2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,insert8211810215607154385at_nat(A2,insert8211810215607154385at_nat(B2,bot_bo2099793752762293965at_nat))),field_4706345496283612622at_nat(R2)) ).

% Field_insert
tff(fact_4432_Field__insert,axiom,
    ! [A2: bool,B2: bool,R2: set_Pr8621012493645324487l_bool] : field_bool(insert3967033758906133175l_bool(produc8297744024737662807l_bool(A2,B2),R2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,insert_bool2(A2,insert_bool2(B2,bot_bot_set_bool))),field_bool(R2)) ).

% Field_insert
tff(fact_4433_Field__insert,axiom,
    ! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] : field_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),R2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,insert_nat2(A2,insert_nat2(B2,bot_bot_set_nat))),field_nat(R2)) ).

% Field_insert
tff(fact_4434_Field__insert,axiom,
    ! [A2: int,B2: int,R2: set_Pr958786334691620121nt_int] : field_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2),R2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,insert_int2(A2,insert_int2(B2,bot_bot_set_int))),field_int(R2)) ).

% Field_insert
tff(fact_4435_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_c,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_size_list_c(Xs)))
     => ( nth_Pr2476257089995772766_nat_c(enumerate_c(N,Xs),M2) = aa_c_P1942785857434746857_nat_c(aa_nat1477143957913088426_nat_c(product_Pair_nat_c,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)),nth_c(Xs,M2)) ) ) ).

% nth_enumerate_eq
tff(fact_4436_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_d,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_size_list_d(Xs)))
     => ( nth_Pr2476257094299001567_nat_d(enumerate_d(N,Xs),M2) = aa_d_P9049729840602276459_nat_d(aa_nat8655121976823735212_nat_d(product_Pair_nat_d,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)),nth_d(Xs,M2)) ) ) ).

% nth_enumerate_eq
tff(fact_4437_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_size_list_nat(Xs)))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs),M2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2)),aa_nat_nat(nth_nat(Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_4438_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_int,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),size_size_list_int(Xs)))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M2) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2),aa_nat_int(nth_int(Xs),M2)) ) ) ).

% nth_enumerate_eq
tff(fact_4439_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(X),Xa),Xb))
     => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(times_assn_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,Xb))))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(times_assn_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,produc7507926704131184380et_nat(H,As)))))
               => ? [As13: set_nat,As24: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                    & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H,As13)))
                    & pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H,As24))) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_4440_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(X),Xa),Xb))
     => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(times_assn_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,Xb))))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(times_assn_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,produc7507926704131184380et_nat(H,As)))))
               => ~ ? [As12: set_nat,As23: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                      & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H,As12)))
                      & pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H,As23))) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_4441_list_Oinject,axiom,
    ! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
      ( ( cons_int(X21,X222) = cons_int(Y21,Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_4442_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
      ( ( cons_nat(X21,X222) = cons_nat(Y21,Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_4443_list_Oinject,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn,Y21: produc6575502325842934193n_assn,Y222: list_P8527749157015355191n_assn] :
      ( ( cons_P2971678138204555879n_assn(X21,X222) = cons_P2971678138204555879n_assn(Y21,Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_4444_Field__Un,axiom,
    ! [R2: set_Pr8621012493645324487l_bool,S2: set_Pr8621012493645324487l_bool] : field_bool(sup_su501663365221149339l_bool(R2,S2)) = aa_set_bool_set_bool(aa_set4734229006438377534t_bool(sup_sup_set_bool,field_bool(R2)),field_bool(S2)) ).

% Field_Un
tff(fact_4445_Field__Un,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] : field_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_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_4446_Field__Un,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,S2: set_Pr5564308138774400199at_nat] : field_2862844103579723188at_nat(sup_su6664651903789602459at_nat(R2,S2)) = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,field_2862844103579723188at_nat(R2)),field_2862844103579723188at_nat(S2)) ).

% Field_Un
tff(fact_4447_Field__Un,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] : field_3145993188331922222at_nat(aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,R2),S2)) = sup_su3642409539654194069at_nat(field_3145993188331922222at_nat(R2),field_3145993188331922222at_nat(S2)) ).

% Field_Un
tff(fact_4448_list__ex__simps_I1_J,axiom,
    ! [P: fun_int_bool,X: int,Xs: list_int] :
      ( pp(aa_list_int_bool(list_ex_int(P),cons_int(X,Xs)))
    <=> ( pp(aa_int_bool(P,X))
        | pp(aa_list_int_bool(list_ex_int(P),Xs)) ) ) ).

% list_ex_simps(1)
tff(fact_4449_list__ex__simps_I1_J,axiom,
    ! [P: fun_nat_bool,X: nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(list_ex_nat(P),cons_nat(X,Xs)))
    <=> ( pp(aa_nat_bool(P,X))
        | pp(aa_list_nat_bool(list_ex_nat(P),Xs)) ) ) ).

% list_ex_simps(1)
tff(fact_4450_list__ex__simps_I1_J,axiom,
    ! [P: fun_Pr8546840118223405289n_bool,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(list_e1087220904399170805n_assn(P),cons_P2971678138204555879n_assn(X,Xs)))
    <=> ( pp(aa_Pro3608213857613363938n_bool(P,X))
        | pp(aa_lis5212129917091495004n_bool(list_e1087220904399170805n_assn(P),Xs)) ) ) ).

% list_ex_simps(1)
tff(fact_4451_length__enumerate,axiom,
    ! [N: nat,Xs: list_c] : size_s385972143775086793_nat_c(enumerate_c(N,Xs)) = size_size_list_c(Xs) ).

% length_enumerate
tff(fact_4452_length__enumerate,axiom,
    ! [N: nat,Xs: list_d] : size_s457006183821432778_nat_d(enumerate_d(N,Xs)) = size_size_list_d(Xs) ).

% length_enumerate
tff(fact_4453_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = size_size_list_nat(Xs) ).

% length_enumerate
tff(fact_4454_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_4455_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] : set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X21,X222)) = insert8211810215607154385at_nat(X21,set_Pr5648618587558075414at_nat(X222)) ).

% list.simps(15)
tff(fact_4456_list_Osimps_I15_J,axiom,
    ! [X21: produc3843707927480180839at_nat,X222: list_P5464809261938338413at_nat] : set_Pr3765526544606949372at_nat(cons_P3940603068885512221at_nat(X21,X222)) = insert9069300056098147895at_nat(X21,set_Pr3765526544606949372at_nat(X222)) ).

% list.simps(15)
tff(fact_4457_list_Osimps_I15_J,axiom,
    ! [X21: bool,X222: list_bool] : set_bool2(cons_bool(X21,X222)) = insert_bool2(X21,set_bool2(X222)) ).

% list.simps(15)
tff(fact_4458_list_Osimps_I15_J,axiom,
    ! [X21: int,X222: list_int] : set_int2(cons_int(X21,X222)) = insert_int2(X21,set_int2(X222)) ).

% list.simps(15)
tff(fact_4459_list_Osimps_I15_J,axiom,
    ! [X21: nat,X222: list_nat] : set_nat2(cons_nat(X21,X222)) = insert_nat2(X21,set_nat2(X222)) ).

% list.simps(15)
tff(fact_4460_list_Osimps_I15_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : set_Pr1139785259514867910n_assn(cons_P2971678138204555879n_assn(X21,X222)) = insert5290817439147925377n_assn(X21,set_Pr1139785259514867910n_assn(X222)) ).

% list.simps(15)
tff(fact_4461_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] : aa_nat_int(nth_int(cons_int(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_4462_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_4463_nth__Cons__0,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_4464_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] : aa_nat_int(nth_int(cons_int(X,Xs)),aa_nat_nat(suc,N)) = aa_nat_int(nth_int(Xs),N) ).

% nth_Cons_Suc
tff(fact_4465_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),aa_nat_nat(suc,N)) = aa_nat_nat(nth_nat(Xs),N) ).

% nth_Cons_Suc
tff(fact_4466_nth__Cons__Suc,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] : nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),aa_nat_nat(suc,N)) = nth_Pr1769885009046257848n_assn(Xs,N) ).

% nth_Cons_Suc
tff(fact_4467_zip__Cons__Cons,axiom,
    ! [X: assn,Xs: list_assn,Y: assn,Ys: list_assn] : zip_assn_assn(cons_assn(X,Xs),cons_assn(Y,Ys)) = cons_P2971678138204555879n_assn(produc118845697133431529n_assn(X,Y),zip_assn_assn(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4468_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] : zip_int_int(cons_int(X,Xs),cons_int(Y,Ys)) = cons_P3334398858971670639nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),zip_int_int(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4469_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys: list_nat] : zip_int_nat(cons_int(X,Xs),cons_nat(Y,Ys)) = cons_P7512249878480867347nt_nat(product_Pair_int_nat(X,Y),zip_int_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4470_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys: list_int] : zip_nat_int(cons_nat(X,Xs),cons_int(Y,Ys)) = cons_P2335045147070616083at_int(product_Pair_nat_int(X,Y),zip_nat_int(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4471_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] : zip_nat_nat(cons_nat(X,Xs),cons_nat(Y,Ys)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),zip_nat_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4472_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] : zip_in7098856637274830732n_assn(cons_int(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)) = cons_P8828502514829229636n_assn(produc3313821056773121708n_assn(X,Y),zip_in7098856637274830732n_assn(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4473_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] : zip_na5338819973521719528n_assn(cons_nat(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)) = cons_P7068465851076118432n_assn(produc1553784393020010504n_assn(X,Y),zip_na5338819973521719528n_assn(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4474_zip__Cons__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int] : zip_Pr1015554993053626726sn_int(cons_P2971678138204555879n_assn(X,Xs),cons_int(Y,Ys)) = cons_P5155970366129889750sn_int(produc6453891449406693510sn_int(X,Y),zip_Pr1015554993053626726sn_int(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4475_zip__Cons__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat] : zip_Pr1018045463562677002sn_nat(cons_P2971678138204555879n_assn(X,Xs),cons_nat(Y,Ys)) = cons_P110449348784310650sn_nat(produc6456381919915743786sn_nat(X,Y),zip_Pr1018045463562677002sn_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4476_zip__Cons__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] : zip_Pr8806919808394484215n_assn(cons_P2971678138204555879n_assn(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)) = cons_P9150528168557782045n_assn(produc6128706242671166167n_assn(X,Y),zip_Pr8806919808394484215n_assn(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4477_Field__empty,axiom,
    field_2862844103579723188at_nat(bot_bo7193250159860175667at_nat) = bot_bo228742789529271731at_nat ).

% Field_empty
tff(fact_4478_Field__empty,axiom,
    field_4706345496283612622at_nat(bot_bo5327735625951526323at_nat) = bot_bo2099793752762293965at_nat ).

% Field_empty
tff(fact_4479_Field__empty,axiom,
    field_bool(bot_bo3663083580490196787l_bool) = bot_bot_set_bool ).

% Field_empty
tff(fact_4480_Field__empty,axiom,
    field_int(bot_bo1796632182523588997nt_int) = bot_bot_set_int ).

% Field_empty
tff(fact_4481_Field__empty,axiom,
    field_3145993188331922222at_nat(bot_bo228742789529271731at_nat) = bot_bo3083307316010499117at_nat ).

% Field_empty
tff(fact_4482_Field__empty,axiom,
    field_nat(bot_bo2099793752762293965at_nat) = bot_bot_set_nat ).

% Field_empty
tff(fact_4483_not__in__set__insert,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ( insert4149901806602606979at_nat(X,Xs) = cons_P3940603068885512221at_nat(X,Xs) ) ) ).

% not_in_set_insert
tff(fact_4484_not__in__set__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( insert8944034826898310173at_nat(X,Xs) = cons_P6512896166579812791at_nat(X,Xs) ) ) ).

% not_in_set_insert
tff(fact_4485_not__in__set__insert,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ( insert451209248500086026t_bool(X,Xs) = cons_f4651180314053712804t_bool(X,Xs) ) ) ).

% not_in_set_insert
tff(fact_4486_not__in__set__insert,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ member_int(X,set_int2(Xs))
     => ( insert_int(X,Xs) = cons_int(X,Xs) ) ) ).

% not_in_set_insert
tff(fact_4487_not__in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ member_nat(X,set_nat2(Xs))
     => ( insert_nat(X,Xs) = cons_nat(X,Xs) ) ) ).

% not_in_set_insert
tff(fact_4488_not__in__set__insert,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ( insert3246601298802261197n_assn(X,Xs) = cons_P2971678138204555879n_assn(X,Xs) ) ) ).

% not_in_set_insert
tff(fact_4489_Cons__listrel1__Cons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Ys)),listre8180072825634266816at_nat(R2))
    <=> ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R2)
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Ys),listre8180072825634266816at_nat(R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4490_Cons__listrel1__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(Y,Ys)),listrel1_int(R2))
    <=> ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4491_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(Y,Ys)),listrel1_nat(R2))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),R2))
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4492_Cons__listrel1__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)),listre4270255209692100112n_assn(R2))
    <=> ( ( member4743710847862083984n_assn(produc6128706242671166167n_assn(X,Y),R2)
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,Ys),listre4270255209692100112n_assn(R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4493_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] : enumerate_int(N,cons_int(X,Xs)) = cons_P2335045147070616083at_int(product_Pair_nat_int(N,X),enumerate_int(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4494_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] : enumerate_nat(N,cons_nat(X,Xs)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),X),enumerate_nat(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4495_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : enumer9200277086812776726n_assn(N,cons_P2971678138204555879n_assn(X,Xs)) = cons_P7068465851076118432n_assn(produc1553784393020010504n_assn(N,X),enumer9200277086812776726n_assn(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_4496_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] : cons_int(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_4497_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] : cons_nat(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_4498_not__Cons__self2,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : cons_P2971678138204555879n_assn(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_4499_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N22: int,R22: list_int] :
      ( ( cons_int(N1,R1) = cons_int(N22,R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_4500_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N22: nat,R22: list_nat] :
      ( ( cons_nat(N1,R1) = cons_nat(N22,R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_4501_list__tail__coinc,axiom,
    ! [N1: produc6575502325842934193n_assn,R1: list_P8527749157015355191n_assn,N22: produc6575502325842934193n_assn,R22: list_P8527749157015355191n_assn] :
      ( ( cons_P2971678138204555879n_assn(N1,R1) = cons_P2971678138204555879n_assn(N22,R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_4502_union__diff__assoc,axiom,
    ! [C2: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat(C2,B) = zero_z1048942125864253310at_nat )
     => ( minus_4286766774447292334at_nat(plus_p7104986032573967614at_nat(A,B),C2) = plus_p7104986032573967614at_nat(A,minus_4286766774447292334at_nat(B,C2)) ) ) ).

% union_diff_assoc
tff(fact_4503_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( zip_int_int(Xs,Ys) = cons_P3334398858971670639nt_int(Xy,Xys) )
     => ~ ! [X2: int,Xs5: list_int] :
            ( ( Xs = cons_int(X2,Xs5) )
           => ! [Y2: int,Ys3: list_int] :
                ( ( Ys = cons_int(Y2,Ys3) )
               => ( ( Xy = aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y2) )
                 => ( Xys != zip_int_int(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4504_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( zip_int_nat(Xs,Ys) = cons_P7512249878480867347nt_nat(Xy,Xys) )
     => ~ ! [X2: int,Xs5: list_int] :
            ( ( Xs = cons_int(X2,Xs5) )
           => ! [Y2: nat,Ys3: list_nat] :
                ( ( Ys = cons_nat(Y2,Ys3) )
               => ( ( Xy = product_Pair_int_nat(X2,Y2) )
                 => ( Xys != zip_int_nat(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4505_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( zip_nat_int(Xs,Ys) = cons_P2335045147070616083at_int(Xy,Xys) )
     => ~ ! [X2: nat,Xs5: list_nat] :
            ( ( Xs = cons_nat(X2,Xs5) )
           => ! [Y2: int,Ys3: list_int] :
                ( ( Ys = cons_int(Y2,Ys3) )
               => ( ( Xy = product_Pair_nat_int(X2,Y2) )
                 => ( Xys != zip_nat_int(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4506_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat(Xs,Ys) = cons_P6512896166579812791at_nat(Xy,Xys) )
     => ~ ! [X2: nat,Xs5: list_nat] :
            ( ( Xs = cons_nat(X2,Xs5) )
           => ! [Y2: nat,Ys3: list_nat] :
                ( ( Ys = cons_nat(Y2,Ys3) )
               => ( ( Xy = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2) )
                 => ( Xys != zip_nat_nat(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4507_zip__eq__ConsE,axiom,
    ! [Xs: list_assn,Ys: list_assn,Xy: produc6575502325842934193n_assn,Xys: list_P8527749157015355191n_assn] :
      ( ( zip_assn_assn(Xs,Ys) = cons_P2971678138204555879n_assn(Xy,Xys) )
     => ~ ! [X2: assn,Xs5: list_assn] :
            ( ( Xs = cons_assn(X2,Xs5) )
           => ! [Y2: assn,Ys3: list_assn] :
                ( ( Ys = cons_assn(Y2,Ys3) )
               => ( ( Xy = produc118845697133431529n_assn(X2,Y2) )
                 => ( Xys != zip_assn_assn(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4508_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn,Xy: produc3167338625816234618n_assn,Xys: list_P1495281457390866698n_assn] :
      ( ( zip_in7098856637274830732n_assn(Xs,Ys) = cons_P8828502514829229636n_assn(Xy,Xys) )
     => ~ ! [X2: int,Xs5: list_int] :
            ( ( Xs = cons_int(X2,Xs5) )
           => ! [Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ( Ys = cons_P2971678138204555879n_assn(Y2,Ys3) )
               => ( ( Xy = produc3313821056773121708n_assn(X2,Y2) )
                 => ( Xys != zip_in7098856637274830732n_assn(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4509_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn,Xy: produc1407301962063123414n_assn,Xys: list_P7711101519850957926n_assn] :
      ( ( zip_na5338819973521719528n_assn(Xs,Ys) = cons_P7068465851076118432n_assn(Xy,Xys) )
     => ~ ! [X2: nat,Xs5: list_nat] :
            ( ( Xs = cons_nat(X2,Xs5) )
           => ! [Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ( Ys = cons_P2971678138204555879n_assn(Y2,Ys3) )
               => ( ( Xy = produc1553784393020010504n_assn(X2,Y2) )
                 => ( Xys != zip_na5338819973521719528n_assn(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4510_zip__eq__ConsE,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int,Xy: produc8718178513971670540sn_int,Xys: list_P421884338359569308sn_int] :
      ( ( zip_Pr1015554993053626726sn_int(Xs,Ys) = cons_P5155970366129889750sn_int(Xy,Xys) )
     => ~ ! [X2: produc6575502325842934193n_assn,Xs5: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Xs5) )
           => ! [Y2: int,Ys3: list_int] :
                ( ( Ys = cons_int(Y2,Ys3) )
               => ( ( Xy = produc6453891449406693510sn_int(X2,Y2) )
                 => ( Xys != zip_Pr1015554993053626726sn_int(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4511_zip__eq__ConsE,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat,Xy: produc3672657496626091440sn_nat,Xys: list_P2911967483291296064sn_nat] :
      ( ( zip_Pr1018045463562677002sn_nat(Xs,Ys) = cons_P110449348784310650sn_nat(Xy,Xys) )
     => ~ ! [X2: produc6575502325842934193n_assn,Xs5: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Xs5) )
           => ! [Y2: nat,Ys3: list_nat] :
                ( ( Ys = cons_nat(Y2,Ys3) )
               => ( ( Xy = produc6456381919915743786sn_nat(X2,Y2) )
                 => ( Xys != zip_Pr1018045463562677002sn_nat(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4512_zip__eq__ConsE,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Xy: produc9053633027152450663n_assn,Xys: list_P28140577958321773n_assn] :
      ( ( zip_Pr8806919808394484215n_assn(Xs,Ys) = cons_P9150528168557782045n_assn(Xy,Xys) )
     => ~ ! [X2: produc6575502325842934193n_assn,Xs5: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Xs5) )
           => ! [Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ( Ys = cons_P2971678138204555879n_assn(Y2,Ys3) )
               => ( ( Xy = produc6128706242671166167n_assn(X2,Y2) )
                 => ( Xys != zip_Pr8806919808394484215n_assn(Xs5,Ys3) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4513_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
      ( member6009668390874684176at_nat(produc2335810615602371415at_nat(I,J),R)
     => member8757157785044589968at_nat(I,field_2862844103579723188at_nat(R)) ) ).

% FieldI1
tff(fact_4514_FieldI1,axiom,
    ! [I: product_prod_nat_nat,J: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(I,J),R)
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(I),field_4706345496283612622at_nat(R))) ) ).

% FieldI1
tff(fact_4515_FieldI1,axiom,
    ! [I: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(I),J),R)
     => member_int(I,field_int(R)) ) ).

% FieldI1
tff(fact_4516_FieldI1,axiom,
    ! [I: fun_Pr6001797050664311790t_bool,J: fun_Pr6001797050664311790t_bool,R: set_Pr2055239900342597383t_bool] :
      ( member9182180520733025616t_bool(produc4314488481105410327t_bool(I,J),R)
     => member4125802036136091927t_bool(I,field_8187305936632580411t_bool(R)) ) ).

% FieldI1
tff(fact_4517_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(I,J),R)
     => member2643936169264416010at_nat(I,field_3145993188331922222at_nat(R)) ) ).

% FieldI1
tff(fact_4518_FieldI1,axiom,
    ! [I: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I),J)),R))
     => member_nat(I,field_nat(R)) ) ).

% FieldI1
tff(fact_4519_FieldI1,axiom,
    ! [I: bool,J: bool,R: set_Pr8621012493645324487l_bool] :
      ( member4869097156692242192l_bool(produc8297744024737662807l_bool(I,J),R)
     => member_bool(I,field_bool(R)) ) ).

% FieldI1
tff(fact_4520_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
      ( member6009668390874684176at_nat(produc2335810615602371415at_nat(I,J),R)
     => member8757157785044589968at_nat(J,field_2862844103579723188at_nat(R)) ) ).

% FieldI2
tff(fact_4521_FieldI2,axiom,
    ! [I: product_prod_nat_nat,J: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(I,J),R)
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(J),field_4706345496283612622at_nat(R))) ) ).

% FieldI2
tff(fact_4522_FieldI2,axiom,
    ! [I: int,J: int,R: set_Pr958786334691620121nt_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(I),J),R)
     => member_int(J,field_int(R)) ) ).

% FieldI2
tff(fact_4523_FieldI2,axiom,
    ! [I: fun_Pr6001797050664311790t_bool,J: fun_Pr6001797050664311790t_bool,R: set_Pr2055239900342597383t_bool] :
      ( member9182180520733025616t_bool(produc4314488481105410327t_bool(I,J),R)
     => member4125802036136091927t_bool(J,field_8187305936632580411t_bool(R)) ) ).

% FieldI2
tff(fact_4524_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(I,J),R)
     => member2643936169264416010at_nat(J,field_3145993188331922222at_nat(R)) ) ).

% FieldI2
tff(fact_4525_FieldI2,axiom,
    ! [I: nat,J: nat,R: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(I),J)),R))
     => member_nat(J,field_nat(R)) ) ).

% FieldI2
tff(fact_4526_FieldI2,axiom,
    ! [I: bool,J: bool,R: set_Pr8621012493645324487l_bool] :
      ( member4869097156692242192l_bool(produc8297744024737662807l_bool(I,J),R)
     => member_bool(J,field_bool(R)) ) ).

% FieldI2
tff(fact_4527_set__ConsD,axiom,
    ! [Y: produc3843707927480180839at_nat,X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(Y,set_Pr3765526544606949372at_nat(cons_P3940603068885512221at_nat(X,Xs)))
     => ( ( Y = X )
        | member8757157785044589968at_nat(Y,set_Pr3765526544606949372at_nat(Xs)) ) ) ).

% set_ConsD
tff(fact_4528_set__ConsD,axiom,
    ! [Y: product_prod_nat_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X,Xs))))
     => ( ( Y = X )
        | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),set_Pr5648618587558075414at_nat(Xs))) ) ) ).

% set_ConsD
tff(fact_4529_set__ConsD,axiom,
    ! [Y: fun_Pr6001797050664311790t_bool,X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(Y,set_fu1437410266168021891t_bool(cons_f4651180314053712804t_bool(X,Xs)))
     => ( ( Y = X )
        | member4125802036136091927t_bool(Y,set_fu1437410266168021891t_bool(Xs)) ) ) ).

% set_ConsD
tff(fact_4530_set__ConsD,axiom,
    ! [Y: int,X: int,Xs: list_int] :
      ( member_int(Y,set_int2(cons_int(X,Xs)))
     => ( ( Y = X )
        | member_int(Y,set_int2(Xs)) ) ) ).

% set_ConsD
tff(fact_4531_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( member_nat(Y,set_nat2(cons_nat(X,Xs)))
     => ( ( Y = X )
        | member_nat(Y,set_nat2(Xs)) ) ) ).

% set_ConsD
tff(fact_4532_set__ConsD,axiom,
    ! [Y: produc6575502325842934193n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(Y,set_Pr1139785259514867910n_assn(cons_P2971678138204555879n_assn(X,Xs)))
     => ( ( Y = X )
        | member7957490590177025114n_assn(Y,set_Pr1139785259514867910n_assn(Xs)) ) ) ).

% set_ConsD
tff(fact_4533_list_Oset__cases,axiom,
    ! [E: produc3843707927480180839at_nat,A2: list_P5464809261938338413at_nat] :
      ( member8757157785044589968at_nat(E,set_Pr3765526544606949372at_nat(A2))
     => ( ! [Z22: list_P5464809261938338413at_nat] : A2 != cons_P3940603068885512221at_nat(E,Z22)
       => ~ ! [Z1: produc3843707927480180839at_nat,Z22: list_P5464809261938338413at_nat] :
              ( ( A2 = cons_P3940603068885512221at_nat(Z1,Z22) )
             => ~ member8757157785044589968at_nat(E,set_Pr3765526544606949372at_nat(Z22)) ) ) ) ).

% list.set_cases
tff(fact_4534_list_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A2: list_P6011104703257516679at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(E),set_Pr5648618587558075414at_nat(A2)))
     => ( ! [Z22: list_P6011104703257516679at_nat] : A2 != cons_P6512896166579812791at_nat(E,Z22)
       => ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
              ( ( A2 = cons_P6512896166579812791at_nat(Z1,Z22) )
             => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(E),set_Pr5648618587558075414at_nat(Z22))) ) ) ) ).

% list.set_cases
tff(fact_4535_list_Oset__cases,axiom,
    ! [E: fun_Pr6001797050664311790t_bool,A2: list_f5857023252961307252t_bool] :
      ( member4125802036136091927t_bool(E,set_fu1437410266168021891t_bool(A2))
     => ( ! [Z22: list_f5857023252961307252t_bool] : A2 != cons_f4651180314053712804t_bool(E,Z22)
       => ~ ! [Z1: fun_Pr6001797050664311790t_bool,Z22: list_f5857023252961307252t_bool] :
              ( ( A2 = cons_f4651180314053712804t_bool(Z1,Z22) )
             => ~ member4125802036136091927t_bool(E,set_fu1437410266168021891t_bool(Z22)) ) ) ) ).

% list.set_cases
tff(fact_4536_list_Oset__cases,axiom,
    ! [E: int,A2: list_int] :
      ( member_int(E,set_int2(A2))
     => ( ! [Z22: list_int] : A2 != cons_int(E,Z22)
       => ~ ! [Z1: int,Z22: list_int] :
              ( ( A2 = cons_int(Z1,Z22) )
             => ~ member_int(E,set_int2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_4537_list_Oset__cases,axiom,
    ! [E: nat,A2: list_nat] :
      ( member_nat(E,set_nat2(A2))
     => ( ! [Z22: list_nat] : A2 != cons_nat(E,Z22)
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A2 = cons_nat(Z1,Z22) )
             => ~ member_nat(E,set_nat2(Z22)) ) ) ) ).

% list.set_cases
tff(fact_4538_list_Oset__cases,axiom,
    ! [E: produc6575502325842934193n_assn,A2: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(E,set_Pr1139785259514867910n_assn(A2))
     => ( ! [Z22: list_P8527749157015355191n_assn] : A2 != cons_P2971678138204555879n_assn(E,Z22)
       => ~ ! [Z1: produc6575502325842934193n_assn,Z22: list_P8527749157015355191n_assn] :
              ( ( A2 = cons_P2971678138204555879n_assn(Z1,Z22) )
             => ~ member7957490590177025114n_assn(E,set_Pr1139785259514867910n_assn(Z22)) ) ) ) ).

% list.set_cases
tff(fact_4539_list_Oset__intros_I1_J,axiom,
    ! [X21: produc3843707927480180839at_nat,X222: list_P5464809261938338413at_nat] : member8757157785044589968at_nat(X21,set_Pr3765526544606949372at_nat(cons_P3940603068885512221at_nat(X21,X222))) ).

% list.set_intros(1)
tff(fact_4540_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] : pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X21),set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X21,X222)))) ).

% list.set_intros(1)
tff(fact_4541_list_Oset__intros_I1_J,axiom,
    ! [X21: fun_Pr6001797050664311790t_bool,X222: list_f5857023252961307252t_bool] : member4125802036136091927t_bool(X21,set_fu1437410266168021891t_bool(cons_f4651180314053712804t_bool(X21,X222))) ).

% list.set_intros(1)
tff(fact_4542_list_Oset__intros_I1_J,axiom,
    ! [X21: int,X222: list_int] : member_int(X21,set_int2(cons_int(X21,X222))) ).

% list.set_intros(1)
tff(fact_4543_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : member_nat(X21,set_nat2(cons_nat(X21,X222))) ).

% list.set_intros(1)
tff(fact_4544_list_Oset__intros_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : member7957490590177025114n_assn(X21,set_Pr1139785259514867910n_assn(cons_P2971678138204555879n_assn(X21,X222))) ).

% list.set_intros(1)
tff(fact_4545_list_Oset__intros_I2_J,axiom,
    ! [Y: produc3843707927480180839at_nat,X222: list_P5464809261938338413at_nat,X21: produc3843707927480180839at_nat] :
      ( member8757157785044589968at_nat(Y,set_Pr3765526544606949372at_nat(X222))
     => member8757157785044589968at_nat(Y,set_Pr3765526544606949372at_nat(cons_P3940603068885512221at_nat(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_4546_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X222: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),set_Pr5648618587558075414at_nat(X222)))
     => pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),set_Pr5648618587558075414at_nat(cons_P6512896166579812791at_nat(X21,X222)))) ) ).

% list.set_intros(2)
tff(fact_4547_list_Oset__intros_I2_J,axiom,
    ! [Y: fun_Pr6001797050664311790t_bool,X222: list_f5857023252961307252t_bool,X21: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(Y,set_fu1437410266168021891t_bool(X222))
     => member4125802036136091927t_bool(Y,set_fu1437410266168021891t_bool(cons_f4651180314053712804t_bool(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_4548_list_Oset__intros_I2_J,axiom,
    ! [Y: int,X222: list_int,X21: int] :
      ( member_int(Y,set_int2(X222))
     => member_int(Y,set_int2(cons_int(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_4549_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X222: list_nat,X21: nat] :
      ( member_nat(Y,set_nat2(X222))
     => member_nat(Y,set_nat2(cons_nat(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_4550_list_Oset__intros_I2_J,axiom,
    ! [Y: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn] :
      ( member7957490590177025114n_assn(Y,set_Pr1139785259514867910n_assn(X222))
     => member7957490590177025114n_assn(Y,set_Pr1139785259514867910n_assn(cons_P2971678138204555879n_assn(X21,X222))) ) ).

% list.set_intros(2)
tff(fact_4551_distinct__length__2__or__more,axiom,
    ! [A2: int,B2: int,Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,cons_int(A2,cons_int(B2,Xs))))
    <=> ( ( A2 != B2 )
        & pp(aa_list_int_bool(distinct_int,cons_int(A2,Xs)))
        & pp(aa_list_int_bool(distinct_int,cons_int(B2,Xs))) ) ) ).

% distinct_length_2_or_more
tff(fact_4552_distinct__length__2__or__more,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,cons_nat(A2,cons_nat(B2,Xs))))
    <=> ( ( A2 != B2 )
        & pp(aa_list_nat_bool(distinct_nat,cons_nat(A2,Xs)))
        & pp(aa_list_nat_bool(distinct_nat,cons_nat(B2,Xs))) ) ) ).

% distinct_length_2_or_more
tff(fact_4553_distinct__length__2__or__more,axiom,
    ! [A2: produc6575502325842934193n_assn,B2: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,cons_P2971678138204555879n_assn(A2,cons_P2971678138204555879n_assn(B2,Xs))))
    <=> ( ( A2 != B2 )
        & pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,cons_P2971678138204555879n_assn(A2,Xs)))
        & pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,cons_P2971678138204555879n_assn(B2,Xs))) ) ) ).

% distinct_length_2_or_more
tff(fact_4554_remove1_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( remove1_int(X,cons_int(Y,Xs)) = Xs ) )
      & ( ( X != Y )
       => ( remove1_int(X,cons_int(Y,Xs)) = cons_int(Y,remove1_int(X,Xs)) ) ) ) ).

% remove1.simps(2)
tff(fact_4555_remove1_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( remove1_nat(X,cons_nat(Y,Xs)) = Xs ) )
      & ( ( X != Y )
       => ( remove1_nat(X,cons_nat(Y,Xs)) = cons_nat(Y,remove1_nat(X,Xs)) ) ) ) ).

% remove1.simps(2)
tff(fact_4556_remove1_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( remove1670527618125605709n_assn(X,cons_P2971678138204555879n_assn(Y,Xs)) = Xs ) )
      & ( ( X != Y )
       => ( remove1670527618125605709n_assn(X,cons_P2971678138204555879n_assn(Y,Xs)) = cons_P2971678138204555879n_assn(Y,remove1670527618125605709n_assn(X,Xs)) ) ) ) ).

% remove1.simps(2)
tff(fact_4557_total__onI,axiom,
    ! [A: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X2,A)
         => ( member2643936169264416010at_nat(Y2,A)
           => ( ( X2 != Y2 )
             => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y2),R2)
                | member8757157785044589968at_nat(produc2922128104949294807at_nat(Y2,X2),R2) ) ) ) )
     => total_7212417256332265445at_nat(A,R2) ) ).

% total_onI
tff(fact_4558_total__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X2: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
          ( member8757157785044589968at_nat(X2,A)
         => ( member8757157785044589968at_nat(Y2,A)
           => ( ( X2 != Y2 )
             => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(X2,Y2),R2)
                | member6009668390874684176at_nat(produc2335810615602371415at_nat(Y2,X2),R2) ) ) ) )
     => total_1513685709984341995at_nat(A,R2) ) ).

% total_onI
tff(fact_4559_total__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X2),A))
         => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y2),A))
           => ( ( X2 != Y2 )
             => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Y2),R2)
                | member8206827879206165904at_nat(produc6161850002892822231at_nat(Y2,X2),R2) ) ) ) )
     => total_3592101749530773125at_nat(A,R2) ) ).

% total_onI
tff(fact_4560_total__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( member_nat(X2,A)
         => ( member_nat(Y2,A)
           => ( ( X2 != Y2 )
             => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)),R2))
                | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y2),X2)),R2)) ) ) ) )
     => pp(aa_set3135850010578026034t_bool(total_on_nat(A),R2)) ) ).

% total_onI
tff(fact_4561_total__onI,axiom,
    ! [A: set_int,R2: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y2: int] :
          ( member_int(X2,A)
         => ( member_int(Y2,A)
           => ( ( X2 != Y2 )
             => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y2),R2)
                | member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y2),X2),R2) ) ) ) )
     => total_on_int(A,R2) ) ).

% total_onI
tff(fact_4562_total__onI,axiom,
    ! [A: set_fu6080891106736489294t_bool,R2: set_Pr2055239900342597383t_bool] :
      ( ! [X2: fun_Pr6001797050664311790t_bool,Y2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(X2,A)
         => ( member4125802036136091927t_bool(Y2,A)
           => ( ( X2 != Y2 )
             => ( member9182180520733025616t_bool(produc4314488481105410327t_bool(X2,Y2),R2)
                | member9182180520733025616t_bool(produc4314488481105410327t_bool(Y2,X2),R2) ) ) ) )
     => total_8389867087861513074t_bool(A,R2) ) ).

% total_onI
tff(fact_4563_total__on__def,axiom,
    ! [A: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( total_7212417256332265445at_nat(A,R2)
    <=> ! [X5: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X5,A)
         => ! [Xa2: set_Pr1261947904930325089at_nat] :
              ( member2643936169264416010at_nat(Xa2,A)
             => ( ( X5 != Xa2 )
               => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Xa2),R2)
                  | member8757157785044589968at_nat(produc2922128104949294807at_nat(Xa2,X5),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_4564_total__on__def,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(total_on_nat(A),R2))
    <=> ! [X5: nat] :
          ( member_nat(X5,A)
         => ! [Xa2: nat] :
              ( member_nat(Xa2,A)
             => ( ( X5 != Xa2 )
               => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Xa2)),R2))
                  | pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa2),X5)),R2)) ) ) ) ) ) ).

% total_on_def
tff(fact_4565_total__on__empty,axiom,
    ! [R2: set_Pr5564308138774400199at_nat] : total_1513685709984341995at_nat(bot_bo228742789529271731at_nat,R2) ).

% total_on_empty
tff(fact_4566_total__on__empty,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] : total_3592101749530773125at_nat(bot_bo2099793752762293965at_nat,R2) ).

% total_on_empty
tff(fact_4567_total__on__empty,axiom,
    ! [R2: set_Pr8621012493645324487l_bool] : total_on_bool(bot_bot_set_bool,R2) ).

% total_on_empty
tff(fact_4568_total__on__empty,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] : pp(aa_set3135850010578026034t_bool(total_on_nat(bot_bot_set_nat),R2)) ).

% total_on_empty
tff(fact_4569_total__on__empty,axiom,
    ! [R2: set_Pr958786334691620121nt_int] : total_on_int(bot_bot_set_int,R2) ).

% total_on_empty
tff(fact_4570_removeAll_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( removeAll_int(X,cons_int(Y,Xs)) = removeAll_int(X,Xs) ) )
      & ( ( X != Y )
       => ( removeAll_int(X,cons_int(Y,Xs)) = cons_int(Y,removeAll_int(X,Xs)) ) ) ) ).

% removeAll.simps(2)
tff(fact_4571_removeAll_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( removeAll_nat(X,cons_nat(Y,Xs)) = removeAll_nat(X,Xs) ) )
      & ( ( X != Y )
       => ( removeAll_nat(X,cons_nat(Y,Xs)) = cons_nat(Y,removeAll_nat(X,Xs)) ) ) ) ).

% removeAll.simps(2)
tff(fact_4572_removeAll_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( remove1763081955827946941n_assn(X,cons_P2971678138204555879n_assn(Y,Xs)) = remove1763081955827946941n_assn(X,Xs) ) )
      & ( ( X != Y )
       => ( remove1763081955827946941n_assn(X,cons_P2971678138204555879n_assn(Y,Xs)) = cons_P2971678138204555879n_assn(Y,remove1763081955827946941n_assn(X,Xs)) ) ) ) ).

% removeAll.simps(2)
tff(fact_4573_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_int_fun_int_int,A2: int,As2: list_int,B2: int,Bs: list_int] : zipf_int_int_int(F3,cons_int(A2,As2),cons_int(B2,Bs)) = cons_int(aa_int_int(aa_int_fun_int_int(F3,A2),B2),zipf_int_int_int(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4574_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_int_fun_int_nat,A2: int,As2: list_int,B2: int,Bs: list_int] : zipf_int_int_nat(F3,cons_int(A2,As2),cons_int(B2,Bs)) = cons_nat(aa_int_nat(aa_int_fun_int_nat(F3,A2),B2),zipf_int_int_nat(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4575_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_int_fun_nat_int,A2: int,As2: list_int,B2: nat,Bs: list_nat] : zipf_int_nat_int(F3,cons_int(A2,As2),cons_nat(B2,Bs)) = cons_int(aa_nat_int(aa_int_fun_nat_int(F3,A2),B2),zipf_int_nat_int(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4576_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_int_fun_nat_nat,A2: int,As2: list_int,B2: nat,Bs: list_nat] : zipf_int_nat_nat(F3,cons_int(A2,As2),cons_nat(B2,Bs)) = cons_nat(aa_nat_nat(aa_int_fun_nat_nat(F3,A2),B2),zipf_int_nat_nat(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4577_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_nat_fun_int_int,A2: nat,As2: list_nat,B2: int,Bs: list_int] : zipf_nat_int_int(F3,cons_nat(A2,As2),cons_int(B2,Bs)) = cons_int(aa_int_int(aa_nat_fun_int_int(F3,A2),B2),zipf_nat_int_int(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4578_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_nat_fun_int_nat,A2: nat,As2: list_nat,B2: int,Bs: list_int] : zipf_nat_int_nat(F3,cons_nat(A2,As2),cons_int(B2,Bs)) = cons_nat(aa_int_nat(aa_nat_fun_int_nat(F3,A2),B2),zipf_nat_int_nat(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4579_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_nat_fun_nat_int,A2: nat,As2: list_nat,B2: nat,Bs: list_nat] : zipf_nat_nat_int(F3,cons_nat(A2,As2),cons_nat(B2,Bs)) = cons_int(aa_nat_int(aa_nat_fun_nat_int(F3,A2),B2),zipf_nat_nat_int(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4580_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_nat_fun_nat_nat,A2: nat,As2: list_nat,B2: nat,Bs: list_nat] : zipf_nat_nat_nat(F3,cons_nat(A2,As2),cons_nat(B2,Bs)) = cons_nat(aa_nat_nat(aa_nat_fun_nat_nat(F3,A2),B2),zipf_nat_nat_nat(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4581_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_in4433485590553487243n_assn,A2: int,As2: list_int,B2: int,Bs: list_int] : zipf_i7756021159461276697n_assn(F3,cons_int(A2,As2),cons_int(B2,Bs)) = cons_P2971678138204555879n_assn(aa_int4912925366867985095n_assn(aa_int4455557282903564020n_assn(F3,A2),B2),zipf_i7756021159461276697n_assn(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4582_zipf_Osimps_I2_J,axiom,
    ! [F3: fun_in1425933616158802663n_assn,A2: int,As2: list_int,B2: nat,Bs: list_nat] : zipf_i5995984495708165493n_assn(F3,cons_int(A2,As2),cons_nat(B2,Bs)) = cons_P2971678138204555879n_assn(aa_nat3152888703114873891n_assn(aa_int2695520619150452816n_assn(F3,A2),B2),zipf_i5995984495708165493n_assn(F3,As2,Bs)) ).

% zipf.simps(2)
tff(fact_4583_null__rec_I1_J,axiom,
    ! [X: int,Xs: list_int] : ~ pp(aa_list_int_bool(null_int,cons_int(X,Xs))) ).

% null_rec(1)
tff(fact_4584_null__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat] : ~ pp(aa_list_nat_bool(null_nat,cons_nat(X,Xs))) ).

% null_rec(1)
tff(fact_4585_null__rec_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : ~ pp(aa_lis5212129917091495004n_bool(null_P7596217980778516671n_assn,cons_P2971678138204555879n_assn(X,Xs))) ).

% null_rec(1)
tff(fact_4586_length__Suc__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( size_s6829681357464350627n_assn(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y4: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
          ( ( Xs = cons_P2971678138204555879n_assn(Y4,Ys4) )
          & ( size_s6829681357464350627n_assn(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_4587_length__Suc__conv,axiom,
    ! [Xs: list_c,N: nat] :
      ( ( size_size_list_c(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y4: c,Ys4: list_c] :
          ( ( Xs = cons_c(Y4,Ys4) )
          & ( size_size_list_c(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_4588_length__Suc__conv,axiom,
    ! [Xs: list_d,N: nat] :
      ( ( size_size_list_d(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y4: d,Ys4: list_d] :
          ( ( Xs = cons_d(Y4,Ys4) )
          & ( size_size_list_d(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_4589_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( size_size_list_nat(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(Y4,Ys4) )
          & ( size_size_list_nat(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_4590_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,N) )
    <=> ? [Y4: int,Ys4: list_int] :
          ( ( Xs = cons_int(Y4,Ys4) )
          & ( size_size_list_int(Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_4591_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( aa_nat_nat(suc,N) = size_s6829681357464350627n_assn(Xs) )
    <=> ? [Y4: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
          ( ( Xs = cons_P2971678138204555879n_assn(Y4,Ys4) )
          & ( size_s6829681357464350627n_assn(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_4592_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_c] :
      ( ( aa_nat_nat(suc,N) = size_size_list_c(Xs) )
    <=> ? [Y4: c,Ys4: list_c] :
          ( ( Xs = cons_c(Y4,Ys4) )
          & ( size_size_list_c(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_4593_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_d] :
      ( ( aa_nat_nat(suc,N) = size_size_list_d(Xs) )
    <=> ? [Y4: d,Ys4: list_d] :
          ( ( Xs = cons_d(Y4,Ys4) )
          & ( size_size_list_d(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_4594_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,N) = size_size_list_nat(Xs) )
    <=> ? [Y4: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(Y4,Ys4) )
          & ( size_size_list_nat(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_4595_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,N) = size_size_list_int(Xs) )
    <=> ? [Y4: int,Ys4: list_int] :
          ( ( Xs = cons_int(Y4,Ys4) )
          & ( size_size_list_int(Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_4596_set__subset__Cons,axiom,
    ! [Xs: list_int,X: int] : ord_less_eq_set_int(set_int2(Xs),set_int2(cons_int(X,Xs))) ).

% set_subset_Cons
tff(fact_4597_set__subset__Cons,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] : ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Xs),set_Pr1139785259514867910n_assn(cons_P2971678138204555879n_assn(X,Xs))) ).

% set_subset_Cons
tff(fact_4598_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ord_less_eq_set_nat(set_nat2(Xs),set_nat2(cons_nat(X,Xs))) ).

% set_subset_Cons
tff(fact_4599_mono__Field,axiom,
    ! [R2: set_Pr8621012493645324487l_bool,S2: set_Pr8621012493645324487l_bool] :
      ( ord_le4880273282464405607l_bool(R2,S2)
     => ord_less_eq_set_bool(field_bool(R2),field_bool(S2)) ) ).

% mono_Field
tff(fact_4600_mono__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(aa_set7594618562247239262t_bool(ord_le3146513528884898305at_nat,R2),S2))
     => ord_less_eq_set_nat(field_nat(R2),field_nat(S2)) ) ).

% mono_Field
tff(fact_4601_impossible__Cons,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_s6829681357464350627n_assn(Xs)),size_s6829681357464350627n_assn(Ys)))
     => ( Xs != cons_P2971678138204555879n_assn(X,Ys) ) ) ).

% impossible_Cons
tff(fact_4602_impossible__Cons,axiom,
    ! [Xs: list_c,Ys: list_c,X: c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_c(Xs)),size_size_list_c(Ys)))
     => ( Xs != cons_c(X,Ys) ) ) ).

% impossible_Cons
tff(fact_4603_impossible__Cons,axiom,
    ! [Xs: list_d,Ys: list_d,X: d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_d(Xs)),size_size_list_d(Ys)))
     => ( Xs != cons_d(X,Ys) ) ) ).

% impossible_Cons
tff(fact_4604_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)))
     => ( Xs != cons_nat(X,Ys) ) ) ).

% impossible_Cons
tff(fact_4605_impossible__Cons,axiom,
    ! [Xs: list_int,Ys: list_int,X: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,size_size_list_int(Xs)),size_size_list_int(Ys)))
     => ( Xs != cons_int(X,Ys) ) ) ).

% impossible_Cons
tff(fact_4606_sorted2,axiom,
    ! [X: nat,Y: nat,Zs: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),cons_nat(X,cons_nat(Y,Zs))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y))
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),cons_nat(Y,Zs))) ) ) ).

% sorted2
tff(fact_4607_sorted2,axiom,
    ! [X: int,Y: int,Zs: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),cons_int(X,cons_int(Y,Zs))))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Y))
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),cons_int(Y,Zs))) ) ) ).

% sorted2
tff(fact_4608_distinct_Osimps_I2_J,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( distin5355857245021852384at_nat(cons_P3940603068885512221at_nat(X,Xs))
    <=> ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
        & distin5355857245021852384at_nat(Xs) ) ) ).

% distinct.simps(2)
tff(fact_4609_distinct_Osimps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( distin6923225563576452346at_nat(cons_P6512896166579812791at_nat(X,Xs))
    <=> ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
        & distin6923225563576452346at_nat(Xs) ) ) ).

% distinct.simps(2)
tff(fact_4610_distinct_Osimps_I2_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( distin2825057960173885031t_bool(cons_f4651180314053712804t_bool(X,Xs))
    <=> ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
        & distin2825057960173885031t_bool(Xs) ) ) ).

% distinct.simps(2)
tff(fact_4611_distinct_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( pp(aa_list_int_bool(distinct_int,cons_int(X,Xs)))
    <=> ( ~ member_int(X,set_int2(Xs))
        & pp(aa_list_int_bool(distinct_int,Xs)) ) ) ).

% distinct.simps(2)
tff(fact_4612_distinct_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( pp(aa_list_nat_bool(distinct_nat,cons_nat(X,Xs)))
    <=> ( ~ member_nat(X,set_nat2(Xs))
        & pp(aa_list_nat_bool(distinct_nat,Xs)) ) ) ).

% distinct.simps(2)
tff(fact_4613_distinct_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,cons_P2971678138204555879n_assn(X,Xs)))
    <=> ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
        & pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,Xs)) ) ) ).

% distinct.simps(2)
tff(fact_4614_list__update__code_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] : list_update_int(cons_int(X,Xs),zero_zero_nat,Y) = cons_int(Y,Xs) ).

% list_update_code(2)
tff(fact_4615_list__update__code_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] : list_update_nat(cons_nat(X,Xs),zero_zero_nat,Y) = cons_nat(Y,Xs) ).

% list_update_code(2)
tff(fact_4616_list__update__code_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] : list_u4534839942911652127n_assn(cons_P2971678138204555879n_assn(X,Xs),zero_zero_nat,Y) = cons_P2971678138204555879n_assn(Y,Xs) ).

% list_update_code(2)
tff(fact_4617_list__update__code_I2_J,axiom,
    ! [X: c,Xs: list_c,Y: c] : list_update_c(cons_c(X,Xs),zero_zero_nat,Y) = cons_c(Y,Xs) ).

% list_update_code(2)
tff(fact_4618_list__update__code_I2_J,axiom,
    ! [X: d,Xs: list_d,Y: d] : list_update_d(cons_d(X,Xs),zero_zero_nat,Y) = cons_d(Y,Xs) ).

% list_update_code(2)
tff(fact_4619_list__update__code_I3_J,axiom,
    ! [X: int,Xs: list_int,I: nat,Y: int] : list_update_int(cons_int(X,Xs),aa_nat_nat(suc,I),Y) = cons_int(X,list_update_int(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4620_list__update__code_I3_J,axiom,
    ! [X: nat,Xs: list_nat,I: nat,Y: nat] : list_update_nat(cons_nat(X,Xs),aa_nat_nat(suc,I),Y) = cons_nat(X,list_update_nat(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4621_list__update__code_I3_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,I: nat,Y: produc6575502325842934193n_assn] : list_u4534839942911652127n_assn(cons_P2971678138204555879n_assn(X,Xs),aa_nat_nat(suc,I),Y) = cons_P2971678138204555879n_assn(X,list_u4534839942911652127n_assn(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4622_list__update__code_I3_J,axiom,
    ! [X: c,Xs: list_c,I: nat,Y: c] : list_update_c(cons_c(X,Xs),aa_nat_nat(suc,I),Y) = cons_c(X,list_update_c(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4623_list__update__code_I3_J,axiom,
    ! [X: d,Xs: list_d,I: nat,Y: d] : list_update_d(cons_d(X,Xs),aa_nat_nat(suc,I),Y) = cons_d(X,list_update_d(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_4624_listrel1I2,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int,X: int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel1_int(R2))
     => member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(X,Ys)),listrel1_int(R2)) ) ).

% listrel1I2
tff(fact_4625_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,X: nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel1_nat(R2))
     => member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(X,Ys)),listrel1_nat(R2)) ) ).

% listrel1I2
tff(fact_4626_listrel1I2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn,X: produc6575502325842934193n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,Ys),listre4270255209692100112n_assn(R2))
     => member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X,Xs),cons_P2971678138204555879n_assn(X,Ys)),listre4270255209692100112n_assn(R2)) ) ).

% listrel1I2
tff(fact_4627_merge_Osimps_I3_J,axiom,
    ! [X1: code_integer,X22: code_integer,L12: list_Code_integer,L23: list_Code_integer] :
      ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X1),X22))
       => ( merge_Code_integer(cons_Code_integer(X1,L12),cons_Code_integer(X22,L23)) = cons_Code_integer(X1,merge_Code_integer(L12,cons_Code_integer(X22,L23))) ) )
      & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X1),X22))
       => ( ( ( X1 = X22 )
           => ( merge_Code_integer(cons_Code_integer(X1,L12),cons_Code_integer(X22,L23)) = cons_Code_integer(X1,merge_Code_integer(L12,L23)) ) )
          & ( ( X1 != X22 )
           => ( merge_Code_integer(cons_Code_integer(X1,L12),cons_Code_integer(X22,L23)) = cons_Code_integer(X22,merge_Code_integer(cons_Code_integer(X1,L12),L23)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_4628_merge_Osimps_I3_J,axiom,
    ! [X1: num,X22: num,L12: list_num,L23: list_num] :
      ( ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X1),X22))
       => ( merge_num(cons_num(X1,L12),cons_num(X22,L23)) = cons_num(X1,merge_num(L12,cons_num(X22,L23))) ) )
      & ( ~ pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X1),X22))
       => ( ( ( X1 = X22 )
           => ( merge_num(cons_num(X1,L12),cons_num(X22,L23)) = cons_num(X1,merge_num(L12,L23)) ) )
          & ( ( X1 != X22 )
           => ( merge_num(cons_num(X1,L12),cons_num(X22,L23)) = cons_num(X22,merge_num(cons_num(X1,L12),L23)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_4629_merge_Osimps_I3_J,axiom,
    ! [X1: nat,X22: nat,L12: list_nat,L23: list_nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X1),X22))
       => ( merge_nat(cons_nat(X1,L12),cons_nat(X22,L23)) = cons_nat(X1,merge_nat(L12,cons_nat(X22,L23))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X1),X22))
       => ( ( ( X1 = X22 )
           => ( merge_nat(cons_nat(X1,L12),cons_nat(X22,L23)) = cons_nat(X1,merge_nat(L12,L23)) ) )
          & ( ( X1 != X22 )
           => ( merge_nat(cons_nat(X1,L12),cons_nat(X22,L23)) = cons_nat(X22,merge_nat(cons_nat(X1,L12),L23)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_4630_merge_Osimps_I3_J,axiom,
    ! [X1: int,X22: int,L12: list_int,L23: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X1),X22))
       => ( merge_int(cons_int(X1,L12),cons_int(X22,L23)) = cons_int(X1,merge_int(L12,cons_int(X22,L23))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X1),X22))
       => ( ( ( X1 = X22 )
           => ( merge_int(cons_int(X1,L12),cons_int(X22,L23)) = cons_int(X1,merge_int(L12,L23)) ) )
          & ( ( X1 != X22 )
           => ( merge_int(cons_int(X1,L12),cons_int(X22,L23)) = cons_int(X22,merge_int(cons_int(X1,L12),L23)) ) ) ) ) ) ).

% merge.simps(3)
tff(fact_4631_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_int_fun_int_assn,A2: int,As2: list_int,C: int,Cs: list_int] : vEBT_L74593716426352029nt_int(P,cons_int(A2,As2),cons_int(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(P,A2),C)),vEBT_L74593716426352029nt_int(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4632_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_int_fun_nat_assn,A2: int,As2: list_int,C: nat,Cs: list_nat] : vEBT_L77084186935402305nt_nat(P,cons_int(A2,As2),cons_nat(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(P,A2),C)),vEBT_L77084186935402305nt_nat(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4633_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_in2917358019923704601n_assn,A2: int,As2: list_int,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] : vEBT_L1620274777744359680n_assn(P,cons_int(A2,As2),cons_P2971678138204555879n_assn(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_int4086841622967611778n_assn(P,A2),C)),vEBT_L1620274777744359680n_assn(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4634_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_nat_fun_int_assn,A2: nat,As2: list_nat,C: int,Cs: list_int] : vEBT_L8298612041380073281at_int(P,cons_nat(A2,As2),cons_int(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(P,A2),C)),vEBT_L8298612041380073281at_int(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4635_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_nat_fun_nat_assn,A2: nat,As2: list_nat,C: nat,Cs: list_nat] : vEBT_L8301102511889123557at_nat(P,cons_nat(A2,As2),cons_nat(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(P,A2),C)),vEBT_L8301102511889123557at_nat(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4636_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_na2667955736958379125n_assn,A2: nat,As2: list_nat,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] : vEBT_L9083610150846024284n_assn(P,cons_nat(A2,As2),cons_P2971678138204555879n_assn(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_nat3837439340002286302n_assn(P,A2),C)),vEBT_L9083610150846024284n_assn(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4637_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_Pr7092578583803811605t_assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: int,Cs: list_int] : vEBT_L4760345170377931482sn_int(P,cons_P2971678138204555879n_assn(A2,As2),cons_int(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_Pro9204279173661784262t_assn(P,A2),C)),vEBT_L4760345170377931482sn_int(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4638_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_Pr2676784290456659641t_assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: nat,Cs: list_nat] : vEBT_L4762835640886981758sn_nat(P,cons_P2971678138204555879n_assn(A2,As2),cons_nat(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_Pro8228743708455606890t_assn(P,A2),C)),vEBT_L4762835640886981758sn_nat(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4639_list__assn_Osimps_I2_J,axiom,
    ! [P: fun_Pr1914754426620969278n_assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] : vEBT_L3108143461992071043n_assn(P,cons_P2971678138204555879n_assn(A2,As2),cons_P2971678138204555879n_assn(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_Pro1520987361448537015n_assn(P,A2),C)),vEBT_L3108143461992071043n_assn(P,As2,Cs)) ).

% list_assn.simps(2)
tff(fact_4640_list__assn__simps_I2_J,axiom,
    ! [P: fun_int_fun_int_assn,A2: int,As2: list_int,C: int,Cs: list_int] : vEBT_L74593716426352029nt_int(P,cons_int(A2,As2),cons_int(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(P,A2),C)),vEBT_L74593716426352029nt_int(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4641_list__assn__simps_I2_J,axiom,
    ! [P: fun_int_fun_nat_assn,A2: int,As2: list_int,C: nat,Cs: list_nat] : vEBT_L77084186935402305nt_nat(P,cons_int(A2,As2),cons_nat(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(P,A2),C)),vEBT_L77084186935402305nt_nat(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4642_list__assn__simps_I2_J,axiom,
    ! [P: fun_in2917358019923704601n_assn,A2: int,As2: list_int,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] : vEBT_L1620274777744359680n_assn(P,cons_int(A2,As2),cons_P2971678138204555879n_assn(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_int4086841622967611778n_assn(P,A2),C)),vEBT_L1620274777744359680n_assn(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4643_list__assn__simps_I2_J,axiom,
    ! [P: fun_nat_fun_int_assn,A2: nat,As2: list_nat,C: int,Cs: list_int] : vEBT_L8298612041380073281at_int(P,cons_nat(A2,As2),cons_int(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(P,A2),C)),vEBT_L8298612041380073281at_int(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4644_list__assn__simps_I2_J,axiom,
    ! [P: fun_nat_fun_nat_assn,A2: nat,As2: list_nat,C: nat,Cs: list_nat] : vEBT_L8301102511889123557at_nat(P,cons_nat(A2,As2),cons_nat(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(P,A2),C)),vEBT_L8301102511889123557at_nat(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4645_list__assn__simps_I2_J,axiom,
    ! [P: fun_na2667955736958379125n_assn,A2: nat,As2: list_nat,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] : vEBT_L9083610150846024284n_assn(P,cons_nat(A2,As2),cons_P2971678138204555879n_assn(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_nat3837439340002286302n_assn(P,A2),C)),vEBT_L9083610150846024284n_assn(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4646_list__assn__simps_I2_J,axiom,
    ! [P: fun_Pr7092578583803811605t_assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: int,Cs: list_int] : vEBT_L4760345170377931482sn_int(P,cons_P2971678138204555879n_assn(A2,As2),cons_int(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_Pro9204279173661784262t_assn(P,A2),C)),vEBT_L4760345170377931482sn_int(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4647_list__assn__simps_I2_J,axiom,
    ! [P: fun_Pr2676784290456659641t_assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: nat,Cs: list_nat] : vEBT_L4762835640886981758sn_nat(P,cons_P2971678138204555879n_assn(A2,As2),cons_nat(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_Pro8228743708455606890t_assn(P,A2),C)),vEBT_L4762835640886981758sn_nat(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4648_list__assn__simps_I2_J,axiom,
    ! [P: fun_Pr1914754426620969278n_assn,A2: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,C: produc6575502325842934193n_assn,Cs: list_P8527749157015355191n_assn] : vEBT_L3108143461992071043n_assn(P,cons_P2971678138204555879n_assn(A2,As2),cons_P2971678138204555879n_assn(C,Cs)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_Pro1520987361448537015n_assn(P,A2),C)),vEBT_L3108143461992071043n_assn(P,As2,Cs)) ).

% list_assn_simps(2)
tff(fact_4649_List_Oinsert__def,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
       => ( insert4149901806602606979at_nat(X,Xs) = Xs ) )
      & ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
       => ( insert4149901806602606979at_nat(X,Xs) = cons_P3940603068885512221at_nat(X,Xs) ) ) ) ).

% List.insert_def
tff(fact_4650_List_Oinsert__def,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => ( insert8944034826898310173at_nat(X,Xs) = Xs ) )
      & ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
       => ( insert8944034826898310173at_nat(X,Xs) = cons_P6512896166579812791at_nat(X,Xs) ) ) ) ).

% List.insert_def
tff(fact_4651_List_Oinsert__def,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ( member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
       => ( insert451209248500086026t_bool(X,Xs) = Xs ) )
      & ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
       => ( insert451209248500086026t_bool(X,Xs) = cons_f4651180314053712804t_bool(X,Xs) ) ) ) ).

% List.insert_def
tff(fact_4652_List_Oinsert__def,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int(X,set_int2(Xs))
       => ( insert_int(X,Xs) = Xs ) )
      & ( ~ member_int(X,set_int2(Xs))
       => ( insert_int(X,Xs) = cons_int(X,Xs) ) ) ) ).

% List.insert_def
tff(fact_4653_List_Oinsert__def,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat(X,set_nat2(Xs))
       => ( insert_nat(X,Xs) = Xs ) )
      & ( ~ member_nat(X,set_nat2(Xs))
       => ( insert_nat(X,Xs) = cons_nat(X,Xs) ) ) ) ).

% List.insert_def
tff(fact_4654_List_Oinsert__def,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
       => ( insert3246601298802261197n_assn(X,Xs) = Xs ) )
      & ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
       => ( insert3246601298802261197n_assn(X,Xs) = cons_P2971678138204555879n_assn(X,Xs) ) ) ) ).

% List.insert_def
tff(fact_4655_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_s6829681357464350627n_assn(Xs)))
    <=> ? [X5: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
          ( ( Xs = cons_P2971678138204555879n_assn(X5,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6829681357464350627n_assn(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_4656_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_c(Xs)))
    <=> ? [X5: c,Ys4: list_c] :
          ( ( Xs = cons_c(X5,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_c(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_4657_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_d(Xs)))
    <=> ? [X5: d,Ys4: list_d] :
          ( ( Xs = cons_d(X5,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_d(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_4658_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_nat(Xs)))
    <=> ? [X5: nat,Ys4: list_nat] :
          ( ( Xs = cons_nat(X5,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_nat(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_4659_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,N)),size_size_list_int(Xs)))
    <=> ? [X5: int,Ys4: list_int] :
          ( ( Xs = cons_int(X5,Ys4) )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_4660_sorted__simps_I2_J,axiom,
    ! [X: nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),cons_nat(X,Ys)))
    <=> ( ! [X5: nat] :
            ( member_nat(X5,set_nat2(Ys))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),X5)) )
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),Ys)) ) ) ).

% sorted_simps(2)
tff(fact_4661_sorted__simps_I2_J,axiom,
    ! [X: int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),cons_int(X,Ys)))
    <=> ( ! [X5: int] :
            ( member_int(X5,set_int2(Ys))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),X5)) )
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),Ys)) ) ) ).

% sorted_simps(2)
tff(fact_4662_strict__sorted__simps_I2_J,axiom,
    ! [X: code_integer,Ys: list_Code_integer] :
      ( sorted710888440204495920nteger(ord_le6747313008572928689nteger,cons_Code_integer(X,Ys))
    <=> ( ! [X5: code_integer] :
            ( member_Code_integer(X5,set_Code_integer2(Ys))
           => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),X5)) )
        & sorted710888440204495920nteger(ord_le6747313008572928689nteger,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_4663_strict__sorted__simps_I2_J,axiom,
    ! [X: num,Ys: list_num] :
      ( sorted_wrt_num(ord_less_num,cons_num(X,Ys))
    <=> ( ! [X5: num] :
            ( member_num(X5,set_num2(Ys))
           => pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,X),X5)) )
        & sorted_wrt_num(ord_less_num,Ys) ) ) ).

% strict_sorted_simps(2)
tff(fact_4664_strict__sorted__simps_I2_J,axiom,
    ! [X: nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),cons_nat(X,Ys)))
    <=> ( ! [X5: nat] :
            ( member_nat(X5,set_nat2(Ys))
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),X5)) )
        & pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),Ys)) ) ) ).

% strict_sorted_simps(2)
tff(fact_4665_strict__sorted__simps_I2_J,axiom,
    ! [X: int,Ys: list_int] :
      ( pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),cons_int(X,Ys)))
    <=> ( ! [X5: int] :
            ( member_int(X5,set_int2(Ys))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),X5)) )
        & pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),Ys)) ) ) ).

% strict_sorted_simps(2)
tff(fact_4666_bijective__def,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( biject3388518990857851981at_nat(R)
    <=> ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Y4),R)
              & member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Z4),R) )
           => ( Y4 = Z4 ) )
        & ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X5,Z4),R)
              & member8757157785044589968at_nat(produc2922128104949294807at_nat(Y4,Z4),R) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4667_bijective__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(bijective_nat_nat,R))
    <=> ( ! [X5: nat,Y4: nat,Z4: nat] :
            ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Y4)),R))
              & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Z4)),R)) )
           => ( Y4 = Z4 ) )
        & ! [X5: nat,Y4: nat,Z4: nat] :
            ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X5),Z4)),R))
              & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y4),Z4)),R)) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4668_bijective__def,axiom,
    ! [R: set_Pr5663546166630115725nteger] :
      ( biject199102631239885075nteger(R)
    <=> ( ! [X5: fun_Pr272337284026993017e_term,Y4: produc8923325533196201883nteger,Z4: produc8923325533196201883nteger] :
            ( ( member6423679177046879214nteger(produc7384524951051686153nteger(X5,Y4),R)
              & member6423679177046879214nteger(produc7384524951051686153nteger(X5,Z4),R) )
           => ( Y4 = Z4 ) )
        & ! [X5: fun_Pr272337284026993017e_term,Y4: fun_Pr272337284026993017e_term,Z4: produc8923325533196201883nteger] :
            ( ( member6423679177046879214nteger(produc7384524951051686153nteger(X5,Z4),R)
              & member6423679177046879214nteger(produc7384524951051686153nteger(Y4,Z4),R) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4669_bijective__def,axiom,
    ! [R: set_Pr1106803562331729365et_nat] :
      ( biject5596389841143135651et_nat(R)
    <=> ( ! [X5: fun_Pr6001797050664311790t_bool,Y4: produc3658429121746597890et_nat,Z4: produc3658429121746597890et_nat] :
            ( ( member8615881266069945910et_nat(produc8148480564485311129et_nat(X5,Y4),R)
              & member8615881266069945910et_nat(produc8148480564485311129et_nat(X5,Z4),R) )
           => ( Y4 = Z4 ) )
        & ! [X5: fun_Pr6001797050664311790t_bool,Y4: fun_Pr6001797050664311790t_bool,Z4: produc3658429121746597890et_nat] :
            ( ( member8615881266069945910et_nat(produc8148480564485311129et_nat(X5,Z4),R)
              & member8615881266069945910et_nat(produc8148480564485311129et_nat(Y4,Z4),R) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4670_bijective__def,axiom,
    ! [R: set_Pr9222568342844664434et_nat] :
      ( biject2138954951139011392et_nat(R)
    <=> ( ! [X5: fun_Pr6001797050664311790t_bool,Y4: produc6108883039684432031et_nat,Z4: produc6108883039684432031et_nat] :
            ( ( member2818932950976722259et_nat(produc3083477295402903094et_nat(X5,Y4),R)
              & member2818932950976722259et_nat(produc3083477295402903094et_nat(X5,Z4),R) )
           => ( Y4 = Z4 ) )
        & ! [X5: fun_Pr6001797050664311790t_bool,Y4: fun_Pr6001797050664311790t_bool,Z4: produc6108883039684432031et_nat] :
            ( ( member2818932950976722259et_nat(produc3083477295402903094et_nat(X5,Z4),R)
              & member2818932950976722259et_nat(produc3083477295402903094et_nat(Y4,Z4),R) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4671_bijective__def,axiom,
    ! [R: set_Pr233528076387193594nt_int] :
      ( biject4655992564398653440nt_int(R)
    <=> ( ! [X5: fun_Pr9164807099015279978e_term,Y4: product_prod_int_int,Z4: product_prod_int_int] :
            ( ( member690604328116145115nt_int(produc1560874099174258678nt_int(X5,Y4),R)
              & member690604328116145115nt_int(produc1560874099174258678nt_int(X5,Z4),R) )
           => ( Y4 = Z4 ) )
        & ! [X5: fun_Pr9164807099015279978e_term,Y4: fun_Pr9164807099015279978e_term,Z4: product_prod_int_int] :
            ( ( member690604328116145115nt_int(produc1560874099174258678nt_int(X5,Z4),R)
              & member690604328116145115nt_int(produc1560874099174258678nt_int(Y4,Z4),R) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4672_bijective__def,axiom,
    ! [R: set_Pr5033440245338114598nt_int] :
      ( biject4479422512835743020nt_int(R)
    <=> ( ! [X5: fun_in6429348457993719614e_term,Y4: product_prod_int_int,Z4: product_prod_int_int] :
            ( ( member500491123954006279nt_int(produc4852315488734385442nt_int(X5,Y4),R)
              & member500491123954006279nt_int(produc4852315488734385442nt_int(X5,Z4),R) )
           => ( Y4 = Z4 ) )
        & ! [X5: fun_in6429348457993719614e_term,Y4: fun_in6429348457993719614e_term,Z4: product_prod_int_int] :
            ( ( member500491123954006279nt_int(produc4852315488734385442nt_int(X5,Z4),R)
              & member500491123954006279nt_int(produc4852315488734385442nt_int(Y4,Z4),R) )
           => ( X5 = Y4 ) ) ) ) ).

% bijective_def
tff(fact_4673_Cons__listrel1E2,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,cons_s6881495754146722583at_nat(Y,Ys)),listre8180072825634266816at_nat(R2))
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( Xs = cons_s6881495754146722583at_nat(X2,Ys) )
           => ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y),R2) )
       => ~ ! [Zs2: list_s1210847774152347623at_nat] :
              ( ( Xs = cons_s6881495754146722583at_nat(Y,Zs2) )
             => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Zs2,Ys),listre8180072825634266816at_nat(R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_4674_Cons__listrel1E2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,cons_int(Y,Ys)),listrel1_int(R2))
     => ( ! [X2: int] :
            ( ( Xs = cons_int(X2,Ys) )
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y),R2) )
       => ~ ! [Zs2: list_int] :
              ( ( Xs = cons_int(Y,Zs2) )
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Zs2,Ys),listrel1_int(R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_4675_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,cons_nat(Y,Ys)),listrel1_nat(R2))
     => ( ! [X2: nat] :
            ( ( Xs = cons_nat(X2,Ys) )
           => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y)),R2)) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs = cons_nat(Y,Zs2) )
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Zs2,Ys),listrel1_nat(R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_4676_Cons__listrel1E2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,cons_P2971678138204555879n_assn(Y,Ys)),listre4270255209692100112n_assn(R2))
     => ( ! [X2: produc6575502325842934193n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Ys) )
           => ~ member4743710847862083984n_assn(produc6128706242671166167n_assn(X2,Y),R2) )
       => ~ ! [Zs2: list_P8527749157015355191n_assn] :
              ( ( Xs = cons_P2971678138204555879n_assn(Y,Zs2) )
             => ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(Zs2,Ys),listre4270255209692100112n_assn(R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_4677_Cons__listrel1E1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),Ys),listre8180072825634266816at_nat(R2))
     => ( ! [Y2: set_Pr1261947904930325089at_nat] :
            ( ( Ys = cons_s6881495754146722583at_nat(Y2,Xs) )
           => ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y2),R2) )
       => ~ ! [Zs2: list_s1210847774152347623at_nat] :
              ( ( Ys = cons_s6881495754146722583at_nat(X,Zs2) )
             => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Zs2),listre8180072825634266816at_nat(R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_4678_Cons__listrel1E1,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),Ys),listrel1_int(R2))
     => ( ! [Y2: int] :
            ( ( Ys = cons_int(Y2,Xs) )
           => ~ member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y2),R2) )
       => ~ ! [Zs2: list_int] :
              ( ( Ys = cons_int(X,Zs2) )
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs,Zs2),listrel1_int(R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_4679_Cons__listrel1E1,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),Ys),listrel1_nat(R2))
     => ( ! [Y2: nat] :
            ( ( Ys = cons_nat(Y2,Xs) )
           => ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y2)),R2)) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys = cons_nat(X,Zs2) )
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Zs2),listrel1_nat(R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_4680_Cons__listrel1E1,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X,Xs),Ys),listre4270255209692100112n_assn(R2))
     => ( ! [Y2: produc6575502325842934193n_assn] :
            ( ( Ys = cons_P2971678138204555879n_assn(Y2,Xs) )
           => ~ member4743710847862083984n_assn(produc6128706242671166167n_assn(X,Y2),R2) )
       => ~ ! [Zs2: list_P8527749157015355191n_assn] :
              ( ( Ys = cons_P2971678138204555879n_assn(X,Zs2) )
             => ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,Zs2),listre4270255209692100112n_assn(R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_4681_listrel1I1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Xs: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R2)
     => member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Xs)),listre8180072825634266816at_nat(R2)) ) ).

% listrel1I1
tff(fact_4682_listrel1I1,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
     => member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(Y,Xs)),listrel1_int(R2)) ) ).

% listrel1I1
tff(fact_4683_listrel1I1,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),R2))
     => member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(Y,Xs)),listrel1_nat(R2)) ) ).

% listrel1I1
tff(fact_4684_listrel1I1,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,R2: set_Pr8116311503512020807n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member4743710847862083984n_assn(produc6128706242671166167n_assn(X,Y),R2)
     => member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X,Xs),cons_P2971678138204555879n_assn(Y,Xs)),listre4270255209692100112n_assn(R2)) ) ).

% listrel1I1
tff(fact_4685_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,cons_int(Y,Ys)),listrel_int_int(R2))
     => ~ ! [X2: int,Xs2: list_int] :
            ( ( Xs = cons_int(X2,Xs2) )
           => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X2),Y),R2)
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs2,Ys),listrel_int_int(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4686_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: int,Ys: list_int,R2: set_Pr7995236796853374141at_int] :
      ( member4850886304473975718st_int(produc7739558402351520821st_int(Xs,cons_int(Y,Ys)),listrel_nat_int(R2))
     => ~ ! [X2: nat,Xs2: list_nat] :
            ( ( Xs = cons_nat(X2,Xs2) )
           => ( member4262671552274231302at_int(product_Pair_nat_int(X2,Y),R2)
             => ~ member4850886304473975718st_int(produc7739558402351520821st_int(Xs2,Ys),listrel_nat_int(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4687_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: nat,Ys: list_nat,R2: set_Pr3448869479623346877nt_nat] :
      ( member9189046780804443046st_nat(produc4542114716404682293st_nat(Xs,cons_nat(Y,Ys)),listrel_int_nat(R2))
     => ~ ! [X2: int,Xs2: list_int] :
            ( ( Xs = cons_int(X2,Xs2) )
           => ( member216504246829706758nt_nat(product_Pair_int_nat(X2,Y),R2)
             => ~ member9189046780804443046st_nat(produc4542114716404682293st_nat(Xs2,Ys),listrel_int_nat(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4688_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,cons_nat(Y,Ys)),listrel_nat_nat(R2))
     => ~ ! [X2: nat,Xs2: list_nat] :
            ( ( Xs = cons_nat(X2,Xs2) )
           => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y)),R2))
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs2,Ys),listrel_nat_nat(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4689_listrel__Cons2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int,R2: set_Pr2181989527757714242sn_int] :
      ( member5295696281703116473st_int(produc1996262876970182428st_int(Xs,cons_int(Y,Ys)),listre8029020076619294380sn_int(R2))
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Xs2) )
           => ( member5792320502188913699sn_int(produc6453891449406693510sn_int(X2,Y),R2)
             => ~ member5295696281703116473st_int(produc1996262876970182428st_int(Xs2,Ys),listre8029020076619294380sn_int(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4690_listrel__Cons2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat,R2: set_Pr4672072672689440998sn_nat] :
      ( member7785779426634843229st_nat(produc6174113896479379136st_nat(Xs,cons_nat(Y,Ys)),listre8031510547128344656sn_nat(R2))
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Xs2) )
           => ( member746799484843334599sn_nat(produc6456381919915743786sn_nat(X2,Y),R2)
             => ~ member7785779426634843229st_nat(produc6174113896479379136st_nat(Xs2,Ys),listre8031510547128344656sn_nat(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4691_listrel__Cons2,axiom,
    ! [Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr3255386646789011632n_assn] :
      ( member490958779117458567n_assn(produc4510294837168919970n_assn(Xs,cons_P2971678138204555879n_assn(Y,Ys)),listre4888949683985722578n_assn(R2))
     => ~ ! [X2: int,Xs2: list_int] :
            ( ( Xs = cons_int(X2,Xs2) )
           => ( member241480614033477777n_assn(produc3313821056773121708n_assn(X2,Y),R2)
             => ~ member490958779117458567n_assn(produc4510294837168919970n_assn(Xs2,Ys),listre4888949683985722578n_assn(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4692_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr247834672394327052n_assn] :
      ( member3687432546746012131n_assn(produc7706768604797473534n_assn(Xs,cons_P2971678138204555879n_assn(Y,Ys)),listre3128913020232611374n_assn(R2))
     => ~ ! [X2: nat,Xs2: list_nat] :
            ( ( Xs = cons_nat(X2,Xs2) )
           => ( member7704815987135142381n_assn(produc1553784393020010504n_assn(X2,Y),R2)
             => ~ member3687432546746012131n_assn(produc7706768604797473534n_assn(Xs2,Ys),listre3128913020232611374n_assn(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4693_listrel__Cons2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,cons_P2971678138204555879n_assn(Y,Ys)),listre5253995379918703921n_assn(R2))
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(X2,Xs2) )
           => ( member4743710847862083984n_assn(produc6128706242671166167n_assn(X2,Y),R2)
             => ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs2,Ys),listre5253995379918703921n_assn(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4694_listrel__Cons2,axiom,
    ! [Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,cons_s6881495754146722583at_nat(Y,Ys)),listre2047417242196832561at_nat(R2))
     => ~ ! [X2: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
            ( ( Xs = cons_s6881495754146722583at_nat(X2,Xs2) )
           => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X2,Y),R2)
             => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs2,Ys),listre2047417242196832561at_nat(R2)) ) ) ) ).

% listrel_Cons2
tff(fact_4695_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(Y,Ys),Xs),listrel_int_int(R2))
     => ~ ! [Y2: int,Ys5: list_int] :
            ( ( Xs = cons_int(Y2,Ys5) )
           => ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(Y),Y2),R2)
             => ~ member6698963635872716290st_int(produc364263696895485585st_int(Ys,Ys5),listrel_int_int(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4696_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_nat,R2: set_Pr3448869479623346877nt_nat] :
      ( member9189046780804443046st_nat(produc4542114716404682293st_nat(cons_int(Y,Ys),Xs),listrel_int_nat(R2))
     => ~ ! [Y2: nat,Ys5: list_nat] :
            ( ( Xs = cons_nat(Y2,Ys5) )
           => ( member216504246829706758nt_nat(product_Pair_int_nat(Y,Y2),R2)
             => ~ member9189046780804443046st_nat(produc4542114716404682293st_nat(Ys,Ys5),listrel_int_nat(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4697_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_int,R2: set_Pr7995236796853374141at_int] :
      ( member4850886304473975718st_int(produc7739558402351520821st_int(cons_nat(Y,Ys),Xs),listrel_nat_int(R2))
     => ~ ! [Y2: int,Ys5: list_int] :
            ( ( Xs = cons_int(Y2,Ys5) )
           => ( member4262671552274231302at_int(product_Pair_nat_int(Y,Y2),R2)
             => ~ member4850886304473975718st_int(produc7739558402351520821st_int(Ys,Ys5),listrel_nat_int(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4698_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(Y,Ys),Xs),listrel_nat_nat(R2))
     => ~ ! [Y2: nat,Ys5: list_nat] :
            ( ( Xs = cons_nat(Y2,Ys5) )
           => ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y),Y2)),R2))
             => ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Ys,Ys5),listrel_nat_nat(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4699_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs: list_P8527749157015355191n_assn,R2: set_Pr3255386646789011632n_assn] :
      ( member490958779117458567n_assn(produc4510294837168919970n_assn(cons_int(Y,Ys),Xs),listre4888949683985722578n_assn(R2))
     => ~ ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(Y2,Ys5) )
           => ( member241480614033477777n_assn(produc3313821056773121708n_assn(Y,Y2),R2)
             => ~ member490958779117458567n_assn(produc4510294837168919970n_assn(Ys,Ys5),listre4888949683985722578n_assn(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4700_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_P8527749157015355191n_assn,R2: set_Pr247834672394327052n_assn] :
      ( member3687432546746012131n_assn(produc7706768604797473534n_assn(cons_nat(Y,Ys),Xs),listre3128913020232611374n_assn(R2))
     => ~ ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(Y2,Ys5) )
           => ( member7704815987135142381n_assn(produc1553784393020010504n_assn(Y,Y2),R2)
             => ~ member3687432546746012131n_assn(produc7706768604797473534n_assn(Ys,Ys5),listre3128913020232611374n_assn(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4701_listrel__Cons1,axiom,
    ! [Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_int,R2: set_Pr2181989527757714242sn_int] :
      ( member5295696281703116473st_int(produc1996262876970182428st_int(cons_P2971678138204555879n_assn(Y,Ys),Xs),listre8029020076619294380sn_int(R2))
     => ~ ! [Y2: int,Ys5: list_int] :
            ( ( Xs = cons_int(Y2,Ys5) )
           => ( member5792320502188913699sn_int(produc6453891449406693510sn_int(Y,Y2),R2)
             => ~ member5295696281703116473st_int(produc1996262876970182428st_int(Ys,Ys5),listre8029020076619294380sn_int(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4702_listrel__Cons1,axiom,
    ! [Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_nat,R2: set_Pr4672072672689440998sn_nat] :
      ( member7785779426634843229st_nat(produc6174113896479379136st_nat(cons_P2971678138204555879n_assn(Y,Ys),Xs),listre8031510547128344656sn_nat(R2))
     => ~ ! [Y2: nat,Ys5: list_nat] :
            ( ( Xs = cons_nat(Y2,Ys5) )
           => ( member746799484843334599sn_nat(produc6456381919915743786sn_nat(Y,Y2),R2)
             => ~ member7785779426634843229st_nat(produc6174113896479379136st_nat(Ys,Ys5),listre8031510547128344656sn_nat(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4703_listrel__Cons1,axiom,
    ! [Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(Y,Ys),Xs),listre5253995379918703921n_assn(R2))
     => ~ ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
            ( ( Xs = cons_P2971678138204555879n_assn(Y2,Ys5) )
           => ( member4743710847862083984n_assn(produc6128706242671166167n_assn(Y,Y2),R2)
             => ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(Ys,Ys5),listre5253995379918703921n_assn(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4704_listrel__Cons1,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Xs: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(Y,Ys),Xs),listre2047417242196832561at_nat(R2))
     => ~ ! [Y2: set_Pr1261947904930325089at_nat,Ys5: list_s1210847774152347623at_nat] :
            ( ( Xs = cons_s6881495754146722583at_nat(Y2,Ys5) )
           => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(Y,Y2),R2)
             => ~ member4080735728053443344at_nat(produc7536900900485677911at_nat(Ys,Ys5),listre2047417242196832561at_nat(R2)) ) ) ) ).

% listrel_Cons1
tff(fact_4705_listrel_OCons,axiom,
    ! [X: int,Y: int,R2: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
     => ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),listrel_int_int(R2))
       => member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(Y,Ys)),listrel_int_int(R2)) ) ) ).

% listrel.Cons
tff(fact_4706_listrel_OCons,axiom,
    ! [X: int,Y: nat,R2: set_Pr3448869479623346877nt_nat,Xs: list_int,Ys: list_nat] :
      ( member216504246829706758nt_nat(product_Pair_int_nat(X,Y),R2)
     => ( member9189046780804443046st_nat(produc4542114716404682293st_nat(Xs,Ys),listrel_int_nat(R2))
       => member9189046780804443046st_nat(produc4542114716404682293st_nat(cons_int(X,Xs),cons_nat(Y,Ys)),listrel_int_nat(R2)) ) ) ).

% listrel.Cons
tff(fact_4707_listrel_OCons,axiom,
    ! [X: nat,Y: int,R2: set_Pr7995236796853374141at_int,Xs: list_nat,Ys: list_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(X,Y),R2)
     => ( member4850886304473975718st_int(produc7739558402351520821st_int(Xs,Ys),listrel_nat_int(R2))
       => member4850886304473975718st_int(produc7739558402351520821st_int(cons_nat(X,Xs),cons_int(Y,Ys)),listrel_nat_int(R2)) ) ) ).

% listrel.Cons
tff(fact_4708_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),R2))
     => ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),listrel_nat_nat(R2))
       => member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(Y,Ys)),listrel_nat_nat(R2)) ) ) ).

% listrel.Cons
tff(fact_4709_listrel_OCons,axiom,
    ! [X: int,Y: produc6575502325842934193n_assn,R2: set_Pr3255386646789011632n_assn,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( member241480614033477777n_assn(produc3313821056773121708n_assn(X,Y),R2)
     => ( member490958779117458567n_assn(produc4510294837168919970n_assn(Xs,Ys),listre4888949683985722578n_assn(R2))
       => member490958779117458567n_assn(produc4510294837168919970n_assn(cons_int(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)),listre4888949683985722578n_assn(R2)) ) ) ).

% listrel.Cons
tff(fact_4710_listrel_OCons,axiom,
    ! [X: nat,Y: produc6575502325842934193n_assn,R2: set_Pr247834672394327052n_assn,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( member7704815987135142381n_assn(produc1553784393020010504n_assn(X,Y),R2)
     => ( member3687432546746012131n_assn(produc7706768604797473534n_assn(Xs,Ys),listre3128913020232611374n_assn(R2))
       => member3687432546746012131n_assn(produc7706768604797473534n_assn(cons_nat(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)),listre3128913020232611374n_assn(R2)) ) ) ).

% listrel.Cons
tff(fact_4711_listrel_OCons,axiom,
    ! [X: produc6575502325842934193n_assn,Y: int,R2: set_Pr2181989527757714242sn_int,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( member5792320502188913699sn_int(produc6453891449406693510sn_int(X,Y),R2)
     => ( member5295696281703116473st_int(produc1996262876970182428st_int(Xs,Ys),listre8029020076619294380sn_int(R2))
       => member5295696281703116473st_int(produc1996262876970182428st_int(cons_P2971678138204555879n_assn(X,Xs),cons_int(Y,Ys)),listre8029020076619294380sn_int(R2)) ) ) ).

% listrel.Cons
tff(fact_4712_listrel_OCons,axiom,
    ! [X: produc6575502325842934193n_assn,Y: nat,R2: set_Pr4672072672689440998sn_nat,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( member746799484843334599sn_nat(produc6456381919915743786sn_nat(X,Y),R2)
     => ( member7785779426634843229st_nat(produc6174113896479379136st_nat(Xs,Ys),listre8031510547128344656sn_nat(R2))
       => member7785779426634843229st_nat(produc6174113896479379136st_nat(cons_P2971678138204555879n_assn(X,Xs),cons_nat(Y,Ys)),listre8031510547128344656sn_nat(R2)) ) ) ).

% listrel.Cons
tff(fact_4713_listrel_OCons,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,R2: set_Pr8116311503512020807n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( member4743710847862083984n_assn(produc6128706242671166167n_assn(X,Y),R2)
     => ( member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,Ys),listre5253995379918703921n_assn(R2))
       => member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)),listre5253995379918703921n_assn(R2)) ) ) ).

% listrel.Cons
tff(fact_4714_listrel_OCons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Xs: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R2)
     => ( member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Ys),listre2047417242196832561at_nat(R2))
       => member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Ys)),listre2047417242196832561at_nat(R2)) ) ) ).

% listrel.Cons
tff(fact_4715_list_Osize_I4_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : size_s6829681357464350627n_assn(cons_P2971678138204555879n_assn(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6829681357464350627n_assn(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4716_list_Osize_I4_J,axiom,
    ! [X21: c,X222: list_c] : size_size_list_c(cons_c(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_c(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4717_list_Osize_I4_J,axiom,
    ! [X21: d,X222: list_d] : size_size_list_d(cons_d(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_d(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4718_list_Osize_I4_J,axiom,
    ! [X21: nat,X222: list_nat] : size_size_list_nat(cons_nat(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4719_list_Osize_I4_J,axiom,
    ! [X21: int,X222: list_int] : size_size_list_int(cons_int(X21,X222)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X222)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_4720_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4721_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4722_nth__Cons_H,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( N = zero_zero_nat )
       => ( nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),N) = X ) )
      & ( ( N != zero_zero_nat )
       => ( nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),N) = nth_Pr1769885009046257848n_assn(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ) ).

% nth_Cons'
tff(fact_4723_nth__equal__first__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat,N: nat] :
      ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s1588839187904960729at_nat(Xs)))
       => ( ( nth_Pr1239524528086697198at_nat(cons_P3940603068885512221at_nat(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4724_nth__equal__first__eq,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s5460976970255530739at_nat(Xs)))
       => ( ( nth_Pr7617993195940197384at_nat(cons_P6512896166579812791at_nat(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4725_nth__equal__first__eq,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool,N: nat] :
      ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s1066996506189871328t_bool(Xs)))
       => ( ( nth_fu2663317684723071093t_bool(cons_f4651180314053712804t_bool(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4726_nth__equal__first__eq,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_s6829681357464350627n_assn(Xs)))
       => ( ( nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4727_nth__equal__first__eq,axiom,
    ! [X: c,Xs: list_c,N: nat] :
      ( ~ member_c(X,set_c2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_c(Xs)))
       => ( ( nth_c(cons_c(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4728_nth__equal__first__eq,axiom,
    ! [X: d,Xs: list_d,N: nat] :
      ( ~ member_d(X,set_d2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_d(Xs)))
       => ( ( nth_d(cons_d(X,Xs),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4729_nth__equal__first__eq,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ~ member_nat(X,set_nat2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_nat(Xs)))
       => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4730_nth__equal__first__eq,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ~ member_int(X,set_int2(Xs))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),size_size_list_int(Xs)))
       => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = X )
        <=> ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_4731_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4732_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4733_nth__non__equal__first__eq,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( X != Y )
     => ( ( nth_Pr1769885009046257848n_assn(cons_P2971678138204555879n_assn(X,Xs),N) = Y )
      <=> ( ( nth_Pr1769885009046257848n_assn(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_4734_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_int(Begin,End,cons_int(X,Xs)) = cons_int(X,slice_int(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_int(Begin,End,cons_int(X,Xs)) = slice_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ) ) ).

% slice_Cons
tff(fact_4735_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_nat(Begin,End,cons_nat(X,Xs)) = cons_nat(X,slice_nat(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_nat(Begin,End,cons_nat(X,Xs)) = slice_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ) ) ).

% slice_Cons
tff(fact_4736_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( ( Begin = zero_zero_nat )
          & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_7964966981652229398n_assn(Begin,End,cons_P2971678138204555879n_assn(X,Xs)) = cons_P2971678138204555879n_assn(X,slice_7964966981652229398n_assn(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),End)) )
       => ( slice_7964966981652229398n_assn(Begin,End,cons_P2971678138204555879n_assn(X,Xs)) = slice_7964966981652229398n_assn(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ) ) ).

% slice_Cons
tff(fact_4737_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(times_assn_raw(X),Xa),Xb))
      <=> pp(Y) )
     => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(times_assn_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,Xb))))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( ( pp(Y)
                <=> ? [As1: set_nat,As22: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                      & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H,As1)))
                      & pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H,As22))) ) )
               => ~ pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(times_assn_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,produc7507926704131184380et_nat(H,As))))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_4738_length__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] : size_s6829681357464350627n_assn(cons_P2971678138204555879n_assn(X,Xs)) = aa_nat_nat(suc,size_s6829681357464350627n_assn(Xs)) ).

% length_Cons
tff(fact_4739_length__Cons,axiom,
    ! [X: c,Xs: list_c] : size_size_list_c(cons_c(X,Xs)) = aa_nat_nat(suc,size_size_list_c(Xs)) ).

% length_Cons
tff(fact_4740_length__Cons,axiom,
    ! [X: d,Xs: list_d] : size_size_list_d(cons_d(X,Xs)) = aa_nat_nat(suc,size_size_list_d(Xs)) ).

% length_Cons
tff(fact_4741_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] : size_size_list_nat(cons_nat(X,Xs)) = aa_nat_nat(suc,size_size_list_nat(Xs)) ).

% length_Cons
tff(fact_4742_length__Cons,axiom,
    ! [X: int,Xs: list_int] : size_size_list_int(cons_int(X,Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ).

% length_Cons
tff(fact_4743_Cons__lenlex__iff,axiom,
    ! [M2: set_Pr1261947904930325089at_nat,Ms: list_s1210847774152347623at_nat,N: set_Pr1261947904930325089at_nat,Ns: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(M2,Ms),cons_s6881495754146722583at_nat(N,Ns)),lenlex1357538814655152620at_nat(R2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s8736152011456118867at_nat(Ms)),size_s8736152011456118867at_nat(Ns)))
        | ( ( size_s8736152011456118867at_nat(Ms) = size_s8736152011456118867at_nat(Ns) )
          & member8757157785044589968at_nat(produc2922128104949294807at_nat(M2,N),R2) )
        | ( ( M2 = N )
          & member4080735728053443344at_nat(produc7536900900485677911at_nat(Ms,Ns),lenlex1357538814655152620at_nat(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4744_Cons__lenlex__iff,axiom,
    ! [M2: produc6575502325842934193n_assn,Ms: list_P8527749157015355191n_assn,N: produc6575502325842934193n_assn,Ns: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(M2,Ms),cons_P2971678138204555879n_assn(N,Ns)),lenlex6671093235567761724n_assn(R2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_s6829681357464350627n_assn(Ms)),size_s6829681357464350627n_assn(Ns)))
        | ( ( size_s6829681357464350627n_assn(Ms) = size_s6829681357464350627n_assn(Ns) )
          & member4743710847862083984n_assn(produc6128706242671166167n_assn(M2,N),R2) )
        | ( ( M2 = N )
          & member3994142547374651152n_assn(produc3014731801419794263n_assn(Ms,Ns),lenlex6671093235567761724n_assn(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4745_Cons__lenlex__iff,axiom,
    ! [M2: c,Ms: list_c,N: c,Ns: list_c,R2: set_Product_prod_c_c] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(cons_c(M2,Ms),cons_c(N,Ns)),lenlex_c(R2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_c(Ms)),size_size_list_c(Ns)))
        | ( ( size_size_list_c(Ms) = size_size_list_c(Ns) )
          & member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,M2),N),R2) )
        | ( ( M2 = N )
          & member7012247553653173456list_c(produc5515445818230282391list_c(Ms,Ns),lenlex_c(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4746_Cons__lenlex__iff,axiom,
    ! [M2: d,Ms: list_d,N: d,Ns: list_d,R2: set_Product_prod_d_d] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(cons_d(M2,Ms),cons_d(N,Ns)),lenlex_d(R2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_d(Ms)),size_size_list_d(Ns)))
        | ( ( size_size_list_d(Ms) = size_size_list_d(Ns) )
          & member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,M2),N),R2) )
        | ( ( M2 = N )
          & member1810801192463332112list_d(produc242965421297323863list_d(Ms,Ns),lenlex_d(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4747_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(M2,Ms),cons_nat(N,Ns)),lenlex_nat(R2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_nat(Ms)),size_size_list_nat(Ns)))
        | ( ( size_size_list_nat(Ms) = size_size_list_nat(Ns) )
          & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N)),R2)) )
        | ( ( M2 = N )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4748_Cons__lenlex__iff,axiom,
    ! [M2: int,Ms: list_int,N: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(M2,Ms),cons_int(N,Ns)),lenlex_int(R2))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,size_size_list_int(Ms)),size_size_list_int(Ns)))
        | ( ( size_size_list_int(Ms) = size_size_list_int(Ns) )
          & member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(M2),N),R2) )
        | ( ( M2 = N )
          & member6698963635872716290st_int(produc364263696895485585st_int(Ms,Ns),lenlex_int(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_4749_Cons__in__lex,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( member4080735728053443344at_nat(produc7536900900485677911at_nat(cons_s6881495754146722583at_nat(X,Xs),cons_s6881495754146722583at_nat(Y,Ys)),lex_se2245640040323279819at_nat(R2))
    <=> ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),R2)
          & ( size_s8736152011456118867at_nat(Xs) = size_s8736152011456118867at_nat(Ys) ) )
        | ( ( X = Y )
          & member4080735728053443344at_nat(produc7536900900485677911at_nat(Xs,Ys),lex_se2245640040323279819at_nat(R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4750_Cons__in__lex,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X,Xs),cons_P2971678138204555879n_assn(Y,Ys)),lex_Pr7559194461235888923n_assn(R2))
    <=> ( ( member4743710847862083984n_assn(produc6128706242671166167n_assn(X,Y),R2)
          & ( size_s6829681357464350627n_assn(Xs) = size_s6829681357464350627n_assn(Ys) ) )
        | ( ( X = Y )
          & member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,Ys),lex_Pr7559194461235888923n_assn(R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4751_Cons__in__lex,axiom,
    ! [X: c,Xs: list_c,Y: c,Ys: list_c,R2: set_Product_prod_c_c] :
      ( member7012247553653173456list_c(produc5515445818230282391list_c(cons_c(X,Xs),cons_c(Y,Ys)),lex_c(R2))
    <=> ( ( member5074992359041316560od_c_c(aa_c_P7556687028265917539od_c_c(aa_c_f8473195554499863982od_c_c(product_Pair_c_c,X),Y),R2)
          & ( size_size_list_c(Xs) = size_size_list_c(Ys) ) )
        | ( ( X = Y )
          & member7012247553653173456list_c(produc5515445818230282391list_c(Xs,Ys),lex_c(R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4752_Cons__in__lex,axiom,
    ! [X: d,Xs: list_d,Y: d,Ys: list_d,R2: set_Product_prod_d_d] :
      ( member1810801192463332112list_d(produc242965421297323863list_d(cons_d(X,Xs),cons_d(Y,Ys)),lex_d(R2))
    <=> ( ( member2287536781371869328od_d_d(aa_d_P4129791150264654178od_d_d(aa_d_f9134427263152513838od_d_d(product_Pair_d_d,X),Y),R2)
          & ( size_size_list_d(Xs) = size_size_list_d(Ys) ) )
        | ( ( X = Y )
          & member1810801192463332112list_d(produc242965421297323863list_d(Xs,Ys),lex_d(R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4753_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(X,Xs),cons_nat(Y,Ys)),lex_nat(R2))
    <=> ( ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),R2))
          & ( size_size_list_nat(Xs) = size_size_list_nat(Ys) ) )
        | ( ( X = Y )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,Ys),lex_nat(R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4754_Cons__in__lex,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(X,Xs),cons_int(Y,Ys)),lex_int(R2))
    <=> ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(product_Pair_int_int(X),Y),R2)
          & ( size_size_list_int(Xs) = size_size_list_int(Ys) ) )
        | ( ( X = Y )
          & member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys),lex_int(R2)) ) ) ) ).

% Cons_in_lex
tff(fact_4755_list__assn_Oelims,axiom,
    ! [X: fun_int_fun_int_assn,Xa: list_int,Xb: list_int,Y: assn] :
      ( ( vEBT_L74593716426352029nt_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: int,As: list_int] :
              ( ( Xa = cons_int(A3,As) )
             => ! [C4: int,Cs2: list_int] :
                  ( ( Xb = cons_int(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_int_fun_int_assn(X,A3),C4)),vEBT_L74593716426352029nt_int(X,As,Cs2)) ) ) )
         => ( ( ? [V3: int,Va: list_int] : Xa = cons_int(V3,Va)
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V3: int,Va: list_int] : Xb = cons_int(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4756_list__assn_Oelims,axiom,
    ! [X: fun_int_fun_nat_assn,Xa: list_int,Xb: list_nat,Y: assn] :
      ( ( vEBT_L77084186935402305nt_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: int,As: list_int] :
              ( ( Xa = cons_int(A3,As) )
             => ! [C4: nat,Cs2: list_nat] :
                  ( ( Xb = cons_nat(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_int_fun_nat_assn(X,A3),C4)),vEBT_L77084186935402305nt_nat(X,As,Cs2)) ) ) )
         => ( ( ? [V3: int,Va: list_int] : Xa = cons_int(V3,Va)
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V3: nat,Va: list_nat] : Xb = cons_nat(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4757_list__assn_Oelims,axiom,
    ! [X: fun_in2917358019923704601n_assn,Xa: list_int,Xb: list_P8527749157015355191n_assn,Y: assn] :
      ( ( vEBT_L1620274777744359680n_assn(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_Pr5671120429643327159n_assn )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: int,As: list_int] :
              ( ( Xa = cons_int(A3,As) )
             => ! [C4: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
                  ( ( Xb = cons_P2971678138204555879n_assn(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_int4086841622967611778n_assn(X,A3),C4)),vEBT_L1620274777744359680n_assn(X,As,Cs2)) ) ) )
         => ( ( ? [V3: int,Va: list_int] : Xa = cons_int(V3,Va)
             => ( ( Xb = nil_Pr5671120429643327159n_assn )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : Xb = cons_P2971678138204555879n_assn(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4758_list__assn_Oelims,axiom,
    ! [X: fun_nat_fun_int_assn,Xa: list_nat,Xb: list_int,Y: assn] :
      ( ( vEBT_L8298612041380073281at_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = cons_nat(A3,As) )
             => ! [C4: int,Cs2: list_int] :
                  ( ( Xb = cons_int(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_nat_fun_int_assn(X,A3),C4)),vEBT_L8298612041380073281at_int(X,As,Cs2)) ) ) )
         => ( ( ? [V3: nat,Va: list_nat] : Xa = cons_nat(V3,Va)
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V3: int,Va: list_int] : Xb = cons_int(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4759_list__assn_Oelims,axiom,
    ! [X: fun_nat_fun_nat_assn,Xa: list_nat,Xb: list_nat,Y: assn] :
      ( ( vEBT_L8301102511889123557at_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = cons_nat(A3,As) )
             => ! [C4: nat,Cs2: list_nat] :
                  ( ( Xb = cons_nat(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_nat_fun_nat_assn(X,A3),C4)),vEBT_L8301102511889123557at_nat(X,As,Cs2)) ) ) )
         => ( ( ? [V3: nat,Va: list_nat] : Xa = cons_nat(V3,Va)
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V3: nat,Va: list_nat] : Xb = cons_nat(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4760_list__assn_Oelims,axiom,
    ! [X: fun_na2667955736958379125n_assn,Xa: list_nat,Xb: list_P8527749157015355191n_assn,Y: assn] :
      ( ( vEBT_L9083610150846024284n_assn(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_Pr5671120429643327159n_assn )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: nat,As: list_nat] :
              ( ( Xa = cons_nat(A3,As) )
             => ! [C4: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
                  ( ( Xb = cons_P2971678138204555879n_assn(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_nat3837439340002286302n_assn(X,A3),C4)),vEBT_L9083610150846024284n_assn(X,As,Cs2)) ) ) )
         => ( ( ? [V3: nat,Va: list_nat] : Xa = cons_nat(V3,Va)
             => ( ( Xb = nil_Pr5671120429643327159n_assn )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : Xb = cons_P2971678138204555879n_assn(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4761_list__assn_Oelims,axiom,
    ! [X: fun_Pr7092578583803811605t_assn,Xa: list_P8527749157015355191n_assn,Xb: list_int,Y: assn] :
      ( ( vEBT_L4760345170377931482sn_int(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_Pr5671120429643327159n_assn )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( Xa = cons_P2971678138204555879n_assn(A3,As) )
             => ! [C4: int,Cs2: list_int] :
                  ( ( Xb = cons_int(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_int_assn(aa_Pro9204279173661784262t_assn(X,A3),C4)),vEBT_L4760345170377931482sn_int(X,As,Cs2)) ) ) )
         => ( ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : Xa = cons_P2971678138204555879n_assn(V3,Va)
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_Pr5671120429643327159n_assn )
               => ( ? [V3: int,Va: list_int] : Xb = cons_int(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4762_list__assn_Oelims,axiom,
    ! [X: fun_Pr2676784290456659641t_assn,Xa: list_P8527749157015355191n_assn,Xb: list_nat,Y: assn] :
      ( ( vEBT_L4762835640886981758sn_nat(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_Pr5671120429643327159n_assn )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( Xa = cons_P2971678138204555879n_assn(A3,As) )
             => ! [C4: nat,Cs2: list_nat] :
                  ( ( Xb = cons_nat(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(aa_Pro8228743708455606890t_assn(X,A3),C4)),vEBT_L4762835640886981758sn_nat(X,As,Cs2)) ) ) )
         => ( ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : Xa = cons_P2971678138204555879n_assn(V3,Va)
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_Pr5671120429643327159n_assn )
               => ( ? [V3: nat,Va: list_nat] : Xb = cons_nat(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4763_list__assn_Oelims,axiom,
    ! [X: fun_Pr1914754426620969278n_assn,Xa: list_P8527749157015355191n_assn,Xb: list_P8527749157015355191n_assn,Y: assn] :
      ( ( vEBT_L3108143461992071043n_assn(X,Xa,Xb) = Y )
     => ( ( ( Xa = nil_Pr5671120429643327159n_assn )
         => ( ( Xb = nil_Pr5671120429643327159n_assn )
           => ( Y != one_one_assn ) ) )
       => ( ! [A3: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( Xa = cons_P2971678138204555879n_assn(A3,As) )
             => ! [C4: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
                  ( ( Xb = cons_P2971678138204555879n_assn(C4,Cs2) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_Pro2979818503705714077n_assn(aa_Pro1520987361448537015n_assn(X,A3),C4)),vEBT_L3108143461992071043n_assn(X,As,Cs2)) ) ) )
         => ( ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : Xa = cons_P2971678138204555879n_assn(V3,Va)
             => ( ( Xb = nil_Pr5671120429643327159n_assn )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_Pr5671120429643327159n_assn )
               => ( ? [V3: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : Xb = cons_P2971678138204555879n_assn(V3,Va)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_4764_count__list_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( aa_int_nat(count_list_int(cons_int(X,Xs)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(count_list_int(Xs),Y)),one_one_nat) ) )
      & ( ( X != Y )
       => ( aa_int_nat(count_list_int(cons_int(X,Xs)),Y) = aa_int_nat(count_list_int(Xs),Y) ) ) ) ).

% count_list.simps(2)
tff(fact_4765_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( aa_nat_nat(count_list_nat(cons_nat(X,Xs)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(count_list_nat(Xs),Y)),one_one_nat) ) )
      & ( ( X != Y )
       => ( aa_nat_nat(count_list_nat(cons_nat(X,Xs)),Y) = aa_nat_nat(count_list_nat(Xs),Y) ) ) ) ).

% count_list.simps(2)
tff(fact_4766_count__list_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( count_2530312006313534765n_assn(cons_P2971678138204555879n_assn(X,Xs),Y) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,count_2530312006313534765n_assn(Xs,Y)),one_one_nat) ) )
      & ( ( X != Y )
       => ( count_2530312006313534765n_assn(cons_P2971678138204555879n_assn(X,Xs),Y) = count_2530312006313534765n_assn(Xs,Y) ) ) ) ).

% count_list.simps(2)
tff(fact_4767_count__notin,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ~ member8757157785044589968at_nat(X,set_Pr3765526544606949372at_nat(Xs))
     => ( count_1483299019173949923at_nat(Xs,X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4768_count__notin,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),set_Pr5648618587558075414at_nat(Xs)))
     => ( count_4203492906077236349at_nat(Xs,X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4769_count__notin,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xs: list_f5857023252961307252t_bool] :
      ( ~ member4125802036136091927t_bool(X,set_fu1437410266168021891t_bool(Xs))
     => ( count_188004415726849386t_bool(Xs,X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4770_count__notin,axiom,
    ! [X: int,Xs: list_int] :
      ( ~ member_int(X,set_int2(Xs))
     => ( aa_int_nat(count_list_int(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4771_count__notin,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ member_nat(X,set_nat2(Xs))
     => ( aa_nat_nat(count_list_nat(Xs),X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4772_count__notin,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ~ member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
     => ( count_2530312006313534765n_assn(Xs,X) = zero_zero_nat ) ) ).

% count_notin
tff(fact_4773_power__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
     => ( pp(aa_rat_bool(ord_less_rat(B2),one_one_rat))
       => ( pp(aa_rat_bool(ord_less_eq_rat(power_power_rat(B2,M2)),power_power_rat(B2,N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_4774_power__decreasing__iff,axiom,
    ! [B2: code_integer,M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_4775_power__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_4776_power__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),M2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2)) ) ) ) ).

% power_decreasing_iff
tff(fact_4777_in__measures_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F3: fun_se4097146774670233028at_nat,Fs: list_f2430256441796990922at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(cons_f718425424816052858at_nat(F3,Fs)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_set4749074711808745205at_nat(F3,X)),aa_set4749074711808745205at_nat(F3,Y)))
        | ( ( aa_set4749074711808745205at_nat(F3,X) = aa_set4749074711808745205at_nat(F3,Y) )
          & member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_4778_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F3: fun_nat_nat,Fs: list_fun_nat_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),measures_nat(cons_fun_nat_nat(F3,Fs))))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(F3,X)),aa_nat_nat(F3,Y)))
        | ( ( aa_nat_nat(F3,X) = aa_nat_nat(F3,Y) )
          & pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),measures_nat(Fs))) ) ) ) ).

% in_measures(2)
tff(fact_4779_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),X))
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_4780_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( zip_int_int(Xs,Ys) = nil_Pr2300489316682597567nt_int )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4781_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( zip_int_nat(Xs,Ys) = nil_Pr6478340336191794275nt_nat )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4782_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_in7098856637274830732n_assn(Xs,Ys) = nil_Pr2440414457004260852n_assn )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4783_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( zip_nat_int(Xs,Ys) = nil_Pr1301135604781543011at_int )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4784_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( zip_nat_nat(Xs,Ys) = nil_Pr5478986624290739719at_nat )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4785_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_na5338819973521719528n_assn(Xs,Ys) = nil_Pr680377793251149648n_assn )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4786_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( zip_Pr1015554993053626726sn_int(Xs,Ys) = nil_Pr7991254345159696774sn_int )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4787_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( zip_Pr1018045463562677002sn_nat(Xs,Ys) = nil_Pr2945733327814117674sn_nat )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4788_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_Pr8806919808394484215n_assn(Xs,Ys) = nil_Pr4327835960982145133n_assn )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4789_zip__eq__Nil__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( zip_assn_assn(Xs,Ys) = nil_Pr5671120429643327159n_assn )
    <=> ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_4790_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_Pr2300489316682597567nt_int = zip_int_int(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4791_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( nil_Pr6478340336191794275nt_nat = zip_int_nat(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4792_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr2440414457004260852n_assn = zip_in7098856637274830732n_assn(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4793_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( nil_Pr1301135604781543011at_int = zip_nat_int(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4794_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat = zip_nat_nat(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4795_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr680377793251149648n_assn = zip_na5338819973521719528n_assn(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4796_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( nil_Pr7991254345159696774sn_int = zip_Pr1015554993053626726sn_int(Xs,Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4797_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( nil_Pr2945733327814117674sn_nat = zip_Pr1018045463562677002sn_nat(Xs,Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4798_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr4327835960982145133n_assn = zip_Pr8806919808394484215n_assn(Xs,Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4799_Nil__eq__zip__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( nil_Pr5671120429643327159n_assn = zip_assn_assn(Xs,Ys) )
    <=> ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_4800_zip__Nil,axiom,
    ! [Ys: list_assn] : zip_assn_assn(nil_assn,Ys) = nil_Pr5671120429643327159n_assn ).

% zip_Nil
tff(fact_4801_Nil__is__rev__conv,axiom,
    ! [Xs: list_int] :
      ( ( nil_int = rev_int(Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_rev_conv
tff(fact_4802_Nil__is__rev__conv,axiom,
    ! [Xs: list_nat] :
      ( ( nil_nat = rev_nat(Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_rev_conv
tff(fact_4803_Nil__is__rev__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn = rev_Pr4855572775806611735n_assn(Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_rev_conv
tff(fact_4804_rev__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( rev_int(Xs) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% rev_is_Nil_conv
tff(fact_4805_rev__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( rev_nat(Xs) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% rev_is_Nil_conv
tff(fact_4806_rev__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( rev_Pr4855572775806611735n_assn(Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rev_is_Nil_conv
tff(fact_4807_list__update__nonempty,axiom,
    ! [Xs: list_int,K3: nat,X: int] :
      ( ( list_update_int(Xs,K3,X) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% list_update_nonempty
tff(fact_4808_list__update__nonempty,axiom,
    ! [Xs: list_nat,K3: nat,X: nat] :
      ( ( list_update_nat(Xs,K3,X) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% list_update_nonempty
tff(fact_4809_list__update__nonempty,axiom,
    ! [Xs: list_P8527749157015355191n_assn,K3: nat,X: produc6575502325842934193n_assn] :
      ( ( list_u4534839942911652127n_assn(Xs,K3,X) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% list_update_nonempty
tff(fact_4810_list__update__nonempty,axiom,
    ! [Xs: list_c,K3: nat,X: c] :
      ( ( list_update_c(Xs,K3,X) = nil_c )
    <=> ( Xs = nil_c ) ) ).

% list_update_nonempty
tff(fact_4811_list__update__nonempty,axiom,
    ! [Xs: list_d,K3: nat,X: d] :
      ( ( list_update_d(Xs,K3,X) = nil_d )
    <=> ( Xs = nil_d ) ) ).

% list_update_nonempty
tff(fact_4812_enumerate__simps_I1_J,axiom,
    ! [N: nat] : enumerate_int(N,nil_int) = nil_Pr1301135604781543011at_int ).

% enumerate_simps(1)
tff(fact_4813_enumerate__simps_I1_J,axiom,
    ! [N: nat] : enumerate_nat(N,nil_nat) = nil_Pr5478986624290739719at_nat ).

% enumerate_simps(1)
tff(fact_4814_enumerate__simps_I1_J,axiom,
    ! [N: nat] : enumer9200277086812776726n_assn(N,nil_Pr5671120429643327159n_assn) = nil_Pr680377793251149648n_assn ).

% enumerate_simps(1)
tff(fact_4815_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( rotate1_int(Xs) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
tff(fact_4816_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( rotate1_nat(Xs) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
tff(fact_4817_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( rotate328796349445179396n_assn(Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rotate1_is_Nil_conv
tff(fact_4818_list__ex__simps_I2_J,axiom,
    ! [P: fun_int_bool] : ~ pp(aa_list_int_bool(list_ex_int(P),nil_int)) ).

% list_ex_simps(2)
tff(fact_4819_list__ex__simps_I2_J,axiom,
    ! [P: fun_nat_bool] : ~ pp(aa_list_nat_bool(list_ex_nat(P),nil_nat)) ).

% list_ex_simps(2)
tff(fact_4820_list__ex__simps_I2_J,axiom,
    ! [P: fun_Pr8546840118223405289n_bool] : ~ pp(aa_lis5212129917091495004n_bool(list_e1087220904399170805n_assn(P),nil_Pr5671120429643327159n_assn)) ).

% list_ex_simps(2)
tff(fact_4821_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_int(Begin,End,nil_int) = nil_int ).

% slice_Nil
tff(fact_4822_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_nat(Begin,End,nil_nat) = nil_nat ).

% slice_Nil
tff(fact_4823_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_7964966981652229398n_assn(Begin,End,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% slice_Nil
tff(fact_4824_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_int] : slice_int(I,I,Xs) = nil_int ).

% slice_eq_bounds_empty
tff(fact_4825_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_nat] : slice_nat(I,I,Xs) = nil_nat ).

% slice_eq_bounds_empty
tff(fact_4826_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn] : slice_7964966981652229398n_assn(I,I,Xs) = nil_Pr5671120429643327159n_assn ).

% slice_eq_bounds_empty
tff(fact_4827_in__measures_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ~ member8757157785044589968at_nat(produc2922128104949294807at_nat(X,Y),measur2694323259624372065at_nat(nil_fu8701418733234493130at_nat)) ).

% in_measures(1)
tff(fact_4828_in__measures_I1_J,axiom,
    ! [X: nat,Y: nat] : ~ pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),measures_nat(nil_fun_nat_nat))) ).

% in_measures(1)
tff(fact_4829_power__inject__exp,axiom,
    ! [A2: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(ord_less_rat(one_one_rat),A2))
     => ( ( power_power_rat(A2,M2) = power_power_rat(A2,N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_4830_power__inject__exp,axiom,
    ! [A2: code_integer,M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),A2))
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),M2) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_4831_power__inject__exp,axiom,
    ! [A2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),A2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_4832_power__inject__exp,axiom,
    ! [A2: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),A2))
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_4833_set__empty,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( set_Pr1139785259514867910n_assn(Xs) = bot_bo1176836662018730877n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% set_empty
tff(fact_4834_set__empty,axiom,
    ! [Xs: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat(Xs) = bot_bo228742789529271731at_nat )
    <=> ( Xs = nil_Pr8341282898164651117at_nat ) ) ).

% set_empty
tff(fact_4835_set__empty,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat(Xs) = bot_bo2099793752762293965at_nat )
    <=> ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% set_empty
tff(fact_4836_set__empty,axiom,
    ! [Xs: list_bool] :
      ( ( set_bool2(Xs) = bot_bot_set_bool )
    <=> ( Xs = nil_bool ) ) ).

% set_empty
tff(fact_4837_set__empty,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2(Xs) = bot_bot_set_nat )
    <=> ( Xs = nil_nat ) ) ).

% set_empty
tff(fact_4838_set__empty,axiom,
    ! [Xs: list_int] :
      ( ( set_int2(Xs) = bot_bot_set_int )
    <=> ( Xs = nil_int ) ) ).

% set_empty
tff(fact_4839_set__empty2,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( bot_bo1176836662018730877n_assn = set_Pr1139785259514867910n_assn(Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% set_empty2
tff(fact_4840_set__empty2,axiom,
    ! [Xs: list_P5464809261938338413at_nat] :
      ( ( bot_bo228742789529271731at_nat = set_Pr3765526544606949372at_nat(Xs) )
    <=> ( Xs = nil_Pr8341282898164651117at_nat ) ) ).

% set_empty2
tff(fact_4841_set__empty2,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( bot_bo2099793752762293965at_nat = set_Pr5648618587558075414at_nat(Xs) )
    <=> ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% set_empty2
tff(fact_4842_set__empty2,axiom,
    ! [Xs: list_bool] :
      ( ( bot_bot_set_bool = set_bool2(Xs) )
    <=> ( Xs = nil_bool ) ) ).

% set_empty2
tff(fact_4843_set__empty2,axiom,
    ! [Xs: list_nat] :
      ( ( bot_bot_set_nat = set_nat2(Xs) )
    <=> ( Xs = nil_nat ) ) ).

% set_empty2
tff(fact_4844_set__empty2,axiom,
    ! [Xs: list_int] :
      ( ( bot_bot_set_int = set_int2(Xs) )
    <=> ( Xs = nil_int ) ) ).

% set_empty2
tff(fact_4845_length__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn(Xs) = zero_zero_nat )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% length_0_conv
tff(fact_4846_length__0__conv,axiom,
    ! [Xs: list_c] :
      ( ( size_size_list_c(Xs) = zero_zero_nat )
    <=> ( Xs = nil_c ) ) ).

% length_0_conv
tff(fact_4847_length__0__conv,axiom,
    ! [Xs: list_d] :
      ( ( size_size_list_d(Xs) = zero_zero_nat )
    <=> ( Xs = nil_d ) ) ).

% length_0_conv
tff(fact_4848_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat(Xs) = zero_zero_nat )
    <=> ( Xs = nil_nat ) ) ).

% length_0_conv
tff(fact_4849_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int(Xs) = zero_zero_nat )
    <=> ( Xs = nil_int ) ) ).

% length_0_conv
tff(fact_4850_singleton__rev__conv,axiom,
    ! [X: int,Xs: list_int] :
      ( ( cons_int(X,nil_int) = rev_int(Xs) )
    <=> ( cons_int(X,nil_int) = Xs ) ) ).

% singleton_rev_conv
tff(fact_4851_singleton__rev__conv,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat(X,nil_nat) = rev_nat(Xs) )
    <=> ( cons_nat(X,nil_nat) = Xs ) ) ).

% singleton_rev_conv
tff(fact_4852_singleton__rev__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn) = rev_Pr4855572775806611735n_assn(Xs) )
    <=> ( cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn) = Xs ) ) ).

% singleton_rev_conv
tff(fact_4853_rev__singleton__conv,axiom,
    ! [Xs: list_int,X: int] :
      ( ( rev_int(Xs) = cons_int(X,nil_int) )
    <=> ( Xs = cons_int(X,nil_int) ) ) ).

% rev_singleton_conv
tff(fact_4854_rev__singleton__conv,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( rev_nat(Xs) = cons_nat(X,nil_nat) )
    <=> ( Xs = cons_nat(X,nil_nat) ) ) ).

% rev_singleton_conv
tff(fact_4855_rev__singleton__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( rev_Pr4855572775806611735n_assn(Xs) = cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn) )
    <=> ( Xs = cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn) ) ) ).

% rev_singleton_conv
tff(fact_4856_Nil__lenlex__iff1,axiom,
    ! [Ns: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(nil_int,Ns),lenlex_int(R2))
    <=> ( Ns != nil_int ) ) ).

% Nil_lenlex_iff1
tff(fact_4857_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(nil_nat,Ns),lenlex_nat(R2))
    <=> ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
tff(fact_4858_Nil__lenlex__iff1,axiom,
    ! [Ns: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] :
      ( member3994142547374651152n_assn(produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,Ns),lenlex6671093235567761724n_assn(R2))
    <=> ( Ns != nil_Pr5671120429643327159n_assn ) ) ).

% Nil_lenlex_iff1
tff(fact_4859_insert__Nil,axiom,
    ! [X: int] : insert_int(X,nil_int) = cons_int(X,nil_int) ).

% insert_Nil
tff(fact_4860_insert__Nil,axiom,
    ! [X: nat] : insert_nat(X,nil_nat) = cons_nat(X,nil_nat) ).

% insert_Nil
tff(fact_4861_insert__Nil,axiom,
    ! [X: produc6575502325842934193n_assn] : insert3246601298802261197n_assn(X,nil_Pr5671120429643327159n_assn) = cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn) ).

% insert_Nil
tff(fact_4862_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_int_fun_int_assn,L3: list_int] : vEBT_L74593716426352029nt_int(P,nil_int,L3) = pure_assn(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,L3),nil_int)) ).

% list_assn_aux_simps(1)
tff(fact_4863_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_int_fun_nat_assn,L3: list_nat] : vEBT_L77084186935402305nt_nat(P,nil_int,L3) = pure_assn(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L3),nil_nat)) ).

% list_assn_aux_simps(1)
tff(fact_4864_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_in2917358019923704601n_assn,L3: list_P8527749157015355191n_assn] : vEBT_L1620274777744359680n_assn(P,nil_int,L3) = pure_assn(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,L3),nil_Pr5671120429643327159n_assn)) ).

% list_assn_aux_simps(1)
tff(fact_4865_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_nat_fun_int_assn,L3: list_int] : vEBT_L8298612041380073281at_int(P,nil_nat,L3) = pure_assn(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,L3),nil_int)) ).

% list_assn_aux_simps(1)
tff(fact_4866_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_nat_fun_nat_assn,L3: list_nat] : vEBT_L8301102511889123557at_nat(P,nil_nat,L3) = pure_assn(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L3),nil_nat)) ).

% list_assn_aux_simps(1)
tff(fact_4867_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_na2667955736958379125n_assn,L3: list_P8527749157015355191n_assn] : vEBT_L9083610150846024284n_assn(P,nil_nat,L3) = pure_assn(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,L3),nil_Pr5671120429643327159n_assn)) ).

% list_assn_aux_simps(1)
tff(fact_4868_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_Pr7092578583803811605t_assn,L3: list_int] : vEBT_L4760345170377931482sn_int(P,nil_Pr5671120429643327159n_assn,L3) = pure_assn(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,L3),nil_int)) ).

% list_assn_aux_simps(1)
tff(fact_4869_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_Pr2676784290456659641t_assn,L3: list_nat] : vEBT_L4762835640886981758sn_nat(P,nil_Pr5671120429643327159n_assn,L3) = pure_assn(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L3),nil_nat)) ).

% list_assn_aux_simps(1)
tff(fact_4870_list__assn__aux__simps_I1_J,axiom,
    ! [P: fun_Pr1914754426620969278n_assn,L3: list_P8527749157015355191n_assn] : vEBT_L3108143461992071043n_assn(P,nil_Pr5671120429643327159n_assn,L3) = pure_assn(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,L3),nil_Pr5671120429643327159n_assn)) ).

% list_assn_aux_simps(1)
tff(fact_4871_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_int_fun_int_assn,L: list_int] : vEBT_L74593716426352029nt_int(P,L,nil_int) = pure_assn(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,L),nil_int)) ).

% list_assn_aux_simps(2)
tff(fact_4872_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_nat_fun_int_assn,L: list_nat] : vEBT_L8298612041380073281at_int(P,L,nil_int) = pure_assn(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L),nil_nat)) ).

% list_assn_aux_simps(2)
tff(fact_4873_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_Pr7092578583803811605t_assn,L: list_P8527749157015355191n_assn] : vEBT_L4760345170377931482sn_int(P,L,nil_int) = pure_assn(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,L),nil_Pr5671120429643327159n_assn)) ).

% list_assn_aux_simps(2)
tff(fact_4874_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_int_fun_nat_assn,L: list_int] : vEBT_L77084186935402305nt_nat(P,L,nil_nat) = pure_assn(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,L),nil_int)) ).

% list_assn_aux_simps(2)
tff(fact_4875_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_nat_fun_nat_assn,L: list_nat] : vEBT_L8301102511889123557at_nat(P,L,nil_nat) = pure_assn(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L),nil_nat)) ).

% list_assn_aux_simps(2)
tff(fact_4876_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_Pr2676784290456659641t_assn,L: list_P8527749157015355191n_assn] : vEBT_L4762835640886981758sn_nat(P,L,nil_nat) = pure_assn(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,L),nil_Pr5671120429643327159n_assn)) ).

% list_assn_aux_simps(2)
tff(fact_4877_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_in2917358019923704601n_assn,L: list_int] : vEBT_L1620274777744359680n_assn(P,L,nil_Pr5671120429643327159n_assn) = pure_assn(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,L),nil_int)) ).

% list_assn_aux_simps(2)
tff(fact_4878_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_na2667955736958379125n_assn,L: list_nat] : vEBT_L9083610150846024284n_assn(P,L,nil_Pr5671120429643327159n_assn) = pure_assn(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,L),nil_nat)) ).

% list_assn_aux_simps(2)
tff(fact_4879_list__assn__aux__simps_I2_J,axiom,
    ! [P: fun_Pr1914754426620969278n_assn,L: list_P8527749157015355191n_assn] : vEBT_L3108143461992071043n_assn(P,L,nil_Pr5671120429643327159n_assn) = pure_assn(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,L),nil_Pr5671120429643327159n_assn)) ).

% list_assn_aux_simps(2)
tff(fact_4880_power__strict__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(ord_less_rat(one_one_rat),B2))
     => ( pp(aa_rat_bool(ord_less_rat(power_power_rat(B2,X)),power_power_rat(B2,Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_4881_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_4882_power__strict__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_4883_power__strict__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ) ).

% power_strict_increasing_iff
tff(fact_4884_power__eq__0__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = zero_z3403309356797280102nteger )
    <=> ( ( A2 = zero_z3403309356797280102nteger )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_4885_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat(A2,N) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_4886_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_4887_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% power_eq_0_iff
tff(fact_4888_length__greater__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_s6829681357464350627n_assn(Xs)))
    <=> ( Xs != nil_Pr5671120429643327159n_assn ) ) ).

% length_greater_0_conv
tff(fact_4889_length__greater__0__conv,axiom,
    ! [Xs: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_c(Xs)))
    <=> ( Xs != nil_c ) ) ).

% length_greater_0_conv
tff(fact_4890_length__greater__0__conv,axiom,
    ! [Xs: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_d(Xs)))
    <=> ( Xs != nil_d ) ) ).

% length_greater_0_conv
tff(fact_4891_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)))
    <=> ( Xs != nil_nat ) ) ).

% length_greater_0_conv
tff(fact_4892_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)))
    <=> ( Xs != nil_int ) ) ).

% length_greater_0_conv
tff(fact_4893_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M2: nat,N: nat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),B2))
     => ( pp(aa_rat_bool(ord_less_rat(B2),one_one_rat))
       => ( pp(aa_rat_bool(ord_less_rat(power_power_rat(B2,M2)),power_power_rat(B2,N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4894_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M2: nat,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,B2),one_one_Code_integer))
       => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4895_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,B2),one_one_nat))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4896_power__strict__decreasing__iff,axiom,
    ! [B2: int,M2: nat,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,B2),one_one_int))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),M2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
        <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2)) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_4897_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)))
          <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_4898_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),A2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(zero_zero_rat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_rat_bool(ord_less_eq_rat(power_power_rat(A2,N)),power_power_rat(B2,N)))
          <=> pp(aa_rat_bool(ord_less_eq_rat(A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_4899_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),A2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,zero_zero_nat),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)))
          <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_4900_power__mono__iff,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),A2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)))
          <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,A2),B2)) ) ) ) ) ).

% power_mono_iff
tff(fact_4901_power__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( pp(aa_rat_bool(ord_less_rat(one_one_rat),B2))
     => ( pp(aa_rat_bool(ord_less_eq_rat(power_power_rat(B2,X)),power_power_rat(B2,Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_4902_power__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,one_one_Code_integer),B2))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),X)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_4903_power__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_4904_power__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),B2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),Y)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,X),Y)) ) ) ).

% power_increasing_iff
tff(fact_4905_length__ge__1__conv,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_s6829681357464350627n_assn(L)))
    <=> ( L != nil_Pr5671120429643327159n_assn ) ) ).

% length_ge_1_conv
tff(fact_4906_length__ge__1__conv,axiom,
    ! [L: list_c] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_c(L)))
    <=> ( L != nil_c ) ) ).

% length_ge_1_conv
tff(fact_4907_length__ge__1__conv,axiom,
    ! [L: list_d] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_d(L)))
    <=> ( L != nil_d ) ) ).

% length_ge_1_conv
tff(fact_4908_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_nat(L)))
    <=> ( L != nil_nat ) ) ).

% length_ge_1_conv
tff(fact_4909_length__ge__1__conv,axiom,
    ! [L: list_int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_int(L)))
    <=> ( L != nil_int ) ) ).

% length_ge_1_conv
tff(fact_4910_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] : X != cons_list_int(nil_int,Xss)
       => ~ ! [X2: int,Xs2: list_int,Xss: list_list_int] : X != cons_list_int(cons_int(X2,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_4911_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] : X != cons_list_nat(nil_nat,Xss)
       => ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] : X != cons_list_nat(cons_nat(X2,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_4912_transpose_Ocases,axiom,
    ! [X: list_l6351802567095793725n_assn] :
      ( ( X != nil_li5476096274760905021n_assn )
     => ( ! [Xss: list_l6351802567095793725n_assn] : X != cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,Xss)
       => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Xss: list_l6351802567095793725n_assn] : X != cons_l2423627976422276333n_assn(cons_P2971678138204555879n_assn(X2,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_4913_Nil__lenlex__iff2,axiom,
    ! [Ns: list_int,R2: set_Pr958786334691620121nt_int] : ~ member6698963635872716290st_int(produc364263696895485585st_int(Ns,nil_int),lenlex_int(R2)) ).

% Nil_lenlex_iff2
tff(fact_4914_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R2: set_Pr1261947904930325089at_nat] : ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Ns,nil_nat),lenlex_nat(R2)) ).

% Nil_lenlex_iff2
tff(fact_4915_Nil__lenlex__iff2,axiom,
    ! [Ns: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] : ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(Ns,nil_Pr5671120429643327159n_assn),lenlex6671093235567761724n_assn(R2)) ).

% Nil_lenlex_iff2
tff(fact_4916_Nil2__notin__lex,axiom,
    ! [Xs: list_int,R2: set_Pr958786334691620121nt_int] : ~ member6698963635872716290st_int(produc364263696895485585st_int(Xs,nil_int),lex_int(R2)) ).

% Nil2_notin_lex
tff(fact_4917_Nil2__notin__lex,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] : ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(Xs,nil_nat),lex_nat(R2)) ).

% Nil2_notin_lex
tff(fact_4918_Nil2__notin__lex,axiom,
    ! [Xs: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] : ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(Xs,nil_Pr5671120429643327159n_assn),lex_Pr7559194461235888923n_assn(R2)) ).

% Nil2_notin_lex
tff(fact_4919_Nil__notin__lex,axiom,
    ! [Ys: list_int,R2: set_Pr958786334691620121nt_int] : ~ member6698963635872716290st_int(produc364263696895485585st_int(nil_int,Ys),lex_int(R2)) ).

% Nil_notin_lex
tff(fact_4920_Nil__notin__lex,axiom,
    ! [Ys: list_nat,R2: set_Pr1261947904930325089at_nat] : ~ member7340969449405702474st_nat(produc2694037385005941721st_nat(nil_nat,Ys),lex_nat(R2)) ).

% Nil_notin_lex
tff(fact_4921_Nil__notin__lex,axiom,
    ! [Ys: list_P8527749157015355191n_assn,R2: set_Pr8116311503512020807n_assn] : ~ member3994142547374651152n_assn(produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,Ys),lex_Pr7559194461235888923n_assn(R2)) ).

% Nil_notin_lex
tff(fact_4922_zip_Osimps_I1_J,axiom,
    ! [Xs: list_assn] : zip_assn_assn(Xs,nil_assn) = nil_Pr5671120429643327159n_assn ).

% zip.simps(1)
tff(fact_4923_mset__distrib,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,M5: multis2468970476368604999at_nat,N7: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat(A,B) = plus_p7104986032573967614at_nat(M5,N7) )
     => ~ ! [Am: multis2468970476368604999at_nat,An: multis2468970476368604999at_nat] :
            ( ( A = plus_p7104986032573967614at_nat(Am,An) )
           => ! [Bm: multis2468970476368604999at_nat,Bn: multis2468970476368604999at_nat] :
                ( ( B = plus_p7104986032573967614at_nat(Bm,Bn) )
               => ( ( M5 = plus_p7104986032573967614at_nat(Am,Bm) )
                 => ( N7 != plus_p7104986032573967614at_nat(An,Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_4924_count__list_Osimps_I1_J,axiom,
    ! [Y: int] : aa_int_nat(count_list_int(nil_int),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_4925_count__list_Osimps_I1_J,axiom,
    ! [Y: nat] : aa_nat_nat(count_list_nat(nil_nat),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_4926_count__list_Osimps_I1_J,axiom,
    ! [Y: produc6575502325842934193n_assn] : count_2530312006313534765n_assn(nil_Pr5671120429643327159n_assn,Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_4927_product_Osimps_I1_J,axiom,
    ! [Uu: list_assn] : product_assn_assn(nil_assn,Uu) = nil_Pr5671120429643327159n_assn ).

% product.simps(1)
tff(fact_4928_power__commutes,axiom,
    ! [A2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_commutes
tff(fact_4929_power__commutes,axiom,
    ! [A2: assn,N: nat] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A2),N)),A2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),aa_nat_assn(power_power_assn(A2),N)) ).

% power_commutes
tff(fact_4930_power__commutes,axiom,
    ! [A2: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(A2,N)),A2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),power_power_rat(A2,N)) ).

% power_commutes
tff(fact_4931_power__commutes,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_commutes
tff(fact_4932_power__commutes,axiom,
    ! [A2: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ).

% power_commutes
tff(fact_4933_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ).

% power_mult_distrib
tff(fact_4934_power__mult__distrib,axiom,
    ! [A2: assn,B2: assn,N: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),N) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A2),N)),aa_nat_assn(power_power_assn(B2),N)) ).

% power_mult_distrib
tff(fact_4935_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N: nat] : power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),B2),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(A2,N)),power_power_rat(B2,N)) ).

% power_mult_distrib
tff(fact_4936_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ).

% power_mult_distrib
tff(fact_4937_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B2),N)) ).

% power_mult_distrib
tff(fact_4938_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),N)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(power_8256067586552552935nteger(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_4939_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),X) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_4940_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: 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,power_power_rat(X,N)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),power_power_rat(X,N)) ) ) ).

% power_commuting_commutes
tff(fact_4941_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: 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(aa_nat_fun_nat_nat(power_power_nat,X),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_4942_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: 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(aa_int_fun_nat_int(power_power_int,X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_4943_successively_Ocases,axiom,
    ! [X: produc1108878098825469532st_int] :
      ( ! [P3: fun_int_fun_int_bool] : X != produc5495881622212842710st_int(P3,nil_int)
     => ( ! [P3: fun_int_fun_int_bool,X2: int] : X != produc5495881622212842710st_int(P3,cons_int(X2,nil_int))
       => ~ ! [P3: fun_int_fun_int_bool,X2: int,Y2: int,Xs2: list_int] : X != produc5495881622212842710st_int(P3,cons_int(X2,cons_int(Y2,Xs2))) ) ) ).

% successively.cases
tff(fact_4944_successively_Ocases,axiom,
    ! [X: produc3430805707523358904st_nat] :
      ( ! [P3: fun_nat_fun_nat_bool] : X != produc282205068633426226st_nat(P3,nil_nat)
     => ( ! [P3: fun_nat_fun_nat_bool,X2: nat] : X != produc282205068633426226st_nat(P3,cons_nat(X2,nil_nat))
       => ~ ! [P3: fun_nat_fun_nat_bool,X2: nat,Y2: nat,Xs2: list_nat] : X != produc282205068633426226st_nat(P3,cons_nat(X2,cons_nat(Y2,Xs2))) ) ) ).

% successively.cases
tff(fact_4945_successively_Ocases,axiom,
    ! [X: produc5668207710305355377n_assn] :
      ( ! [P3: fun_Pr6563606323009366189n_bool] : X != produc7632362218354018657n_assn(P3,nil_Pr5671120429643327159n_assn)
     => ( ! [P3: fun_Pr6563606323009366189n_bool,X2: produc6575502325842934193n_assn] : X != produc7632362218354018657n_assn(P3,cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn))
       => ~ ! [P3: fun_Pr6563606323009366189n_bool,X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != produc7632362218354018657n_assn(P3,cons_P2971678138204555879n_assn(X2,cons_P2971678138204555879n_assn(Y2,Xs2))) ) ) ).

% successively.cases
tff(fact_4946_sorted__wrt_Ocases,axiom,
    ! [X: produc1108878098825469532st_int] :
      ( ! [P3: fun_int_fun_int_bool] : X != produc5495881622212842710st_int(P3,nil_int)
     => ~ ! [P3: fun_int_fun_int_bool,X2: int,Ys5: list_int] : X != produc5495881622212842710st_int(P3,cons_int(X2,Ys5)) ) ).

% sorted_wrt.cases
tff(fact_4947_sorted__wrt_Ocases,axiom,
    ! [X: produc3430805707523358904st_nat] :
      ( ! [P3: fun_nat_fun_nat_bool] : X != produc282205068633426226st_nat(P3,nil_nat)
     => ~ ! [P3: fun_nat_fun_nat_bool,X2: nat,Ys5: list_nat] : X != produc282205068633426226st_nat(P3,cons_nat(X2,Ys5)) ) ).

% sorted_wrt.cases
tff(fact_4948_sorted__wrt_Ocases,axiom,
    ! [X: produc5668207710305355377n_assn] :
      ( ! [P3: fun_Pr6563606323009366189n_bool] : X != produc7632362218354018657n_assn(P3,nil_Pr5671120429643327159n_assn)
     => ~ ! [P3: fun_Pr6563606323009366189n_bool,X2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : X != produc7632362218354018657n_assn(P3,cons_P2971678138204555879n_assn(X2,Ys5)) ) ).

% sorted_wrt.cases
tff(fact_4949_neq__NilE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [X2: int,Xs2: list_int] : L != cons_int(X2,Xs2) ) ).

% neq_NilE
tff(fact_4950_neq__NilE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] : L != cons_nat(X2,Xs2) ) ).

% neq_NilE
tff(fact_4951_neq__NilE,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( L != nil_Pr5671120429643327159n_assn )
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : L != cons_P2971678138204555879n_assn(X2,Xs2) ) ).

% neq_NilE
tff(fact_4952_list__2pre__induct,axiom,
    ! [P: fun_li582272206764200913t_bool,W1: list_int,W2: list_int] :
      ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),nil_int))
     => ( ! [E2: int,W12: list_int,W22: list_int] :
            ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W12),W22))
           => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(E2,W12)),W22)) )
       => ( ! [E2: int,W13: list_int,W23: list_int] :
              ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W13),W23))
             => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W13),cons_int(E2,W23))) )
         => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4953_list__2pre__induct,axiom,
    ! [P: fun_li7501238334638356269t_bool,W1: list_int,W2: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),nil_nat))
     => ( ! [E2: int,W12: list_int,W22: list_nat] :
            ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W12),W22))
           => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(E2,W12)),W22)) )
       => ( ! [E2: nat,W13: list_int,W23: list_nat] :
              ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W13),W23))
             => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W13),cons_nat(E2,W23))) )
         => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4954_list__2pre__induct,axiom,
    ! [P: fun_li6284172134459303836n_bool,W1: list_int,W2: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,nil_int),nil_Pr5671120429643327159n_assn))
     => ( ! [E2: int,W12: list_int,W22: list_P8527749157015355191n_assn] :
            ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,W12),W22))
           => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,cons_int(E2,W12)),W22)) )
       => ( ! [E2: produc6575502325842934193n_assn,W13: list_int,W23: list_P8527749157015355191n_assn] :
              ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,W13),W23))
             => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,W13),cons_P2971678138204555879n_assn(E2,W23))) )
         => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4955_list__2pre__induct,axiom,
    ! [P: fun_li1679355187408199469t_bool,W1: list_nat,W2: list_int] :
      ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),nil_int))
     => ( ! [E2: nat,W12: list_nat,W22: list_int] :
            ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W12),W22))
           => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(E2,W12)),W22)) )
       => ( ! [E2: int,W13: list_nat,W23: list_int] :
              ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W13),W23))
             => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W13),cons_int(E2,W23))) )
         => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4956_list__2pre__induct,axiom,
    ! [P: fun_li8598321315282354825t_bool,W1: list_nat,W2: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),nil_nat))
     => ( ! [E2: nat,W12: list_nat,W22: list_nat] :
            ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W12),W22))
           => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(E2,W12)),W22)) )
       => ( ! [E2: nat,W13: list_nat,W23: list_nat] :
              ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W13),W23))
             => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W13),cons_nat(E2,W23))) )
         => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4957_list__2pre__induct,axiom,
    ! [P: fun_li3458975838448298304n_bool,W1: list_nat,W2: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,nil_nat),nil_Pr5671120429643327159n_assn))
     => ( ! [E2: nat,W12: list_nat,W22: list_P8527749157015355191n_assn] :
            ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,W12),W22))
           => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,cons_nat(E2,W12)),W22)) )
       => ( ! [E2: produc6575502325842934193n_assn,W13: list_nat,W23: list_P8527749157015355191n_assn] :
              ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,W13),W23))
             => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,W13),cons_P2971678138204555879n_assn(E2,W23))) )
         => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4958_list__2pre__induct,axiom,
    ! [P: fun_li1781568775383035136t_bool,W1: list_P8527749157015355191n_assn,W2: list_int] :
      ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,nil_Pr5671120429643327159n_assn),nil_int))
     => ( ! [E2: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_int] :
            ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,W12),W22))
           => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,cons_P2971678138204555879n_assn(E2,W12)),W22)) )
       => ( ! [E2: int,W13: list_P8527749157015355191n_assn,W23: list_int] :
              ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,W13),W23))
             => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,W13),cons_int(E2,W23))) )
         => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4959_list__2pre__induct,axiom,
    ! [P: fun_li8700534903257190492t_bool,W1: list_P8527749157015355191n_assn,W2: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,nil_Pr5671120429643327159n_assn),nil_nat))
     => ( ! [E2: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_nat] :
            ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,W12),W22))
           => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,cons_P2971678138204555879n_assn(E2,W12)),W22)) )
       => ( ! [E2: nat,W13: list_P8527749157015355191n_assn,W23: list_nat] :
              ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,W13),W23))
             => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,W13),cons_nat(E2,W23))) )
         => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4960_list__2pre__induct,axiom,
    ! [P: fun_li4979294636826588077n_bool,W1: list_P8527749157015355191n_assn,W2: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,nil_Pr5671120429643327159n_assn),nil_Pr5671120429643327159n_assn))
     => ( ! [E2: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_P8527749157015355191n_assn] :
            ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,W12),W22))
           => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,cons_P2971678138204555879n_assn(E2,W12)),W22)) )
       => ( ! [E2: produc6575502325842934193n_assn,W13: list_P8527749157015355191n_assn,W23: list_P8527749157015355191n_assn] :
              ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,W13),W23))
             => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,W13),cons_P2971678138204555879n_assn(E2,W23))) )
         => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,W1),W2)) ) ) ) ).

% list_2pre_induct
tff(fact_4961_list__induct__first2,axiom,
    ! [P: fun_list_int_bool,Xs: list_int] :
      ( pp(aa_list_int_bool(P,nil_int))
     => ( ! [X2: int] : pp(aa_list_int_bool(P,cons_int(X2,nil_int)))
       => ( ! [X12: int,X23: int,Xs2: list_int] :
              ( pp(aa_list_int_bool(P,Xs2))
             => pp(aa_list_int_bool(P,cons_int(X12,cons_int(X23,Xs2)))) )
         => pp(aa_list_int_bool(P,Xs)) ) ) ) ).

% list_induct_first2
tff(fact_4962_list__induct__first2,axiom,
    ! [P: fun_list_nat_bool,Xs: list_nat] :
      ( pp(aa_list_nat_bool(P,nil_nat))
     => ( ! [X2: nat] : pp(aa_list_nat_bool(P,cons_nat(X2,nil_nat)))
       => ( ! [X12: nat,X23: nat,Xs2: list_nat] :
              ( pp(aa_list_nat_bool(P,Xs2))
             => pp(aa_list_nat_bool(P,cons_nat(X12,cons_nat(X23,Xs2)))) )
         => pp(aa_list_nat_bool(P,Xs)) ) ) ) ).

% list_induct_first2
tff(fact_4963_list__induct__first2,axiom,
    ! [P: fun_li2741412277222221411n_bool,Xs: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(P,nil_Pr5671120429643327159n_assn))
     => ( ! [X2: produc6575502325842934193n_assn] : pp(aa_lis5212129917091495004n_bool(P,cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn)))
       => ( ! [X12: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( pp(aa_lis5212129917091495004n_bool(P,Xs2))
             => pp(aa_lis5212129917091495004n_bool(P,cons_P2971678138204555879n_assn(X12,cons_P2971678138204555879n_assn(X23,Xs2)))) )
         => pp(aa_lis5212129917091495004n_bool(P,Xs)) ) ) ) ).

% list_induct_first2
tff(fact_4964_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li582272206764200913t_bool,R: fun_int_fun_int_bool,Xs: list_int,Ys: list_int] :
      ( ! [Xs2: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs2),nil_int))
     => ( ! [Ys5: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),Ys5))
       => ( ! [X2: int,Xs2: list_int,Y2: int,Ys5: list_int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(R,X2),Y2))
             => ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs2),cons_int(Y2,Ys5)))
               => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),cons_int(Y2,Ys5))) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: int,Ys5: list_int] :
                ( ~ pp(aa_int_bool(aa_int_fun_int_bool(R,X2),Y2))
               => ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),Ys5))
                 => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),cons_int(Y2,Ys5))) ) )
           => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4965_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li1679355187408199469t_bool,R: fun_nat_fun_int_bool,Xs: list_nat,Ys: list_int] :
      ( ! [Xs2: list_nat] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs2),nil_int))
     => ( ! [Ys5: list_int] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),Ys5))
       => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int] :
              ( pp(aa_int_bool(aa_nat_fun_int_bool(R,X2),Y2))
             => ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs2),cons_int(Y2,Ys5)))
               => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5))) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int] :
                ( ~ pp(aa_int_bool(aa_nat_fun_int_bool(R,X2),Y2))
               => ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),Ys5))
                 => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5))) ) )
           => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4966_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li1781568775383035136t_bool,R: fun_Pr1996806492818889366t_bool,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,Xs2),nil_int))
     => ( ! [Ys5: list_int] : pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,nil_Pr5671120429643327159n_assn),Ys5))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys5: list_int] :
              ( pp(aa_int_bool(aa_Pro3157246541785891599t_bool(R,X2),Y2))
             => ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,Xs2),cons_int(Y2,Ys5)))
               => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_int(Y2,Ys5))) ) )
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys5: list_int] :
                ( ~ pp(aa_int_bool(aa_Pro3157246541785891599t_bool(R,X2),Y2))
               => ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),Ys5))
                 => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_int(Y2,Ys5))) ) )
           => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4967_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li7501238334638356269t_bool,R: fun_int_fun_nat_bool,Xs: list_int,Ys: list_nat] :
      ( ! [Xs2: list_int] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs2),nil_nat))
     => ( ! [Ys5: list_nat] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),Ys5))
       => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys5: list_nat] :
              ( pp(aa_nat_bool(aa_int_fun_nat_bool(R,X2),Y2))
             => ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs2),cons_nat(Y2,Ys5)))
               => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys5: list_nat] :
                ( ~ pp(aa_nat_bool(aa_int_fun_nat_bool(R,X2),Y2))
               => ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),Ys5))
                 => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
           => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4968_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li8598321315282354825t_bool,R: fun_nat_fun_nat_bool,Xs: list_nat,Ys: list_nat] :
      ( ! [Xs2: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs2),nil_nat))
     => ( ! [Ys5: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),Ys5))
       => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X2),Y2))
             => ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs2),cons_nat(Y2,Ys5)))
               => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat] :
                ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(R,X2),Y2))
               => ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),Ys5))
                 => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
           => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4969_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li8700534903257190492t_bool,R: fun_Pr2457089444465535986t_bool,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,Xs2),nil_nat))
     => ( ! [Ys5: list_nat] : pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,nil_Pr5671120429643327159n_assn),Ys5))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys5: list_nat] :
              ( pp(aa_nat_bool(aa_Pro6281537508254515819t_bool(R,X2),Y2))
             => ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,Xs2),cons_nat(Y2,Ys5)))
               => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys5: list_nat] :
                ( ~ pp(aa_nat_bool(aa_Pro6281537508254515819t_bool(R,X2),Y2))
               => ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),Ys5))
                 => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
           => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4970_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li6284172134459303836n_bool,R: fun_in1269341073776488338n_bool,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_int] : pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,Xs2),nil_Pr5671120429643327159n_assn))
     => ( ! [Ys5: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,nil_int),Ys5))
       => ( ! [X2: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
              ( pp(aa_Pro3608213857613363938n_bool(aa_int8683589726377689427n_bool(R,X2),Y2))
             => ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,Xs2),cons_P2971678138204555879n_assn(Y2,Ys5)))
               => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,cons_int(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ~ pp(aa_Pro3608213857613363938n_bool(aa_int8683589726377689427n_bool(R,X2),Y2))
               => ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,cons_int(X2,Xs2)),Ys5))
                 => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,cons_int(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) ) )
           => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4971_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li3458975838448298304n_bool,R: fun_na3584559861018862902n_bool,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_nat] : pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,Xs2),nil_Pr5671120429643327159n_assn))
     => ( ! [Ys5: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,nil_nat),Ys5))
       => ( ! [X2: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
              ( pp(aa_Pro3608213857613363938n_bool(aa_nat1775436476765288183n_bool(R,X2),Y2))
             => ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,Xs2),cons_P2971678138204555879n_assn(Y2,Ys5)))
               => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,cons_nat(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ~ pp(aa_Pro3608213857613363938n_bool(aa_nat1775436476765288183n_bool(R,X2),Y2))
               => ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,cons_nat(X2,Xs2)),Ys5))
                 => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,cons_nat(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) ) )
           => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4972_mergesort__by__rel__merge__induct,axiom,
    ! [P: fun_li4979294636826588077n_bool,R: fun_Pr6563606323009366189n_bool,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,Xs2),nil_Pr5671120429643327159n_assn))
     => ( ! [Ys5: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,nil_Pr5671120429643327159n_assn),Ys5))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
              ( pp(aa_Pro3608213857613363938n_bool(aa_Pro3155263873353247326n_bool(R,X2),Y2))
             => ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,Xs2),cons_P2971678138204555879n_assn(Y2,Ys5)))
               => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) ) )
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( ~ pp(aa_Pro3608213857613363938n_bool(aa_Pro3155263873353247326n_bool(R,X2),Y2))
               => ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),Ys5))
                 => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) ) )
           => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,Xs),Ys)) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4973_list_Odistinct_I1_J,axiom,
    ! [X21: int,X222: list_int] : nil_int != cons_int(X21,X222) ).

% list.distinct(1)
tff(fact_4974_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : nil_nat != cons_nat(X21,X222) ).

% list.distinct(1)
tff(fact_4975_list_Odistinct_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : nil_Pr5671120429643327159n_assn != cons_P2971678138204555879n_assn(X21,X222) ).

% list.distinct(1)
tff(fact_4976_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X222: list_int] :
      ( ( List = cons_int(X21,X222) )
     => ( List != nil_int ) ) ).

% list.discI
tff(fact_4977_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List = cons_nat(X21,X222) )
     => ( List != nil_nat ) ) ).

% list.discI
tff(fact_4978_list_OdiscI,axiom,
    ! [List: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] :
      ( ( List = cons_P2971678138204555879n_assn(X21,X222) )
     => ( List != nil_Pr5671120429643327159n_assn ) ) ).

% list.discI
tff(fact_4979_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X223: list_int] : Y != cons_int(X212,X223) ) ).

% list.exhaust
tff(fact_4980_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] : Y != cons_nat(X212,X223) ) ).

% list.exhaust
tff(fact_4981_list_Oexhaust,axiom,
    ! [Y: list_P8527749157015355191n_assn] :
      ( ( Y != nil_Pr5671120429643327159n_assn )
     => ~ ! [X212: produc6575502325842934193n_assn,X223: list_P8527749157015355191n_assn] : Y != cons_P2971678138204555879n_assn(X212,X223) ) ).

% list.exhaust
tff(fact_4982_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X2: int,Xs2: list_int] : X != cons_int(X2,Xs2)
     => ( X = nil_int ) ) ).

% min_list.cases
tff(fact_4983_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X2: nat,Xs2: list_nat] : X != cons_nat(X2,Xs2)
     => ( X = nil_nat ) ) ).

% min_list.cases
tff(fact_4984_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X2: int] : X != cons_int(X2,nil_int)
       => ~ ! [X2: int,Y2: int,Xs2: list_int] : X != cons_int(X2,cons_int(Y2,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_4985_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X2: nat] : X != cons_nat(X2,nil_nat)
       => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] : X != cons_nat(X2,cons_nat(Y2,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_4986_remdups__adj_Ocases,axiom,
    ! [X: list_P8527749157015355191n_assn] :
      ( ( X != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : X != cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn)
       => ~ ! [X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != cons_P2971678138204555879n_assn(X2,cons_P2971678138204555879n_assn(Y2,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_4987_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
    <=> ? [Y4: int,Ys4: list_int] : Xs = cons_int(Y4,Ys4) ) ).

% neq_Nil_conv
tff(fact_4988_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
    <=> ? [Y4: nat,Ys4: list_nat] : Xs = cons_nat(Y4,Ys4) ) ).

% neq_Nil_conv
tff(fact_4989_neq__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
    <=> ? [Y4: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] : Xs = cons_P2971678138204555879n_assn(Y4,Ys4) ) ).

% neq_Nil_conv
tff(fact_4990_list__induct2_H,axiom,
    ! [P: fun_li582272206764200913t_bool,Xs: list_int,Ys: list_int] :
      ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),nil_int))
     => ( ! [X2: int,Xs2: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),nil_int))
       => ( ! [Y2: int,Ys5: list_int] : pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,nil_int),cons_int(Y2,Ys5)))
         => ( ! [X2: int,Xs2: list_int,Y2: int,Ys5: list_int] :
                ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,cons_int(X2,Xs2)),cons_int(Y2,Ys5))) )
           => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4991_list__induct2_H,axiom,
    ! [P: fun_li7501238334638356269t_bool,Xs: list_int,Ys: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),nil_nat))
     => ( ! [X2: int,Xs2: list_int] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),nil_nat))
       => ( ! [Y2: nat,Ys5: list_nat] : pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,nil_int),cons_nat(Y2,Ys5)))
         => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys5: list_nat] :
                ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,cons_int(X2,Xs2)),cons_nat(Y2,Ys5))) )
           => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4992_list__induct2_H,axiom,
    ! [P: fun_li6284172134459303836n_bool,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,nil_int),nil_Pr5671120429643327159n_assn))
     => ( ! [X2: int,Xs2: list_int] : pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,cons_int(X2,Xs2)),nil_Pr5671120429643327159n_assn))
       => ( ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,nil_int),cons_P2971678138204555879n_assn(Y2,Ys5)))
         => ( ! [X2: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,Xs2),Ys5))
               => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,cons_int(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) )
           => pp(aa_lis5212129917091495004n_bool(aa_lis7405485056535563357n_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4993_list__induct2_H,axiom,
    ! [P: fun_li1679355187408199469t_bool,Xs: list_nat,Ys: list_int] :
      ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),nil_int))
     => ( ! [X2: nat,Xs2: list_nat] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),nil_int))
       => ( ! [Y2: int,Ys5: list_int] : pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,nil_nat),cons_int(Y2,Ys5)))
         => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int] :
                ( pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5))) )
           => pp(aa_list_int_bool(aa_lis4307785153233063318t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4994_list__induct2_H,axiom,
    ! [P: fun_li8598321315282354825t_bool,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),nil_nat))
     => ( ! [X2: nat,Xs2: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),nil_nat))
       => ( ! [Y2: nat,Ys5: list_nat] : pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,nil_nat),cons_nat(Y2,Ys5)))
         => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat] :
                ( pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5))) )
           => pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4995_list__induct2_H,axiom,
    ! [P: fun_li3458975838448298304n_bool,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,nil_nat),nil_Pr5671120429643327159n_assn))
     => ( ! [X2: nat,Xs2: list_nat] : pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,cons_nat(X2,Xs2)),nil_Pr5671120429643327159n_assn))
       => ( ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,nil_nat),cons_P2971678138204555879n_assn(Y2,Ys5)))
         => ( ! [X2: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,Xs2),Ys5))
               => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,cons_nat(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) )
           => pp(aa_lis5212129917091495004n_bool(aa_lis4580288760524557825n_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4996_list__induct2_H,axiom,
    ! [P: fun_li1781568775383035136t_bool,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,nil_Pr5671120429643327159n_assn),nil_int))
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),nil_int))
       => ( ! [Y2: int,Ys5: list_int] : pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,nil_Pr5671120429643327159n_assn),cons_int(Y2,Ys5)))
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys5: list_int] :
                ( pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_int(Y2,Ys5))) )
           => pp(aa_list_int_bool(aa_lis3708800635922548601t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4997_list__induct2_H,axiom,
    ! [P: fun_li8700534903257190492t_bool,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,nil_Pr5671120429643327159n_assn),nil_nat))
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),nil_nat))
       => ( ! [Y2: nat,Ys5: list_nat] : pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,nil_Pr5671120429643327159n_assn),cons_nat(Y2,Ys5)))
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys5: list_nat] :
                ( pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_nat(Y2,Ys5))) )
           => pp(aa_list_nat_bool(aa_lis4169083587569195221t_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4998_list__induct2_H,axiom,
    ! [P: fun_li4979294636826588077n_bool,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,nil_Pr5671120429643327159n_assn),nil_Pr5671120429643327159n_assn))
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),nil_Pr5671120429643327159n_assn))
       => ( ! [Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] : pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,nil_Pr5671120429643327159n_assn),cons_P2971678138204555879n_assn(Y2,Ys5)))
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys5: list_P8527749157015355191n_assn] :
                ( pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,Xs2),Ys5))
               => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,cons_P2971678138204555879n_assn(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys5))) )
           => pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_4999_list__nonempty__induct,axiom,
    ! [Xs: list_int,P: fun_list_int_bool] :
      ( ( Xs != nil_int )
     => ( ! [X2: int] : pp(aa_list_int_bool(P,cons_int(X2,nil_int)))
       => ( ! [X2: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( pp(aa_list_int_bool(P,Xs2))
               => pp(aa_list_int_bool(P,cons_int(X2,Xs2))) ) )
         => pp(aa_list_int_bool(P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_5000_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: fun_list_nat_bool] :
      ( ( Xs != nil_nat )
     => ( ! [X2: nat] : pp(aa_list_nat_bool(P,cons_nat(X2,nil_nat)))
       => ( ! [X2: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( pp(aa_list_nat_bool(P,Xs2))
               => pp(aa_list_nat_bool(P,cons_nat(X2,Xs2))) ) )
         => pp(aa_list_nat_bool(P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_5001_list__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_li2741412277222221411n_bool] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : pp(aa_lis5212129917091495004n_bool(P,cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn)))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( pp(aa_lis5212129917091495004n_bool(P,Xs2))
               => pp(aa_lis5212129917091495004n_bool(P,cons_P2971678138204555879n_assn(X2,Xs2))) ) )
         => pp(aa_lis5212129917091495004n_bool(P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_5002_distinct_Osimps_I1_J,axiom,
    pp(aa_list_int_bool(distinct_int,nil_int)) ).

% distinct.simps(1)
tff(fact_5003_distinct_Osimps_I1_J,axiom,
    pp(aa_list_nat_bool(distinct_nat,nil_nat)) ).

% distinct.simps(1)
tff(fact_5004_distinct_Osimps_I1_J,axiom,
    pp(aa_lis5212129917091495004n_bool(distin4809818325527169450n_assn,nil_Pr5671120429643327159n_assn)) ).

% distinct.simps(1)
tff(fact_5005_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: fun_Pr6563606323009366189n_bool] : pp(aa_lis5212129917091495004n_bool(sorted3986126766855596574n_assn(P),nil_Pr5671120429643327159n_assn)) ).

% sorted_wrt.simps(1)
tff(fact_5006_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: fun_nat_fun_nat_bool] : pp(aa_list_nat_bool(sorted_wrt_nat(P),nil_nat)) ).

% sorted_wrt.simps(1)
tff(fact_5007_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: fun_int_fun_int_bool] : pp(aa_list_int_bool(sorted_wrt_int(P),nil_int)) ).

% sorted_wrt.simps(1)
tff(fact_5008_rev_Osimps_I1_J,axiom,
    rev_int(nil_int) = nil_int ).

% rev.simps(1)
tff(fact_5009_rev_Osimps_I1_J,axiom,
    rev_nat(nil_nat) = nil_nat ).

% rev.simps(1)
tff(fact_5010_rev_Osimps_I1_J,axiom,
    rev_Pr4855572775806611735n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% rev.simps(1)
tff(fact_5011_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: int] : list_update_int(nil_int,I,V2) = nil_int ).

% list_update.simps(1)
tff(fact_5012_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: nat] : list_update_nat(nil_nat,I,V2) = nil_nat ).

% list_update.simps(1)
tff(fact_5013_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: produc6575502325842934193n_assn] : list_u4534839942911652127n_assn(nil_Pr5671120429643327159n_assn,I,V2) = nil_Pr5671120429643327159n_assn ).

% list_update.simps(1)
tff(fact_5014_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: c] : list_update_c(nil_c,I,V2) = nil_c ).

% list_update.simps(1)
tff(fact_5015_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V2: d] : list_update_d(nil_d,I,V2) = nil_d ).

% list_update.simps(1)
tff(fact_5016_list__update__code_I1_J,axiom,
    ! [I: nat,Y: int] : list_update_int(nil_int,I,Y) = nil_int ).

% list_update_code(1)
tff(fact_5017_list__update__code_I1_J,axiom,
    ! [I: nat,Y: nat] : list_update_nat(nil_nat,I,Y) = nil_nat ).

% list_update_code(1)
tff(fact_5018_list__update__code_I1_J,axiom,
    ! [I: nat,Y: produc6575502325842934193n_assn] : list_u4534839942911652127n_assn(nil_Pr5671120429643327159n_assn,I,Y) = nil_Pr5671120429643327159n_assn ).

% list_update_code(1)
tff(fact_5019_list__update__code_I1_J,axiom,
    ! [I: nat,Y: c] : list_update_c(nil_c,I,Y) = nil_c ).

% list_update_code(1)
tff(fact_5020_list__update__code_I1_J,axiom,
    ! [I: nat,Y: d] : list_update_d(nil_d,I,Y) = nil_d ).

% list_update_code(1)
tff(fact_5021_butlast_Osimps_I1_J,axiom,
    butlast_int(nil_int) = nil_int ).

% butlast.simps(1)
tff(fact_5022_butlast_Osimps_I1_J,axiom,
    butlast_nat(nil_nat) = nil_nat ).

% butlast.simps(1)
tff(fact_5023_butlast_Osimps_I1_J,axiom,
    butlas3012047794866324995n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% butlast.simps(1)
tff(fact_5024_remove1_Osimps_I1_J,axiom,
    ! [X: int] : remove1_int(X,nil_int) = nil_int ).

% remove1.simps(1)
tff(fact_5025_remove1_Osimps_I1_J,axiom,
    ! [X: nat] : remove1_nat(X,nil_nat) = nil_nat ).

% remove1.simps(1)
tff(fact_5026_remove1_Osimps_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn] : remove1670527618125605709n_assn(X,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% remove1.simps(1)
tff(fact_5027_removeAll_Osimps_I1_J,axiom,
    ! [X: int] : removeAll_int(X,nil_int) = nil_int ).

% removeAll.simps(1)
tff(fact_5028_removeAll_Osimps_I1_J,axiom,
    ! [X: nat] : removeAll_nat(X,nil_nat) = nil_nat ).

% removeAll.simps(1)
tff(fact_5029_removeAll_Osimps_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn] : remove1763081955827946941n_assn(X,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% removeAll.simps(1)
tff(fact_5030_merge_Osimps_I1_J,axiom,
    ! [L23: list_int] : merge_int(nil_int,L23) = L23 ).

% merge.simps(1)
tff(fact_5031_merge_Osimps_I1_J,axiom,
    ! [L23: list_nat] : merge_nat(nil_nat,L23) = L23 ).

% merge.simps(1)
tff(fact_5032_rotate1_Osimps_I1_J,axiom,
    rotate1_int(nil_int) = nil_int ).

% rotate1.simps(1)
tff(fact_5033_rotate1_Osimps_I1_J,axiom,
    rotate1_nat(nil_nat) = nil_nat ).

% rotate1.simps(1)
tff(fact_5034_rotate1_Osimps_I1_J,axiom,
    rotate328796349445179396n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% rotate1.simps(1)
tff(fact_5035_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_int_fun_int_int] : zipf_int_int_int(F3,nil_int,nil_int) = nil_int ).

% zipf.simps(1)
tff(fact_5036_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_int_fun_int_nat] : zipf_int_int_nat(F3,nil_int,nil_int) = nil_nat ).

% zipf.simps(1)
tff(fact_5037_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_int_fun_nat_int] : zipf_int_nat_int(F3,nil_int,nil_nat) = nil_int ).

% zipf.simps(1)
tff(fact_5038_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_int_fun_nat_nat] : zipf_int_nat_nat(F3,nil_int,nil_nat) = nil_nat ).

% zipf.simps(1)
tff(fact_5039_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_nat_fun_int_int] : zipf_nat_int_int(F3,nil_nat,nil_int) = nil_int ).

% zipf.simps(1)
tff(fact_5040_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_nat_fun_int_nat] : zipf_nat_int_nat(F3,nil_nat,nil_int) = nil_nat ).

% zipf.simps(1)
tff(fact_5041_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_nat_fun_nat_int] : zipf_nat_nat_int(F3,nil_nat,nil_nat) = nil_int ).

% zipf.simps(1)
tff(fact_5042_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_nat_fun_nat_nat] : zipf_nat_nat_nat(F3,nil_nat,nil_nat) = nil_nat ).

% zipf.simps(1)
tff(fact_5043_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_in4433485590553487243n_assn] : zipf_i7756021159461276697n_assn(F3,nil_int,nil_int) = nil_Pr5671120429643327159n_assn ).

% zipf.simps(1)
tff(fact_5044_zipf_Osimps_I1_J,axiom,
    ! [F3: fun_in1425933616158802663n_assn] : zipf_i5995984495708165493n_assn(F3,nil_int,nil_nat) = nil_Pr5671120429643327159n_assn ).

% zipf.simps(1)
tff(fact_5045_null__rec_I2_J,axiom,
    pp(aa_list_int_bool(null_int,nil_int)) ).

% null_rec(2)
tff(fact_5046_null__rec_I2_J,axiom,
    pp(aa_list_nat_bool(null_nat,nil_nat)) ).

% null_rec(2)
tff(fact_5047_null__rec_I2_J,axiom,
    pp(aa_lis5212129917091495004n_bool(null_P7596217980778516671n_assn,nil_Pr5671120429643327159n_assn)) ).

% null_rec(2)
tff(fact_5048_eq__Nil__null,axiom,
    ! [Xs: list_int] :
      ( ( Xs = nil_int )
    <=> pp(aa_list_int_bool(null_int,Xs)) ) ).

% eq_Nil_null
tff(fact_5049_eq__Nil__null,axiom,
    ! [Xs: list_nat] :
      ( ( Xs = nil_nat )
    <=> pp(aa_list_nat_bool(null_nat,Xs)) ) ).

% eq_Nil_null
tff(fact_5050_eq__Nil__null,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs = nil_Pr5671120429643327159n_assn )
    <=> pp(aa_lis5212129917091495004n_bool(null_P7596217980778516671n_assn,Xs)) ) ).

% eq_Nil_null
tff(fact_5051_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( pp(aa_rat_bool(ord_less_rat(zero_zero_rat),A2))
     => pp(aa_rat_bool(ord_less_rat(zero_zero_rat),power_power_rat(A2,N))) ) ).

% zero_less_power
tff(fact_5052_zero__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2))
     => pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% zero_less_power
tff(fact_5053_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),A2))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% zero_less_power
tff(fact_5054_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),A2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N))) ) ).

% zero_less_power
tff(fact_5055_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: 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),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_5056_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = one_one_assn )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_5057_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: 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,power_power_rat(X,N)),power_power_rat(Y,N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_5058_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: 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(aa_nat_fun_nat_nat(power_power_nat,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_5059_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: 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(aa_int_fun_nat_int(power_power_int,X),N)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_5060_power__Suc,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_Suc
tff(fact_5061_power__Suc,axiom,
    ! [A2: assn,N: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),aa_nat_assn(power_power_assn(A2),N)) ).

% power_Suc
tff(fact_5062_power__Suc,axiom,
    ! [A2: rat,N: nat] : power_power_rat(A2,aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A2),power_power_rat(A2,N)) ).

% power_Suc
tff(fact_5063_power__Suc,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_Suc
tff(fact_5064_power__Suc,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ).

% power_Suc
tff(fact_5065_power__Suc2,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),A2) ).

% power_Suc2
tff(fact_5066_power__Suc2,axiom,
    ! [A2: assn,N: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A2),N)),A2) ).

% power_Suc2
tff(fact_5067_power__Suc2,axiom,
    ! [A2: rat,N: nat] : power_power_rat(A2,aa_nat_nat(suc,N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(A2,N)),A2) ).

% power_Suc2
tff(fact_5068_power__Suc2,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),A2) ).

% power_Suc2
tff(fact_5069_power__Suc2,axiom,
    ! [A2: int,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)),A2) ).

% power_Suc2
tff(fact_5070_power__add,axiom,
    ! [A2: code_integer,M2: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M2)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_add
tff(fact_5071_power__add,axiom,
    ! [A2: assn,M2: nat,N: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A2),M2)),aa_nat_assn(power_power_assn(A2),N)) ).

% power_add
tff(fact_5072_power__add,axiom,
    ! [A2: rat,M2: nat,N: nat] : power_power_rat(A2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(A2,M2)),power_power_rat(A2,N)) ).

% power_add
tff(fact_5073_power__add,axiom,
    ! [A2: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_add
tff(fact_5074_power__add,axiom,
    ! [A2: int,M2: nat,N: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),M2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A2),N)) ).

% power_add
tff(fact_5075_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),I))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_5076_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_c,P: fun_li6164063741086132077c_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(P,nil_c),nil_c))
       => ( ! [X2: c,Xs2: list_c,Y2: c,Ys5: list_c] :
              ( ( size_size_list_c(Xs2) = size_size_list_c(Ys5) )
             => ( pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(P,Xs2),Ys5))
               => pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(P,cons_c(X2,Xs2)),cons_c(Y2,Ys5))) ) )
         => pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5077_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_d,P: fun_li7441736792402676908d_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(P,nil_c),nil_d))
       => ( ! [X2: c,Xs2: list_c,Y2: d,Ys5: list_d] :
              ( ( size_size_list_c(Xs2) = size_size_list_d(Ys5) )
             => ( pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(P,Xs2),Ys5))
               => pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(P,cons_c(X2,Xs2)),cons_d(Y2,Ys5))) ) )
         => pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5078_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_nat,P: fun_li2431554908110534861t_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(P,nil_c),nil_nat))
       => ( ! [X2: c,Xs2: list_c,Y2: nat,Ys5: list_nat] :
              ( ( size_size_list_c(Xs2) = size_size_list_nat(Ys5) )
             => ( pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(P,cons_c(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
         => pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5079_list__induct2,axiom,
    ! [Xs: list_c,Ys: list_int,P: fun_li4735960817091155313t_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(P,nil_c),nil_int))
       => ( ! [X2: c,Xs2: list_c,Y2: int,Ys5: list_int] :
              ( ( size_size_list_c(Xs2) = size_size_list_int(Ys5) )
             => ( pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(P,cons_c(X2,Xs2)),cons_int(Y2,Ys5))) ) )
         => pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5080_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_c,P: fun_li880640290856192494c_bool] :
      ( ( size_size_list_d(Xs) = size_size_list_c(Ys) )
     => ( pp(aa_list_c_bool(aa_lis314501548457165087c_bool(P,nil_d),nil_c))
       => ( ! [X2: d,Xs2: list_d,Y2: c,Ys5: list_c] :
              ( ( size_size_list_d(Xs2) = size_size_list_c(Ys5) )
             => ( pp(aa_list_c_bool(aa_lis314501548457165087c_bool(P,Xs2),Ys5))
               => pp(aa_list_c_bool(aa_lis314501548457165087c_bool(P,cons_d(X2,Xs2)),cons_c(Y2,Ys5))) ) )
         => pp(aa_list_c_bool(aa_lis314501548457165087c_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5081_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_d,P: fun_li2158313342172737325d_bool] :
      ( ( size_size_list_d(Xs) = size_size_list_d(Ys) )
     => ( pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(P,nil_d),nil_d))
       => ( ! [X2: d,Xs2: list_d,Y2: d,Ys5: list_d] :
              ( ( size_size_list_d(Xs2) = size_size_list_d(Ys5) )
             => ( pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(P,Xs2),Ys5))
               => pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(P,cons_d(X2,Xs2)),cons_d(Y2,Ys5))) ) )
         => pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5082_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_nat,P: fun_li3672680081608402956t_bool] :
      ( ( size_size_list_d(Xs) = size_size_list_nat(Ys) )
     => ( pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(P,nil_d),nil_nat))
       => ( ! [X2: d,Xs2: list_d,Y2: nat,Ys5: list_nat] :
              ( ( size_size_list_d(Xs2) = size_size_list_nat(Ys5) )
             => ( pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(P,Xs2),Ys5))
               => pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(P,cons_d(X2,Xs2)),cons_nat(Y2,Ys5))) ) )
         => pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5083_list__induct2,axiom,
    ! [Xs: list_d,Ys: list_int,P: fun_li5977085990589023408t_bool] :
      ( ( size_size_list_d(Xs) = size_size_list_int(Ys) )
     => ( pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(P,nil_d),nil_int))
       => ( ! [X2: d,Xs2: list_d,Y2: int,Ys5: list_int] :
              ( ( size_size_list_d(Xs2) = size_size_list_int(Ys5) )
             => ( pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(P,Xs2),Ys5))
               => pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(P,cons_d(X2,Xs2)),cons_int(Y2,Ys5))) ) )
         => pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5084_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_c,P: fun_li1676132496845444273c_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_c(Ys) )
     => ( pp(aa_list_c_bool(aa_lis748638795178973170c_bool(P,nil_nat),nil_c))
       => ( ! [X2: nat,Xs2: list_nat,Y2: c,Ys5: list_c] :
              ( ( size_size_list_nat(Xs2) = size_size_list_c(Ys5) )
             => ( pp(aa_list_c_bool(aa_lis748638795178973170c_bool(P,Xs2),Ys5))
               => pp(aa_list_c_bool(aa_lis748638795178973170c_bool(P,cons_nat(X2,Xs2)),cons_c(Y2,Ys5))) ) )
         => pp(aa_list_c_bool(aa_lis748638795178973170c_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5085_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_d,P: fun_li2953805548161989104d_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_d(Ys) )
     => ( pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(P,nil_nat),nil_d))
       => ( ! [X2: nat,Xs2: list_nat,Y2: d,Ys5: list_d] :
              ( ( size_size_list_nat(Xs2) = size_size_list_d(Ys5) )
             => ( pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(P,Xs2),Ys5))
               => pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(P,cons_nat(X2,Xs2)),cons_d(Y2,Ys5))) ) )
         => pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5086_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_c,P: fun_li7775044557710271146c_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( ( size_size_list_c(Ys) = size_size_list_c(Zs) )
       => ( pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis2093744656504115619c_bool(P,nil_c),nil_c),nil_c))
         => ( ! [X2: c,Xs2: list_c,Y2: c,Ys5: list_c,Z3: c,Zs2: list_c] :
                ( ( size_size_list_c(Xs2) = size_size_list_c(Ys5) )
               => ( ( size_size_list_c(Ys5) = size_size_list_c(Zs2) )
                 => ( pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis2093744656504115619c_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis2093744656504115619c_bool(P,cons_c(X2,Xs2)),cons_c(Y2,Ys5)),cons_c(Z3,Zs2))) ) ) )
           => pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis2093744656504115619c_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5087_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_d,P: fun_li4889974048426743913d_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( ( size_size_list_c(Ys) = size_size_list_d(Zs) )
       => ( pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis3371417707820660450d_bool(P,nil_c),nil_c),nil_d))
         => ( ! [X2: c,Xs2: list_c,Y2: c,Ys5: list_c,Z3: d,Zs2: list_d] :
                ( ( size_size_list_c(Xs2) = size_size_list_c(Ys5) )
               => ( ( size_size_list_c(Ys5) = size_size_list_d(Zs2) )
                 => ( pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis3371417707820660450d_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis3371417707820660450d_bool(P,cons_c(X2,Xs2)),cons_c(Y2,Ys5)),cons_d(Z3,Zs2))) ) ) )
           => pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis3371417707820660450d_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5088_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_nat,P: fun_li3771098141839667046t_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( ( size_size_list_c(Ys) = size_size_list_nat(Zs) )
       => ( pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis7814392122336824599t_bool(P,nil_c),nil_c),nil_nat))
         => ( ! [X2: c,Xs2: list_c,Y2: c,Ys5: list_c,Z3: nat,Zs2: list_nat] :
                ( ( size_size_list_c(Xs2) = size_size_list_c(Ys5) )
               => ( ( size_size_list_c(Ys5) = size_size_list_nat(Zs2) )
                 => ( pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis7814392122336824599t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis7814392122336824599t_bool(P,cons_c(X2,Xs2)),cons_c(Y2,Ys5)),cons_nat(Z3,Zs2))) ) ) )
           => pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis7814392122336824599t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5089_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_c,Zs: list_int,P: fun_li4457674174880217610t_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_c(Ys) )
     => ( ( size_size_list_c(Ys) = size_size_list_int(Zs) )
       => ( pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis895425994462669243t_bool(P,nil_c),nil_c),nil_int))
         => ( ! [X2: c,Xs2: list_c,Y2: c,Ys5: list_c,Z3: int,Zs2: list_int] :
                ( ( size_size_list_c(Xs2) = size_size_list_c(Ys5) )
               => ( ( size_size_list_c(Ys5) = size_size_list_int(Zs2) )
                 => ( pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis895425994462669243t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis895425994462669243t_bool(P,cons_c(X2,Xs2)),cons_c(Y2,Ys5)),cons_int(Z3,Zs2))) ) ) )
           => pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis895425994462669243t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5090_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_c,P: fun_li3531615669003998379c_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( ( size_size_list_d(Ys) = size_size_list_c(Zs) )
       => ( pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis6033693243128951844c_bool(P,nil_c),nil_d),nil_c))
         => ( ! [X2: c,Xs2: list_c,Y2: d,Ys5: list_d,Z3: c,Zs2: list_c] :
                ( ( size_size_list_c(Xs2) = size_size_list_d(Ys5) )
               => ( ( size_size_list_d(Ys5) = size_size_list_c(Zs2) )
                 => ( pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis6033693243128951844c_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis6033693243128951844c_bool(P,cons_c(X2,Xs2)),cons_d(Y2,Ys5)),cons_c(Z3,Zs2))) ) ) )
           => pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis6033693243128951844c_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5091_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_d,P: fun_li646545159720471146d_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( ( size_size_list_d(Ys) = size_size_list_d(Zs) )
       => ( pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis7311366294445496675d_bool(P,nil_c),nil_d),nil_d))
         => ( ! [X2: c,Xs2: list_c,Y2: d,Ys5: list_d,Z3: d,Zs2: list_d] :
                ( ( size_size_list_c(Xs2) = size_size_list_d(Ys5) )
               => ( ( size_size_list_d(Ys5) = size_size_list_d(Zs2) )
                 => ( pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis7311366294445496675d_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis7311366294445496675d_bool(P,cons_c(X2,Xs2)),cons_d(Y2,Ys5)),cons_d(Z3,Zs2))) ) ) )
           => pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis7311366294445496675d_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5092_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_nat,P: fun_li7477572284639489829t_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( ( size_size_list_d(Ys) = size_size_list_nat(Zs) )
       => ( pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(aa_lis9055517295834692694t_bool(P,nil_c),nil_d),nil_nat))
         => ( ! [X2: c,Xs2: list_c,Y2: d,Ys5: list_d,Z3: nat,Zs2: list_nat] :
                ( ( size_size_list_c(Xs2) = size_size_list_d(Ys5) )
               => ( ( size_size_list_d(Ys5) = size_size_list_nat(Zs2) )
                 => ( pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(aa_lis9055517295834692694t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(aa_lis9055517295834692694t_bool(P,cons_c(X2,Xs2)),cons_d(Y2,Ys5)),cons_nat(Z3,Zs2))) ) ) )
           => pp(aa_list_nat_bool(aa_lis6769324676173542533t_bool(aa_lis9055517295834692694t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5093_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_d,Zs: list_int,P: fun_li8164148317680040393t_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_d(Ys) )
     => ( ( size_size_list_d(Ys) = size_size_list_int(Zs) )
       => ( pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(aa_lis2136551167960537338t_bool(P,nil_c),nil_d),nil_int))
         => ( ! [X2: c,Xs2: list_c,Y2: d,Ys5: list_d,Z3: int,Zs2: list_int] :
                ( ( size_size_list_c(Xs2) = size_size_list_d(Ys5) )
               => ( ( size_size_list_d(Ys5) = size_size_list_int(Zs2) )
                 => ( pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(aa_lis2136551167960537338t_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(aa_lis2136551167960537338t_bool(P,cons_c(X2,Xs2)),cons_d(Y2,Ys5)),cons_int(Z3,Zs2))) ) ) )
           => pp(aa_list_int_bool(aa_lis6309041724526895913t_bool(aa_lis2136551167960537338t_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5094_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_nat,Zs: list_c,P: fun_li3341451312573339978c_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_c(Zs) )
       => ( pp(aa_list_c_bool(aa_lis748638795178973170c_bool(aa_lis7058969711071734011c_bool(P,nil_c),nil_nat),nil_c))
         => ( ! [X2: c,Xs2: list_c,Y2: nat,Ys5: list_nat,Z3: c,Zs2: list_c] :
                ( ( size_size_list_c(Xs2) = size_size_list_nat(Ys5) )
               => ( ( size_size_list_nat(Ys5) = size_size_list_c(Zs2) )
                 => ( pp(aa_list_c_bool(aa_lis748638795178973170c_bool(aa_lis7058969711071734011c_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_c_bool(aa_lis748638795178973170c_bool(aa_lis7058969711071734011c_bool(P,cons_c(X2,Xs2)),cons_nat(Y2,Ys5)),cons_c(Z3,Zs2))) ) ) )
           => pp(aa_list_c_bool(aa_lis748638795178973170c_bool(aa_lis7058969711071734011c_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5095_list__induct3,axiom,
    ! [Xs: list_c,Ys: list_nat,Zs: list_d,P: fun_li456380803289812745d_bool] :
      ( ( size_size_list_c(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_d(Zs) )
       => ( pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(aa_lis8336642762388278842d_bool(P,nil_c),nil_nat),nil_d))
         => ( ! [X2: c,Xs2: list_c,Y2: nat,Ys5: list_nat,Z3: d,Zs2: list_d] :
                ( ( size_size_list_c(Xs2) = size_size_list_nat(Ys5) )
               => ( ( size_size_list_nat(Ys5) = size_size_list_d(Zs2) )
                 => ( pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(aa_lis8336642762388278842d_bool(P,Xs2),Ys5),Zs2))
                   => pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(aa_lis8336642762388278842d_bool(P,cons_c(X2,Xs2)),cons_nat(Y2,Ys5)),cons_d(Z3,Zs2))) ) ) )
           => pp(aa_list_d_bool(aa_lis7663137664432106673d_bool(aa_lis8336642762388278842d_bool(P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5096_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_c,P: fun_li2514166507294599495c_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_c(Ws) )
         => ( pp(aa_list_c_bool(aa_lis1956755928107353166c_bool(aa_lis3863201514048117931c_bool(aa_lis6676963690625055624c_bool(P,nil_nat),nil_nat),nil_int),nil_c))
           => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z3: int,Zs2: list_int,W3: c,Ws2: list_c] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys5) )
                 => ( ( size_size_list_nat(Ys5) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_c(Ws2) )
                     => ( pp(aa_list_c_bool(aa_lis1956755928107353166c_bool(aa_lis3863201514048117931c_bool(aa_lis6676963690625055624c_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_c_bool(aa_lis1956755928107353166c_bool(aa_lis3863201514048117931c_bool(aa_lis6676963690625055624c_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5)),cons_int(Z3,Zs2)),cons_c(W3,Ws2))) ) ) ) )
             => pp(aa_list_c_bool(aa_lis1956755928107353166c_bool(aa_lis3863201514048117931c_bool(aa_lis6676963690625055624c_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5097_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_d,P: fun_li3858631544104087430d_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_d(Ws) )
         => ( pp(aa_list_d_bool(aa_lis8871254797360486669d_bool(aa_lis5140874565364662762d_bool(aa_lis3791893181341528391d_bool(P,nil_nat),nil_nat),nil_int),nil_d))
           => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z3: int,Zs2: list_int,W3: d,Ws2: list_d] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys5) )
                 => ( ( size_size_list_nat(Ys5) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_d(Ws2) )
                     => ( pp(aa_list_d_bool(aa_lis8871254797360486669d_bool(aa_lis5140874565364662762d_bool(aa_lis3791893181341528391d_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_d_bool(aa_lis8871254797360486669d_bool(aa_lis5140874565364662762d_bool(aa_lis3791893181341528391d_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5)),cons_int(Z3,Zs2)),cons_d(W3,Ws2))) ) ) ) )
             => pp(aa_list_d_bool(aa_lis8871254797360486669d_bool(aa_lis5140874565364662762d_bool(aa_lis3791893181341528391d_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5098_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_nat,P: fun_li4602333441959630751t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_nat(Ws) )
         => ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,nil_nat),nil_nat),nil_int),nil_nat))
           => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z3: int,Zs2: list_int,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys5) )
                 => ( ( size_size_list_nat(Ys5) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_nat(Ws2) )
                     => ( pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5)),cons_int(Z3,Zs2)),cons_nat(W3,Ws2))) ) ) ) )
             => pp(aa_list_nat_bool(aa_lis3670985124235711382t_bool(aa_lis7037195574095253519t_bool(aa_lis8207810475382258440t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5099_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_int,Ws: list_int,P: fun_li6998123803554500675t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_int(Ws) )
         => ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,nil_nat),nil_nat),nil_int),nil_int))
           => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys5: list_nat,Z3: int,Zs2: list_int,W3: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_nat(Ys5) )
                 => ( ( size_size_list_nat(Ys5) = size_size_list_int(Zs2) )
                   => ( ( size_size_list_int(Zs2) = size_size_list_int(Ws2) )
                     => ( pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,cons_nat(X2,Xs2)),cons_nat(Y2,Ys5)),cons_int(Z3,Zs2)),cons_int(W3,Ws2))) ) ) ) )
             => pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(aa_lis118229446221098163t_bool(aa_lis8894386508422809004t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5100_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_c,P: fun_li2225018760373637251c_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_c(Zs) )
       => ( ( size_size_list_c(Zs) = size_size_list_c(Ws) )
         => ( pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis4240944861120901107c_bool(aa_lis1076721955647843308c_bool(P,nil_nat),nil_int),nil_c),nil_c))
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z3: c,Zs2: list_c,W3: c,Ws2: list_c] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys5) )
                 => ( ( size_size_list_int(Ys5) = size_size_list_c(Zs2) )
                   => ( ( size_size_list_c(Zs2) = size_size_list_c(Ws2) )
                     => ( pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis4240944861120901107c_bool(aa_lis1076721955647843308c_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis4240944861120901107c_bool(aa_lis1076721955647843308c_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5)),cons_c(Z3,Zs2)),cons_c(W3,Ws2))) ) ) ) )
             => pp(aa_list_c_bool(aa_lis5597924998687104670c_bool(aa_lis4240944861120901107c_bool(aa_lis1076721955647843308c_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5101_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_d,P: fun_li3569483797183125186d_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_c(Zs) )
       => ( ( size_size_list_c(Zs) = size_size_list_d(Ws) )
         => ( pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis5518617912437445938d_bool(aa_lis7415023483219091883d_bool(P,nil_nat),nil_int),nil_c),nil_d))
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z3: c,Zs2: list_c,W3: d,Ws2: list_d] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys5) )
                 => ( ( size_size_list_int(Ys5) = size_size_list_c(Zs2) )
                   => ( ( size_size_list_c(Zs2) = size_size_list_d(Ws2) )
                     => ( pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis5518617912437445938d_bool(aa_lis7415023483219091883d_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis5518617912437445938d_bool(aa_lis7415023483219091883d_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5)),cons_c(Z3,Zs2)),cons_d(W3,Ws2))) ) ) ) )
             => pp(aa_list_d_bool(aa_lis3289051831085462365d_bool(aa_lis5518617912437445938d_bool(aa_lis7415023483219091883d_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5102_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_nat,P: fun_li3224250543824542051t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_c(Zs) )
       => ( ( size_size_list_c(Zs) = size_size_list_nat(Ws) )
         => ( pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis2067709983372802759t_bool(aa_lis6047864785249466788t_bool(P,nil_nat),nil_int),nil_c),nil_nat))
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z3: c,Zs2: list_c,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys5) )
                 => ( ( size_size_list_int(Ys5) = size_size_list_c(Zs2) )
                   => ( ( size_size_list_c(Zs2) = size_size_list_nat(Ws2) )
                     => ( pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis2067709983372802759t_bool(aa_lis6047864785249466788t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis2067709983372802759t_bool(aa_lis6047864785249466788t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5)),cons_c(Z3,Zs2)),cons_nat(W3,Ws2))) ) ) ) )
             => pp(aa_list_nat_bool(aa_lis5528199502675674438t_bool(aa_lis2067709983372802759t_bool(aa_lis6047864785249466788t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5103_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_c,Ws: list_int,P: fun_li5620040905419411975t_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_c(Zs) )
       => ( ( size_size_list_c(Zs) = size_size_list_int(Ws) )
         => ( pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis4372115892353423211t_bool(aa_lis6734440818290017352t_bool(P,nil_nat),nil_int),nil_c),nil_int))
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z3: c,Zs2: list_c,W3: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys5) )
                 => ( ( size_size_list_int(Ys5) = size_size_list_c(Zs2) )
                   => ( ( size_size_list_c(Zs2) = size_size_list_int(Ws2) )
                     => ( pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis4372115892353423211t_bool(aa_lis6734440818290017352t_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis4372115892353423211t_bool(aa_lis6734440818290017352t_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5)),cons_c(Z3,Zs2)),cons_int(W3,Ws2))) ) ) ) )
             => pp(aa_list_int_bool(aa_lis5067916551029027818t_bool(aa_lis4372115892353423211t_bool(aa_lis6734440818290017352t_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5104_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_d,Ws: list_c,P: fun_li8813649075853578756c_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_d(Zs) )
       => ( ( size_size_list_d(Zs) = size_size_list_c(Ws) )
         => ( pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis8180893447745737332c_bool(aa_lis6056665103796346349c_bool(P,nil_nat),nil_int),nil_d),nil_c))
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z3: d,Zs2: list_d,W3: c,Ws2: list_c] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys5) )
                 => ( ( size_size_list_int(Ys5) = size_size_list_d(Zs2) )
                   => ( ( size_size_list_d(Zs2) = size_size_list_c(Ws2) )
                     => ( pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis8180893447745737332c_bool(aa_lis6056665103796346349c_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis8180893447745737332c_bool(aa_lis6056665103796346349c_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5)),cons_d(Z3,Zs2)),cons_c(W3,Ws2))) ) ) ) )
             => pp(aa_list_c_bool(aa_lis314501548457165087c_bool(aa_lis8180893447745737332c_bool(aa_lis6056665103796346349c_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5105_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_d,Ws: list_d,P: fun_li934742075808290883d_bool] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_d(Zs) )
       => ( ( size_size_list_d(Zs) = size_size_list_d(Ws) )
         => ( pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis235194462207506355d_bool(aa_lis3171594594512819116d_bool(P,nil_nat),nil_int),nil_d),nil_d))
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys5: list_int,Z3: d,Zs2: list_d,W3: d,Ws2: list_d] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys5) )
                 => ( ( size_size_list_int(Ys5) = size_size_list_d(Zs2) )
                   => ( ( size_size_list_d(Zs2) = size_size_list_d(Ws2) )
                     => ( pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis235194462207506355d_bool(aa_lis3171594594512819116d_bool(P,Xs2),Ys5),Zs2),Ws2))
                       => pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis235194462207506355d_bool(aa_lis3171594594512819116d_bool(P,cons_nat(X2,Xs2)),cons_int(Y2,Ys5)),cons_d(Z3,Zs2)),cons_d(W3,Ws2))) ) ) ) )
             => pp(aa_list_d_bool(aa_lis7229000417710298590d_bool(aa_lis235194462207506355d_bool(aa_lis3171594594512819116d_bool(P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5106_power__gt__expt,axiom,
    ! [N: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K3))) ) ).

% power_gt_expt
tff(fact_5107_greaterThan__Suc,axiom,
    ! [K3: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_or1210151606488870762an_nat,K3)),insert_nat2(aa_nat_nat(suc,K3),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_5108_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) ) ).

% nat_int_comparison(2)
tff(fact_5109_zdiff__int__split,axiom,
    ! [P: fun_int_bool,X: nat,Y: nat] :
      ( pp(aa_int_bool(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y))))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Y),X))
         => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y)))) )
        & ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
         => pp(aa_int_bool(P,zero_zero_int)) ) ) ) ).

% zdiff_int_split
tff(fact_5110_imp__le__cong,axiom,
    ! [X: int,X4: int,P: bool,P2: bool] :
      ( ( X = X4 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
           => pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
           => pp(P2) ) ) ) ) ).

% imp_le_cong
tff(fact_5111_conj__le__cong,axiom,
    ! [X: int,X4: int,P: bool,P2: bool] :
      ( ( X = X4 )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
         => ( pp(P)
          <=> pp(P2) ) )
       => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X))
            & pp(P) )
        <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),X4))
            & pp(P2) ) ) ) ) ).

% conj_le_cong
tff(fact_5112_incr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_bool,K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X2: int] :
            ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D2))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K3))
         => ! [X6: int] :
              ( pp(aa_int_bool(P,X6))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D2)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_5113_decr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_bool,K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X2: int] :
            ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D2))) )
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),K3))
         => ! [X6: int] :
              ( pp(aa_int_bool(P,X6))
             => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D2)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5114_minusinfinity,axiom,
    ! [D2: int,P1: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X2: int,K: int] :
            ( pp(aa_int_bool(P1,X2))
          <=> pp(aa_int_bool(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2)))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z2))
             => ( pp(aa_int_bool(P,X2))
              <=> pp(aa_int_bool(P1,X2)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P1,X_13))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% minusinfinity
tff(fact_5115_plusinfinity,axiom,
    ! [D2: int,P2: fun_int_bool,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X2: int,K: int] :
            ( pp(aa_int_bool(P2,X2))
          <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2)))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X2))
             => ( pp(aa_int_bool(P,X2))
              <=> pp(aa_int_bool(P2,X2)) ) )
         => ( ? [X_13: int] : pp(aa_int_bool(P2,X_13))
           => ? [X_1: int] : pp(aa_int_bool(P,X_1)) ) ) ) ) ).

% plusinfinity
tff(fact_5116_pos__int__cases,axiom,
    ! [K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K3))
     => ~ ! [N5: nat] :
            ( ( K3 = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5)) ) ) ).

% pos_int_cases
tff(fact_5117_zero__less__imp__eq__int,axiom,
    ! [K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K3))
     => ? [N5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
          & ( K3 = aa_nat_int(semiri1314217659103216013at_int,N5) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_5118_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K3: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,I),J))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K3)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K3)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_5119_eucl__rel__int__iff,axiom,
    ! [K3: int,L: int,Q5: int,R2: int] :
      ( eucl_rel_int(K3,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
    <=> ( ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q5)),R2) )
        & ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),R2))
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,R2),L)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),L))
         => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),R2))
                & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,R2),zero_zero_int)) ) )
            & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,L),zero_zero_int))
             => ( Q5 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5120_nat0__intermed__int__val,axiom,
    ! [N: nat,F3: fun_nat_int,K3: int] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I5),one_one_nat))),aa_nat_int(F3,I5)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,zero_zero_nat)),K3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K3),aa_nat_int(F3,N)))
         => ? [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
              & ( aa_nat_int(F3,I5) = K3 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_5121_unique__remainder,axiom,
    ! [A2: int,B2: int,Q5: int,R2: int,Q6: int,R3: int] :
      ( eucl_rel_int(A2,B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
     => ( eucl_rel_int(A2,B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R3))
       => ( R2 = R3 ) ) ) ).

% unique_remainder
tff(fact_5122_unique__quotient,axiom,
    ! [A2: int,B2: int,Q5: int,R2: int,Q6: int,R3: int] :
      ( eucl_rel_int(A2,B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
     => ( eucl_rel_int(A2,B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q6),R3))
       => ( Q5 = Q6 ) ) ) ).

% unique_quotient
tff(fact_5123_eucl__rel__int__by0,axiom,
    ! [K3: int] : eucl_rel_int(K3,zero_zero_int,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K3)) ).

% eucl_rel_int_by0
tff(fact_5124_eucl__rel__int__dividesI,axiom,
    ! [L: int,K3: int,Q5: int] :
      ( ( L != zero_zero_int )
     => ( ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L) )
       => eucl_rel_int(K3,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5125_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F3: fun_nat_int,K3: int] :
      ( ! [I5: nat] :
          ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),I5))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N)) )
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F3,aa_nat_nat(suc,I5))),aa_nat_int(F3,I5)))),one_one_int)) )
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,M2)),K3))
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K3),aa_nat_int(F3,N)))
           => ? [I5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),I5))
                & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
                & ( aa_nat_int(F3,I5) = K3 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5126_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D2)))) ) ).

% incr_lemma
tff(fact_5127_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z))),one_one_int)),D2))),Z)) ) ).

% decr_lemma
tff(fact_5128_nat__ivt__aux,axiom,
    ! [N: nat,F3: fun_nat_int,K3: int] :
      ( ! [I5: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),N))
         => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F3,aa_nat_nat(suc,I5))),aa_nat_int(F3,I5)))),one_one_int)) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(F3,zero_zero_nat)),K3))
       => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,K3),aa_nat_int(F3,N)))
         => ? [I5: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I5),N))
              & ( aa_nat_int(F3,I5) = K3 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_5129_upto__aux__rec,axiom,
    ! [J: int,I: int,Js: list_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = Js ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( upto_aux(I,J,Js) = upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js)) ) ) ) ).

% upto_aux_rec
tff(fact_5130_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K4: int] :
            ( ( A1 = K4 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),K4) ) )
        | ? [L4: int,K4: int,Q7: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q7),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K4 = aa_int_int(aa_int_fun_int_int(times_times_int,Q7),L4) ) )
        | ? [R4: int,L4: int,K4: int,Q7: int] :
            ( ( A1 = K4 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q7),R4) )
            & ( aa_int_int(sgn_sgn_int,R4) = aa_int_int(sgn_sgn_int,L4) )
            & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R4)),aa_int_int(abs_abs_int,L4)))
            & ( K4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q7),L4)),R4) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5131_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A1) ) )
       => ( ! [Q4: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22) ) ) )
         => ~ ! [R5: int,Q4: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(product_Pair_int_int(Q4),R5) )
               => ( ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,A22) )
                 => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,A22)))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),A22)),R5) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5132_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K3: int,Q5: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L)))
       => ( ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L)),R2) )
         => eucl_rel_int(K3,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5133_int__power__div__base,axiom,
    ! [M2: nat,K3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),K3))
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K3),M2)),K3) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_5134_merge__true__star,axiom,
    aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_5135_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_5136_entt__true,axiom,
    ! [A: assn] : pp(aa_assn_bool(entailst(A),top_top_assn)) ).

% entt_true
tff(fact_5137_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_5138_entt__star__true__simp_I1_J,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),top_top_assn)))
    <=> pp(aa_assn_bool(entailst(A),B)) ) ).

% entt_star_true_simp(1)
tff(fact_5139_entt__star__true__simp_I2_J,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),top_top_assn)),B))
    <=> pp(aa_assn_bool(entailst(A),B)) ) ).

% entt_star_true_simp(2)
tff(fact_5140_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),P)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_5141_ent__true,axiom,
    ! [P: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),top_top_assn)) ).

% ent_true
tff(fact_5142_norm__assertion__simps_I12_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),top_top_assn) = top_top_assn ).

% norm_assertion_simps(12)
tff(fact_5143_norm__assertion__simps_I11_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X) = top_top_assn ).

% norm_assertion_simps(11)
tff(fact_5144_norm__assertion__simps_I4_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).

% norm_assertion_simps(4)
tff(fact_5145_norm__assertion__simps_I3_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),X) = X ).

% norm_assertion_simps(3)
tff(fact_5146_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_5147_mod__star__trueI,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
     => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H2)) ) ).

% mod_star_trueI
tff(fact_5148_mod__star__trueE,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H2))
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H5)) ) ).

% mod_star_trueE
tff(fact_5149_ent__star__mono__true,axiom,
    ! [A: assn,A4: assn,B: assn,B8: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),top_top_assn)))
     => ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B8),top_top_assn)))
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A4),B8)),top_top_assn))) ) ) ).

% ent_star_mono_true
tff(fact_5150_ent__refl__true,axiom,
    ! [A: assn] : pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),top_top_assn))) ).

% ent_refl_true
tff(fact_5151_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),R)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))) ) ).

% ent_true_drop(1)
tff(fact_5152_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),Q))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))) ) ).

% ent_true_drop(2)
tff(fact_5153_div__int__unique,axiom,
    ! [K3: int,L: int,Q5: int,R2: int] :
      ( eucl_rel_int(K3,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),L) = Q5 ) ) ).

% div_int_unique
tff(fact_5154_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,top_top_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat))) ).

% mod_h_bot_iff(2)
tff(fact_5155_entt__def__true,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(entailst(P),Q))
    <=> pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))) ) ).

% entt_def_true
tff(fact_5156_entailst__def,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(A),B))
    <=> pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),top_top_assn))) ) ).

% entailst_def
tff(fact_5157_enttI__true,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn)))
     => pp(aa_assn_bool(entailst(P),Q)) ) ).

% enttI_true
tff(fact_5158_enttI,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),top_top_assn)))
     => pp(aa_assn_bool(entailst(A),B)) ) ).

% enttI
tff(fact_5159_enttD,axiom,
    ! [A: assn,B: assn] :
      ( pp(aa_assn_bool(entailst(A),B))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),top_top_assn))) ) ).

% enttD
tff(fact_5160_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( pp(aa_assn_bool(fI_QUERY(P,Q),top_top_assn))
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q),top_top_assn))) ) ).

% entails_solve_init(1)
tff(fact_5161_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_5162_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),N) = M2 ) ) ).

% div_mult_self_is_m
tff(fact_5163_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),N) = M2 ) ) ).

% div_mult_self1_is_m
tff(fact_5164_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_5165_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N)))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),I)) ) ).

% less_mult_imp_div_less
tff(fact_5166_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ) ).

% div_greater_zero_iff
tff(fact_5167_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K3),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K3),M2))) ) ) ).

% div_le_mono2
tff(fact_5168_div__less__iff__less__mult,axiom,
    ! [Q5: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),Q5)),N))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q5))) ) ) ).

% div_less_iff_less_mult
tff(fact_5169_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = M2 )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_5170_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),M2)) ) ) ).

% div_less_dividend
tff(fact_5171_div__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
          | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = zero_zero_nat ) )
      & ( ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
            | ( N = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N)) ) ) ) ).

% div_if
tff(fact_5172_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q5: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Q5))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q5)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),Q5)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_5173_div__nat__eqI,axiom,
    ! [N: nat,Q5: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q5)),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q5))))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = Q5 ) ) ) ).

% div_nat_eqI
tff(fact_5174_split__div,axiom,
    ! [P: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,zero_zero_nat)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J2) )
               => pp(aa_nat_bool(P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_5175_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_5176_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N))))) ) ).

% dividend_less_times_div
tff(fact_5177_nat__mult__div__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_5178_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),M2))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_5179_split__div_H,axiom,
    ! [P: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
          & pp(aa_nat_bool(P,zero_zero_nat)) )
        | ? [Q7: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q7)),M2))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q7))))
            & pp(aa_nat_bool(P,Q7)) ) ) ) ).

% split_div'
tff(fact_5180_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N)) ) ) ) ).

% div_geq
tff(fact_5181_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = M2 ) ) ).

% mod_less
tff(fact_5182_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),N)) ) ).

% mod_less_divisor
tff(fact_5183_div__less__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N) = zero_zero_nat )
           => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),N))) ) ) ) ) ).

% div_less_mono
tff(fact_5184_mod__induct,axiom,
    ! [P: fun_nat_bool,N: nat,P4: nat,M2: nat] :
      ( pp(aa_nat_bool(P,N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),P4))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),P4))
         => ( ! [N5: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N5),P4))
               => ( pp(aa_nat_bool(P,N5))
                 => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N5)),P4))) ) )
           => pp(aa_nat_bool(P,M2)) ) ) ) ) ).

% mod_induct
tff(fact_5185_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ).

% mod_geq
tff(fact_5186_mod__if,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = M2 ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),N) ) ) ) ).

% mod_if
tff(fact_5187_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)),N)) ) ).

% mod_le_divisor
tff(fact_5188_split__mod,axiom,
    ! [P: fun_nat_bool,M2: nat,N: nat] :
      ( pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))
    <=> ( ( ( N = zero_zero_nat )
         => pp(aa_nat_bool(P,M2)) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,J2),N))
             => ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J2) )
               => pp(aa_nat_bool(P,J2)) ) ) ) ) ) ).

% split_mod
tff(fact_5189_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A),B))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(if_nat(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),N)),zero_zero_nat)),one_one_nat),zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),N))) ) ) ).

% verit_le_mono_div
tff(fact_5190_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N))),M2) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5191_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_bool,M2: nat,N: nat] :
      ( ! [M6: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M6),zero_zero_nat))
     => ( ! [M6: nat,N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(P,N5),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M6),N5)))
             => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M6),N5)) ) )
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(P,M2),N)) ) ) ).

% gcd_nat_induct
tff(fact_5192_mod__int__unique,axiom,
    ! [K3: int,L: int,Q5: int,R2: int] :
      ( eucl_rel_int(K3,L,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K3),L) = R2 ) ) ).

% mod_int_unique
tff(fact_5193_eucl__rel__int,axiom,
    ! [K3: int,L: int] : eucl_rel_int(K3,L,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K3),L))) ).

% eucl_rel_int
tff(fact_5194_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_5195_minus__assn__def,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A2),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),aa_assn_assn(uminus_uminus_assn,B2)) ).

% minus_assn_def
tff(fact_5196_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N5: nat] : M2 != aa_nat_int(semiri1314217659103216013at_int,N5)
     => ~ ! [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
           => ( M2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) ) ) ) ).

% int_cases4
tff(fact_5197_int__cases3,axiom,
    ! [K3: int] :
      ( ( K3 != zero_zero_int )
     => ( ! [N5: nat] :
            ( ( K3 = aa_nat_int(semiri1314217659103216013at_int,N5) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5)) )
       => ~ ! [N5: nat] :
              ( ( K3 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) )
             => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5)) ) ) ) ).

% int_cases3
tff(fact_5198_neg__int__cases,axiom,
    ! [K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,K3),zero_zero_int))
     => ~ ! [N5: nat] :
            ( ( K3 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N5)) )
           => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5)) ) ) ).

% neg_int_cases
tff(fact_5199_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q5: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
     => ( ( B2 != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A2),B2,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)),aa_int_int(uminus_uminus_int,Q5)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q5)),one_one_int))),aa_int_int(aa_int_fun_int_int(if_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)),zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_5200_entails__solve__finalize_I1_J,axiom,
    ! [M5: list_P8527749157015355191n_assn,P: assn] : pp(aa_assn_bool(fI_RESULT(M5,P,one_one_assn),top_top_assn)) ).

% entails_solve_finalize(1)
tff(fact_5201_frame__inference__finalize,axiom,
    ! [M5: list_P8527749157015355191n_assn,F: assn] : pp(aa_assn_bool(fI_RESULT(M5,F,one_one_assn),F)) ).

% frame_inference_finalize
tff(fact_5202_entails__solve__finalize_I2_J,axiom,
    ! [M5: list_P8527749157015355191n_assn] : pp(aa_assn_bool(fI_RESULT(M5,one_one_assn,one_one_assn),one_one_assn)) ).

% entails_solve_finalize(2)
tff(fact_5203_Sup__nat__empty,axiom,
    complete_Sup_Sup_nat(bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_5204_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(numeral_numeral_int(M2),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = cons_int(numeral_numeral_int(M2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(numeral_numeral_int(M2),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = nil_int ) ) ) ).

% upto_rec_numeral(2)
tff(fact_5205_upto__empty,axiom,
    ! [J: int,I: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_5206_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil2
tff(fact_5207_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I)) ) ).

% upto_Nil
tff(fact_5208_upto__single,axiom,
    ! [I: int] : upto(I,I) = cons_int(I,nil_int) ).

% upto_single
tff(fact_5209_nth__upto,axiom,
    ! [I: int,K3: nat,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K3))),J))
     => ( aa_nat_int(nth_int(upto(I,J)),K3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K3)) ) ) ).

% nth_upto
tff(fact_5210_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M2)),numeral_numeral_int(N)))
       => ( upto(numeral_numeral_int(M2),numeral_numeral_int(N)) = cons_int(numeral_numeral_int(M2),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),one_one_int),numeral_numeral_int(N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(M2)),numeral_numeral_int(N)))
       => ( upto(numeral_numeral_int(M2),numeral_numeral_int(N)) = nil_int ) ) ) ).

% upto_rec_numeral(1)
tff(fact_5211_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = nil_int ) ) ) ).

% upto_rec_numeral(4)
tff(fact_5212_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),numeral_numeral_int(N)) = cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int),numeral_numeral_int(N))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)))
       => ( upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),numeral_numeral_int(N)) = nil_int ) ) ) ).

% upto_rec_numeral(3)
tff(fact_5213_distinct__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(distinct_int,upto(I,J))) ).

% distinct_upto
tff(fact_5214_sorted__upto,axiom,
    ! [M2: int,N: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_eq_int),upto(M2,N))) ).

% sorted_upto
tff(fact_5215_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : pp(aa_list_int_bool(sorted_wrt_int(ord_less_int),upto(I,J))) ).

% sorted_wrt_upto
tff(fact_5216_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list_int] : upto_aux(I,J,Js) = append_int(upto(I,J),Js) ).

% upto_aux_def
tff(fact_5217_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil_int) ).

% upto_code
tff(fact_5218_upto__split2,axiom,
    ! [I: int,J: int,K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K3))
       => ( upto(I,K3) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K3)) ) ) ) ).

% upto_split2
tff(fact_5219_upto__split1,axiom,
    ! [I: int,J: int,K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K3))
       => ( upto(I,K3) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K3)) ) ) ) ).

% upto_split1
tff(fact_5220_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_5221_upto_Osimps,axiom,
    ! [I: int,J: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
       => ( upto(I,J) = nil_int ) ) ) ).

% upto.simps
tff(fact_5222_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
         => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
tff(fact_5223_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_5224_upto__split3,axiom,
    ! [I: int,J: int,K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,J),K3))
       => ( upto(I,K3) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K3))) ) ) ) ).

% upto_split3
tff(fact_5225_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(X)),N)),aa_int_nat(nat2,A2)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,numeral_numeral_int(X)),N)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_5226_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(X)),N)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,A2),aa_nat_int(aa_int_fun_nat_int(power_power_int,numeral_numeral_int(X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_5227_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
    <=> ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
        & pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% zless_nat_conj
tff(fact_5228_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z)) ) ).

% zero_less_nat_eq
tff(fact_5229_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_5230_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,one_one_int),Z)) ) ).

% one_less_nat_eq
tff(fact_5231_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Z))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_mono_iff
tff(fact_5232_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_int_nat(nat2,Z)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M2)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_5233_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z)))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_5234_nat__less__iff,axiom,
    ! [W: int,M2: nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),W))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_int_nat(nat2,W)),M2))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M2))) ) ) ).

% nat_less_iff
tff(fact_5235_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q5: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,B2),zero_zero_int))
     => ( eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int),B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5236_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q5: int,R2: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,zero_zero_int),B2))
     => ( eucl_rel_int(A2,B2,aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),B2),aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5237_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),numeral_numeral_nat(aa_num_num(bit0,one)))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),numeral_numeral_nat(aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_5238_less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% less_exp
tff(fact_5239_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),numeral_numeral_nat(aa_num_num(bit0,one))))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_5240_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),numeral_numeral_nat(aa_num_num(bit0,one))))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_5241_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% div_2_gt_zero
tff(fact_5242_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_5243_ex__power__ivl1,axiom,
    ! [B2: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,one_one_nat),K3))
       => ? [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N5)),K3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),one_one_nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_5244_ex__power__ivl2,axiom,
    ! [B2: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K3))
       => ? [N5: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N5)),K3))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N5),one_one_nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_5245_nat__bit__induct,axiom,
    ! [P: fun_nat_bool,N: nat] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [N5: nat] :
            ( pp(aa_nat_bool(P,N5))
           => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
             => pp(aa_nat_bool(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N5))) ) )
       => ( ! [N5: nat] :
              ( pp(aa_nat_bool(P,N5))
             => pp(aa_nat_bool(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N5)))) )
         => pp(aa_nat_bool(P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_5246_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_5247_uminus__dvd__conv_I1_J,axiom,
    ! [D2: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(uminus_uminus_int,D2)),T)) ) ).

% uminus_dvd_conv(1)
tff(fact_5248_uminus__dvd__conv_I2_J,axiom,
    ! [D2: int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),T))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(uminus_uminus_int,T))) ) ).

% uminus_dvd_conv(2)
tff(fact_5249_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
       => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)) ) ) ).

% nat_dvd_not_less
tff(fact_5250_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2)) ) ) ).

% dvd_pos_nat
tff(fact_5251_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% dvd_minus_self
tff(fact_5252_zdvd__mono,axiom,
    ! [K3: int,M2: int,T: int] :
      ( ( K3 != zero_zero_int )
     => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,M2),T))
      <=> pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K3),M2)),aa_int_int(aa_int_fun_int_int(times_times_int,K3),T))) ) ) ).

% zdvd_mono
tff(fact_5253_dvd__imp__le,axiom,
    ! [K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,K3),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N)) ) ) ).

% dvd_imp_le
tff(fact_5254_nat__mult__dvd__cancel1,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)))
      <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_5255_dvd__mult__cancel,axiom,
    ! [K3: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),M2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,M2),N)) ) ) ).

% dvd_mult_cancel
tff(fact_5256_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),M2)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_5257_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),M2))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5258_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),M2))
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5259_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))))) ) ).

% even_even_mod_4_iff
tff(fact_5260_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,R2),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,R2),M2))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),R2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5261_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,one_one_nat),I))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ) ).

% power_dvd_imp_le
tff(fact_5262_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2) = M2 ) ) ).

% take_bit_nat_eq_self
tff(fact_5263_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% take_bit_nat_less_exp
tff(fact_5264_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2) = M2 )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5265_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M2)),M2))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),M2)) ) ).

% take_bit_nat_less_self_iff
tff(fact_5266_take__bit__int__less__eq,axiom,
    ! [N: nat,K3: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,numeral_numeral_int(aa_num_num(bit0,one))),N)),K3))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,N),K3)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K3),aa_nat_int(aa_int_fun_nat_int(power_power_int,numeral_numeral_int(aa_num_num(bit0,one))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_5267_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% even_mod_4_div_2
tff(fact_5268_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N))
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N))) ) ) ).

% even_diff_nat
tff(fact_5269_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% odd_pos
tff(fact_5270_Divides_Oadjust__div__eq,axiom,
    ! [Q5: int,R2: int] : adjust_div(aa_int7462565939943104719nt_int(product_Pair_int_int(Q5),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q5),zero_n2684676970156552555ol_int(aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(fequal_int,R2),zero_zero_int)))) ).

% Divides.adjust_div_eq
tff(fact_5271_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),numeral_numeral_nat(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_nat(aa_num_num(bit1,one)) )
     => ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,numeral_numeral_nat(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),numeral_numeral_nat(aa_num_num(bit0,one))))) ) ).

% odd_mod_4_div_2
tff(fact_5272_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != product_Pair_num_num(one,one) )
     => ( ! [N5: num] : X != product_Pair_num_num(one,aa_num_num(bit0,N5))
       => ( ! [N5: num] : X != product_Pair_num_num(one,aa_num_num(bit1,N5))
         => ( ! [M6: num] : X != product_Pair_num_num(aa_num_num(bit0,M6),one)
           => ( ! [M6: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit0,M6),aa_num_num(bit0,N5))
             => ( ! [M6: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit0,M6),aa_num_num(bit1,N5))
               => ( ! [M6: num] : X != product_Pair_num_num(aa_num_num(bit1,M6),one)
                 => ( ! [M6: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit1,M6),aa_num_num(bit0,N5))
                   => ~ ! [M6: num,N5: num] : X != product_Pair_num_num(aa_num_num(bit1,M6),aa_num_num(bit1,N5)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5273_divmod_H__nat__def,axiom,
    ! [M2: num,N: num] : unique5055182867167087721od_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N))) ).

% divmod'_nat_def
tff(fact_5274_divmod__int__def,axiom,
    ! [M2: num,N: num] : unique5052692396658037445od_int(M2,N) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,numeral_numeral_int(M2)),numeral_numeral_int(N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,numeral_numeral_int(M2)),numeral_numeral_int(N))) ).

% divmod_int_def
tff(fact_5275_divmod__BitM__2__eq,axiom,
    ! [M2: num] : unique5052692396658037445od_int(aa_num_num(bitM,M2),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_5276_less__numeral__Suc,axiom,
    ! [K3: num,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(K3)),aa_nat_nat(suc,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,pred_numeral(K3)),N)) ) ).

% less_numeral_Suc
tff(fact_5277_less__Suc__numeral,axiom,
    ! [N: nat,K3: num] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(K3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),pred_numeral(K3))) ) ).

% less_Suc_numeral
tff(fact_5278_divmod__nat__def,axiom,
    ! [M2: nat,N: nat] : divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N)) ).

% divmod_nat_def
tff(fact_5279_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N)) ) ).

% mask_nat_positive_iff
tff(fact_5280_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ) ).

% less_mask
tff(fact_5281_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N))) ).

% mask_nat_less_exp
tff(fact_5282_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa))
       => ~ ( ( ( ( member_int(X,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xa,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => ( Y = aa_int_int(uminus_uminus_int,zero_n2684676970156552555ol_int(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),Xa))))) ) )
              & ( ~ ( member_int(X,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(Xa,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => ( Y = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),X))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),Xa))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),numeral_numeral_int(aa_num_num(bit0,one)))))) ) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_5283_and__int_Opsimps,axiom,
    ! [K3: int,L: int] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(K3),L))
     => ( ( ( member_int(K3,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(L,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K3),L) = aa_int_int(uminus_uminus_int,zero_n2684676970156552555ol_int(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),K3))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),L))))) ) )
        & ( ~ ( member_int(K3,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
              & member_int(L,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
         => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K3),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),K3))),aa_bool_bool(fNot,aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,numeral_numeral_int(aa_num_num(bit0,one))),L))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K3),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(aa_num_num(bit0,one)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_5284_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K3: int] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
     => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K3),N))
       => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M2),K3))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_5285_int__bit__bound,axiom,
    ! [K3: int] :
      ~ ! [N5: nat] :
          ( ! [M3: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N5),M3))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K3),M3))
              <=> pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K3),N5)) ) )
         => ~ ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N5))
             => ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N5),one_one_nat)))
              <=> ~ pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K3),N5)) ) ) ) ).

% int_bit_bound
tff(fact_5286_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [K: int,L2: int] :
            ( accp_P1096762738010456898nt_int(bit_and_int_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(K),L2))
           => ( ( ~ ( member_int(K,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L2,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),numeral_numeral_int(aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),numeral_numeral_int(aa_num_num(bit0,one))))) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,K),L2)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_5287_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(I),J))
     => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) )
        & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I),J))
         => ( upto(I,J) = nil_int ) ) ) ) ).

% upto.psimps
tff(fact_5288_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( upto(X,Xa) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa))
       => ~ ( ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
               => ( Y = cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)) ) )
              & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X),Xa))
               => ( Y = nil_int ) ) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_5289_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_bool] :
      ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(A0),A1))
     => ( ! [I5: int,J3: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,aa_int7462565939943104719nt_int(product_Pair_int_int(I5),J3))
           => ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,I5),J3))
               => pp(aa_int_bool(aa_int_fun_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I5),one_one_int)),J3)) )
             => pp(aa_int_bool(aa_int_fun_int_bool(P,I5),J3)) ) )
       => pp(aa_int_bool(aa_int_fun_int_bool(P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_5290_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = aa_num_num(inc,aa_nat_num(num_of_nat,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = one ) ) ) ).

% num_of_nat.simps(2)
tff(fact_5291_binomial__code,axiom,
    ! [N: nat,K3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3) = zero_zero_nat ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K3)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K3)) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),K3)))
           => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K3)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K3)) ) ) ) ) ) ).

% binomial_code
tff(fact_5292_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ member_nat(N,nat_set_decode(Z))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),Z)) = insert_nat2(N,nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5293_binomial__eq__0__iff,axiom,
    ! [N: nat,K3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3) = zero_zero_nat )
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3)) ) ).

% binomial_eq_0_iff
tff(fact_5294_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_5295_zero__less__binomial__iff,axiom,
    ! [N: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N)) ) ).

% zero_less_binomial_iff
tff(fact_5296_binomial__eq__0,axiom,
    ! [N: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),K3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_5297_zero__less__binomial,axiom,
    ! [K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K3),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3))) ) ).

% zero_less_binomial
tff(fact_5298_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M6: nat] : X != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K),M6) ).

% prod_decode_aux.cases
tff(fact_5299_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( numeral_numeral_nat(aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_5300_choose__reduce__nat,axiom,
    ! [N: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K3),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K3)) ) ) ) ).

% choose_reduce_nat
tff(fact_5301_times__binomial__minus1__eq,axiom,
    ! [K3: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),K3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K3),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K3),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_5302_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_5303_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,M2)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_5304_binomial__addition__formula,axiom,
    ! [N: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K3))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K3)) ) ) ).

% binomial_addition_formula
tff(fact_5305_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
         => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)) ) )
        & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
         => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_5306_prod__decode__aux_Osimps,axiom,
    ! [M2: nat,K3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),K3))
       => ( nat_prod_decode_aux(K3,M2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K3),M2)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),K3))
       => ( nat_prod_decode_aux(K3,M2) = nat_prod_decode_aux(aa_nat_nat(suc,K3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(suc,K3))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_5307_cppi,axiom,
    ! [D: int,P: fun_int_bool,P2: fun_int_bool,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Z2),X2))
           => ( pp(aa_int_bool(P,X2))
            <=> pp(aa_int_bool(P2,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
             => ( pp(aa_int_bool(P,X2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D))) ) )
         => ( ! [X2: int,K: int] :
                ( pp(aa_int_bool(P2,X2))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D))
                    & pp(aa_int_bool(P2,X5)) )
                | ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D))
                    & ? [Xa2: int] :
                        ( member_int(Xa2,A)
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X5))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5308_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1266510415728281911st_int(I,J) = set_int2(upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_5309_bset_I1_J,axiom,
    ! [D: int,B: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D))) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D))) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ) ).

% bset(1)
tff(fact_5310_bset_I2_J,axiom,
    ! [D: int,B: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D))) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D))) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ) ).

% bset(2)
tff(fact_5311_aset_I1_J,axiom,
    ! [D: int,A: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D))) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D))) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                & pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)))
                & pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ) ) ).

% aset(1)
tff(fact_5312_aset_I2_J,axiom,
    ! [D: int,A: set_int,P: fun_int_bool,Q: fun_int_bool] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( pp(aa_int_bool(P,X2))
           => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D))) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( pp(aa_int_bool(Q,X2))
             => pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D))) ) )
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( pp(aa_int_bool(P,X6))
                | pp(aa_int_bool(Q,X6)) )
             => ( pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)))
                | pp(aa_int_bool(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ) ) ).

% aset(2)
tff(fact_5313_aset_I10_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T))) ) ) ) ).

% aset(10)
tff(fact_5314_aset_I9_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T))) ) ) ) ).

% aset(9)
tff(fact_5315_bset_I10_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => ~ pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T))) ) ) ) ).

% bset(10)
tff(fact_5316_bset_I9_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),T)))
           => pp(aa_int_bool(aa_int_fun_int_bool(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T))) ) ) ) ).

% bset(9)
tff(fact_5317_periodic__finite__ex,axiom,
    ! [D2: int,P: fun_int_bool] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D2))
     => ( ! [X2: int,K: int] :
            ( pp(aa_int_bool(P,X2))
          <=> pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D2)))) )
       => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
        <=> ? [X5: int] :
              ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D2))
              & pp(aa_int_bool(P,X5)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5318_aset_I7_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ).

% aset(7)
tff(fact_5319_aset_I5_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(T,A)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T)) ) ) ) ) ).

% aset(5)
tff(fact_5320_aset_I4_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(T,A)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X6 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D) != T ) ) ) ) ) ).

% aset(4)
tff(fact_5321_aset_I3_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X6 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D) = T ) ) ) ) ) ).

% aset(3)
tff(fact_5322_bset_I7_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(T,B)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),X6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ).

% bset(7)
tff(fact_5323_bset_I5_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X6),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T)) ) ) ) ).

% bset(5)
tff(fact_5324_bset_I4_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(T,B)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X6 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D) != T ) ) ) ) ) ).

% bset(4)
tff(fact_5325_bset_I3_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X6 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D) = T ) ) ) ) ) ).

% bset(3)
tff(fact_5326_simp__from__to,axiom,
    ! [J: int,I: int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = bot_bot_set_int ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,J),I))
       => ( set_or1266510415728281911st_int(I,J) = insert_int2(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ) ).

% simp_from_to
tff(fact_5327_bset_I6_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D)),T)) ) ) ) ).

% bset(6)
tff(fact_5328_bset_I8_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X6),D))) ) ) ) ) ).

% bset(8)
tff(fact_5329_aset_I6_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A)
       => ! [X6: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A)
                   => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,X6),T))
             => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D)),T)) ) ) ) ) ).

% aset(6)
tff(fact_5330_aset_I8_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ! [X6: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A)
                 => ( X6 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),X6))
           => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X6),D))) ) ) ) ).

% aset(8)
tff(fact_5331_cpmi,axiom,
    ! [D: int,P: fun_int_bool,P2: fun_int_bool,B: set_int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),D))
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X2),Z2))
           => ( pp(aa_int_bool(P,X2))
            <=> pp(aa_int_bool(P2,X2)) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
             => ( pp(aa_int_bool(P,X2))
               => pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D))) ) )
         => ( ! [X2: int,K: int] :
                ( pp(aa_int_bool(P2,X2))
              <=> pp(aa_int_bool(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K),D)))) )
           => ( ? [X_12: int] : pp(aa_int_bool(P,X_12))
            <=> ( ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D))
                    & pp(aa_int_bool(P2,X5)) )
                | ? [X5: int] :
                    ( member_int(X5,set_or1266510415728281911st_int(one_one_int,D))
                    & ? [Xa2: int] :
                        ( member_int(Xa2,B)
                        & pp(aa_int_bool(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X5))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5332_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N)),transi6264000038957366511cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) ) ).

% less_eq
tff(fact_5333_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
         => pp(aa_nat_bool(P,M)) )
    <=> ! [X5: nat] :
          ( member_nat(X5,set_or1269000886237332187st_nat(zero_zero_nat,N))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less
tff(fact_5334_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M),N))
          & pp(aa_nat_bool(P,M)) )
    <=> ? [X5: nat] :
          ( member_nat(X5,set_or1269000886237332187st_nat(zero_zero_nat,N))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less
tff(fact_5335_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat2(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(zero_zero_nat,N)) ).

% atLeast0_atMost_Suc
tff(fact_5336_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( insert_nat2(M2,set_or1269000886237332187st_nat(aa_nat_nat(suc,M2),N)) = set_or1269000886237332187st_nat(M2,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_5337_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),aa_nat_nat(suc,N)))
     => ( set_or1269000886237332187st_nat(M2,aa_nat_nat(suc,N)) = insert_nat2(aa_nat_nat(suc,N),set_or1269000886237332187st_nat(M2,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_5338_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
     => ( set_or1269000886237332187st_nat(M2,N) = insert_nat2(M2,set_or1269000886237332187st_nat(aa_nat_nat(suc,M2),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_5339_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_5340_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_atMost_nat,N)),insert_nat2(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_5341_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat2(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_5342_atMost__Suc,axiom,
    ! [K3: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K3)) = insert_nat2(aa_nat_nat(suc,K3),aa_nat_set_nat(set_ord_atMost_nat,K3)) ).

% atMost_Suc
tff(fact_5343_atMost__nat__numeral,axiom,
    ! [K3: num] : aa_nat_set_nat(set_ord_atMost_nat,numeral_numeral_nat(K3)) = insert_nat2(numeral_numeral_nat(K3),aa_nat_set_nat(set_ord_atMost_nat,pred_numeral(K3))) ).

% atMost_nat_numeral
tff(fact_5344_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I))))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5345_sorted__list__of__set__atMost__Suc,axiom,
    ! [K3: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K3))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K3)),cons_nat(aa_nat_nat(suc,K3),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_5346_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_5347_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5348_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_5349_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5350_finite__nat__set__iff__bounded,axiom,
    ! [N7: set_nat] :
      ( finite_finite_nat(N7)
    <=> ? [M: nat] :
        ! [X5: nat] :
          ( member_nat(X5,N7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),M)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_5351_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N: nat] :
      ( ! [X2: nat] :
          ( member_nat(X2,N7)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),N)) )
     => finite_finite_nat(N7) ) ).

% bounded_nat_set_is_finite
tff(fact_5352_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(suc,I)),J))
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_5353_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( finite_finite_nat(M5)
     => ( gcd_Gcd_nat(M5) = gcd_Gcd_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,M5),insert_nat2(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_5354_set__encode__insert,axiom,
    ! [A: set_nat,N: nat] :
      ( finite_finite_nat(A)
     => ( ~ member_nat(N,A)
       => ( nat_set_encode(insert_nat2(N,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),nat_set_encode(A)) ) ) ) ).

% set_encode_insert
tff(fact_5355_set__encode__empty,axiom,
    nat_set_encode(bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_5356_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_5357_unbounded__k__infinite,axiom,
    ! [K3: nat,S: set_nat] :
      ( ! [M6: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,K3),M6))
         => ? [N8: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M6),N8))
              & member_nat(N8,S) ) )
     => ~ finite_finite_nat(S) ) ).

% unbounded_k_infinite
tff(fact_5358_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ finite_finite_nat(S)
    <=> ! [M: nat] :
        ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
          & member_nat(N2,S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_5359_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K3: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K3))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K3)),cons_nat(K3,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_5360_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_5361_atLeastLessThan__singleton,axiom,
    ! [M2: nat] : set_or4665077453230672383an_nat(M2,aa_nat_nat(suc,M2)) = insert_nat2(M2,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_5362_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ? [M: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
          & pp(aa_nat_bool(P,M)) )
    <=> ? [X5: nat] :
          ( member_nat(X5,set_or4665077453230672383an_nat(zero_zero_nat,N))
          & pp(aa_nat_bool(P,X5)) ) ) ).

% ex_nat_less_eq
tff(fact_5363_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_bool] :
      ( ! [M: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N))
         => pp(aa_nat_bool(P,M)) )
    <=> ! [X5: nat] :
          ( member_nat(X5,set_or4665077453230672383an_nat(zero_zero_nat,N))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_nat_less_eq
tff(fact_5364_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_5365_lessThan__Suc,axiom,
    ! [K3: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K3)) = insert_nat2(K3,aa_nat_set_nat(set_ord_lessThan_nat,K3)) ).

% lessThan_Suc
tff(fact_5366_atLeastLessThan0,axiom,
    ! [M2: nat] : set_or4665077453230672383an_nat(M2,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_5367_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),N) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,N)),insert_nat2(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_5368_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat2(N,set_or4665077453230672383an_nat(zero_zero_nat,N)) ).

% atLeast0_lessThan_Suc
tff(fact_5369_lessThan__nat__numeral,axiom,
    ! [K3: num] : aa_nat_set_nat(set_ord_lessThan_nat,numeral_numeral_nat(K3)) = insert_nat2(pred_numeral(K3),aa_nat_set_nat(set_ord_lessThan_nat,pred_numeral(K3))) ).

% lessThan_nat_numeral
tff(fact_5370_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5371_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( set_or4665077453230672383an_nat(M2,aa_nat_nat(suc,N)) = insert_nat2(N,set_or4665077453230672383an_nat(M2,N)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
       => ( set_or4665077453230672383an_nat(M2,aa_nat_nat(suc,N)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
tff(fact_5372_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,N)) = insert_nat2(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5373_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K3: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),pred_numeral(K3)))
       => ( set_or4665077453230672383an_nat(M2,numeral_numeral_nat(K3)) = insert_nat2(pred_numeral(K3),set_or4665077453230672383an_nat(M2,pred_numeral(K3))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),pred_numeral(K3)))
       => ( set_or4665077453230672383an_nat(M2,numeral_numeral_nat(K3)) = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_5374_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_5375_atLeast__Suc,axiom,
    ! [K3: nat] : set_ord_atLeast_nat(aa_nat_nat(suc,K3)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,set_ord_atLeast_nat(K3)),insert_nat2(K3,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_5376_card__Collect__less__nat,axiom,
    ! [N: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,N))) = N ).

% card_Collect_less_nat
tff(fact_5377_finite__Collect__less__nat,axiom,
    ! [K3: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,K3))) ).

% finite_Collect_less_nat
tff(fact_5378_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ member_nat(zero_zero_nat,M5)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af(M5),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ag(M5),I))) ) ) ).

% card_less_Suc2
tff(fact_5379_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( member_nat(zero_zero_nat,M5)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_af(M5),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ag(M5),I))) ) ) ).

% card_less_Suc
tff(fact_5380_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( member_nat(zero_zero_nat,M5)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ag(M5),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_5381_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_bool,I: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ah(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_5382_nat__less__as__int,axiom,
    ! [X6: nat,Xa3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),Xa3))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X6)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ) ).

% nat_less_as_int
tff(fact_5383_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => finite_finite_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ai,M2))) ) ).

% finite_divisors_nat
tff(fact_5384_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aj,N),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,N) ) ) ).

% range_mod
tff(fact_5385_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ak,C),set_or4665077453230672383an_nat(X,Y)) = set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,C),Y))
       => ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ak,C),set_or4665077453230672383an_nat(X,Y)) = insert_nat2(zero_zero_nat,bot_bot_set_nat) ) )
          & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y))
           => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ak,C),set_or4665077453230672383an_nat(X,Y)) = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5386_Sup__nat__def,axiom,
    ! [X3: set_nat] :
      ( ( ( X3 = bot_bot_set_nat )
       => ( complete_Sup_Sup_nat(X3) = zero_zero_nat ) )
      & ( ( X3 != bot_bot_set_nat )
       => ( complete_Sup_Sup_nat(X3) = lattic8265883725875713057ax_nat(X3) ) ) ) ).

% Sup_nat_def
tff(fact_5387_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,N))) ).

% mask_eq_sum_exp_nat
tff(fact_5388_divmod__nat__if,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( ( N = zero_zero_nat )
          | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) )
       => ( divmod_nat(M2,N) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M2) ) )
      & ( ~ ( ( N = zero_zero_nat )
            | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M2),N)) )
       => ( divmod_nat(M2,N) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_al),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_5389_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_am(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5390_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_an(L),Qr) ).

% divmod_step_int_def
tff(fact_5391_wf__less,axiom,
    pp(aa_set3135850010578026034t_bool(wf_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)))) ).

% wf_less
tff(fact_5392_natLeq__on__wo__rel,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(bNF_We3818239936649020644el_nat,collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N))))) ).

% natLeq_on_wo_rel
tff(fact_5393_Field__natLeq__on,axiom,
    ! [N: nat] : field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N)))) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,N)) ).

% Field_natLeq_on
tff(fact_5394_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc8955392012077580105t_bool(ord_less_nat)) ).

% natLess_def
tff(fact_5395_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_ap,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_5396_remdups__upt,axiom,
    ! [M2: nat,N: nat] : remdups_nat(upt(M2,N)) = upt(M2,N) ).

% remdups_upt
tff(fact_5397_sort__upt,axiom,
    ! [M2: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_aq,upt(M2,N)) = upt(M2,N) ).

% sort_upt
tff(fact_5398_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl_nat(upt(M2,N)) = upt(aa_nat_nat(suc,M2),N) ).

% tl_upt
tff(fact_5399_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_5400_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] : drop_nat(M2,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2),J) ).

% drop_upt
tff(fact_5401_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)),N))
     => ( take_nat(M2,upt(I,N)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M2)) ) ) ).

% take_upt
tff(fact_5402_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_5403_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I))
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_5404_upt__merge,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),K3)) )
     => ( append_nat(upt(I,J),upt(J,K3)) = upt(I,K3) ) ) ).

% upt_merge
tff(fact_5405_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(M2,N)) = upt(M2,N) ).

% sorted_list_of_set_range
tff(fact_5406_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_5407_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_5408_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_5409_nth__upt,axiom,
    ! [I: nat,K3: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3)),J))
     => ( aa_nat_nat(nth_nat(upt(I,J)),K3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K3) ) ) ).

% nth_upt
tff(fact_5410_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)))
       => ( upt(numeral_numeral_nat(M2),numeral_numeral_nat(N)) = cons_nat(numeral_numeral_nat(M2),upt(aa_nat_nat(suc,numeral_numeral_nat(M2)),numeral_numeral_nat(N))) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)))
       => ( upt(numeral_numeral_nat(M2),numeral_numeral_nat(N)) = nil_nat ) ) ) ).

% upt_rec_numeral
tff(fact_5411_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or4665077453230672383an_nat(I,J) = set_nat2(upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_5412_distinct__upt,axiom,
    ! [I: nat,J: nat] : pp(aa_list_nat_bool(distinct_nat,upt(I,J))) ).

% distinct_upt
tff(fact_5413_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,Ofs),upt(A2,B2)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),Ofs)) ).

% map_add_upt'
tff(fact_5414_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : map_nat_nat(suc,upt(M2,N)) = upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_5415_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : set_or1269000886237332187st_nat(N,M2) = set_nat2(upt(N,aa_nat_nat(suc,M2))) ).

% atLeastAtMost_upt
tff(fact_5416_butlast__upt,axiom,
    ! [M2: nat,N: nat] : butlast_nat(upt(M2,N)) = upt(M2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% butlast_upt
tff(fact_5417_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_5418_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q5: nat] :
      ( ( cons_nat(M2,cons_nat(N,Ns)) = upt(M2,Q5) )
    <=> ( cons_nat(N,Ns) = upt(aa_nat_nat(suc,M2),Q5) ) ) ).

% upt_conv_Cons_Cons
tff(fact_5419_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_5420_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_nat),upt(M2,N))) ).

% sorted_wrt_upt
tff(fact_5421_map__add__upt,axiom,
    ! [N: nat,M2: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,N),upt(zero_zero_nat,M2)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% map_add_upt
tff(fact_5422_sorted__upt,axiom,
    ! [M2: nat,N: nat] : pp(aa_list_nat_bool(sorted_wrt_nat(ord_less_eq_nat),upt(M2,N))) ).

% sorted_upt
tff(fact_5423_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : set_or6659071591806873216st_nat(N,M2) = set_nat2(upt(aa_nat_nat(suc,N),aa_nat_nat(suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_5424_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( ( upt(I,J) = append_nat(Xs,Ys) )
      <=> ? [K4: nat] :
            ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),K4))
            & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,K4),J))
            & ( upt(I,K4) = Xs )
            & ( upt(K4,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_5425_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : set_or5834768355832116004an_nat(N,M2) = set_nat2(upt(aa_nat_nat(suc,N),M2)) ).

% greaterThanLessThan_upt
tff(fact_5426_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_5427_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_5428_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: fun_nat_bool] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),U3))
     => ( ! [I5: nat] :
            ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,U),I5))
              & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),U3)) )
           => ~ pp(aa_nat_bool(P,I5)) )
       => ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U3)) ) ) ) ).

% upt_filter_extend
tff(fact_5429_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : map_nat_nat(aTP_Lamm_as,upt(aa_nat_nat(suc,M2),aa_nat_nat(suc,N))) = upt(M2,N) ).

% map_decr_upt
tff(fact_5430_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_5431_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K3: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K3)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K3))) ) ) ).

% upt_add_eq_append
tff(fact_5432_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(X,Xs) )
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
        & ( I = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_5433_upt__rec,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),J))
       => ( upt(I,J) = nil_nat ) ) ) ).

% upt_rec
tff(fact_5434_upt__eq__lel__conv,axiom,
    ! [L: nat,H2: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H2) = append_nat(Is1,cons_nat(I,Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H2) )
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,L),I))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I),H2)) ) ) ).

% upt_eq_lel_conv
tff(fact_5435_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) )
      & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
       => ( upt(I,aa_nat_nat(suc,J)) = nil_nat ) ) ) ).

% upt_Suc
tff(fact_5436_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,I),J))
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_5437_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_at(L),Qr) ).

% divmod_step_integer_def
tff(fact_5438_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_5439_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5440_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_5441_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_au(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_5442_divmod__integer_H__def,axiom,
    ! [M2: num,N: num] : unique3479559517661332726nteger(M2,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_num_Code_integer(numera6620942414471956472nteger,M2)),aa_num_Code_integer(numera6620942414471956472nteger,N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_num_Code_integer(numera6620942414471956472nteger,M2)),aa_num_Code_integer(numera6620942414471956472nteger,N))) ).

% divmod_integer'_def
tff(fact_5443_finite__enumerate,axiom,
    ! [S: set_nat] :
      ( finite_finite_nat(S)
     => ? [R5: fun_nat_nat] :
          ( strict1292158309912662752at_nat(R5,aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S)))
          & ! [N8: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N8),finite_card_nat(S)))
             => member_nat(aa_nat_nat(R5,N8),S) ) ) ) ).

% finite_enumerate
tff(fact_5444_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_5445_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_5446_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or4266950643985792945nteger(L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_5447_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite6017078050557962740nteger(set_or2715278749043346189nteger(L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_5448_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : finite6017078050557962740nteger(set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_5449_bit__cut__integer__def,axiom,
    ! [K3: code_integer] : code_bit_cut_integer(K3) = produc5135875973608410785r_bool(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K3),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_bool_bool(fNot,aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K3))) ).

% bit_cut_integer_def
tff(fact_5450_divmod__integer__def,axiom,
    ! [K3: code_integer,L: code_integer] : code_divmod_integer(K3,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K3),L),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(K3),L)) ).

% divmod_integer_def
tff(fact_5451_bit__cut__integer__code,axiom,
    ! [K3: code_integer] :
      ( ( ( K3 = zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K3) = produc5135875973608410785r_bool(zero_z3403309356797280102nteger,fFalse) ) )
      & ( ( K3 != zero_z3403309356797280102nteger )
       => ( code_bit_cut_integer(K3) = produc6117575242473779287r_bool(aTP_Lamm_av(K3),code_divmod_abs(K3,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_5452_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U))
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_5453_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_5454_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).

% divmod_abs_code(5)
tff(fact_5455_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X))
     => ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y))
       => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,X)),aa_Code_integer_nat(code_nat_of_integer,Y)))
        <=> pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,X),Y)) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_5456_divmod__abs__def,axiom,
    ! [K3: code_integer,L: code_integer] : code_divmod_abs(K3,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K3)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K3)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ).

% divmod_abs_def
tff(fact_5457_divmod__integer__code,axiom,
    ! [K3: code_integer,L: code_integer] :
      ( ( ( K3 = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K3,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K3 != zero_z3403309356797280102nteger )
       => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K3))
               => ( code_divmod_integer(K3,L) = code_divmod_abs(K3,L) ) )
              & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K3))
               => ( code_divmod_integer(K3,L) = produc6916734918728496179nteger(aTP_Lamm_aw(L),code_divmod_abs(K3,L)) ) ) ) )
          & ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L))
           => ( ( ( L = zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K3,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K3) ) )
              & ( ( L != zero_z3403309356797280102nteger )
               => ( code_divmod_integer(K3,L) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,K3),zero_z3403309356797280102nteger),code_divmod_abs(K3,L),produc6916734918728496179nteger(aTP_Lamm_ax(L),code_divmod_abs(K3,L)))) ) ) ) ) ) ) ) ).

% divmod_integer_code
tff(fact_5458_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( finite_finite_nat(M5)
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ member_nat(zero_zero_nat,M5)
         => ( gcd_Gcd_nat(M5) = lattic8265883725875713057ax_nat(comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_ay,M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_5459_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_5460_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_ba),Xa),X)) ).

% times_int.abs_eq
tff(fact_5461_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => member_nat(complete_Inf_Inf_nat(K5),K5) ) ).

% Inf_nat_def1
tff(fact_5462_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y2: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y2)) ).

% eq_Abs_Integ
tff(fact_5463_zero__int__def,axiom,
    zero_zero_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_5464_int__def,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,N) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N),zero_zero_nat)) ).

% int_def
tff(fact_5465_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_bb),X)) ).

% uminus_int.abs_eq
tff(fact_5466_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_5467_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_bd),Xa),X)) ) ).

% less_int.abs_eq
tff(fact_5468_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_bf),Xa),X)) ).

% plus_int.abs_eq
tff(fact_5469_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_bh),Xa),X)) ).

% minus_int.abs_eq
tff(fact_5470_divmod__integer__eq__cases,axiom,
    ! [K3: code_integer,L: code_integer] :
      ( ( ( K3 = zero_z3403309356797280102nteger )
       => ( code_divmod_integer(K3,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ) )
      & ( ( K3 != zero_z3403309356797280102nteger )
       => ( ( ( L = zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K3,L) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,K3) ) )
          & ( ( L != zero_z3403309356797280102nteger )
           => ( code_divmod_integer(K3,L) = aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K3)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L)),code_divmod_abs(K3,L),produc6916734918728496179nteger(aTP_Lamm_bi(L),code_divmod_abs(K3,L)))) ) ) ) ) ) ).

% divmod_integer_eq_cases
tff(fact_5471_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,X),Xa))
    <=> pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(produc5023667602222594023t_bool(aTP_Lamm_bd),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa))) ) ).

% less_int.rep_eq
tff(fact_5472_assn__aci_I11_J,axiom,
    ! [X: assn,Y: assn,A2: assn,B2: assn] :
      ( pp(aa_assn_bool(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),A2) ) ) ).

% assn_aci(11)
tff(fact_5473_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_bb)) ).

% uminus_int_def
tff(fact_5474_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ba)) ).

% times_int_def
tff(fact_5475_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_bh)) ).

% minus_int_def
tff(fact_5476_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_bf)) ).

% plus_int_def
tff(fact_5477_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_bj,aTP_Lamm_ae) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5478_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N)) ) ).

% mono_times_nat
tff(fact_5479_less__int__def,axiom,
    ord_less_int = map_fu2743658913401090412t_bool(rep_Integ,map_fu6608131713810608381l_bool(rep_Integ,id_bool),produc5023667602222594023t_bool(aTP_Lamm_bd)) ).

% less_int_def
tff(fact_5480_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa))))) ) ) ) ) ).

% bezw.elims
tff(fact_5481_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero_nat )
       => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
      & ( ( Y != zero_zero_nat )
       => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ) ).

% bezw.simps
tff(fact_5482_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),Y))
     => ( bezw(X,Y) = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y))))) ) ) ).

% bezw_non_0
tff(fact_5483_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)))
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),zero_zero_int) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y = aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa))))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))) ) ) ) ).

% bezw.pelims
tff(fact_5484_mod__pure,axiom,
    ! [B2: bool,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,pure_assn(B2)),H2))
    <=> ( ( produc8586169260539613262et_nat(H2) = bot_bot_set_nat )
        & pp(B2) ) ) ).

% mod_pure
tff(fact_5485_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,one_one_assn),H2))
    <=> ( produc8586169260539613262et_nat(H2) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_5486_fi__match__entails,axiom,
    ! [M2: list_P8527749157015355191n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(M2))
         => pp(aa_Pro3608213857613363938n_bool(produc6933888896912550553n_bool(entails),X2)) )
     => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M2)),one_one_assn)),aa_assn_assn(foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M2)),one_one_assn))) ) ).

% fi_match_entails
tff(fact_5487_mod__star__trueE_H,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),top_top_assn)),H2))
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( produc1824681642469235216et_nat(H5) = produc1824681642469235216et_nat(H2) )
           => ( ord_less_eq_set_nat(produc8586169260539613262et_nat(H5),produc8586169260539613262et_nat(H2))
             => ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H5)) ) ) ) ).

% mod_star_trueE'
tff(fact_5488_FI__RESULT__def,axiom,
    ! [M5: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F: assn] :
      ( pp(aa_assn_bool(fI_RESULT(M5,UP,UQ),F))
    <=> ( ! [X5: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X5,set_Pr1139785259514867910n_assn(M5))
           => pp(aa_Pro3608213857613363938n_bool(produc6933888896912550553n_bool(entails),X5)) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M5)),one_one_assn)),UP)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M5)),one_one_assn)),UQ)),F))) ) ) ).

% FI_RESULT_def
tff(fact_5489_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)))
       => ~ ( ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
               => ( Y = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)) ) )
              & ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Xa),X))
               => ( Y = nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X))) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_5490_FI__def,axiom,
    ! [M2: list_P8527749157015355191n_assn,P4: assn,Q5: assn,Up: assn,Uq: assn,F3: assn] :
      ( pp(aa_assn_bool(fi(M2,P4,Q5,Up,Uq),F3))
    <=> ( ! [X5: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X5,set_Pr1139785259514867910n_assn(M2))
           => pp(aa_Pro3608213857613363938n_bool(produc6933888896912550553n_bool(entails),X5)) )
       => pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M2)),one_one_assn)),P4)),Up)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M2)),one_one_assn)),Q5)),Uq)),F3))) ) ) ).

% FI_def
tff(fact_5491_FI__p__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Ps2: assn,Qs: assn,Q5: assn,P4: assn,Up: assn,Uq: assn,F3: assn] :
      ( pp(aa_assn_bool(fi(M2,Ps2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q5),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P4),Up),Uq),F3))
     => pp(aa_assn_bool(fi(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps2),P4),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q5),Up,Uq),F3)) ) ).

% FI_p_nomatch
tff(fact_5492_FI__finalize,axiom,
    ! [M2: list_P8527749157015355191n_assn,P4: assn,Up: assn,Q5: assn,Uq: assn,F3: assn] :
      ( pp(aa_assn_bool(fI_RESULT(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P4),Up),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q5),Uq)),F3))
     => pp(aa_assn_bool(fi(M2,P4,Q5,Up,Uq),F3)) ) ).

% FI_finalize
tff(fact_5493_FI__init,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( pp(aa_assn_bool(fi(nil_Pr5671120429643327159n_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),P),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Q),sln,sln),F))
     => pp(aa_assn_bool(fI_QUERY(P,Q),F)) ) ).

% FI_init
tff(fact_5494_SLN__right,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),sln) = P ).

% SLN_right
tff(fact_5495_SLN__left,axiom,
    ! [P: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),P) = P ).

% SLN_left
tff(fact_5496_SLN__def,axiom,
    sln = one_one_assn ).

% SLN_def
tff(fact_5497_FI__q__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Up: assn,Qs: assn,Q5: assn,Uq: assn,F3: assn] :
      ( pp(aa_assn_bool(fi(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Up),Qs,sln,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Q5),Uq)),F3))
     => pp(aa_assn_bool(fi(M2,sln,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q5),Up,Uq),F3)) ) ).

% FI_q_nomatch
tff(fact_5498_FI__match,axiom,
    ! [P4: assn,Q5: assn,M2: list_P8527749157015355191n_assn,Ps2: assn,Up: assn,Qs: assn,Uq: assn,F3: assn] :
      ( pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,P4),Q5))
     => ( pp(aa_assn_bool(fi(cons_P2971678138204555879n_assn(produc118845697133431529n_assn(P4,Q5),M2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps2),Up),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Uq),sln,sln),F3))
       => pp(aa_assn_bool(fi(M2,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Ps2),P4),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qs),Q5),Up,Uq),F3)) ) ) ).

% FI_match
tff(fact_5499_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M2),N)),transi2905341329935302413cl_nat(pred_nat)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_5500_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A: set_nat] : vimage_nat_nat(suc,insert_nat2(aa_nat_nat(suc,N),A)) = insert_nat2(N,vimage_nat_nat(suc,A)) ).

% vimage_Suc_insert_Suc
tff(fact_5501_vimage__Suc__insert__0,axiom,
    ! [A: set_nat] : vimage_nat_nat(suc,insert_nat2(zero_zero_nat,A)) = vimage_nat_nat(suc,A) ).

% vimage_Suc_insert_0
tff(fact_5502_finite__le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( finite_finite_nat(S)
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),finite_card_nat(S)))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N),aa_nat_nat(infini8530281810654367211te_nat(S),N))) ) ) ).

% finite_le_enumerate
tff(fact_5503_less__natural_Orsp,axiom,
    bNF_re1480548845665215743t_bool(fequal_nat,bNF_re6223507645442431167l_bool(fequal_nat,fequal_bool),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_5504_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_ba),times_times_int) ).

% times_int.transfer
tff(fact_5505_zero__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int)) ).

% zero_int.transfer
tff(fact_5506_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_bk,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_5507_uminus__int_Otransfer,axiom,
    pp(aa_fun_int_int_bool(aa_fun7527181275634263258t_bool(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_bb)),uminus_uminus_int)) ).

% uminus_int.transfer
tff(fact_5508_one__int_Otransfer,axiom,
    pp(aa_int_bool(aa_Pro2928840338864870335t_bool(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int)) ).

% one_int.transfer
tff(fact_5509_less__int_Otransfer,axiom,
    bNF_re2632563231991741785t_bool(pcr_int,bNF_re1498568104593929552l_bool(pcr_int,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_bd),ord_less_int) ).

% less_int.transfer
tff(fact_5510_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_bf),plus_plus_int) ).

% plus_int.transfer
tff(fact_5511_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_bh),minus_minus_int) ).

% minus_int.transfer
tff(fact_5512_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_ba),produc4528391901023682385at_nat(aTP_Lamm_ba)) ).

% times_int.rsp
tff(fact_5513_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V2)))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),V2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_5514_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_5515_zero__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat))) ).

% zero_int.rsp
tff(fact_5516_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_5517_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_5518_uminus__int_Orsp,axiom,
    pp(aa_fun4519828053337421206t_bool(aa_fun2399787666232550060t_bool(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_bb)),produc2626176000494625587at_nat(aTP_Lamm_bb))) ).

% uminus_int.rsp
tff(fact_5519_one__int_Orsp,axiom,
    pp(aa_Pro2232559814647311250t_bool(aa_Pro595088089689268830t_bool(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat))) ).

% one_int.rsp
tff(fact_5520_less__int_Orsp,axiom,
    bNF_re2219331435504962769t_bool(intrel,bNF_re3189209160384574813l_bool(intrel,fequal_bool),produc5023667602222594023t_bool(aTP_Lamm_bd),produc5023667602222594023t_bool(aTP_Lamm_bd)) ).

% less_int.rsp
tff(fact_5521_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_bh),produc4528391901023682385at_nat(aTP_Lamm_bh)) ).

% minus_int.rsp
tff(fact_5522_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_bf),produc4528391901023682385at_nat(aTP_Lamm_bf)) ).

% plus_int.rsp
tff(fact_5523_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_5524_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(aTP_Lamm_bl(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_5525_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_bool,K3: nat] :
      ( eventually_nat(aa_nat_fun_nat_bool(aTP_Lamm_bm(P),K3),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_5526_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_bn,at_top_nat) ).

% eventually_False_sequentially
tff(fact_5527_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_bool] :
      ( ! [X2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,C),X2))
         => pp(aa_nat_bool(P,X2)) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_5528_eventually__sequentially,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N2))
         => pp(aa_nat_bool(P,N2)) ) ) ).

% eventually_sequentially
tff(fact_5529_le__sequentially,axiom,
    ! [F: filter_nat] :
      ( ord_le2510731241096832064er_nat(F,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),F) ) ).

% le_sequentially
tff(fact_5530_bit__concat__bit__iff,axiom,
    ! [M2: nat,K3: int,L: int,N: nat] :
      ( pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M2),K3),L)),N))
    <=> ( ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,N),M2))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,K3),N)) )
        | ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,M2),N))
          & pp(aa_nat_bool(aa_int_fun_nat_bool(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5531_bot__nat__0_Oordering__top__axioms,axiom,
    pp(aa_nat_bool(ordering_top_nat(aTP_Lamm_bj,aTP_Lamm_ae),zero_zero_nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_5532_pair__lessI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,S2),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_5533_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Z)),fun_pair_less)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Y),Z)) ) ).

% pair_less_iff1
tff(fact_5534_pair__lessI1,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_5535_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_5536_pair__leqI2,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),B2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,S2),T))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_5537_smin__emptyI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( X3 != bot_bo2099793752762293965at_nat )
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,bot_bo2099793752762293965at_nat),fun_min_strict) ) ).

% smin_emptyI
tff(fact_5538_pair__leqI1,axiom,
    ! [A2: nat,B2: nat,S2: nat,T: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,A2),B2))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),S2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_5539_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),YS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_max_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,XS),YS),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_5540_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(X),XS))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,YS),fun_min_weak)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(XS,insert8211810215607154385at_nat(Y,YS)),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_5541_wmax__emptyI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,X3))
     => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,X3),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_5542_wmin__emptyI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,bot_bo2099793752762293965at_nat),fun_min_weak) ).

% wmin_emptyI
tff(fact_5543_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_min_strict,fun_min_weak)) ).

% min_rpair_set
tff(fact_5544_max__weak__def,axiom,
    fun_max_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,max_ex8135407076693332796at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_5545_min__weak__def,axiom,
    fun_min_weak = aa_set4883462369968598652at_nat(aa_set7704514097189807246at_nat(sup_su5525570899277871387at_nat,min_ex6901939911449802026at_nat(fun_pair_leq)),insert9069300056098147895at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,bot_bo2099793752762293965at_nat),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_5546_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X3: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Y),Y5))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_less)
       => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(X3,Y5),fun_max_strict)
         => member8757157785044589968at_nat(produc2922128104949294807at_nat(insert8211810215607154385at_nat(X,X3),Y5),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_5547_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat(produc9060074326276436823at_nat(fun_max_strict,fun_max_weak)) ).

% max_rpair_set
tff(fact_5548_smax__emptyI,axiom,
    ! [Y5: set_Pr1261947904930325089at_nat] :
      ( pp(aa_set3135850010578026034t_bool(finite6177210948735845034at_nat,Y5))
     => ( ( Y5 != bot_bo2099793752762293965at_nat )
       => member8757157785044589968at_nat(produc2922128104949294807at_nat(bot_bo2099793752762293965at_nat,Y5),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_5549_wmsI,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A),set_ms8126754132646256062at_nat(B)),fun_max_strict)
        | ( ( A = zero_z1048942125864253310at_nat )
          & ( B = zero_z1048942125864253310at_nat ) ) )
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(plus_p7104986032573967614at_nat(Z5,A),plus_p7104986032573967614at_nat(Z5,B)),ms_weak) ) ).

% wmsI
tff(fact_5550_smsI,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,Z5: multis2468970476368604999at_nat] :
      ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A),set_ms8126754132646256062at_nat(B)),fun_max_strict)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(plus_p7104986032573967614at_nat(Z5,A),plus_p7104986032573967614at_nat(Z5,B)),ms_strict) ) ).

% smsI
tff(fact_5551_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat(produc5245064249948416855at_nat(ms_strict,ms_weak)) ).

% ms_reduction_pair
tff(fact_5552_ms__weakI1,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( pw_leq(Z5,Z6)
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A),set_ms8126754132646256062at_nat(B)),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(plus_p7104986032573967614at_nat(Z5,A),plus_p7104986032573967614at_nat(Z6,B)),ms_weak) ) ) ).

% ms_weakI1
tff(fact_5553_ms__strictI,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( pw_leq(Z5,Z6)
     => ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A),set_ms8126754132646256062at_nat(B)),fun_max_strict)
       => member6689249552917799696at_nat(produc4348348721325984599at_nat(plus_p7104986032573967614at_nat(Z5,A),plus_p7104986032573967614at_nat(Z6,B)),ms_strict) ) ) ).

% ms_strictI
tff(fact_5554_pw__leq__lstep,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
     => pw_leq(add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat),add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat)) ) ).

% pw_leq_lstep
tff(fact_5555_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X3: multis2468970476368604999at_nat,Y5: multis2468970476368604999at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y),fun_pair_leq)
     => ( pw_leq(X3,Y5)
       => pw_leq(plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X,zero_z1048942125864253310at_nat),X3),plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y,zero_z1048942125864253310at_nat),Y5)) ) ) ).

% pw_leq_step
tff(fact_5556_pw__leq_Osimps,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
    <=> ( ( ( A1 = zero_z1048942125864253310at_nat )
          & ( A22 = zero_z1048942125864253310at_nat ) )
        | ? [X5: product_prod_nat_nat,Y4: product_prod_nat_nat,X7: multis2468970476368604999at_nat,Y6: multis2468970476368604999at_nat] :
            ( ( A1 = plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X5,zero_z1048942125864253310at_nat),X7) )
            & ( A22 = plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y4,zero_z1048942125864253310at_nat),Y6) )
            & member8206827879206165904at_nat(produc6161850002892822231at_nat(X5,Y4),fun_pair_leq)
            & pw_leq(X7,Y6) ) ) ) ).

% pw_leq.simps
tff(fact_5557_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
      ( pw_leq(A1,A22)
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A22 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,X8: multis2468970476368604999at_nat] :
              ( ( A1 = plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(X2,zero_z1048942125864253310at_nat),X8) )
             => ! [Y7: multis2468970476368604999at_nat] :
                  ( ( A22 = plus_p7104986032573967614at_nat(add_ms2612439473150266591at_nat(Y2,zero_z1048942125864253310at_nat),Y7) )
                 => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X2,Y2),fun_pair_leq)
                   => ~ pw_leq(X8,Y7) ) ) ) ) ) ).

% pw_leq.cases
tff(fact_5558_ms__weakI2,axiom,
    ! [Z5: multis2468970476368604999at_nat,Z6: multis2468970476368604999at_nat] :
      ( pw_leq(Z5,Z6)
     => member6689249552917799696at_nat(produc4348348721325984599at_nat(plus_p7104986032573967614at_nat(Z5,zero_z1048942125864253310at_nat),plus_p7104986032573967614at_nat(Z6,zero_z1048942125864253310at_nat)),ms_weak) ) ).

% ms_weakI2
tff(fact_5559_pw__leq__split,axiom,
    ! [X3: multis2468970476368604999at_nat,Y5: multis2468970476368604999at_nat] :
      ( pw_leq(X3,Y5)
     => ? [A8: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
          ( ( X3 = plus_p7104986032573967614at_nat(A8,Z7) )
          & ( Y5 = plus_p7104986032573967614at_nat(B4,Z7) )
          & ( member8757157785044589968at_nat(produc2922128104949294807at_nat(set_ms8126754132646256062at_nat(A8),set_ms8126754132646256062at_nat(B4)),fun_max_strict)
            | ( ( B4 = zero_z1048942125864253310at_nat )
              & ( A8 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
tff(fact_5560_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y)),less_than))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X),Y)) ) ).

% less_than_iff
tff(fact_5561_natLeq__on__well__order__on,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(order_2888998067076097458on_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,N))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N))))) ).

% natLeq_on_well_order_on
tff(fact_5562_natLeq__on__Well__order,axiom,
    ! [N: nat] : pp(aa_set3135850010578026034t_bool(order_2888998067076097458on_nat(field_nat(collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N))))),collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N))))) ).

% natLeq_on_Well_order
tff(fact_5563_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr7757163693017731481t_bool] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N9: nat] :
        ! [M: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),M))
         => ! [N2: nat] :
              ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,N9),N2))
             => pp(aa_Pro2232559814647311250t_bool(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N2),M))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5564_UNIV__bool,axiom,
    top_top_set_bool = insert_bool2(fFalse,insert_bool2(fTrue,bot_bot_set_bool)) ).

% UNIV_bool
tff(fact_5565_ctwo__Cnotzero,axiom,
    ( ~ member1730920086087385872l_bool(produc5355662241087601495l_bool(bNF_Cardinal_ctwo,bNF_Ca1500629177841160323o_bool),bNF_We9067914441000329236l_bool)
    & bNF_Ca3869711322664823701n_bool(field_bool(bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).

% ctwo_Cnotzero
tff(fact_5566_ctwo__ordLess__natLeq,axiom,
    member3042533333095794346at_nat(produc4079257842390172913at_nat(bNF_Cardinal_ctwo,bNF_Ca8665028551170535155natLeq),bNF_We6144460630316186525ol_nat) ).

% ctwo_ordLess_natLeq
tff(fact_5567_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,N) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,N)) ).

% natLeq_underS_less
tff(fact_5568_Restr__natLeq2,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,N),aTP_Lamm_bo(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N))) ).

% Restr_natLeq2
tff(fact_5569_Restr__natLeq,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,N)),aTP_Lamm_bp(N))) = collec3392354462482085612at_nat(produc8955392012077580105t_bool(aTP_Lamm_ao(N))) ).

% Restr_natLeq
tff(fact_5570_card__of__nat,axiom,
    member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(top_top_set_nat),bNF_Ca8665028551170535155natLeq),bNF_We5258908940166488438at_nat) ).

% card_of_nat
tff(fact_5571_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat(produc2922128104949294807at_nat(bNF_Ca3793111618940312692of_nat(field_nat(bNF_Ca8665028551170535155natLeq)),bNF_Ca8665028551170535155natLeq),bNF_We5258908940166488438at_nat) ).

% card_of_Field_natLeq
tff(fact_5572_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_bq,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_5573_one__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),one_one_rat)) ).

% one_rat.transfer
tff(fact_5574_zero__rat_Otransfer,axiom,
    pp(aa_rat_bool(aa_Pro6332413336323567195t_bool(pcr_rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),zero_zero_rat)) ).

% zero_rat.transfer
tff(fact_5575_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_br,times_times_rat) ).

% times_rat.transfer
tff(fact_5576_uminus__rat_Otransfer,axiom,
    pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_bs),uminus_uminus_rat)) ).

% uminus_rat.transfer
tff(fact_5577_normalize__negative,axiom,
    ! [Q5: int,P4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,Q5),zero_zero_int))
     => ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,P4)),aa_int_int(uminus_uminus_int,Q5))) ) ) ).

% normalize_negative
tff(fact_5578_inverse__rat_Otransfer,axiom,
    pp(aa_fun_rat_rat_bool(aa_fun2103967415933125506t_bool(bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_bt),inverse_inverse_rat)) ).

% inverse_rat.transfer
tff(fact_5579_normalize__denom__zero,axiom,
    ! [P4: int] : normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P4),zero_zero_int)) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_5580_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P4: int,Q5: int] :
      ( ( normalize(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q5)) ) ).

% normalize_denom_pos
tff(fact_5581_normalize__crossproduct,axiom,
    ! [Q5: int,S2: int,P4: int,R2: int] :
      ( ( Q5 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P4),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q5) ) ) ) ) ).

% normalize_crossproduct
tff(fact_5582_rat__inverse__code,axiom,
    ! [P4: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P4)) = produc4245557441103728435nt_int(aTP_Lamm_bu,quotient_of(P4)) ).

% rat_inverse_code
tff(fact_5583_normalize__def,axiom,
    ! [P4: product_prod_int_int] :
      ( ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P4)))
       => ( normalize(P4) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P4)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P4)),product_snd_int_int(P4)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P4)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P4)),product_snd_int_int(P4)))) ) )
      & ( ~ pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),product_snd_int_int(P4)))
       => ( ( ( product_snd_int_int(P4) = zero_zero_int )
           => ( normalize(P4) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ) )
          & ( ( product_snd_int_int(P4) != zero_zero_int )
           => ( normalize(P4) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P4)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P4)),product_snd_int_int(P4))))),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P4)),aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,product_fst_int_int(P4)),product_snd_int_int(P4))))) ) ) ) ) ) ).

% normalize_def
tff(fact_5584_quotient__of__number_I3_J,axiom,
    ! [K3: num] : quotient_of(numeral_numeral_rat(K3)) = aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(K3)),one_one_int) ).

% quotient_of_number(3)
tff(fact_5585_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int) ).

% rat_one_code
tff(fact_5586_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_5587_quotient__of__number_I5_J,axiom,
    ! [K3: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,numeral_numeral_rat(K3))) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(K3))),one_one_int) ).

% quotient_of_number(5)
tff(fact_5588_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_5589_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Gcd_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_gcd_int,Xs),zero_zero_int) ).

% Gcd_int_set_eq_fold
tff(fact_5590_quotient__of__div,axiom,
    ! [R2: rat,N: int,D2: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(N),D2) )
     => ( R2 = divide_divide_rat(ring_1_of_int_rat(N),ring_1_of_int_rat(D2)) ) ) ).

% quotient_of_div
tff(fact_5591_quotient__of__denom__pos,axiom,
    ! [R2: rat,P4: int,Q5: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) )
     => pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q5)) ) ).

% quotient_of_denom_pos
tff(fact_5592_rat__abs__code,axiom,
    ! [P4: rat] : quotient_of(abs_abs_rat(P4)) = produc4245557441103728435nt_int(aTP_Lamm_bv,quotient_of(P4)) ).

% rat_abs_code
tff(fact_5593_rat__sgn__code,axiom,
    ! [P4: rat] : quotient_of(sgn_sgn_rat(P4)) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(sgn_sgn_int,product_fst_int_int(quotient_of(P4)))),one_one_int) ).

% rat_sgn_code
tff(fact_5594_rat__uminus__code,axiom,
    ! [P4: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P4)) = produc4245557441103728435nt_int(aTP_Lamm_bw,quotient_of(P4)) ).

% rat_uminus_code
tff(fact_5595_rat__divide__code,axiom,
    ! [P4: rat,Q5: rat] : quotient_of(divide_divide_rat(P4,Q5)) = produc4245557441103728435nt_int(aTP_Lamm_by(Q5),quotient_of(P4)) ).

% rat_divide_code
tff(fact_5596_rat__minus__code,axiom,
    ! [P4: rat,Q5: rat] : quotient_of(minus_minus_rat(P4,Q5)) = produc4245557441103728435nt_int(aTP_Lamm_ca(Q5),quotient_of(P4)) ).

% rat_minus_code
tff(fact_5597_rat__times__code,axiom,
    ! [P4: rat,Q5: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P4),Q5)) = produc4245557441103728435nt_int(aTP_Lamm_cc(Q5),quotient_of(P4)) ).

% rat_times_code
tff(fact_5598_rat__plus__code,axiom,
    ! [P4: rat,Q5: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P4),Q5)) = produc4245557441103728435nt_int(aTP_Lamm_ce(Q5),quotient_of(P4)) ).

% rat_plus_code
tff(fact_5599_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(A2),one_one_int) ).

% quotient_of_int
tff(fact_5600_Frct__code__post_I5_J,axiom,
    ! [K3: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),numeral_numeral_int(K3))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K3)) ).

% Frct_code_post(5)
tff(fact_5601_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N)))
    <=> ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_5602_Gcd__in,axiom,
    ! [A: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( member_nat(A3,A)
         => ( member_nat(B3,A)
           => member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A3),B3),A) ) )
     => ( ( A != bot_bot_set_nat )
       => member_nat(gcd_Gcd_nat(A),A) ) ) ).

% Gcd_in
tff(fact_5603_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : gcd_Gcd_nat(set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_5604_Frct__code__post_I6_J,axiom,
    ! [K3: num,L: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(K3)),numeral_numeral_int(L))) = divide_divide_rat(numeral_numeral_rat(K3),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_5605_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_cf(N),M2))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_5606_Frct__code__post_I4_J,axiom,
    ! [K3: num] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(numeral_numeral_int(K3)),one_one_int)) = numeral_numeral_rat(K3) ).

% Frct_code_post(4)
tff(fact_5607_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_5608_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),A2)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_5609_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_5610_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),aa_int_int(uminus_uminus_int,B2))) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2))) ).

% Frct_code_post(8)
tff(fact_5611_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] : frct(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,A2)),B2)) = aa_rat_rat(uminus_uminus_rat,frct(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2))) ).

% Frct_code_post(7)
tff(fact_5612_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y )
     => ( pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa)))
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)) ) ) )
           => ~ pp(aa_Pro2232559814647311250t_bool(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa))) ) ) ) ).

% gcd_nat.pelims
tff(fact_5613_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_bq,aTP_Lamm_bq) ).

% plus_rat.rsp
tff(fact_5614_one__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int))) ).

% one_rat.rsp
tff(fact_5615_zero__rat_Orsp,axiom,
    pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int))) ).

% zero_rat.rsp
tff(fact_5616_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_cg,aTP_Lamm_cg) ).

% Fract.rsp
tff(fact_5617_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_br,aTP_Lamm_br) ).

% times_rat.rsp
tff(fact_5618_uminus__rat_Orsp,axiom,
    pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_bs),aTP_Lamm_bs)) ).

% uminus_rat.rsp
tff(fact_5619_inverse__rat_Orsp,axiom,
    pp(aa_fun1403452029515102102t_bool(aa_fun8298786006102058668t_bool(bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_bt),aTP_Lamm_bt)) ).

% inverse_rat.rsp
tff(fact_5620_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa),Xa))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa)),product_snd_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X)),product_snd_int_int(Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa)),product_snd_int_int(X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_5621_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(X)),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(X)),product_fst_int_int(X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_5622_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(one_one_int),one_one_int)) ).

% one_rat_def
tff(fact_5623_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int)) ).

% zero_rat_def
tff(fact_5624_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,X),X))
     => ( aa_rat_rat(uminus_uminus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,product_fst_int_int(X))),product_snd_int_int(X))) ) ) ).

% uminus_rat.abs_eq
tff(fact_5625_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(ratrel,Xa),Xa))
     => ( pp(aa_Pro4501881432599481050t_bool(aa_Pro1534211998248933470t_bool(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,product_fst_int_int(Xa)),product_fst_int_int(X))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xa)),product_snd_int_int(X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_5626_inverse__rat__def,axiom,
    inverse_inverse_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_bt) ).

% inverse_rat_def
tff(fact_5627_uminus__rat__def,axiom,
    uminus_uminus_rat = aa_fun7041730090766645614at_rat(map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_bs) ).

% uminus_rat_def
tff(fact_5628_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_bq) ).

% plus_rat_def
tff(fact_5629_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_br) ).

% times_rat_def
tff(fact_5630_Rep__assn__inverse,axiom,
    ! [X: assn] : aa_fun1993714536301460826l_assn(abs_assn,aa_ass8975924923219239432t_bool(rep_assn,X)) = X ).

% Rep_assn_inverse
tff(fact_5631_Abs__assn__eqI_I2_J,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(P,H))
        <=> pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Pr),H)) )
     => ( Pr = aa_fun1993714536301460826l_assn(abs_assn,P) ) ) ).

% Abs_assn_eqI(2)
tff(fact_5632_Abs__assn__eqI_I1_J,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Pr: assn] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(P,H))
        <=> pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Pr),H)) )
     => ( aa_fun1993714536301460826l_assn(abs_assn,P) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_5633_pure__assn__def,axiom,
    ! [B2: bool] : pure_assn(B2) = aa_fun1993714536301460826l_assn(abs_assn,pure_a825153325127701367it_nat(B2)) ).

% pure_assn_def
tff(fact_5634_bot__assn__def,axiom,
    bot_bot_assn = aa_fun1993714536301460826l_assn(abs_assn,aTP_Lamm_ch) ).

% bot_assn_def
tff(fact_5635_one__assn__def,axiom,
    one_one_assn = aa_fun1993714536301460826l_assn(abs_assn,one_assn_raw) ).

% one_assn_def
tff(fact_5636_sup__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,P),Q) = aa_fun1993714536301460826l_assn(abs_assn,aa_ass8975924923219239432t_bool(aTP_Lamm_ci(P),Q)) ).

% sup_assn_def
tff(fact_5637_inf__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P),Q) = aa_fun1993714536301460826l_assn(abs_assn,aa_ass8975924923219239432t_bool(aTP_Lamm_cj(P),Q)) ).

% inf_assn_def
tff(fact_5638_times__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q) = aa_fun1993714536301460826l_assn(abs_assn,aa_fun6808777976510360636t_bool(times_assn_raw(aa_ass8975924923219239432t_bool(rep_assn,P)),aa_ass8975924923219239432t_bool(rep_assn,Q))) ).

% times_assn_def
tff(fact_5639_wand__assn__def,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,P),Q) = aa_fun1993714536301460826l_assn(abs_assn,aa_fun6808777976510360636t_bool(wand_raw(aa_ass8975924923219239432t_bool(rep_assn,P)),aa_ass8975924923219239432t_bool(rep_assn,Q))) ).

% wand_assn_def
tff(fact_5640_uminus__assn__def,axiom,
    ! [P: assn] : aa_assn_assn(uminus_uminus_assn,P) = aa_fun1993714536301460826l_assn(abs_assn,aTP_Lamm_ck(P)) ).

% uminus_assn_def
tff(fact_5641_in__range__empty,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,bot_bot_set_nat))) ).

% in_range_empty
tff(fact_5642_mod__true,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,top_top_assn),H2))
    <=> pp(aa_Pro2991721470636411351t_bool(in_range,H2)) ) ).

% mod_true
tff(fact_5643_in__range__dist__union,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,As3: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3))))
    <=> ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2)))
        & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As3))) ) ) ).

% in_range_dist_union
tff(fact_5644_mod__not__dist,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(uminus_uminus_assn,P)),H2))
    <=> ( pp(aa_Pro2991721470636411351t_bool(in_range,H2))
        & ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2)) ) ) ).

% mod_not_dist
tff(fact_5645_in__range__subset,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( ord_less_eq_set_nat(As2,As3)
     => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As3)))
       => pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2))) ) ) ).

% in_range_subset
tff(fact_5646_models__in__range,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H2))
     => pp(aa_Pro2991721470636411351t_bool(in_range,H2)) ) ).

% models_in_range
tff(fact_5647_top__assn__def,axiom,
    top_top_assn = aa_fun1993714536301460826l_assn(abs_assn,in_range) ).

% top_assn_def
tff(fact_5648_wand__assnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,R: assn] :
      ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,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 )
           => ( relH(As2,H2,H5)
             => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H5,As2)))
               => ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Q),produc7507926704131184380et_nat(H5,As4)))
                 => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,R),produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As4)))) ) ) ) )
       => pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Q),R)),produc7507926704131184380et_nat(H2,As2))) ) ) ).

% wand_assnI
tff(fact_5649_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(X),Xa),Xb))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,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 )
                    & relH(As,H,H5)
                    & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H5,As)))
                    & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H5,As4))) )
                 => pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4)))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_5650_relH__dist__union,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3),H2,H4)
    <=> ( relH(As2,H2,H4)
        & relH(As3,H2,H4) ) ) ).

% relH_dist_union
tff(fact_5651_relH__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2)))
     => relH(As2,H2,H2) ) ).

% relH_refl
tff(fact_5652_relH__in__rangeI_I1_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H4)
     => pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2))) ) ).

% relH_in_rangeI(1)
tff(fact_5653_relH__in__rangeI_I2_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H4)
     => pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H4,As2))) ) ).

% relH_in_rangeI(2)
tff(fact_5654_mod__relH,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,P: assn] :
      ( relH(As2,H2,H4)
     => ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H2,As2)))
      <=> pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),produc7507926704131184380et_nat(H4,As2))) ) ) ).

% mod_relH
tff(fact_5655_relH__trans,axiom,
    ! [As2: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( relH(As2,H12,H23)
     => ( relH(As2,H23,H32)
       => relH(As2,H12,H32) ) ) ).

% relH_trans
tff(fact_5656_relH__sym,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H4)
     => relH(As2,H4,H2) ) ).

% relH_sym
tff(fact_5657_relH__subset,axiom,
    ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( relH(Bs,H2,H4)
     => ( ord_less_eq_set_nat(As2,Bs)
       => relH(As2,H2,H4) ) ) ).

% relH_subset
tff(fact_5658_wand__raw_Osimps,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Q: fun_Pr6001797050664311790t_bool,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(P),Q),produc7507926704131184380et_nat(H2,As2)))
    <=> ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2)))
        & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As5) = bot_bot_set_nat )
              & relH(As2,H2,H6)
              & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H6,As2)))
              & pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H6,As5))) )
           => pp(aa_Pro2991721470636411351t_bool(Q,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As5)))) ) ) ) ).

% wand_raw.simps
tff(fact_5659_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(X),Xa),Xb))
      <=> pp(Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ( pp(Y)
            <=> ~ ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,As)))
                  & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                        & relH(As,H,H6)
                        & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H6,As)))
                        & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H6,As5))) )
                     => pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5)))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_5660_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(X),Xa),Xb))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H,As) )
           => ~ ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,As)))
                & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                      & relH(As,H,H7)
                      & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H7,As)))
                      & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H7,As6))) )
                   => pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6)))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_5661_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(X),Xa),Xb))
     => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(wand_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,Xb))))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(wand_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,produc7507926704131184380et_nat(H,As)))))
               => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,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 )
                        & relH(As,H,H5)
                        & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H5,As)))
                        & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H5,As4))) )
                     => pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H5,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4)))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_5662_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(X),Xa),Xb))
     => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(wand_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,Xb))))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(wand_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,produc7507926704131184380et_nat(H,As)))))
               => ~ ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,As)))
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                          & relH(As,H,H7)
                          & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H7,As)))
                          & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H7,As6))) )
                       => pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6)))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_5663_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Xa: fun_Pr6001797050664311790t_bool,Xb: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(wand_raw(X),Xa),Xb))
      <=> pp(Y) )
     => ( pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(wand_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,Xb))))
       => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H,As) )
             => ( ( pp(Y)
                <=> ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,As)))
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                          & relH(As,H,H6)
                          & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H6,As)))
                          & pp(aa_Pro2991721470636411351t_bool(X,produc7507926704131184380et_nat(H6,As5))) )
                       => pp(aa_Pro2991721470636411351t_bool(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5)))) ) ) )
               => ~ pp(aa_Pro6092667153363046813t_bool(accp_P9010229746810818675et_nat(wand_raw_rel),produc3083477295402903094et_nat(X,produc8148480564485311129et_nat(Xa,produc7507926704131184380et_nat(H,As))))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_5664_normalize__stable,axiom,
    ! [Q5: int,P4: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(ord_less_int,zero_zero_int),Q5))
     => ( pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P4),Q5))
       => ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) ) ) ) ).

% normalize_stable
tff(fact_5665_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P4: int,Q5: int] :
      ( ( normalize(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) )
     => pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P4),Q5)) ) ).

% normalize_coprime
tff(fact_5666_quotient__of__coprime,axiom,
    ! [R2: rat,P4: int,Q5: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) )
     => pp(aa_int_bool(aa_int_fun_int_bool(algebr932160517623751201me_int,P4),Q5)) ) ).

% quotient_of_coprime
tff(fact_5667_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_5668_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).

% coprime_diff_one_left_nat
tff(fact_5669_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_cg,fract) ).

% Fract.transfer
tff(fact_5670_quotient__of__eq,axiom,
    ! [A2: int,B2: int,P4: int,Q5: int] :
      ( ( quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A2),B2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P4),Q5) = aa_int_rat(aa_int_fun_int_rat(fract,A2),B2) ) ) ).

% quotient_of_eq
tff(fact_5671_normalize__eq,axiom,
    ! [A2: int,B2: int,P4: int,Q5: int] :
      ( ( normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2)) = aa_int7462565939943104719nt_int(product_Pair_int_int(P4),Q5) )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,P4),Q5) = aa_int_rat(aa_int_fun_int_rat(fract,A2),B2) ) ) ).

% normalize_eq
tff(fact_5672_quotient__of__Fract,axiom,
    ! [A2: int,B2: int] : quotient_of(aa_int_rat(aa_int_fun_int_rat(fract,A2),B2)) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(A2),B2)) ).

% quotient_of_Fract
tff(fact_5673_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa_int_rat(aa_int_fun_int_rat(fract,Xa),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Xa),X))) ).

% Fract.abs_eq
tff(fact_5674_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc5207004643709237345nteger] :
      ~ ! [F4: fun_Co6545587727236015215e_term,D3: code_integer,I5: code_integer] : X != produc4282985567423050387nteger(F4,produc1086072967326762835nteger(D3,I5)) ).

% exhaustive_integer'.cases
tff(fact_5675_exhaustive__int_H_Ocases,axiom,
    ! [X: produc6768802789813714928nt_int] :
      ~ ! [F4: fun_in6429348457993719614e_term,D3: int,I5: int] : X != produc4852315488734385442nt_int(F4,aa_int7462565939943104719nt_int(product_Pair_int_int(D3),I5)) ).

% exhaustive_int'.cases
tff(fact_5676_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc4441460161354545860nt_int] :
      ~ ! [F4: fun_Pr9164807099015279978e_term,D3: int,I5: int] : X != produc1560874099174258678nt_int(F4,aa_int7462565939943104719nt_int(product_Pair_int_int(D3),I5)) ).

% full_exhaustive_int'.cases
tff(fact_5677_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1616345218596728535nteger] :
      ~ ! [F4: fun_Pr272337284026993017e_term,D3: code_integer,I5: code_integer] : X != produc7384524951051686153nteger(F4,produc1086072967326762835nteger(D3,I5)) ).

% full_exhaustive_integer'.cases
tff(fact_5678_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X2: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X2,Xs2) )
             => ( Y != aa_nat_nat(suc,nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_5679_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] : nat_list_encode(cons_nat(X,Xs)) = aa_nat_nat(suc,nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_5680_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,A2),nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)))) ).

% le_prod_encode_1
tff(fact_5681_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,B2),nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2)))) ).

% le_prod_encode_2
tff(fact_5682_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),X))
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),nil_nat)) ) )
         => ~ ! [X2: nat,Xs2: list_nat] :
                ( ( X = cons_nat(X2,Xs2) )
               => ( ( Y = aa_nat_nat(suc,nat_prod_encode(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),nat_list_encode(Xs2)))) )
                 => ~ pp(aa_list_nat_bool(accp_list_nat(nat_list_encode_rel),cons_nat(X2,Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_5683_INFM__nat,axiom,
    ! [P: fun_nat_bool] :
      ( frequently_nat(P,cofinite_nat)
    <=> ! [M: nat] :
        ? [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
          & pp(aa_nat_bool(P,N2)) ) ) ).

% INFM_nat
tff(fact_5684_INFM__nat__inductI,axiom,
    ! [P: fun_nat_bool,Q: fun_nat_bool] :
      ( pp(aa_nat_bool(P,zero_zero_nat))
     => ( ! [I5: nat] :
            ( pp(aa_nat_bool(P,I5))
           => ? [J4: nat] :
                ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,I5),J4))
                & pp(aa_nat_bool(P,J4))
                & pp(aa_nat_bool(Q,J4)) ) )
       => frequently_nat(Q,cofinite_nat) ) ) ).

% INFM_nat_inductI
tff(fact_5685_MOST__nat,axiom,
    ! [P: fun_nat_bool] :
      ( eventually_nat(P,cofinite_nat)
    <=> ? [M: nat] :
        ! [N2: nat] :
          ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,M),N2))
         => pp(aa_nat_bool(P,N2)) ) ) ).

% MOST_nat
tff(fact_5686_cofinite__eq__sequentially,axiom,
    cofinite_nat = at_top_nat ).

% cofinite_eq_sequentially
tff(fact_5687_list__ex__iff__not__all__inverval__int,axiom,
    ! [P: fun_int_bool,I: int,J: int] :
      ( pp(aa_list_int_bool(list_ex_int(P),upto(I,J)))
    <=> ~ pp(aa_int_bool(aa_int_fun_int_bool(all_interval_int(comp_bool_bool_int(fNot,P)),I),J)) ) ).

% list_ex_iff_not_all_inverval_int
tff(fact_5688_all__interval__int__def,axiom,
    ! [P: fun_int_bool,I: int,J: int] :
      ( pp(aa_int_bool(aa_int_fun_int_bool(all_interval_int(P),I),J))
    <=> ! [X5: int] :
          ( member_int(X5,set_or1266510415728281911st_int(I,J))
         => pp(aa_int_bool(P,X5)) ) ) ).

% all_interval_int_def
tff(fact_5689_list__all__iff__all__interval__int,axiom,
    ! [P: fun_int_bool,I: int,J: int] :
      ( pp(aa_list_int_bool(list_all_int(P),upto(I,J)))
    <=> pp(aa_int_bool(aa_int_fun_int_bool(all_interval_int(P),I),J)) ) ).

% list_all_iff_all_interval_int
tff(fact_5690_is__pure__assn__pure,axiom,
    ! [P: bool] : pp(aa_assn_bool(is_pure_assn,pure_assn(P))) ).

% is_pure_assn_pure
tff(fact_5691_is__pure__assn__starI,axiom,
    ! [A2: assn,B2: assn] :
      ( pp(aa_assn_bool(is_pure_assn,A2))
     => ( pp(aa_assn_bool(is_pure_assn,B2))
       => pp(aa_assn_bool(is_pure_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2))) ) ) ).

% is_pure_assn_starI
tff(fact_5692_Lcm__nat__empty,axiom,
    gcd_Lcm_nat(bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_5693_is__pure__assn__basic__simps_I2_J,axiom,
    pp(aa_assn_bool(is_pure_assn,one_one_assn)) ).

% is_pure_assn_basic_simps(2)
tff(fact_5694_is__pure__assn__def,axiom,
    ! [A2: assn] :
      ( pp(aa_assn_bool(is_pure_assn,A2))
    <=> ? [P5: bool] : A2 = pure_assn(P5) ) ).

% is_pure_assn_def
tff(fact_5695_is__pure__assnE,axiom,
    ! [A2: assn] :
      ( pp(aa_assn_bool(is_pure_assn,A2))
     => ~ ! [P3: bool] : A2 != pure_assn(P3) ) ).

% is_pure_assnE
tff(fact_5696_is__pure__assn__basic__simps_I1_J,axiom,
    pp(aa_assn_bool(is_pure_assn,bot_bot_assn)) ).

% is_pure_assn_basic_simps(1)
tff(fact_5697_Lcm__eq__Max__nat,axiom,
    ! [M5: set_nat] :
      ( finite_finite_nat(M5)
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ member_nat(zero_zero_nat,M5)
         => ( ! [M6: nat,N5: nat] :
                ( member_nat(M6,M5)
               => ( member_nat(N5,M5)
                 => member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M6),N5),M5) ) )
           => ( gcd_Lcm_nat(M5) = lattic8265883725875713057ax_nat(M5) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_5698_Lcm__nat__insert,axiom,
    ! [N: nat,M5: set_nat] : gcd_Lcm_nat(insert_nat2(N,M5)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,N),gcd_Lcm_nat(M5)) ).

% Lcm_nat_insert
tff(fact_5699_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M5: set_nat] :
      ( finite_finite_nat(M5)
     => ( ( M5 != bot_bot_set_nat )
       => ( ! [M6: nat,N5: nat] :
              ( member_nat(M6,M5)
             => ( member_nat(N5,M5)
               => member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M6),N5),M5) ) )
         => member_nat(gcd_Lcm_nat(M5),M5) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_5700_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),M2))
     => ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
       => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,M2),N))) ) ) ).

% lcm_pos_nat
tff(fact_5701_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : gcd_Lcm_nat(set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_lcm_nat,Xs),one_one_nat) ).

% Lcm_nat_set_eq_fold
tff(fact_5702_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Lcm_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_lcm_int,Xs),one_one_int) ).

% Lcm_int_set_eq_fold
tff(fact_5703_properI,axiom,
    ! [P: fun_Pr6001797050664311790t_bool] :
      ( ! [As: set_nat,H: heap_e7401611519738050253t_unit] :
          ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H,As)))
         => pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H,As))) )
     => ( ! [As: set_nat,H: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
            ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H,As)))
           => ( relH(As,H,H5)
             => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H5,As)))
               => pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H5,As))) ) ) )
       => pp(aa_fun5588795374382764133l_bool(proper,P)) ) ) ).

% properI
tff(fact_5704_properD2,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,H2: heap_e7401611519738050253t_unit,As2: set_nat,H4: heap_e7401611519738050253t_unit] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H2,As2)))
       => ( relH(As2,H2,H4)
         => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H4,As2)))
           => pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H4,As2))) ) ) ) ) ).

% properD2
tff(fact_5705_bool__assn__proper_I4_J,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Q: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => ( pp(aa_fun5588795374382764133l_bool(proper,Q))
       => pp(aa_fun5588795374382764133l_bool(proper,aa_fun6808777976510360636t_bool(aTP_Lamm_cl(P),Q))) ) ) ).

% bool_assn_proper(4)
tff(fact_5706_bool__assn__proper_I3_J,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Q: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => ( pp(aa_fun5588795374382764133l_bool(proper,Q))
       => pp(aa_fun5588795374382764133l_bool(proper,aa_fun6808777976510360636t_bool(aTP_Lamm_cm(P),Q))) ) ) ).

% bool_assn_proper(3)
tff(fact_5707_bool__assn__proper_I2_J,axiom,
    pp(aa_fun5588795374382764133l_bool(proper,aTP_Lamm_ch)) ).

% bool_assn_proper(2)
tff(fact_5708_bool__assn__proper_I1_J,axiom,
    pp(aa_fun5588795374382764133l_bool(proper,in_range)) ).

% bool_assn_proper(1)
tff(fact_5709_one__assn__proper,axiom,
    pp(aa_fun5588795374382764133l_bool(proper,one_assn_raw)) ).

% one_assn_proper
tff(fact_5710_times__assn__proper,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Q: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => ( pp(aa_fun5588795374382764133l_bool(proper,Q))
       => pp(aa_fun5588795374382764133l_bool(proper,aa_fun6808777976510360636t_bool(times_assn_raw(P),Q))) ) ) ).

% times_assn_proper
tff(fact_5711_wand__proper,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,Q: fun_Pr6001797050664311790t_bool] : pp(aa_fun5588795374382764133l_bool(proper,aa_fun6808777976510360636t_bool(wand_raw(P),Q))) ).

% wand_proper
tff(fact_5712_bool__assn__proper_I5_J,axiom,
    ! [P: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => pp(aa_fun5588795374382764133l_bool(proper,aa_fun6808777976510360636t_bool(aTP_Lamm_cn,P))) ) ).

% bool_assn_proper(5)
tff(fact_5713_pure__assn__proper,axiom,
    ! [B2: bool] : pp(aa_fun5588795374382764133l_bool(proper,pure_a825153325127701367it_nat(B2))) ).

% pure_assn_proper
tff(fact_5714_Abs__assn__inverse,axiom,
    ! [Y: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(Y,collec3007137136614167897t_bool(proper))
     => ( aa_ass8975924923219239432t_bool(rep_assn,aa_fun1993714536301460826l_assn(abs_assn,Y)) = Y ) ) ).

% Abs_assn_inverse
tff(fact_5715_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: fun_Pr6001797050664311790t_bool] :
          ( ( X = aa_fun1993714536301460826l_assn(abs_assn,Y2) )
         => ~ member4125802036136091927t_bool(Y2,collec3007137136614167897t_bool(proper)) ) ).

% Abs_assn_cases
tff(fact_5716_Abs__assn__induct,axiom,
    ! [P: fun_assn_bool,X: assn] :
      ( ! [Y2: fun_Pr6001797050664311790t_bool] :
          ( member4125802036136091927t_bool(Y2,collec3007137136614167897t_bool(proper))
         => pp(aa_assn_bool(P,aa_fun1993714536301460826l_assn(abs_assn,Y2))) )
     => pp(aa_assn_bool(P,X)) ) ).

% Abs_assn_induct
tff(fact_5717_Abs__assn__inject,axiom,
    ! [X: fun_Pr6001797050664311790t_bool,Y: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(X,collec3007137136614167897t_bool(proper))
     => ( member4125802036136091927t_bool(Y,collec3007137136614167897t_bool(proper))
       => ( ( aa_fun1993714536301460826l_assn(abs_assn,X) = aa_fun1993714536301460826l_assn(abs_assn,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_5718_Rep__assn__induct,axiom,
    ! [Y: fun_Pr6001797050664311790t_bool,P: fun_fu9160419385450930796l_bool] :
      ( member4125802036136091927t_bool(Y,collec3007137136614167897t_bool(proper))
     => ( ! [X2: assn] : pp(aa_fun5588795374382764133l_bool(P,aa_ass8975924923219239432t_bool(rep_assn,X2)))
       => pp(aa_fun5588795374382764133l_bool(P,Y)) ) ) ).

% Rep_assn_induct
tff(fact_5719_Rep__assn__cases,axiom,
    ! [Y: fun_Pr6001797050664311790t_bool] :
      ( member4125802036136091927t_bool(Y,collec3007137136614167897t_bool(proper))
     => ~ ! [X2: assn] : Y != aa_ass8975924923219239432t_bool(rep_assn,X2) ) ).

% Rep_assn_cases
tff(fact_5720_Rep__assn,axiom,
    ! [X: assn] : member4125802036136091927t_bool(aa_ass8975924923219239432t_bool(rep_assn,X),collec3007137136614167897t_bool(proper)) ).

% Rep_assn
tff(fact_5721_properD1,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H2,As2)))
       => pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2))) ) ) ).

% properD1
tff(fact_5722_proper__iff,axiom,
    ! [P: fun_Pr6001797050664311790t_bool,As2: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
     => ( relH(As2,H2,H4)
       => ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H4,As2)))
         => ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H2,As2)))
          <=> pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H4,As2))) ) ) ) ) ).

% proper_iff
tff(fact_5723_proper__def,axiom,
    ! [P: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(proper,P))
    <=> ! [H3: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,As7: set_nat] :
          ( ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H3,As7)))
           => pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H3,As7))) )
          & ( ( pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H3,As7)))
              & relH(As7,H3,H6)
              & pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H6,As7))) )
           => pp(aa_Pro2991721470636411351t_bool(P,produc7507926704131184380et_nat(H6,As7))) ) ) ) ).

% proper_def
tff(fact_5724_list__ex__iff__not__all__inverval__nat,axiom,
    ! [P: fun_nat_bool,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(list_ex_nat(P),upt(I,J)))
    <=> ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(all_interval_nat(comp_bool_bool_nat(fNot,P)),I),J)) ) ).

% list_ex_iff_not_all_inverval_nat
tff(fact_5725_list__all__iff__all__interval__nat,axiom,
    ! [P: fun_nat_bool,I: nat,J: nat] :
      ( pp(aa_list_nat_bool(list_all_nat(P),upt(I,J)))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(all_interval_nat(P),I),J)) ) ).

% list_all_iff_all_interval_nat
tff(fact_5726_all__interval__nat__def,axiom,
    ! [P: fun_nat_bool,I: nat,J: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(all_interval_nat(P),I),J))
    <=> ! [X5: nat] :
          ( member_nat(X5,set_or4665077453230672383an_nat(I,J))
         => pp(aa_nat_bool(P,X5)) ) ) ).

% all_interval_nat_def
tff(fact_5727_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),N))
     => ( nat_of_num(aa_nat_num(num_of_nat,N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_5728_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,zero_zero_nat),nat_of_num(X))) ).

% nat_of_num_pos
tff(fact_5729_less__num__def,axiom,
    ! [M2: num,N: num] :
      ( pp(aa_num_bool(aa_num_fun_num_bool(ord_less_num,M2),N))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,nat_of_num(M2)),nat_of_num(N))) ) ).

% less_num_def
tff(fact_5730_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,code_nat_of_natural(N)),aa_nat_nat(suc,zero_zero_nat))),code_nat_of_natural(N))) ) ).

% natural_decr
tff(fact_5731_less__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ord_le5570908160329646204atural(X,Xa)
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,code_nat_of_natural(X)),code_nat_of_natural(Xa))) ) ).

% less_natural.rep_eq
tff(fact_5732_next_Osimps,axiom,
    ! [V2: code_natural,W: code_natural] : aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V2,W)) = aa_Pro7165575656943818355atural(produc6639722614265839536atural(plus_p4538020629002901425atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),plus_p4538020629002901425atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))),one_one_Code_natural)),one_one_Code_natural)),produc3574140220909816553atural(minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(V2,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))))))))))))),minus_shift(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),times_2397367101498566445atural(modulo8411746178871703098atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))),times_2397367101498566445atural(divide5121882707175180666atural(W,numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))))))))))))))),numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))) ).

% next.simps
tff(fact_5733_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc651309673056299192atural] :
      ~ ! [F4: fun_Pr8336614522058983534e_term,D3: code_natural,I5: code_natural] : X != produc2566709293081654762atural(F4,produc3574140220909816553atural(D3,I5)) ).

% full_exhaustive_natural'.cases
tff(fact_5734_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc6284619029570389740atural] :
      ~ ! [F4: fun_Co621691207274083130e_term,D3: code_natural,I5: code_natural] : X != produc1507819840854600734atural(F4,produc3574140220909816553atural(D3,I5)) ).

% exhaustive_natural'.cases
tff(fact_5735_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B3: code_natural,I5: code_natural] : X != produc3574140220909816553atural(B3,I5) ).

% log.cases
tff(fact_5736_split__seed__def,axiom,
    ! [S2: produc7822875418678951345atural] : split_seed(S2) = produc8282080750456430313atural(aTP_Lamm_cp(S2),S2) ).

% split_seed_def
tff(fact_5737_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D3: int,I5: int] : X != aa_int7462565939943104719nt_int(product_Pair_int_int(D3),I5) ).

% small_lazy'.cases
tff(fact_5738_Random_Orange__def,axiom,
    ! [K3: code_natural] : range(K3) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),K3),aTP_Lamm_cr,one_one_Code_natural),aTP_Lamm_cs(K3)) ).

% Random.range_def
tff(fact_5739_less__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ord_le5570908160329646204atural(code_natural_of_nat(Xa),code_natural_of_nat(X))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Xa),X)) ) ).

% less_natural.abs_eq
tff(fact_5740_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa))
       => ~ ( ( ( ( ord_le1926595141338095240atural(X,one_one_Code_natural)
                  | ord_le5570908160329646204atural(Xa,X) )
               => ( Y = one_one_Code_natural ) )
              & ( ~ ( ord_le1926595141338095240atural(X,one_one_Code_natural)
                    | ord_le5570908160329646204atural(Xa,X) )
               => ( Y = plus_p4538020629002901425atural(one_one_Code_natural,log(X,divide5121882707175180666atural(Xa,X))) ) ) )
           => ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa)) ) ) ) ).

% log.pelims
tff(fact_5741_in__range_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(in_range,produc7507926704131184380et_nat(H2,As2)))
    <=> ! [X5: nat] :
          ( member_nat(X5,As2)
         => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),lim_Product_unit(H2))) ) ) ).

% in_range.simps
tff(fact_5742_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ pp(aa_Pro2991721470636411351t_bool(in_range,X))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ! [X2: nat] :
                ( member_nat(X2,As)
               => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X2),lim_Product_unit(H))) ) ) ) ).

% in_range.elims(3)
tff(fact_5743_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(in_range,X))
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ~ ! [X6: nat] :
                  ( member_nat(X6,As)
                 => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X6),lim_Product_unit(H))) ) ) ) ).

% in_range.elims(2)
tff(fact_5744_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: bool] :
      ( ( pp(aa_Pro2991721470636411351t_bool(in_range,X))
      <=> pp(Y) )
     => ~ ! [H: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H,As) )
           => ( pp(Y)
            <=> ~ ! [X5: nat] :
                    ( member_nat(X5,As)
                   => pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,X5),lim_Product_unit(H))) ) ) ) ) ).

% in_range.elims(1)
tff(fact_5745_new__addrs__def,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,H4: heap_e7401611519738050253t_unit] : hoare_new_addrs(H2,As2,H4) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),collect_nat(aa_hea5076648075882977743t_bool(aTP_Lamm_ct(H2),H4))) ).

% new_addrs_def
tff(fact_5746_execute__assert_H,axiom,
    ! [P: bool,H2: heap_e7401611519738050253t_unit] :
      ( ( pp(P)
       => ( heap_T875086893843062177t_unit(refine_Imp_assert(P),H2) = some_P1914260805536162275it_nat(produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(H2,zero_zero_nat))) ) )
      & ( ~ pp(P)
       => ( heap_T875086893843062177t_unit(refine_Imp_assert(P),H2) = none_P9117596204409417319it_nat ) ) ) ).

% execute_assert'
tff(fact_5747_assert_H__rule,axiom,
    ! [P: assn,Phi: bool] :
      ( ! [H: produc3658429121746597890et_nat] :
          ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,P),H))
         => pp(Phi) )
     => hoare_8945653483474564448t_unit(P,refine_Imp_assert(Phi),aTP_Lamm_cu(P)) ) ).

% assert'_rule
tff(fact_5748_wait__def,axiom,
    ! [N: nat] : heap_Time_wait(N) = heap_T6183433275982383450t_unit(aTP_Lamm_cv(N)) ).

% wait_def
tff(fact_5749_less__nat__rel,axiom,
    ord_less_nat = transi2163837189807498211lp_nat(aTP_Lamm_cw) ).

% less_nat_rel
tff(fact_5750_type__definition__assn,axiom,
    type_d18088346645320984t_bool(rep_assn,abs_assn,collec3007137136614167897t_bool(proper)) ).

% type_definition_assn
tff(fact_5751_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_bt,Uu) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,product_fst_int_int(Uu)),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(product_snd_int_int(Uu)),product_fst_int_int(Uu))) ).

% ATP.lambda_1
tff(fact_5752_ATP_Olambda__2,axiom,
    ! [Uu: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_bs,Uu) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(uminus_uminus_int,product_fst_int_int(Uu))),product_snd_int_int(Uu)) ).

% ATP.lambda_2
tff(fact_5753_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_as,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_3
tff(fact_5754_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_bk,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_4
tff(fact_5755_ATP_Olambda__5,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_cr,Uu) = produc5538323210962509403atural(next,aTP_Lamm_cq(Uu)) ).

% ATP.lambda_5
tff(fact_5756_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_ay,Uu) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ai,Uu)) ).

% ATP.lambda_6
tff(fact_5757_ATP_Olambda__7,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_bw,Uu) = product_Pair_int_int(aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_7
tff(fact_5758_ATP_Olambda__8,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_bv,Uu) = product_Pair_int_int(aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_8
tff(fact_5759_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_al,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_9
tff(fact_5760_ATP_Olambda__10,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bu,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,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_10
tff(fact_5761_ATP_Olambda__11,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_cg,Uu),Uua) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(fequal_int,Uua),zero_zero_int),aa_int7462565939943104719nt_int(product_Pair_int_int(zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(product_Pair_int_int(Uu),Uua)) ).

% ATP.lambda_11
tff(fact_5762_ATP_Olambda__12,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_bq,Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_12
tff(fact_5763_ATP_Olambda__13,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_br,Uu),Uua) = aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_fst_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu)),product_snd_int_int(Uua))) ).

% ATP.lambda_13
tff(fact_5764_ATP_Olambda__14,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aTP_Lamm_ck(Uu),Uua))
    <=> ( pp(aa_Pro2991721470636411351t_bool(in_range,Uua))
        & ~ pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Uu),Uua)) ) ) ).

% ATP.lambda_14
tff(fact_5765_ATP_Olambda__15,axiom,
    ! [Uu: fun_Pr6001797050664311790t_bool,Uua: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(aTP_Lamm_cn,Uu),Uua))
    <=> ( pp(aa_Pro2991721470636411351t_bool(in_range,Uua))
        & ~ pp(aa_Pro2991721470636411351t_bool(Uu,Uua)) ) ) ).

% ATP.lambda_15
tff(fact_5766_ATP_Olambda__16,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cw,Uu),Uua))
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_16
tff(fact_5767_ATP_Olambda__17,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bj,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uu)) ) ).

% ATP.lambda_17
tff(fact_5768_ATP_Olambda__18,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aj,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_18
tff(fact_5769_ATP_Olambda__19,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ae,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu)) ) ).

% ATP.lambda_19
tff(fact_5770_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ak,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_20
tff(fact_5771_ATP_Olambda__21,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_au(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu) ).

% ATP.lambda_21
tff(fact_5772_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ar,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_22
tff(fact_5773_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ai,Uu),Uua))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uua),Uu)) ) ).

% ATP.lambda_23
tff(fact_5774_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_bb,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_24
tff(fact_5775_ATP_Olambda__25,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( pp(aa_Pro1809015098595223084t_bool(aTP_Lamm_a(Uu),Uua))
    <=> member8757157785044589968at_nat(Uua,Uu) ) ).

% ATP.lambda_25
tff(fact_5776_ATP_Olambda__26,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( pp(aa_Pro2232559814647311250t_bool(aTP_Lamm_ac(Uu),Uua))
    <=> pp(aa_set3135850010578026034t_bool(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_26
tff(fact_5777_ATP_Olambda__27,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_ab(Uu),Uua))
    <=> member_nat(Uua,Uu) ) ).

% ATP.lambda_27
tff(fact_5778_ATP_Olambda__28,axiom,
    ! [Uu: set_int,Uua: int] :
      ( pp(aa_int_bool(aTP_Lamm_aa(Uu),Uua))
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_28
tff(fact_5779_ATP_Olambda__29,axiom,
    ! [Uu: set_fu6080891106736489294t_bool,Uua: fun_Pr6001797050664311790t_bool] :
      ( pp(aa_fun5588795374382764133l_bool(aTP_Lamm_ad(Uu),Uua))
    <=> member4125802036136091927t_bool(Uua,Uu) ) ).

% ATP.lambda_29
tff(fact_5780_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_bo(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_30
tff(fact_5781_ATP_Olambda__31,axiom,
    ! [Uu: fun_nat_bool,Uua: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_bl(Uu),Uua))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(suc,Uua))) ) ).

% ATP.lambda_31
tff(fact_5782_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_bh,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_bg(Uu),Uua)) ).

% ATP.lambda_32
tff(fact_5783_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_bf,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_be(Uu),Uua)) ).

% ATP.lambda_33
tff(fact_5784_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat4401263874371387303t_bool(aa_nat4909001760120733684t_bool(aTP_Lamm_bd,Uu),Uua) = produc8955392012077580105t_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_bc(Uu),Uua)) ).

% ATP.lambda_34
tff(fact_5785_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ba,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_az(Uu),Uua)) ).

% ATP.lambda_35
tff(fact_5786_ATP_Olambda__36,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_cq(Uu),Uua) = produc6639722614265839536atural(plus_p4538020629002901425atural(Uua,times_2397367101498566445atural(Uu,numera5444537566228673987atural(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_36
tff(fact_5787_ATP_Olambda__37,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_cs(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_37
tff(fact_5788_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: heap_e7401611519738050253t_unit] : aa_hea684056619475940329it_nat(aTP_Lamm_cv(Uu),Uua) = some_P1914260805536162275it_nat(produc7133225469290080770it_nat(product_Unity,produc584006145561248582it_nat(Uua,Uu))) ).

% ATP.lambda_38
tff(fact_5789_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_bp(Uu),Uua) = collect_nat(aa_nat_fun_nat_bool(aTP_Lamm_ae,Uu)) ).

% ATP.lambda_39
tff(fact_5790_ATP_Olambda__40,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_at(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_40
tff(fact_5791_ATP_Olambda__41,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_am(Uu),Uua),Uub) = if_Pro6206227464963214023at_nat(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_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(aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_41
tff(fact_5792_ATP_Olambda__42,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_an(Uu),Uua),Uub) = if_Pro3027730157355071871nt_int(aa_int_bool(aa_int_fun_int_bool(ord_less_eq_int,numeral_numeral_int(Uu)),Uub),aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_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(aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_42
tff(fact_5793_ATP_Olambda__43,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ax(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_43
tff(fact_5794_ATP_Olambda__44,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bi(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).

% ATP.lambda_44
tff(fact_5795_ATP_Olambda__45,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aw(Uu),Uua),Uub) = if_Pro6119634080678213985nteger(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),Uub))) ).

% ATP.lambda_45
tff(fact_5796_ATP_Olambda__46,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa_Cod2825494507867081354r_bool(aa_Cod4981436349852597092r_bool(aTP_Lamm_av(Uu),Uua),Uub) = produc5135875973608410785r_bool(aa_Cod1783974652533396728nteger(if_Code_integer(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,Uub),one_one_Code_integer)) ).

% ATP.lambda_46
tff(fact_5797_ATP_Olambda__47,axiom,
    ! [Uu: produc7822875418678951345atural,Uua: code_natural,Uub: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aTP_Lamm_cp(Uu),Uua),Uub) = produc8282080750456430313atural(aa_Cod8811915439870959971atural(aTP_Lamm_co(Uua),Uub),produc6591912806276919810atural(aa_Pro7165575656943818355atural(next,Uu))) ).

% ATP.lambda_47
tff(fact_5798_ATP_Olambda__48,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ce(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_cd(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_48
tff(fact_5799_ATP_Olambda__49,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_cc(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_cb(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_49
tff(fact_5800_ATP_Olambda__50,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ca(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bz(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_50
tff(fact_5801_ATP_Olambda__51,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_by(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bx(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_51
tff(fact_5802_ATP_Olambda__52,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: heap_e7401611519738050253t_unit,Uub: nat] :
      ( pp(aa_nat_bool(aa_hea5076648075882977743t_bool(aTP_Lamm_ct(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,lim_Product_unit(Uu)),Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),lim_Product_unit(Uua))) ) ) ).

% ATP.lambda_52
tff(fact_5803_ATP_Olambda__53,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_af(Uu),Uua),Uub))
    <=> ( member_nat(aa_nat_nat(suc,Uub),Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_53
tff(fact_5804_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ao(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uua),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uu))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_eq_nat,Uua),Uub)) ) ) ).

% ATP.lambda_54
tff(fact_5805_ATP_Olambda__55,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ag(Uu),Uua),Uub))
    <=> ( member_nat(Uub,Uu)
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),aa_nat_nat(suc,Uua))) ) ) ).

% ATP.lambda_55
tff(fact_5806_ATP_Olambda__56,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(aTP_Lamm_ci(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Uu),Uub))
        | pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Uua),Uub)) ) ) ).

% ATP.lambda_56
tff(fact_5807_ATP_Olambda__57,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(aTP_Lamm_cj(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Uu),Uub))
        & pp(aa_Pro2991721470636411351t_bool(aa_ass8975924923219239432t_bool(rep_assn,Uua),Uub)) ) ) ).

% ATP.lambda_57
tff(fact_5808_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_cf(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uua))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(dvd_dvd_nat,Uub),Uu)) ) ) ).

% ATP.lambda_58
tff(fact_5809_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_ah(Uu),Uua),Uub))
    <=> ( pp(aa_nat_bool(Uu,Uub))
        & pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,Uub),Uua)) ) ) ).

% ATP.lambda_59
tff(fact_5810_ATP_Olambda__60,axiom,
    ! [Uu: fun_Pr6001797050664311790t_bool,Uua: fun_Pr6001797050664311790t_bool,Uub: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(aTP_Lamm_cm(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2991721470636411351t_bool(Uu,Uub))
        | pp(aa_Pro2991721470636411351t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_60
tff(fact_5811_ATP_Olambda__61,axiom,
    ! [Uu: fun_Pr6001797050664311790t_bool,Uua: fun_Pr6001797050664311790t_bool,Uub: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aa_fun6808777976510360636t_bool(aTP_Lamm_cl(Uu),Uua),Uub))
    <=> ( pp(aa_Pro2991721470636411351t_bool(Uu,Uub))
        & pp(aa_Pro2991721470636411351t_bool(Uua,Uub)) ) ) ).

% ATP.lambda_61
tff(fact_5812_ATP_Olambda__62,axiom,
    ! [Uu: fun_nat_bool,Uua: nat,Uub: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aTP_Lamm_bm(Uu),Uua),Uub))
    <=> pp(aa_nat_bool(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_62
tff(fact_5813_ATP_Olambda__63,axiom,
    ! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa_Cod6808739660676082561atural(aa_Cod4292056611156568242atural(aa_Cod8811915439870959971atural(aTP_Lamm_co(Uu),Uua),Uub),Uuc) = produc4480994950612372183atural(produc3574140220909816553atural(inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uu),Uuc),produc3574140220909816553atural(Uub,inc_shift(numera5444537566228673987atural(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))),Uua))) ).

% ATP.lambda_63
tff(fact_5814_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_az(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_64
tff(fact_5815_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa_nat_bool(aa_nat_fun_nat_bool(aa_nat8637530788439456283t_bool(aTP_Lamm_bc(Uu),Uua),Uub),Uuc))
    <=> pp(aa_nat_bool(aa_nat_fun_nat_bool(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua))) ) ).

% ATP.lambda_65
tff(fact_5816_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_be(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_66
tff(fact_5817_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_bg(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_67
tff(fact_5818_ATP_Olambda__68,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bz(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_68
tff(fact_5819_ATP_Olambda__69,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_cd(Uu),Uua),Uub),Uuc) = normalize(aa_int7462565939943104719nt_int(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_69
tff(fact_5820_ATP_Olambda__70,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_cb(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_70
tff(fact_5821_ATP_Olambda__71,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_bx(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_71
tff(fact_5822_ATP_Olambda__72,axiom,
    ! [Uu: assn,Uua: product_unit] : aa_Product_unit_assn(aTP_Lamm_cu(Uu),Uua) = Uu ).

% ATP.lambda_72
tff(fact_5823_ATP_Olambda__73,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aq,Uu) = Uu ).

% ATP.lambda_73
tff(fact_5824_ATP_Olambda__74,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_74
tff(fact_5825_ATP_Olambda__75,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( pp(aa_Pro2991721470636411351t_bool(aTP_Lamm_ch,Uu))
    <=> $false ) ).

% ATP.lambda_75
tff(fact_5826_ATP_Olambda__76,axiom,
    ! [Uu: nat] :
      ( pp(aa_nat_bool(aTP_Lamm_bn,Uu))
    <=> $false ) ).

% ATP.lambda_76

% Helper facts (37)
tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa_bool_bool(fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa_bool_bool(fNot,P))
      | ~ pp(P) ) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fconj,P),Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fdisj,P),Q)) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(if_int(fTrue),X),Y) = X ).

tff(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fFalse),X),Y) = Y ).

tff(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(if_nat(fTrue),X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ pp(aa_int_bool(aa_int_fun_int_bool(fequal_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      | pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ~ pp(aa_nat_bool(aa_nat_fun_nat_bool(fequal_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ( X != Y )
      | pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__HOL__Obool_T,axiom,
    ! [X: bool,Y: bool] :
      ( ~ pp(aa_bool_bool(aa_boo8970175072276269406l_bool(fequal_bool,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fFalse,X),Y) = Y ).

tff(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Cod1783974652533396728nteger(if_Code_integer(fTrue,X),Y) = X ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
      | pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ~ pp(aa_Code_integer_bool(aa_Cod7250484176784686200r_bool(fequal_Code_integer,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( X != Y )
      | pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ~ pp(aa_list_int_bool(aa_lis3210702172589064762t_bool(fequal_list_int,X),Y))
      | ( X = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( X != Y )
      | pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ~ pp(aa_list_nat_bool(aa_lis4768068104879709938t_bool(fequal_list_nat,X),Y))
      | ( X = Y ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] : if_Pro3027730157355071871nt_int(fTrue,X,Y) = X ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] : if_Pro6206227464963214023at_nat(fTrue,X,Y) = X ).

tff(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fFalse,X,Y) = Y ).

tff(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] : if_Pro6119634080678213985nteger(fTrue,X,Y) = X ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_T,axiom,
    ! [X: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( X != Y )
      | pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,X),Y)) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J_T,axiom,
    ! [X: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ~ pp(aa_lis5212129917091495004n_bool(aa_lis724820132743370846n_bool(fequal6748768051200047756n_assn,X),Y))
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa_assn_bool(aa_ass5323576183532441186n_bool(entails,p),q)) ).

%------------------------------------------------------------------------------