%------------------------------------------------------------------------------
% File     : ITP206_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Imperative_HOL_Add 00014_000455
% 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    : 0017_Imperative_HOL_Add_00014_000455 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.1.0
% Syntax   : Number of formulae    : 10575 (2184 unt;5446 typ;   0 def)
%            Number of atoms       : 12088 (4855 equ)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 7521 ( 930   ~; 108   |; 534   &)
%                                         ( 892 <=>;5057  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   5 avg)
%            Maximal term depth    :   12 (   2 avg)
%            Number of FOOLs       : 1750 ( 419 fml;1331 var)
%            Number of X terms     :  320 (   0  []; 315 ite;   5 let)
%            Number of types       : 1338 (1337 usr)
%            Number of type conns  : 6347 (3618   >;2729   *;   0   +;   0  <<)
%            Number of predicates  :  423 ( 420 usr;   3 prp; 0-5 aty)
%            Number of functors    : 3691 (3691 usr; 492 con; 0-4 aty)
%            Number of variables   : 15837 (15565   !; 272   ?;15837   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 13:39:21.492
%------------------------------------------------------------------------------
% Could-be-implicit typings (1337)
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    set_o: $tType ).

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

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

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

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

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

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

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

% Explicit typings (4109)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: fun_se5282722391638144526_nat_o ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: fun_set_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: set_num > fun_fu8556913022524534882at_num ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: set_num > fun_fu4519675068721578778nt_num ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: set_o > fun_fu2540046296233003972un_o_o ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: set_o > fun_fu2924390228791184986_nat_o ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: set_o > fun_fu686522788921195354_int_o ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: set_nat > fun_fu3480060562565761428at_nat ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: set_nat > fun_fu7464972141212037478_o_nat ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: set_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: set_nat > fun_fu376754086352608316nt_nat ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: fun_Product_unit_o > fun_se5026263784410539374t_unit ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: fun_o_o > fun_se3991331202441734329un_o_o ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: fun_nat_o > fun_se231366166962891095_o_nat ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: fun_Pr4311113161502339697t_char > fun_se7382767483982833867t_unit ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: fun_o_char > fun_se6051290791390349078char_o ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: fun_nat_char > fun_se6343395635247293550ar_nat ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: fun_Product_unit_rat > fun_se3943670388958440324t_unit ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: fun_o_rat > fun_se852627502931137359_rat_o ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: fun_nat_rat > fun_se5255405988999443061at_nat ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: fun_int_nat > fun_se464509122368715137at_int ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: fun_nat_Product_unit > fun_fu2272389550985054554t_unit ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: fun_nat_Product_unit > fun_fu2129427594523859474t_unit ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: fun_int_nat > fun_fu3633881621402835147nt_nat ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_int_nat > fun_fu4722942173471469543it_nat ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: fun_int_nat > fun_fu207300434625979687_o_nat ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_int_nat > fun_fu1731011121079923219at_nat ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: fun_fu5809151745113873117at_nat ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: fun_int_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: fun_fu8143749503082306781nt_int ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: fun_re9091073251222685225t_unit ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_nat_Product_unit > fun_Pr1628945989756059814et_nat ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_na367238361360896496at_nat > fun_Pr6764980218223570944et_nat ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_na1476630412504255114at_nat > fun_Pr9038118721296872730et_nat ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_o > fun_o_set_nat ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_nat_nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_Product_unit_nat > fun_fu3412095402257024966nt_nat ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: fun_Product_unit_nat > fun_fu5850005074487364102it_nat ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: fun_Product_unit_nat > fun_fu235524729905330332_o_nat ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_Product_unit_nat > fun_fu2636998066579878598at_nat ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: set_Pr2560585780119916871nt_int > fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: set_Pr7459493094073627847at_nat > fun_se5672428230568430382_nat_o ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: set_Pr4329608150637261639at_nat > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: set_int > fun_fu2583724240955387574_int_o ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: set_o > fun_fu7138923643431704310un_o_o ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: set_int > fun_fu4838637399144961554_int_o ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: set_nat > fun_fu3388977059355398174_nat_o ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: set_nat > fun_fu4988404976222497994_nat_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: fun_se374060482380105870_nat_o ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: set_nat > fun_fu7243318134412071974_nat_o ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: set_o > fun_fu9204702599725184795un_o_o ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: set_int > fun_fu5959969181158722695_int_o ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: set_Pr958786334691620121nt_int > fun_fu5648273804738094549_int_o ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: set_Pr1261947904930325089at_nat > fun_fu7618582134647032973_nat_o ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: fun_char_nat > fun_fu1676310012736982328ar_nat ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_rat_nat > fun_fu2602254339009987519at_nat ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_nat_rat > fun_fu3314643129243502543at_rat ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_nat_rat > fun_fu3450081077967528567at_int ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: fun_nat_int > fun_fu8044810989908836387at_rat ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: fun_nat_int > fun_fu8180248938632862411at_int ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: fun_nat_rat > fun_fu1988719594483546047at_nat ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: fun_nat_int > fun_fu6718887455148879891at_nat ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_nat_nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_nat_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_Pr363642936170290612_nat_o ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_int_nat ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: fun_na819858167508000762st_nat ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: fun_pr6291093682560481277atural ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_nat_fun_nat_o > fun_fu3845992429478311890_nat_o ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_nat_fun_nat_o > fun_fu6523524854190375001_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_fun_nat_o > fun_fu4130307860071370321_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_fun_nat_o > fun_fu6097738348958820013_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_num_fun_num_o > fun_fu7174430410164640793_num_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_int_fun_int_o > fun_fu6313733283296143130_int_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_int_fun_int_o > fun_fu8991265708008206241_int_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_int_fun_int_o > fun_fu6598048713889201561_int_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_int_fun_int_o > fun_fu8565479202776651253_int_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_fun_nat_o > fun_fu5673074405570184289_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_int_int ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_na1668100129766719005at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_na5914473681949881652_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_in1696471567231710611nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_int_fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: fun_rat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_rat_nat > fun_fu5485476364901847999at_nat ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_rat_int > fun_fu2995393219970121243at_int ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_int_rat > fun_fu4574396820457021467at_rat ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_rat > fun_fu622952192041312191at_rat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_nat_rat > fun_fu2525822692364224119nt_rat ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: fun_nat_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: fun_nat_int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_o_o > fun_se7410564775176163214un_o_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_o_o > fun_se7410564775176163214un_o_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_int_o > fun_se7788308268708561250un_o_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_int_o > fun_se7460340907868704146_int_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_o_int > fun_se5483692266196779150_int_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_o_int > fun_se1227706266003314534un_o_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_int_int > fun_se7515062963244431458_int_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_int_int > fun_se7515062963244431458_int_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_nat_rat > fun_se7137282183165283078_rat_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_nat_rat > fun_se5770093131577469710_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_nat_o > fun_se6580191135780181254un_o_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_nat_o > fun_se5710877183863993418_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_nat_int > fun_se8357335538579958278_int_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_nat_int > fun_se8285385046356996962_nat_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_o_nat > fun_se659592270355342_nat_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_o_nat > fun_se8896292107381019842un_o_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_int_nat > fun_se2032030289318007650_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_int_nat > fun_se2103980781540968966_int_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_nat_nat > fun_se2874302864653534470_nat_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_nat > fun_se2874302864653534470_nat_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_Pr4796943543331071514_nat_o > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_Pr3238374608701176162_int_o > fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_Pr4329608150637261639at_nat > fun_fu1322491050725084261_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_Pr1261947904930325089at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_Pr958786334691620121nt_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_o_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_na1668100129766719005at_nat > fun_Pr557098546508347797at_nat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_na5914473681949881652_nat_o > fun_Pr3455331679158536364_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_rat > fun_fu4118273423298180673ar_rat ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_nat > fun_fu1274887723921082425ar_nat ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_nat_int > fun_fu8008176615844131477ar_int ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_int_nat > fun_fu2170221585441270933ar_nat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_int_int > fun_fu8903510477364319985ar_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_nat_rat > fun_fu431421103351642063at_rat ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_nat_nat > fun_fu6811407440829319623at_nat ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_nat_int > fun_fu4321324295897592867at_int ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_int_nat > fun_fu1539480032390253091at_nat ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_int_int > fun_fu8272768924313302143at_int ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_num_fun_num_o > fun_fu7488796153234266514_num_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_num_fun_num_o > fun_fu964304068329126142_num_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_num_fun_num_o > fun_fu2498393115218153618_nat_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_num_fun_num_o > fun_fu964304068329126142_num_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_re4636080666289656953_ref_a ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_re918128527241130269t_unit ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_int_int > fun_fu6952078531906779223nt_int ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_na1668100129766719005at_nat > fun_fu3700668756932941393at_nat ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_na5914473681949881652_nat_o > fun_fu9128415864042724456_nat_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_in1696471567231710611nt_int > fun_fu7131445012926115099nt_int ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_int_fun_int_o > fun_fu4714799948997380346_int_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_int_fun_int_int > fun_fu1406360978810026402nt_int ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_se5221685241404922237at_nat ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_se6990371478945942781at_nat ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_nu6538312362126568467at_num ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_int_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_nat_o > fun_fu8257668495121851639_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_int_nat > fun_fu1756172458508103847_int_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: fun_nat_int > fun_fu6128684156180612191_nat_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_Pr7647638413817135296at_int > fun_fu3357602431878479730_nat_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_Pr982841511892225912nt_int > fun_fu5825343285696310970_int_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_Pr2602117396471556196at_nat > fun_fu3869517070857141966_nat_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_se3385054562097667690_nat_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_se8406536441587994090_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_se1056357380201108920_num_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_se4014881416909572940_nat_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_se5614109888078709212_int_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_o_4732642864545671890od_o_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nu5148503272595741012um_num ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_se1438234007300843299at_nat ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_se4714833218632934537at_nat ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_in4211310926729683942nt_int ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: ref_a > fun_he6342020308413958580unit_a ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_re7427564137345809141_ref_a ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_he4132455388531741535t_unit ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_re3268817776988127511_ref_a ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_he586462870606553245t_unit ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_fu5818728815147633517nteger > fun_fu570797399075587616eger_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_o > fun_fun_nat_o_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_nat > fun_fun_nat_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_nat > fun_fun_int_nat_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_int > fun_fun_int_int_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_num > fun_fun_int_num_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: num > fun_nat_num ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: nat > fun_int_nat ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_Pr5876161306168176259nteger > fun_fu4800041728187187127nteger ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_rat > fun_fu4152799434893411935at_nat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: set_int > fun_nat_set_int ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_int_nat > fun_se7318352605367997114et_nat ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_rat > fun_fu265603220170922335at_int ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_int_int > fun_se2504804186580078358et_int ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_nat_nat > fun_fu4152238915952267095at_int ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_int_int > fun_se2974063760341720334et_int ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_nat_int > fun_fu3694974300620294323nt_nat ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_int_nat > fun_se4573049793270438158et_nat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_int_nat > fun_fu3812433851890366899at_rat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_nat > fun_int_set_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_rat > fun_se2816373030513818206et_rat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_int_nat > fun_fu1046249110317253299at_int ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_int > fun_se7804972798165601458et_int ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_int_nat > fun_fu2073167325106465787at_int ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_int > fun_int_set_int ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_int_int > fun_fu8691345058690680335nt_rat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_rat > fun_se8825182681509393082et_rat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_int_int > fun_fu588984494985280527nt_nat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_nat_nat > fun_se180586794239543474et_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_int_int > fun_fu1615902709774493015nt_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_rat > fun_nat_set_rat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_Pr1261947904930325089at_nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_Pr1133966238863718811t_unit > fun_re9091073251222685225t_unit ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_Pr7459493094073627847at_nat > fun_fu2013277659925970333_nat_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_Pr4329608150637261639at_nat > fun_fu7121398387511592835_nat_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_Pr1261947904930325089at_nat > fun_fu7243318134412071974_nat_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: set_Pr958786334691620121nt_int > fun_fu4988404976222497994_nat_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_Pr7459493094073627847at_nat > fun_fu1710127741857456961_int_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: set_Pr4329608150637261639at_nat > fun_fu6413789880000431911_int_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_Pr1261947904930325089at_nat > fun_fu4838637399144961554_int_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: set_Pr958786334691620121nt_int > fun_fu2583724240955387574_int_o ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_re5167877524888533480unit_o > fun_Pr5834887685177840844unit_o ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Pr4796943543331071514_nat_o > fun_fu1712994508427853293_nat_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_Pr4796943543331071514_nat_o > fun_Pr1469247711798767971at_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_Pr4796943543331071514_nat_o > fun_fu1103891411754596823_nat_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_nat_o > fun_Pr9038118721296872730et_nat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_Pr4796943543331071514_nat_o > fun_fu1190392236513288173_int_o ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_Pr3238374608701176162_int_o > fun_Pr9097774924029995035nt_int ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_o > fun_fu2248797666949362174_nat_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_Pr4796943543331071514_nat_o > fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_nat_o > fun_fu8937908729881586830_int_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_Pr3238374608701176162_int_o > fun_na5449828587677346466nt_int ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_Pr3238374608701176162_int_o > fun_fu3638655979462329509_nat_o ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_Pr4796943543331071514_nat_o > fun_Pr5398690294390750379at_nat ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_Pr3238374608701176162_int_o > fun_fu2790114129539692831_nat_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_o > fun_Pr3513738403344720850et_nat ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_Pr3238374608701176162_int_o > fun_fu3116053707547764389_int_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_Pr3238374608701176162_int_o > fun_Pr3803845469767201635nt_int ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_o > fun_fu2106003848923686156_nat_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_o > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_int_o > fun_fu7333115222899818672_int_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_int_o > fun_int_set_int ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_char_set_char ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_char_set_rat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_char_set_nat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_char_set_int ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_rat_set_char ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_rat_set_rat ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_rat_set_nat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_rat_set_int ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_set_char ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_set_rat ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_Pr4329608150637261639at_nat > fun_se7373623061493298922_nat_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_Pr1261947904930325089at_nat > fun_se6502775985981750992_nat_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: nat > fun_nu2628545618185965215_num_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_Pr4796943543331071514_nat_o > fun_fu512629329099166147_nat_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_Pr2028404973800837934_nat_o > fun_Pr1469247711798767971at_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_Pr4796943543331071514_nat_o > fun_fu7755011012798512067_nat_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_Pr6984107991879650211_nat_o > fun_Pr9038118721296872730et_nat ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_Pr4796943543331071514_nat_o > fun_fu4235351074729334723_int_o ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_Pr4496145827618669174_int_o > fun_Pr9097774924029995035nt_int ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_o > fun_fu3504456128069025355_nat_o ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_na3970208099681795815_nat_o > fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_o > fun_fu7234456906075575515_int_o ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_na6437948953499627055_int_o > fun_na5449828587677346466nt_int ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_Pr3238374608701176162_int_o > fun_fu2208996459339058115_nat_o ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_Pr6550376307508196838_nat_o > fun_Pr5398690294390750379at_nat ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_Pr3238374608701176162_int_o > fun_fu2270615362525379155_nat_o ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_Pr8286563733112057579_nat_o > fun_Pr3513738403344720850et_nat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_Pr3238374608701176162_int_o > fun_fu5931718204969226691_int_o ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_Pr9018117161326028078_int_o > fun_Pr3803845469767201635nt_int ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_nat_o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_fun_nat_o > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_int_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_int_fun_int_o > fun_int_set_int ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: set_o > fun_o_set_o ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: set_nat > fun_o_set_nat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: set_int > fun_o_set_int ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: set_o > fun_nat_set_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: set_o > fun_int_set_o ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: set_o > fun_Pr8467005832278515962_set_o ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: set_se5479201105039950375at_nat > fun_se3430100612660269923at_nat ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: set_se7855581050983116737at_nat > fun_se953073389798102371at_nat ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: set_nat > fun_num_set_nat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: set_num > fun_nat_set_num ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: set_nat > fun_rat_set_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: set_int > fun_rat_set_int ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_rat > fun_int_set_rat ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_Product_unit > fun_Pr3013102738158121955t_unit ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_o > fun_Pr6739565440344758894_set_o ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: set_nat > fun_Pr1628945989756059814et_nat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_Product_unit > fun_o_1833192751255691672t_unit ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: set_Product_unit > fun_na896991816813928310t_unit ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: set_he8644174446132254637t_unit > fun_re9091073251222685225t_unit ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_se953073389798102371at_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: set_Pr1261947904930325089at_nat > fun_Pr3332065415989118679at_nat ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: $o > fun_se5639961202347202350_nat_o ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: $o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: $o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: set_o > fun_fu503842305785720280un_o_o ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: set_int > fun_fu4462626833368551022_int_o ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: set_nat > fun_fu8233823608780690286_nat_o ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: set_o > fun_fu5370137289309429475un_o_o ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: set_int > fun_fu7671276271812522531_int_o ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: set_nat > fun_fu8708304106554588451_nat_o ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: set_o > fun_fu4049601875756261639un_o_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: set_int > fun_fu3900186759307340847_int_o ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: set_nat > fun_fu4467818495196037351_nat_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: set_ref_a > fun_fu2089683210858310382ef_a_o ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: product_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: $o > fun_char_o ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: nat > fun_char_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: product_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: $o > fun_rat_o ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: nat > fun_rat_nat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: product_unit > fun_nat_Product_unit ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: set_char > fun_Pr533032343101338535t_char ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: set_char > fun_o_set_char ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: set_char > fun_nat_set_char ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: set_rat > fun_Pr4472331689133158062et_rat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: set_rat > fun_o_set_rat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: set_int > fun_Pr8362234881679108866et_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: set_Pr4329608150637261639at_nat > fun_Pr5191738088909055843at_nat ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: set_Pr1261947904930325089at_nat > fun_Pr1469247711798767971at_nat ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_Pr2191293291710260995t_unit ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_nat > fun_Pr6764980218223570944et_nat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: set_nat > fun_Pr9038118721296872730et_nat ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: product_unit > fun_Pr2191293291710260995t_unit ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: product_unit > fun_Pr8319229360018371625t_unit ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: product_unit > fun_Pr2481279687482350479t_unit ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: product_unit > fun_o_Product_unit ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: product_unit > fun_fu4476336116263203907t_unit ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: $o > fun_fu7451344543480372697_set_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: product_unit > fun_fu501496177993531693t_unit ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: $o > fun_fu4482994518552752579_set_o ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: product_unit > fun_fu6135282915725538755t_unit ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: $o > fun_fu3096191502022873091_set_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: nat > fun_fu3364353395890117059et_nat ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: nat > fun_fu5908311470514908163et_nat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: product_prod_nat_nat > fun_fu8514662043097244971at_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_Pr3013102738158121955t_unit ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_Pr5191738088909055843at_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_Pr1469247711798767971at_nat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_o_set_o ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: product_unit > fun_fu3802732836198117571t_unit ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: $o > fun_fu3112348720009954563_set_o ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: nat > fun_fu2074493022567586523et_nat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: product_unit > fun_fu4089560697581542491t_unit ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_nat_set_nat > fun_se3320465564477198516at_nat ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: set_Pr4329608150637261639at_nat > fun_se9134159295793868547at_nat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_se3430100612660269923at_nat > fun_se9134159295793868547at_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: nat > fun_nu5900142480299520106at_num ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_set_num > fun_na4015238086716081972at_num ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: int > fun_in7209865251430457670nt_int ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_int_set_int > fun_in7209865251430457670nt_int ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_Pr1261947904930325089at_nat > fun_se1551789852802198377at_nat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_se953073389798102371at_nat > fun_se1551789852802198377at_nat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: nat > fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_nat_set_nat > fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_set_nat > fun_na6128946301132549280et_nat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_set_nat > fun_fu7530206860547196985at_nat ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_set_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_set_nat > fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_se9134159295793868547at_nat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_in7209865251430457670nt_int ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_se1551789852802198377at_nat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_na7044673412300895210at_nat ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_rat > fun_nat_set_rat ).

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

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_int_int > fun_int_set_int ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_int_nat > fun_int_set_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_rat_set_nat > fun_fu4382068692248749429et_nat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_set_nat > fun_fu5004605035024130925et_nat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_int_set_nat > fun_fu173695997200249801et_nat ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_set_nat > fun_fu947770073829403173et_nat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_int_set_nat > fun_fu5340233072860297857et_nat ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: $o > fun_Product_unit_o ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: nat > fun_Product_unit_nat ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: nat > fun_o_nat ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: product_unit > fun_int_Product_unit ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_o_fun_int_o ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_nat_set_nat > fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: int > fun_Product_unit_int ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: int > fun_o_int ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_ch5514019212406764701t_unit > fun_ch5514019212406764701t_unit ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_ch5514019212406764701t_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_ch5514019212406764701t_unit > fun_fu3244842278976105679t_unit ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_ch5514019212406764701t_unit > fun_fu313385910871358727t_unit ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_ch7403422310664315094t_unit > fun_ra2707149721221939542t_unit ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_ch7403422310664315094t_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_ch7403422310664315094t_unit > fun_fu8900287769237569217t_unit ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_ch7403422310664315094t_unit > fun_fu4537078353777772224t_unit ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_ch9120586758495803614t_unit > fun_na6561490796278613342t_unit ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_ch9120586758495803614t_unit > fun_nat_Product_unit ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_ch9120586758495803614t_unit > fun_fu3638106572463594193t_unit ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_ch9120586758495803614t_unit > fun_fu2196479924495653576t_unit ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_ch1452000917118098306t_unit > fun_in4306577638089039362t_unit ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_ch1452000917118098306t_unit > fun_int_Product_unit ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_ch1452000917118098306t_unit > fun_fu952724092823889433t_unit ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_ch1452000917118098306t_unit > fun_fu2005774009263389548t_unit ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_ra2707149721221939542t_unit > fun_ch7403422310664315094t_unit ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_ra2707149721221939542t_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_ra2707149721221939542t_unit > fun_fu1124013290187893256t_unit ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_ra2707149721221939542t_unit > fun_fu5211608675627952448t_unit ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_ra3623852323372631695t_unit > fun_ra3623852323372631695t_unit ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_ra3623852323372631695t_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_ra3623852323372631695t_unit > fun_fu4006739739138223610t_unit ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_ra3623852323372631695t_unit > fun_fu4794418277351132153t_unit ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_ra5341016771204120215t_unit > fun_na1234882596965831319t_unit ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_ra5341016771204120215t_unit > fun_nat_Product_unit ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_ra5341016771204120215t_unit > fun_fu3660932303109722634t_unit ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_ra5341016771204120215t_unit > fun_fu3941614162935670785t_unit ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_ra6895802966681190715t_unit > fun_in8866373506301333307t_unit ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_ra6895802966681190715t_unit > fun_int_Product_unit ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_ra6895802966681190715t_unit > fun_fu3803894259570917714t_unit ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_ra6895802966681190715t_unit > fun_fu7775853263819697317t_unit ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_na6561490796278613342t_unit > fun_ch9120586758495803614t_unit ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_na6561490796278613342t_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_na6561490796278613342t_unit > fun_fu7017555856346147344t_unit ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_na6561490796278613342t_unit > fun_fu6928773123459440968t_unit ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_na1234882596965831319t_unit > fun_ra5341016771204120215t_unit ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_na1234882596965831319t_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_na1234882596965831319t_unit > fun_fu2475235030552360450t_unit ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_na1234882596965831319t_unit > fun_fu6511582725182620673t_unit ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_ch5514019212406764701t_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_ch5514019212406764701t_unit > fun_fu313385910871358727t_unit ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_ch7403422310664315094t_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_ch7403422310664315094t_unit > fun_fu4537078353777772224t_unit ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_ch9120586758495803614t_unit > fun_nat_Product_unit ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_ch9120586758495803614t_unit > fun_fu2196479924495653576t_unit ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_ch1452000917118098306t_unit > fun_int_Product_unit ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_ch1452000917118098306t_unit > fun_fu2005774009263389548t_unit ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_ra2707149721221939542t_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_ra2707149721221939542t_unit > fun_fu5211608675627952448t_unit ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_ra3623852323372631695t_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_ra3623852323372631695t_unit > fun_fu4794418277351132153t_unit ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_ra5341016771204120215t_unit > fun_nat_Product_unit ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_ra5341016771204120215t_unit > fun_fu3941614162935670785t_unit ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_ra6895802966681190715t_unit > fun_int_Product_unit ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_ra6895802966681190715t_unit > fun_fu7775853263819697317t_unit ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_na6561490796278613342t_unit > fun_ch5401695571711387349t_unit ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_na6561490796278613342t_unit > fun_fu6928773123459440968t_unit ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_na1234882596965831319t_unit > fun_rat_Product_unit ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_na1234882596965831319t_unit > fun_fu6511582725182620673t_unit ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: set_char > fun_fu1823366087498536067t_char ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: set_char > fun_fu5985699018863926339t_char ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: set_char > fun_fu2421111455328269989t_char ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: set_rat > fun_fu541550854055878595et_rat ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: set_rat > fun_fu3688842493855139331et_rat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: set_rat > fun_fu5953382810611313715et_rat ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: set_int > fun_fu3534008338840198851et_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: set_int > fun_fu1332340193377564419et_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: set_int > fun_fu4581781630362247899et_int ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: set_nat > fun_fu3364353395890117059et_nat ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_Co1981900195452921617nteger ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_Pr533032343101338535t_char > fun_se2345829390869815737t_char ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_o_set_char > fun_se35625083398921668t_char ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_nat_set_char > fun_se8005896535649485400t_char ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_Pr4472331689133158062et_rat > fun_se683497417899989625et_rat ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_o_set_rat > fun_se7007405147008830212et_rat ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_set_rat > fun_se2193836687738436710et_rat ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_Pr8362234881679108866et_int > fun_se7289271276603518841et_int ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_o_set_int > fun_se241328533291454980et_int ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_nat_set_int > fun_se2974063760341720334et_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_Pr1628945989756059814et_nat > fun_se8085683455625549945et_nat ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: set_se5479201105039950375at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: set_set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: set_se7855581050983116737at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: set_se6260736226359567993nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: set_set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: set_se5479201105039950375at_nat > fun_Pr3059359313727237556_nat_o ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: set_set_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: set_set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: set_se7855581050983116737at_nat > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: set_se6260736226359567993nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: set_set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: set_nat > fun_fu3094553867479596099et_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: set_nat > fun_fu5014701096895771203et_nat ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: set_nat > fun_fu5908311470514908163et_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_se5672428230568430382_nat_o > fun_se8139871178935176202_nat_o ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_se5639961202347202350_nat_o > fun_se2210973626355406192_nat_o ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_fun_nat_o > fun_se4095769911529315581_nat_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_int_fun_int_o > fun_se4642411815161519337_int_o ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: product_unit > fun_se7528283263925519628t_unit ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: $o > fun_se5315385049746747468_set_o ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: nat > fun_se8085683455625549945et_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: product_unit > fun_se6479263725144128739t_unit ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: $o > fun_se4053726005633169529_set_o ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: nat > fun_se5245582867311637764et_nat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: product_unit > fun_se2298057445283529624t_unit ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: $o > fun_se3315603611433934638_set_o ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: num > fun_se1398106661611058410et_num ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: int > fun_se8982873411337295210et_int ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: set_char > fun_fu1823366087498536067t_char ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_char > fun_fu5985699018863926339t_char ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_char > fun_fu2421111455328269989t_char ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: set_rat > fun_fu541550854055878595et_rat ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: set_rat > fun_fu3688842493855139331et_rat ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: set_rat > fun_fu5953382810611313715et_rat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: set_int > fun_fu3534008338840198851et_int ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: set_int > fun_fu1332340193377564419et_int ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: set_int > fun_fu4581781630362247899et_int ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: set_nat > fun_fu3364353395890117059et_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_Pr533032343101338535t_char > fun_se2345829390869815737t_char ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_o_set_char > fun_se35625083398921668t_char ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_set_char > fun_se8005896535649485400t_char ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_Pr4472331689133158062et_rat > fun_se683497417899989625et_rat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_o_set_rat > fun_se7007405147008830212et_rat ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_nat_set_rat > fun_se2193836687738436710et_rat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_Pr8362234881679108866et_int > fun_se7289271276603518841et_int ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_o_set_int > fun_se241328533291454980et_int ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_set_int > fun_se2974063760341720334et_int ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_Pr1628945989756059814et_nat > fun_se8085683455625549945et_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_int_set_int > fun_fu526684654072379101et_int ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_int_set_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_in4731230917697954842t_unit > fun_fu5624083597678921450et_int ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_in4731230917697954842t_unit > fun_fu8837881197197444916nt_int ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_int_set_o > fun_fu908641882492701557et_int ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_int_set_o > fun_fu987901723025782719nt_int ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_Pr8362234881679108866et_int > fun_fu6798134381489791942et_int ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_Pr8362234881679108866et_int > fun_fu8943522082879247376it_int ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_Pr3013102738158121955t_unit > fun_fu7031575659533622777et_int ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_Pr3013102738158121955t_unit > fun_fu2232859028539742787it_int ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_Pr6739565440344758894_set_o > fun_fu4027353925470698798et_int ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_Pr6739565440344758894_set_o > fun_fu6370921397939457400it_int ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_o_set_int > fun_fu2060400330212408785et_int ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_o_set_int > fun_fu3242287236610103835_o_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_o_1833192751255691672t_unit > fun_fu7209406626366242948et_int ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_o_1833192751255691672t_unit > fun_fu5811362835885304014_o_int ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_o_set_o > fun_fu6190897981441692473et_int ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_o_set_o > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_nat_set_int > fun_fu3741247039931580033et_int ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_set_int > fun_fu8180248938632862411at_int ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_int_set_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_in4731230917697954842t_unit > fun_fu8837881197197444916nt_int ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_int_set_o > fun_fu987901723025782719nt_int ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_Pr8362234881679108866et_int > fun_fu8943522082879247376it_int ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_Pr3013102738158121955t_unit > fun_fu2232859028539742787it_int ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_Pr6739565440344758894_set_o > fun_fu6370921397939457400it_int ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_o_set_int > fun_fu3242287236610103835_o_int ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_o_1833192751255691672t_unit > fun_fu5811362835885304014_o_int ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_o_set_o > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_set_int > fun_fu8180248938632862411at_int ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_int_o > fun_fu973771059875708133nt_nat ).

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

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_int_o > fun_fu5221552698205305849nt_int ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_o > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_nat_o > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_o > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_o > fun_fu8135962457450383814at_nat ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_Pr3238374608701176162_int_o > fun_fu3905284967227325581at_nat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_Pr4796943543331071514_nat_o > fun_fu6463799022293586613at_nat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_nat_o > fun_fu8337820376756068806at_nat ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_Pr3238374608701176162_int_o > fun_fu4684172282582160525at_nat ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: set_Product_unit > fun_fu4476336116263203907t_unit ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: set_o > fun_fu7451344543480372697_set_o ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: set_Product_unit > fun_fu501496177993531693t_unit ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: set_o > fun_fu4482994518552752579_set_o ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: set_Product_unit > fun_fu6135282915725538755t_unit ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: set_o > fun_fu3096191502022873091_set_o ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: set_nat > fun_fu3364353395890117059et_nat ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: set_nat > fun_fu5908311470514908163et_nat ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: set_Pr1261947904930325089at_nat > fun_fu8514662043097244971at_nat ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_Pr3013102738158121955t_unit > fun_se6479263725144128739t_unit ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_Pr6739565440344758894_set_o > fun_se4053726005633169529_set_o ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_o_1833192751255691672t_unit > fun_se2298057445283529624t_unit ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_o_set_o > fun_se3315603611433934638_set_o ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_na896991816813928310t_unit > fun_se7528283263925519628t_unit ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_set_o > fun_se5315385049746747468_set_o ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_Pr1628945989756059814et_nat > fun_se8085683455625549945et_nat ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_o_set_nat > fun_se5245582867311637764et_nat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_set_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_Pr3332065415989118679at_nat > fun_se4804000464841151819at_nat ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: set_Code_integer > fun_Co744628173305925485nteger ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_rat > fun_se656839901185093834at_rat ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_int > fun_se2894038184755951474at_int ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_int_int > fun_se7329855882545492614nt_int ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_o_Product_unit > fun_se5546929288235092472t_unit ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_Product_unit > fun_se8713449004847062726t_unit ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_int_Product_unit > fun_se8416637778297840674t_unit ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_o_o > fun_se7410564775176163214un_o_o ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_o > fun_se2874302864653534470_nat_o ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_int_o > fun_se7515062963244431458_int_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_o_nat > fun_se6216328805891991566_o_nat ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_nat_set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_Pr1469247711798767971at_nat > fun_fu3854441050363238723at_nat ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_se953073389798102371at_nat > fun_fu7445151911848907075at_nat ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: set_Pr1261947904930325089at_nat > fun_se9034565410000537175at_nat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: set_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_nat_set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_se6846035212209387651at_nat > fun_se1438234007300843299at_nat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_se320340636488258563at_nat > fun_se4714833218632934537at_nat ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_nat_num > fun_na6415222038798377684at_num ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: fun_nat_nat > fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_int_int > fun_in4211310926729683942nt_int ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: set_Pr8693737435421807431at_nat > fun_se5689047571578729036at_nat ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: set_Pr4329608150637261639at_nat > fun_se4816296336258376268at_nat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: set_nat > fun_se5245582867311637764et_nat ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_Product_unit_int > fun_fu2232859028539742787it_int ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: fun_o_int > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_Pr2191293291710260995t_unit > fun_fu3471618943627436675t_unit ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_o_Product_unit > fun_fu39903228990708589t_unit ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_nat_Product_unit > fun_fu4407618052490078659t_unit ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_Pr1628945989756059814et_nat > fun_fu3364353395890117059et_nat ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_Product_unit_int > fun_fu2232859028539742787it_int ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_o_int > fun_fu6940589326549028739_o_int ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_Pr2191293291710260995t_unit > fun_fu3471618943627436675t_unit ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_o_Product_unit > fun_fu39903228990708589t_unit ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_nat_Product_unit > fun_fu4407618052490078659t_unit ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_Pr1628945989756059814et_nat > fun_fu3364353395890117059et_nat ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: set_Pr1261947904930325089at_nat > fun_se320340636488258563at_nat ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: set_nat > fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: product_unit > fun_Pr2191293291710260995t_unit ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: set_Pr8693737435421807431at_nat > fun_se6131316820643522691at_nat ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: set_Pr4329608150637261639at_nat > fun_se6846035212209387651at_nat ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: set_nat > fun_set_nat_set_nat ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: product_unit > fun_Pr2191293291710260995t_unit ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_Pr2191293291710260995t_unit > fun_Pr3553901264588531491t_unit ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_o_Product_unit > fun_Pr2727576175602020824t_unit ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_Product_unit > fun_Pr6742640159767676172t_unit ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_o_set_nat > fun_se5245582867311637764et_nat ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_set_nat > fun_se3395149180098744406et_nat ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_Pr2481279687482350479t_unit > fun_Pr2249568190214445103t_unit ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_Pr3332065415989118679at_nat > fun_se4804000464841151819at_nat ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_o_4630291819367765772at_nat > fun_se475724006023232640at_nat ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_na7044673412300895210at_nat > fun_se6112554743067240204at_nat ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: product_unit > fun_fu3471618943627436675t_unit ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: product_unit > fun_fu39903228990708589t_unit ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: product_unit > fun_fu4407618052490078659t_unit ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: set_nat > fun_fu5908311470514908163et_nat ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: set_nat > fun_fu945287533868614691et_nat ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: product_unit > fun_fu672158849875421851t_unit ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: set_Pr1261947904930325089at_nat > fun_fu8514662043097244971at_nat ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: set_Pr1261947904930325089at_nat > fun_fu3488651239929255829at_nat ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: set_Pr1261947904930325089at_nat > fun_fu2153439384680343107at_nat ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: fun_Pr9038118721296872730et_nat > fun_se7838879063450412781et_nat ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_Pr53831763054929981at_nat > fun_se1290235323266432407at_nat ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_o_7500966680246433010at_nat > fun_se5689047571578729036at_nat ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: fun_na317059118361677008at_nat > fun_se8580171928748181388at_nat ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: fun_Pr1686037257031381053at_nat > fun_se6422601326696732055at_nat ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: fun_Pr6108405906935189705at_nat > fun_se6906297397824016291at_nat ).

tff(sy_c_ATP_058Lamm__yf____,type,
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tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: set_nat > fun_fu5014701096895771203et_nat ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: set_Pr8693737435421807431at_nat > fun_fu6738464136248977143at_nat ).

tff(sy_c_ATP_058Lamm__yi____,type,
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tff(sy_c_ATP_058Lamm__yj____,type,
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tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: set_Pr4329608150637261639at_nat > fun_fu5586957678748882679at_nat ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: set_Pr8693737435421807431at_nat > fun_fu7334305174532356111at_nat ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: set_Pr4329608150637261639at_nat > fun_fu5591933921004034017at_nat ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_Pr6212429452034653949at_nat > fun_se7687844536836009329at_nat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_Pr1469247711798767971at_nat > fun_se9034565410000537175at_nat ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_Pr6764980218223570944et_nat > fun_se8712464678868890323et_nat ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_Pr9038118721296872730et_nat > fun_se7838879063450412781et_nat ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: set_Pr1261947904930325089at_nat > fun_fu6163922802919523959at_nat ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: set_Pr1261947904930325089at_nat > fun_fu3854441050363238723at_nat ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: set_nat > fun_fu3094553867479596099et_nat ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: set_nat > fun_fu5014701096895771203et_nat ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_int_int > fun_fu1332340193377564419et_int ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_int_int > fun_fu4581781630362247899et_int ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_int_int > fun_fu5710987119061219731et_int ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_int_nat > fun_fu6145888612165483175et_nat ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_int_nat > fun_fu171958012295390847et_nat ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_int_nat > fun_fu1301163500994362679et_nat ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_nat_rat > fun_fu5329535320929981963et_rat ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_nat_int > fun_fu1094763051726989407et_int ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: fun_nat_rat > fun_fu366511384283688491et_rat ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_nat_int > fun_fu5355111151935471743et_int ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_nat_rat > fun_fu6202111536596174990at_rat ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_nat_rat > fun_fu3162970995458829064et_rat ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: fun_nat_rat > fun_fu4136008861349390990at_nat ).

tff(sy_c_ATP_058Lamm__zi____,type,
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tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_nat_rat > fun_fu6466760888570446734at_int ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_nat_rat > fun_fu4060156567689697712et_int ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_nat_int > fun_fu3524591565614576866nt_rat ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_nat_int > fun_fu1759800260107257692et_rat ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_nat_int > fun_fu1458488890367792866nt_nat ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: fun_nat_int > fun_fu7499689513045208908et_nat ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_int_int > fun_fu6690865562528016938nt_nat ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_int_int > fun_fu803196889506662404et_nat ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_nat_int > fun_fu3789240917588848610nt_int ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_nat_int > fun_fu2656985832338126340et_int ).

tff(sy_c_ATP_058Lamm__zt____,type,
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tff(sy_c_ATP_058Lamm__zu____,type,
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tff(sy_c_ATP_058Lamm__zv____,type,
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tff(sy_c_ATP_058Lamm__zw____,type,
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tff(sy_c_ATP_058Lamm__zx____,type,
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tff(sy_c_ATP_058Lamm__zy____,type,
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tff(sy_c_ATP_058Lamm__zz____,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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tff(sy_c_BNF__Def_OGrp_001_Eo_001_Eo,type,
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    comp_P5051845089628979463um_nat: ( fun_Pr5328756427386567416um_num * fun_Pr8941987727687675267at_num * product_prod_num_nat ) > num ).

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    comp_P3192427084145116251at_num: ( fun_Pr8275629403677631605_o_nat * fun_nu209911646301618442um_nat * num ) > product_prod_o_nat ).

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tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
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tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Nat__Onat_J_001t__Int__Oint,type,
    comp_c912979988581399656at_int: ( fun_char_set_nat * fun_int_char ) > fun_int_set_nat ).

tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    comp_c915470459090449932at_nat: ( fun_char_set_nat * fun_nat_char ) > fun_nat_set_nat ).

tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Nat__Onat_J_001t__Num__Onum,type,
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tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Rat__Orat_J_001_Eo,type,
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tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Rat__Orat_J_001t__Int__Oint,type,
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tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Rat__Orat_J_001t__Nat__Onat,type,
    comp_c5738158305436223508at_nat: ( fun_char_set_rat * fun_nat_char ) > fun_nat_set_rat ).

tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__Set__Oset_It__Rat__Orat_J_001t__Num__Onum,type,
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tff(sy_c_Fun_Ocomp_001t__String__Ochar_001t__String__Ochar_001t__String__Ochar,type,
    comp_char_char_char: ( fun_char_char * fun_char_char ) > fun_char_char ).

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tff(sy_c_Fun_Ofun__upd_001_Eo_001_Eo,type,
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tff(sy_c_Fun_Oid_001_Eo,type,
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tff(sy_c_Fun_Oid_001t__Rat__Orat,type,
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    id_set_int: fun_set_int_set_int ).

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    id_set_nat: fun_set_nat_set_nat ).

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    id_set_num: fun_set_num_set_num ).

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    id_fun8244602250740870226nteger: fun_fu4688806437979560835nteger ).

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    inj_on_o_nat: fun_o_nat > fun_set_o_o ).

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    inj_on_o_char: fun_o_char > fun_set_o_o ).

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    inj_on_int_nat: fun_int_nat > fun_set_int_o ).

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    inj_on_nat_nat: fun_nat_nat > fun_set_nat_o ).

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tff(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: fun_nat_char > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Num__Onum_001_Eo,type,
    inj_on_num_o: ( fun_num_o * set_num ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Num__Onum_001t__Int__Oint,type,
    inj_on_num_int: ( fun_num_int * set_num ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Num__Onum_001t__Nat__Onat,type,
    inj_on_num_nat: ( fun_num_nat * set_num ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    inj_on20428373856169009nteger: ( fun_Pr5876161306168176259nteger * set_Pr4811707699266497531nteger ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    inj_on6454360653267994289nt_int: ( fun_Pr2478283820680522243nt_int * set_Pr958786334691620121nt_int ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    inj_on2178005380612969504at_nat: fun_Pr2602117396471556196at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on8969904277767023793at_nat: fun_Pr2118491707810114563at_nat > fun_se4782206882984618170_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Num__Onum_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    inj_on2339680571444523825at_num: ( fun_Pr8941987727687675267at_num * set_Pr3280342998896498603um_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    inj_on991952103320416098_nat_o: ( fun_Pr363642936170290612_nat_o * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    inj_on490725545064285830at_nat: ( fun_Pr2871540613828922186at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on229527429751349963at_nat: ( fun_Pr3189796393087291805at_nat * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Ounit,type,
    inj_on1041401102088758999t_unit: ( fun_Pr5236084471349978153t_unit * set_Pr8693737435421807431at_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inj_on8430303779476321073at_nat: fun_Pr9034069120582568067at_nat > fun_se7788542033685991124_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Ounit,type,
    inj_on3108384905993948887t_unit: fun_Pr8319229360018371625t_unit > fun_se7788542033685991124_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    inj_on3942866938762682161at_nat: ( fun_Pr5176915617721616003at_nat * set_Pr7459493094073627847at_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001_Eo,type,
    inj_on8854722734121536060unit_o: ( fun_Product_unit_o * set_Product_unit ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Int__Oint,type,
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tff(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Nat__Onat,type,
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tff(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
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tff(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Rat__Orat,type,
    inj_on7795309031694339124it_rat: ( fun_Product_unit_rat * set_Product_unit ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__String__Ochar,type,
    inj_on1959294024837303853t_char: ( fun_Pr4311113161502339697t_char * set_Product_unit ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Rat__Orat_001_Eo,type,
    inj_on_rat_o: fun_rat_o > fun_set_rat_o ).

tff(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Int__Oint,type,
    inj_on_rat_int: fun_rat_int > fun_set_rat_o ).

tff(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Nat__Onat,type,
    inj_on_rat_nat: fun_rat_nat > fun_set_rat_o ).

tff(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    inj_on8867882244352206326at_nat: fun_ra4806459883200776834at_nat > fun_set_rat_o ).

tff(sy_c_Fun_Oinj__on_001t__Rat__Orat_001t__Rat__Orat,type,
    inj_on_rat_rat: fun_rat_rat > fun_set_rat_o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on6435365835345961783et_int: ( fun_set_int_set_int * set_set_int ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
    inj_on1389844818000382683et_nat: ( fun_set_int_set_nat * set_set_int ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on426556184350386907et_int: ( fun_set_nat_set_int * set_set_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Rat__Orat_J,type,
    inj_on1096178645466186887et_rat: ( fun_set_nat_set_rat * set_set_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    inj_on699426629415580343at_nat: ( fun_se4714833218632934537at_nat * set_se7855581050983116737at_nat ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
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tff(sy_c_Fun_Oinj__on_001t__String__Ochar_001_Eo,type,
    inj_on_char_o: ( fun_char_o * set_char ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__String__Ochar_001t__Int__Oint,type,
    inj_on_char_int: ( fun_char_int * set_char ) > $o ).

tff(sy_c_Fun_Oinj__on_001t__String__Ochar_001t__Nat__Onat,type,
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tff(sy_c_Fun_Oinj__on_001t__String__Ochar_001t__String__Ochar,type,
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    inj_on5093811489028909101nteger: ( fun_fu4800041728187187127nteger * set_fu5818728815147633517nteger ) > $o ).

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tff(sy_c_Fun_Oinj__on_001t__fun_It__Int__Oint_Mt__Int__Oint_J_001t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(sy_c_Fun_Oinj__on_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_001t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(sy_c_Fun_Oinj__on_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Fun__Def_Omin__strict,type,
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tff(sy_c_Fun__Def_Opair__leq,type,
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tff(sy_c_Fun__Def_Opair__less,type,
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tff(sy_c_GCD_OGcd__class_OLcm_001t__Int__Oint,type,
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tff(sy_c_GCD_OGcd__class_OLcm_001t__Nat__Onat,type,
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    inf_inf_set_char: ( set_char * set_char ) > set_char ).

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    inf_in8908015913983109984_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_int_fun_int_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    inf_in375002649321709592_nat_o: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_nat_fun_nat_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    inf_in1133183212291220558_num_o: ( fun_nat_fun_num_o * fun_nat_fun_num_o ) > fun_nat_fun_num_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    inf_in484816107290947356_nat_o: ( fun_se5639961202347202350_nat_o * fun_se5639961202347202350_nat_o ) > fun_se5639961202347202350_nat_o ).

tff(sy_c_Lattices_Oinf__class_Oinf_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    inf_in2372659476514261276_nat_o: ( fun_se5672428230568430382_nat_o * fun_se5672428230568430382_nat_o ) > fun_se5672428230568430382_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: fun_int_fun_int_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
    sup_sup_Product_unit: product_unit > fun_Pr2191293291710260995t_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
    sup_sup_rat: rat > fun_rat_rat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
    sup_sup_set_o: ( set_o * set_o ) > set_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
    sup_sup_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > fun_set_nat_set_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    sup_su5769328420594410459od_o_o: ( set_Product_prod_o_o * set_Product_prod_o_o ) > set_Product_prod_o_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    sup_su8679140417860611193_o_nat: ( set_Pr2101469702781467981_o_nat * set_Pr2101469702781467981_o_nat ) > set_Pr2101469702781467981_o_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Ounit_J_J,type,
    sup_su4058753791541417424t_unit: ( set_Pr6671815714118170108t_unit * set_Pr6671815714118170108t_unit ) > set_Pr6671815714118170108t_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    sup_su6024340866399070445nt_int: ( set_Pr958786334691620121nt_int * set_Pr958786334691620121nt_int ) > set_Pr958786334691620121nt_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    sup_su503371503184139039_nat_o: ( set_Pr3149072824959771635_nat_o * set_Pr3149072824959771635_nat_o ) > set_Pr3149072824959771635_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > fun_se320340636488258563at_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    sup_su2042722026077122175at_num: ( set_Pr6200539531224447659at_num * set_Pr6200539531224447659at_num ) > set_Pr6200539531224447659at_num ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
    sup_su4209953369189666580t_unit: ( set_Pr4334478416066269672t_unit * set_Pr4334478416066269672t_unit ) > set_Pr4334478416066269672t_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > fun_se6131316820643522691at_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_M_Eo_J_J,type,
    sup_su2851927382625014182unit_o: ( set_Pr5464989305201766866unit_o * set_Pr5464989305201766866unit_o ) > set_Pr5464989305201766866unit_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
    sup_su1639320892072265582it_nat: ( set_Pr1763845938948868674it_nat * set_Pr1763845938948868674it_nat ) > set_Pr1763845938948868674it_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
    sup_su6481247232241170203t_unit: ( set_Pr5094982260447487303t_unit * set_Pr5094982260447487303t_unit ) > set_Pr5094982260447487303t_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > fun_se6846035212209387651at_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    sup_su5395584912770621083at_nat: ( set_Pr7459493094073627847at_nat * set_Pr7459493094073627847at_nat ) > set_Pr7459493094073627847at_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    sup_su793286257634532545t_unit: ( set_Product_unit * set_Product_unit ) > set_Product_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
    sup_sup_set_rat: ( set_rat * set_rat ) > set_rat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    sup_sup_set_set_o: ( set_set_o * set_set_o ) > set_set_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: ( set_set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    sup_su3642409539654194069at_nat: ( set_se7855581050983116737at_nat * set_se7855581050983116737at_nat ) > set_se7855581050983116737at_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    sup_su474284620922681339at_nat: ( set_se5479201105039950375at_nat * set_se5479201105039950375at_nat ) > set_se5479201105039950375at_nat ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
    sup_su5975557051335651745t_unit: ( set_set_Product_unit * set_set_Product_unit ) > set_set_Product_unit ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Int__Oint_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
    sup_su3989259347931230805eger_o: ( fun_in4891666191172872833eger_o * fun_in4891666191172872833eger_o ) > fun_in4891666191172872833eger_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    sup_su7393609867835711046_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > fun_int_fun_int_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    sup_su8083968640029086462_nat_o: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_nat_fun_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    sup_su8842149202998597428_num_o: ( fun_nat_fun_num_o * fun_nat_fun_num_o ) > fun_nat_fun_num_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    sup_su556009932677587827_int_o: ( fun_Pr4738963024209204295_int_o * fun_Pr4738963024209204295_int_o ) > fun_Pr4738963024209204295_int_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    sup_su52541223548345602_nat_o: ( fun_Pr2028404973800837934_nat_o * fun_Pr2028404973800837934_nat_o ) > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    sup_su5424986631308460290_nat_o: ( fun_se5639961202347202350_nat_o * fun_se5639961202347202350_nat_o ) > fun_se5639961202347202350_nat_o ).

tff(sy_c_Lattices_Osup__class_Osup_001t__fun_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__fun_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    sup_su4361989354738342146_nat_o: ( fun_se5672428230568430382_nat_o * fun_se5672428230568430382_nat_o ) > fun_se5672428230568430382_nat_o ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: fun_set_nat_nat ).

tff(sy_c_Lifting_OQuotient_001_Eo_001_Eo,type,
    quotient_o_o: ( fun_o_fun_o_o * fun_o_o * fun_o_o * fun_o_fun_o_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001_Eo_001t__Nat__Onat,type,
    quotient_o_nat: ( fun_o_fun_o_o * fun_o_nat * fun_nat_o * fun_o_fun_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    quotie2291698863146621601nteger: ( fun_int_fun_int_o * fun_int_Code_integer * fun_Code_integer_int * fun_in4891666191172872833eger_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Int__Oint,type,
    quotient_int_int: ( fun_int_fun_int_o * fun_int_int * fun_int_int * fun_int_fun_int_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Nat__Onat,type,
    quotient_int_nat: ( fun_int_fun_int_o * fun_int_nat * fun_nat_int * fun_int_fun_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Num__Onum,type,
    quotient_int_num: ( fun_int_fun_int_o * fun_int_num * fun_num_int * fun_int_fun_num_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Nat__Onat_001_Eo,type,
    quotient_nat_o: ( fun_nat_fun_nat_o * fun_nat_o * fun_o_nat * fun_nat_fun_o_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Int__Oint,type,
    quotient_nat_int: ( fun_nat_fun_nat_o * fun_nat_int * fun_int_nat * fun_nat_fun_int_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Nat__Onat,type,
    quotient_nat_nat: ( fun_nat_fun_nat_o * fun_nat_nat * fun_nat_nat * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Num__Onum,type,
    quotient_nat_num: ( fun_nat_fun_nat_o * fun_nat_num * fun_num_nat * fun_nat_fun_num_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Rat__Orat,type,
    quotient_nat_rat: ( fun_nat_fun_nat_o * fun_nat_rat * fun_rat_nat * fun_nat_fun_rat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__String__Ochar,type,
    quotient_nat_char: ( fun_nat_fun_nat_o * fun_nat_char * fun_char_nat * fun_nat_fun_char_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Num__Onum_001t__Int__Oint,type,
    quotient_num_int: ( fun_num_fun_num_o * fun_num_int * fun_int_num * fun_num_fun_int_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Num__Onum_001t__Nat__Onat,type,
    quotient_num_nat: ( fun_num_fun_num_o * fun_num_nat * fun_nat_num * fun_num_fun_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
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tff(sy_c_Lifting_OQuotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    quotie4166942682936818134at_nat: ( fun_Pr2028404973800837934_nat_o * fun_Pr2118491707810114563at_nat * fun_Pr2118491707810114563at_nat * fun_Pr2028404973800837934_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Int__Oint,type,
    quotient_rat_int: ( fun_rat_fun_rat_o * fun_rat_int * fun_int_rat * fun_rat_fun_int_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Nat__Onat,type,
    quotient_rat_nat: ( fun_rat_fun_rat_o * fun_rat_nat * fun_nat_rat * fun_rat_fun_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__Rat__Orat,type,
    quotient_rat_rat: ( fun_rat_fun_rat_o * fun_rat_rat * fun_rat_rat * fun_rat_fun_rat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__Rat__Orat_001t__String__Ochar,type,
    quotient_rat_char: ( fun_rat_fun_rat_o * fun_rat_char * fun_char_rat * fun_rat_fun_char_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__Int__Oint,type,
    quotient_char_int: ( fun_char_fun_char_o * fun_char_int * fun_int_char * fun_char_fun_int_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__Nat__Onat,type,
    quotient_char_nat: ( fun_char_fun_char_o * fun_char_nat * fun_nat_char * fun_char_fun_nat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__Rat__Orat,type,
    quotient_char_rat: ( fun_char_fun_char_o * fun_char_rat * fun_rat_char * fun_char_fun_rat_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__String__Ochar_001t__String__Ochar,type,
    quotient_char_char: ( fun_char_fun_char_o * fun_char_char * fun_char_char * fun_char_fun_char_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
    quotie2414878292700455852nteger: ( fun_fu4788845708739168348eger_o * fun_fu5689865433563736081nteger * fun_Co1981900195452921617nteger * fun_fu855086506144988854eger_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    quotie3207804130827357816nteger: ( fun_fu7883101333820356676eger_o * fun_fu9117715662874197469nteger * fun_Co1561183069634336733nteger * fun_fu8559083954280392170eger_o ) > $o ).

tff(sy_c_Lifting_OQuotient_001t__fun_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__fun_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    quotie2859788831201447250nteger: ( fun_fu7883101333820356676eger_o * fun_fu7265242236430298039nteger * fun_fu4800041728187187127nteger * fun_fu7956894566369240976eger_o ) > $o ).

tff(sy_c_List_Oall__interval__nat,type,
    all_interval_nat: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > fun_li7731370096534215997st_nat ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist__ex_001t__Nat__Onat,type,
    list_ex_nat: ( fun_nat_o * list_nat ) > $o ).

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

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

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    member_fun_rat_o: ( fun_rat_o * set_fun_rat_o ) > $o ).

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    member_fun_char_o: ( fun_char_o * set_fun_char_o ) > $o ).

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    member_fun_char_nat: ( fun_char_nat * set_fun_char_nat ) > $o ).

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tff(sy_v_h,type,
    h: heap_e7401611519738050253t_unit ).

tff(sy_v_h_H,type,
    h2: heap_e7401611519738050253t_unit ).

tff(sy_v_r,type,
    r: ref_a ).

tff(sy_v_thesis,type,
    thesis: $o ).

tff(sy_v_x,type,
    x: a ).

% Relevant facts (5097)
tff(fact_0_that,axiom,
    ( ( r = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(x),h)) )
   => ( ( h2 = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(alloc_a(x),h)) )
     => thesis ) ) ).

% that
tff(fact_1_assms,axiom,
    aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_a_Heap_Heap_ref_a(ref_a2,x),h),h2),r) ).

% assms
tff(fact_2_prod_Oexpand,axiom,
    ! [Prod: produc7812382424243267045t_unit,Prod2: produc7812382424243267045t_unit] :
      ( ( ( aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Prod) = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Prod2) )
        & ( aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Prod) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_3_prod__eqI,axiom,
    ! [P: produc7812382424243267045t_unit,Q: produc7812382424243267045t_unit] :
      ( ( aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P) = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Q) )
     => ( ( aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Q) )
       => ( P = Q ) ) ) ).

% prod_eqI
tff(fact_4_exE__realizer_H,axiom,
    ! [P2: fun_he4684787894021915082ef_a_o,P: produc7812382424243267045t_unit] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(P2,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P)),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P))
     => ~ ! [X: ref_a,Y: heap_e7401611519738050253t_unit] : ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(P2,Y),X) ) ).

% exE_realizer'
tff(fact_5_prod__eq__iff,axiom,
    ! [S: produc7812382424243267045t_unit,T: produc7812382424243267045t_unit] :
      ( ( S = T )
    <=> ( ( aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,S) = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,T) )
        & ( aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,S) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,T) ) ) ) ).

% prod_eq_iff
tff(fact_6_Ref_Oget__alloc,axiom,
    ! [Xa: a,Ha: heap_e7401611519738050253t_unit,X2: a] : get_a(aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(alloc_a(Xa),Ha)),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(X2),Ha))) = Xa ).

% Ref.get_alloc
tff(fact_7_set__alloc,axiom,
    ! [V: a,Ha: heap_e7401611519738050253t_unit,V2: a] : aa_hea2136485429539994620t_unit(set_a(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha)),V2),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha))) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(alloc_a(V2),Ha)) ).

% set_alloc
tff(fact_8_Ref_Opresent__alloc,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Ra: ref_a,V: a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),Ra)
     => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha))),Ra) ) ).

% Ref.present_alloc
tff(fact_9_Ref_Onot__present__alloc,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,V: a] : ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha))) ).

% Ref.not_present_alloc
tff(fact_10_Ref_Oset__same,axiom,
    ! [Ra: ref_a,Xa: a,Y2: a,Ha: heap_e7401611519738050253t_unit] : aa_hea2136485429539994620t_unit(set_a(Ra,Xa),aa_hea2136485429539994620t_unit(set_a(Ra,Y2),Ha)) = aa_hea2136485429539994620t_unit(set_a(Ra,Xa),Ha) ).

% Ref.set_same
tff(fact_11_Ref_Oget__set__eq,axiom,
    ! [Ra: ref_a,Xa: a,Ha: heap_e7401611519738050253t_unit] : get_a(aa_hea2136485429539994620t_unit(set_a(Ra,Xa),Ha),Ra) = Xa ).

% Ref.get_set_eq
tff(fact_12_present__set,axiom,
    ! [Ra: ref_a,V: a,Ha: heap_e7401611519738050253t_unit] : aa_hea7918659527173910083ef_a_o(present_a,aa_hea2136485429539994620t_unit(set_a(Ra,V),Ha)) = aa_hea7918659527173910083ef_a_o(present_a,Ha) ).

% present_set
tff(fact_13_effect__refE,axiom,
    ! [V: a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_a_Heap_Heap_ref_a(ref_a2,V),Ha),H_a),Ra)
     => ~ ( ( get_a(H_a,Ra) = V )
         => ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,H_a),Ra)
           => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),Ra) ) ) ) ).

% effect_refE
tff(fact_14_alloc__set,axiom,
    ! [Xa: a,Ra: ref_a,X2: a,Ha: heap_e7401611519738050253t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(Xa),aa_hea2136485429539994620t_unit(set_a(Ra,X2),Ha))) = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(Xa),Ha)) ).

% alloc_set
tff(fact_15_ref_H__def,axiom,
    ref_a3 = ref_a2 ).

% ref'_def
tff(fact_16_get__alloc__neq,axiom,
    ! [Ra: ref_a,V: a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_ref_a_o(aa_ref_a_fun_ref_a_o(noteq_a_a,Ra),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha)))
     => ( get_a(aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha)),Ra) = get_a(Ha,Ra) ) ) ).

% get_alloc_neq
tff(fact_17_effect__changeE,axiom,
    ! [F: fun_a_a,R: ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: a] :
      ( heap_effect_a(change_a(F,R),Ha,H_a,Ra)
     => ~ ( ( H_a = aa_hea2136485429539994620t_unit(set_a(R,aa_a_a(F,get_a(Ha,R))),Ha) )
         => ( Ra != aa_a_a(F,get_a(Ha,R)) ) ) ) ).

% effect_changeE
tff(fact_18_effect__changeE,axiom,
    ! [F: fun_ref_a_ref_a,R: ref_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(change_ref_a(F,R),Ha),H_a),Ra)
     => ~ ( ( H_a = aa_hea2136485429539994620t_unit(set_ref_a2(R,aa_ref_a_ref_a(F,get_ref_a(Ha,R))),Ha) )
         => ( Ra != aa_ref_a_ref_a(F,get_ref_a(Ha,R)) ) ) ) ).

% effect_changeE
tff(fact_19_effect__changeI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,Ra: ref_a,F: fun_a_a,Ha: heap_e7401611519738050253t_unit,Xa: a] :
      ( ( H_a = aa_hea2136485429539994620t_unit(set_a(Ra,aa_a_a(F,get_a(Ha,Ra))),Ha) )
     => ( ( Xa = aa_a_a(F,get_a(Ha,Ra)) )
       => heap_effect_a(change_a(F,Ra),Ha,H_a,Xa) ) ) ).

% effect_changeI
tff(fact_20_effect__changeI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,Ra: ref_ref_a,F: fun_ref_a_ref_a,Ha: heap_e7401611519738050253t_unit,Xa: ref_a] :
      ( ( H_a = aa_hea2136485429539994620t_unit(set_ref_a2(Ra,aa_ref_a_ref_a(F,get_ref_a(Ha,Ra))),Ha) )
     => ( ( Xa = aa_ref_a_ref_a(F,get_ref_a(Ha,Ra)) )
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(change_ref_a(F,Ra),Ha),H_a),Xa) ) ) ).

% effect_changeI
tff(fact_21_present__alloc__neq,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Ra: ref_a,V: a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),Ra)
     => aa_ref_a_o(aa_ref_a_fun_ref_a_o(noteq_a_a,Ra),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(alloc_a(V),Ha))) ) ).

% present_alloc_neq
tff(fact_22_Ref_Oget__set__neq,axiom,
    ! [Ra: ref_a,S: ref_a,Xa: a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_ref_a_o(aa_ref_a_fun_ref_a_o(noteq_a_a,Ra),S)
     => ( get_a(aa_hea2136485429539994620t_unit(set_a(S,Xa),Ha),Ra) = get_a(Ha,Ra) ) ) ).

% Ref.get_set_neq
tff(fact_23_ref__def,axiom,
    ! [V: a] : aa_a_Heap_Heap_ref_a(ref_a2,V) = heap_heap_ref_a(alloc_a(V)) ).

% ref_def
tff(fact_24_effect__lookupE,axiom,
    ! [Ra: ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Xa: a] :
      ( heap_effect_a(lookup_a(Ra),Ha,H_a,Xa)
     => ~ ( ( H_a = Ha )
         => ( Xa != get_a(Ha,Ra) ) ) ) ).

% effect_lookupE
tff(fact_25_effect__lookupE,axiom,
    ! [Ra: ref_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Xa: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(lookup_ref_a(Ra),Ha),H_a),Xa)
     => ~ ( ( H_a = Ha )
         => ( Xa != get_ref_a(Ha,Ra) ) ) ) ).

% effect_lookupE
tff(fact_26_effect__lookupI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,Ha: heap_e7401611519738050253t_unit,Xa: a,Ra: ref_a] :
      ( ( H_a = Ha )
     => ( ( Xa = get_a(Ha,Ra) )
       => heap_effect_a(lookup_a(Ra),Ha,H_a,Xa) ) ) ).

% effect_lookupI
tff(fact_27_effect__lookupI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,Ha: heap_e7401611519738050253t_unit,Xa: ref_a,Ra: ref_ref_a] :
      ( ( H_a = Ha )
     => ( ( Xa = get_ref_a(Ha,Ra) )
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(lookup_ref_a(Ra),Ha),H_a),Xa) ) ) ).

% effect_lookupI
tff(fact_28_effect__refI,axiom,
    ! [Ra: ref_a,H_a: heap_e7401611519738050253t_unit,V: a,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Ra),H_a) = aa_hea1409956391221964662t_unit(alloc_a(V),Ha) )
     => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_a_Heap_Heap_ref_a(ref_a2,V),Ha),H_a),Ra) ) ).

% effect_refI
tff(fact_29_snd__swap,axiom,
    ! [Xa: produc7812382424243267045t_unit] : aa_Pro7573530150736202740_ref_a(produc6369869760173755749_ref_a,produc970745769005243767t_unit(Xa)) = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa) ).

% snd_swap
tff(fact_30_snd__swap,axiom,
    ! [Xa: produc1799199577659733785_ref_a] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,produc6214465446503547763_ref_a(Xa)) = aa_Pro7100128628477207734t_unit(produc5717040794327095847_ref_a,Xa) ).

% snd_swap
tff(fact_31_fst__swap,axiom,
    ! [Xa: produc1799199577659733785_ref_a] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,produc6214465446503547763_ref_a(Xa)) = aa_Pro7573530150736202740_ref_a(produc6369869760173755749_ref_a,Xa) ).

% fst_swap
tff(fact_32_fst__swap,axiom,
    ! [Xa: produc7812382424243267045t_unit] : aa_Pro7100128628477207734t_unit(produc5717040794327095847_ref_a,produc970745769005243767t_unit(Xa)) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa) ).

% fst_swap
tff(fact_33_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y22: set_Pr4329608150637261639at_nat] :
      ( ( aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X1),X22) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_34_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y22: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X1),X22) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_35_prod_Oinject,axiom,
    ! [X1: nat,X22: num,Y1: nat,Y22: num] :
      ( ( aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X1),X22) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_36_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_37_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_38_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_39_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_40_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_41_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_42_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_43_swap__simp,axiom,
    ! [Xa: num,Y2: nat] : aa_Pro2855824045875849084at_num(product_swap_num_nat,product_Pair_num_nat(Xa,Y2)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Y2),Xa) ).

% swap_simp
tff(fact_44_swap__simp,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : aa_Pro5334082655537895548at_nat(produc5396952528824381687at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Y2),Xa) ).

% swap_simp
tff(fact_45_swap__simp,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_Pro9111386209873968764at_nat(produc3946833401584108663at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Y2),Xa) ).

% swap_simp
tff(fact_46_swap__simp,axiom,
    ! [Xa: nat,Y2: num] : aa_Pro6665103621288092028um_nat(product_swap_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = product_Pair_num_nat(Y2,Xa) ).

% swap_simp
tff(fact_47_swap__simp,axiom,
    ! [Xa: nat,Y2: nat] : aa_Pro262675715343573244at_nat(product_swap_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y2),Xa) ).

% swap_simp
tff(fact_48_swap__simp,axiom,
    ! [Xa: int,Y2: int] : aa_Pro6970504127699319548nt_int(product_swap_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y2),Xa) ).

% swap_simp
tff(fact_49_prod_Ocollapse,axiom,
    ! [Prod: produc1319942482725812455at_nat] : aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,Prod)),aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_50_prod_Ocollapse,axiom,
    ! [Prod: produc3843707927480180839at_nat] : aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,Prod)),aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_51_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_num] : aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_Pro6261568864010571487um_nat(product_fst_nat_num,Prod)),aa_Pro2818900650166350121um_num(product_snd_nat_num,Prod)) = Prod ).

% prod.collapse
tff(fact_52_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_53_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) = Prod ).

% prod.collapse
tff(fact_54_prod_Ocollapse,axiom,
    ! [Prod: produc7812382424243267045t_unit] : aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Prod)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Prod)) = Prod ).

% prod.collapse
tff(fact_55_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc1319942482725812455at_nat] :
      ~ ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : Y2 != aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_56_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc3843707927480180839at_nat] :
      ~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : Y2 != aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_57_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_num] :
      ~ ! [A3: nat,B3: num] : Y2 != aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3) ).

% old.prod.exhaust
tff(fact_58_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] : Y2 != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) ).

% old.prod.exhaust
tff(fact_59_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_int_int] :
      ~ ! [A3: int,B3: int] : Y2 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) ).

% old.prod.exhaust
tff(fact_60_surj__pair,axiom,
    ! [P: produc1319942482725812455at_nat] :
    ? [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : P = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),Y) ).

% surj_pair
tff(fact_61_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : P = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),Y) ).

% surj_pair
tff(fact_62_surj__pair,axiom,
    ! [P: product_prod_nat_num] :
    ? [X: nat,Y: num] : P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) ).

% surj_pair
tff(fact_63_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X: nat,Y: nat] : P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) ).

% surj_pair
tff(fact_64_surj__pair,axiom,
    ! [P: product_prod_int_int] :
    ? [X: int,Y: int] : P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) ).

% surj_pair
tff(fact_65_prod__cases,axiom,
    ! [P2: fun_Pr2734351334351715892_nat_o,P: produc1319942482725812455at_nat] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : aa_Pro4518350200472837805_nat_o(P2,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A3),B3))
     => aa_Pro4518350200472837805_nat_o(P2,P) ) ).

% prod_cases
tff(fact_66_prod__cases,axiom,
    ! [P2: fun_Pr3059359313727237556_nat_o,P: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : aa_Pro5947980084102686253_nat_o(P2,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A3),B3))
     => aa_Pro5947980084102686253_nat_o(P2,P) ) ).

% prod_cases
tff(fact_67_prod__cases,axiom,
    ! [P2: fun_Pr5555124106300582480_num_o,P: product_prod_nat_num] :
      ( ! [A3: nat,B3: num] : aa_Pro6908818167415475913_num_o(P2,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3))
     => aa_Pro6908818167415475913_num_o(P2,P) ) ).

% prod_cases
tff(fact_68_prod__cases,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,P: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P2,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3))
     => aa_Pro6150637604445964947_nat_o(P2,P) ) ).

% prod_cases
tff(fact_69_prod__cases,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,P: product_prod_int_int] :
      ( ! [A3: int,B3: int] : aa_Pro4592068669816069595_int_o(P2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3))
     => aa_Pro4592068669816069595_int_o(P2,P) ) ).

% prod_cases
tff(fact_70_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_71_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_72_Pair__inject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_73_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_74_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_75_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P2: fun_Pr3059359313727237556_nat_o] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,P2))
    <=> aa_Pro5947980084102686253_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_76_mem__Collect__eq,axiom,
    ! [A: $o,P2: fun_o_o] :
      ( aa_set_o_o(member_o((A)),aa_fun_o_o_set_o(collect_o,P2))
    <=> aa_o_o(P2,(A)) ) ).

% mem_Collect_eq
tff(fact_77_mem__Collect__eq,axiom,
    ! [A: int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(A),aa_fun_int_o_set_int(collect_int,P2))
    <=> aa_int_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_78_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2))
    <=> aa_Pro6150637604445964947_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_79_mem__Collect__eq,axiom,
    ! [A: nat,P2: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),aa_fun_nat_o_set_nat(collect_nat,P2))
    <=> aa_nat_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_80_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P2: fun_Pr3238374608701176162_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(A),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2))
    <=> aa_Pro4592068669816069595_int_o(P2,A) ) ).

% mem_Collect_eq
tff(fact_81_Collect__mem__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_set6353399069419443519_nat_o(aTP_Lamm_a,A4)) = A4 ).

% Collect_mem_eq
tff(fact_82_Collect__mem__eq,axiom,
    ! [A4: set_o] : aa_fun_o_o_set_o(collect_o,aa_set_o_fun_o_o(aTP_Lamm_aa,A4)) = A4 ).

% Collect_mem_eq
tff(fact_83_Collect__mem__eq,axiom,
    ! [A4: set_int] : aa_fun_int_o_set_int(collect_int,aa_set_int_fun_int_o(aTP_Lamm_ab,A4)) = A4 ).

% Collect_mem_eq
tff(fact_84_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_set1857767853180004543_nat_o(aTP_Lamm_ac,A4)) = A4 ).

% Collect_mem_eq
tff(fact_85_Collect__mem__eq,axiom,
    ! [A4: set_nat] : aa_fun_nat_o_set_nat(collect_nat,aa_set_nat_fun_nat_o(aTP_Lamm_ad,A4)) = A4 ).

% Collect_mem_eq
tff(fact_86_Collect__mem__eq,axiom,
    ! [A4: set_Pr958786334691620121nt_int] : aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aTP_Lamm_ae(A4)) = A4 ).

% Collect_mem_eq
tff(fact_87_Collect__cong,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_Pr4796943543331071514_nat_o] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P2,X)
        <=> aa_Pro6150637604445964947_nat_o(Q2,X) )
     => ( aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,Q2) ) ) ).

% Collect_cong
tff(fact_88_Collect__cong,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
        <=> aa_nat_o(Q2,X) )
     => ( aa_fun_nat_o_set_nat(collect_nat,P2) = aa_fun_nat_o_set_nat(collect_nat,Q2) ) ) ).

% Collect_cong
tff(fact_89_Collect__cong,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_Pr3238374608701176162_int_o] :
      ( ! [X: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P2,X)
        <=> aa_Pro4592068669816069595_int_o(Q2,X) )
     => ( aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,Q2) ) ) ).

% Collect_cong
tff(fact_90_fst__eqD,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_91_fst__eqD,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_92_fst__eqD,axiom,
    ! [Xa: nat,Y2: num,A: nat] :
      ( ( aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_93_fst__eqD,axiom,
    ! [Xa: nat,Y2: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_94_fst__eqD,axiom,
    ! [Xa: int,Y2: int,A: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_95_fst__eqD,axiom,
    ! [Xa: ref_a,Y2: heap_e7401611519738050253t_unit,A: ref_a] :
      ( ( aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Xa),Y2)) = A )
     => ( Xa = A ) ) ).

% fst_eqD
tff(fact_96_fst__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] : aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_97_fst__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] : aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_98_fst__conv,axiom,
    ! [X1: nat,X22: num] : aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X1),X22)) = X1 ).

% fst_conv
tff(fact_99_fst__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_100_fst__conv,axiom,
    ! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = X1 ).

% fst_conv
tff(fact_101_fst__conv,axiom,
    ! [X1: ref_a,X22: heap_e7401611519738050253t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(X1),X22)) = X1 ).

% fst_conv
tff(fact_102_snd__eqD,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = A )
     => ( Y2 = A ) ) ).

% snd_eqD
tff(fact_103_snd__eqD,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = A )
     => ( Y2 = A ) ) ).

% snd_eqD
tff(fact_104_snd__eqD,axiom,
    ! [Xa: nat,Y2: num,A: num] :
      ( ( aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = A )
     => ( Y2 = A ) ) ).

% snd_eqD
tff(fact_105_snd__eqD,axiom,
    ! [Xa: nat,Y2: nat,A: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = A )
     => ( Y2 = A ) ) ).

% snd_eqD
tff(fact_106_snd__eqD,axiom,
    ! [Xa: int,Y2: int,A: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = A )
     => ( Y2 = A ) ) ).

% snd_eqD
tff(fact_107_snd__eqD,axiom,
    ! [Xa: ref_a,Y2: heap_e7401611519738050253t_unit,A: heap_e7401611519738050253t_unit] :
      ( ( aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Xa),Y2)) = A )
     => ( Y2 = A ) ) ).

% snd_eqD
tff(fact_108_snd__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] : aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_109_snd__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] : aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_110_snd__conv,axiom,
    ! [X1: nat,X22: num] : aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X1),X22)) = X22 ).

% snd_conv
tff(fact_111_snd__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_112_snd__conv,axiom,
    ! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = X22 ).

% snd_conv
tff(fact_113_snd__conv,axiom,
    ! [X1: ref_a,X22: heap_e7401611519738050253t_unit] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(X1),X22)) = X22 ).

% snd_conv
tff(fact_114_prod_Oswap__def,axiom,
    ! [P: produc1319942482725812455at_nat] : aa_Pro5334082655537895548at_nat(produc5396952528824381687at_nat,P) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,P)),aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,P)) ).

% prod.swap_def
tff(fact_115_prod_Oswap__def,axiom,
    ! [P: produc3843707927480180839at_nat] : aa_Pro9111386209873968764at_nat(produc3946833401584108663at_nat,P) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,P)),aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,P)) ).

% prod.swap_def
tff(fact_116_prod_Oswap__def,axiom,
    ! [P: product_prod_num_nat] : aa_Pro2855824045875849084at_num(product_swap_num_nat,P) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,product_snd_num_nat(P)),product_fst_num_nat(P)) ).

% prod.swap_def
tff(fact_117_prod_Oswap__def,axiom,
    ! [P: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(product_swap_nat_nat,P) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P)) ).

% prod.swap_def
tff(fact_118_prod_Oswap__def,axiom,
    ! [P: product_prod_int_int] : aa_Pro6970504127699319548nt_int(product_swap_int_int,P) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P)) ).

% prod.swap_def
tff(fact_119_prod_Oswap__def,axiom,
    ! [P: produc7812382424243267045t_unit] : produc970745769005243767t_unit(P) = produc2564672488694626579_ref_a(aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P)) ).

% prod.swap_def
tff(fact_120_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1319942482725812455at_nat] : Prod = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,Prod)),aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_121_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3843707927480180839at_nat] : Prod = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,Prod)),aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_122_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_num] : Prod = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_Pro6261568864010571487um_nat(product_fst_nat_num,Prod)),aa_Pro2818900650166350121um_num(product_snd_nat_num,Prod)) ).

% prod.exhaust_sel
tff(fact_123_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] : Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_124_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% prod.exhaust_sel
tff(fact_125_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7812382424243267045t_unit] : Prod = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Prod)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Prod)) ).

% prod.exhaust_sel
tff(fact_126_exI__realizer,axiom,
    ! [P2: fun_se5672428230568430382_nat_o,Y2: set_Pr4329608150637261639at_nat,Xa: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(P2,Y2),Xa)
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(P2,aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2))),aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2))) ) ).

% exI_realizer
tff(fact_127_exI__realizer,axiom,
    ! [P2: fun_se5639961202347202350_nat_o,Y2: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(P2,Y2),Xa)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(P2,aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2))),aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2))) ) ).

% exI_realizer
tff(fact_128_exI__realizer,axiom,
    ! [P2: fun_num_fun_nat_o,Y2: num,Xa: nat] :
      ( aa_nat_o(aa_num_fun_nat_o(P2,Y2),Xa)
     => aa_nat_o(aa_num_fun_nat_o(P2,aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2))),aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2))) ) ).

% exI_realizer
tff(fact_129_exI__realizer,axiom,
    ! [P2: fun_nat_fun_nat_o,Y2: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P2,Y2),Xa)
     => aa_nat_o(aa_nat_fun_nat_o(P2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2))) ) ).

% exI_realizer
tff(fact_130_exI__realizer,axiom,
    ! [P2: fun_int_fun_int_o,Y2: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(P2,Y2),Xa)
     => aa_int_o(aa_int_fun_int_o(P2,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2))) ) ).

% exI_realizer
tff(fact_131_exI__realizer,axiom,
    ! [P2: fun_he4684787894021915082ef_a_o,Y2: heap_e7401611519738050253t_unit,Xa: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(P2,Y2),Xa)
     => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(P2,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Xa),Y2))),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Xa),Y2))) ) ).

% exI_realizer
tff(fact_132_conjI__realizer,axiom,
    ! [P2: fun_se7788542033685991124_nat_o,P: set_Pr4329608150637261639at_nat,Q2: fun_se7788542033685991124_nat_o,Q: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(P2,P)
     => ( aa_set5844364881195198285_nat_o(Q2,Q)
       => ( aa_set5844364881195198285_nat_o(P2,aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,P),Q)))
          & aa_set5844364881195198285_nat_o(Q2,aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,P),Q))) ) ) ) ).

% conjI_realizer
tff(fact_133_conjI__realizer,axiom,
    ! [P2: fun_se4782206882984618170_nat_o,P: set_Pr1261947904930325089at_nat,Q2: fun_se4782206882984618170_nat_o,Q: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(P2,P)
     => ( aa_set3702741264012733747_nat_o(Q2,Q)
       => ( aa_set3702741264012733747_nat_o(P2,aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,P),Q)))
          & aa_set3702741264012733747_nat_o(Q2,aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,P),Q))) ) ) ) ).

% conjI_realizer
tff(fact_134_conjI__realizer,axiom,
    ! [P2: fun_nat_o,P: nat,Q2: fun_num_o,Q: num] :
      ( aa_nat_o(P2,P)
     => ( aa_num_o(Q2,Q)
       => ( aa_nat_o(P2,aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,P),Q)))
          & aa_num_o(Q2,aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,P),Q))) ) ) ) ).

% conjI_realizer
tff(fact_135_conjI__realizer,axiom,
    ! [P2: fun_nat_o,P: nat,Q2: fun_nat_o,Q: nat] :
      ( aa_nat_o(P2,P)
     => ( aa_nat_o(Q2,Q)
       => ( aa_nat_o(P2,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P),Q)))
          & aa_nat_o(Q2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P),Q))) ) ) ) ).

% conjI_realizer
tff(fact_136_conjI__realizer,axiom,
    ! [P2: fun_int_o,P: int,Q2: fun_int_o,Q: int] :
      ( aa_int_o(P2,P)
     => ( aa_int_o(Q2,Q)
       => ( aa_int_o(P2,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q)))
          & aa_int_o(Q2,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P),Q))) ) ) ) ).

% conjI_realizer
tff(fact_137_conjI__realizer,axiom,
    ! [P2: fun_ref_a_o,P: ref_a,Q2: fun_he8413527710729349838unit_o,Q: heap_e7401611519738050253t_unit] :
      ( aa_ref_a_o(P2,P)
     => ( aa_hea7994211679605476167unit_o(Q2,Q)
       => ( aa_ref_a_o(P2,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(P),Q)))
          & aa_hea7994211679605476167unit_o(Q2,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(P),Q))) ) ) ) ).

% conjI_realizer
tff(fact_138_surjective__pairing,axiom,
    ! [T: produc1319942482725812455at_nat] : T = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,T)),aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,T)) ).

% surjective_pairing
tff(fact_139_surjective__pairing,axiom,
    ! [T: produc3843707927480180839at_nat] : T = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,T)),aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,T)) ).

% surjective_pairing
tff(fact_140_surjective__pairing,axiom,
    ! [T: product_prod_nat_num] : T = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_Pro6261568864010571487um_nat(product_fst_nat_num,T)),aa_Pro2818900650166350121um_num(product_snd_nat_num,T)) ).

% surjective_pairing
tff(fact_141_surjective__pairing,axiom,
    ! [T: product_prod_nat_nat] : T = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,T)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,T)) ).

% surjective_pairing
tff(fact_142_surjective__pairing,axiom,
    ! [T: product_prod_int_int] : T = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,T)),aa_Pro6799258434465214121nt_int(product_snd_int_int,T)) ).

% surjective_pairing
tff(fact_143_surjective__pairing,axiom,
    ! [T: produc7812382424243267045t_unit] : T = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,T)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,T)) ).

% surjective_pairing
tff(fact_144_Ref_Oset__set__swap,axiom,
    ! [Ra: ref_a,R: ref_a,Xa: a,X2: a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_ref_a_o(aa_ref_a_fun_ref_a_o(noteq_a_a,Ra),R)
     => ( aa_hea2136485429539994620t_unit(set_a(Ra,Xa),aa_hea2136485429539994620t_unit(set_a(R,X2),Ha)) = aa_hea2136485429539994620t_unit(set_a(R,X2),aa_hea2136485429539994620t_unit(set_a(Ra,Xa),Ha)) ) ) ).

% Ref.set_set_swap
tff(fact_145_next__fresh,axiom,
    ! [Ra: ref_a,H_a: heap_e7401611519738050253t_unit,Xa: a,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Ra),H_a) = aa_hea1409956391221964662t_unit(alloc_a(Xa),Ha) )
     => ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),Ra) ) ).

% next_fresh
tff(fact_146_next__present,axiom,
    ! [Ra: ref_a,H_a: heap_e7401611519738050253t_unit,Xa: a,Ha: heap_e7401611519738050253t_unit] :
      ( ( aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Ra),H_a) = aa_hea1409956391221964662t_unit(alloc_a(Xa),Ha) )
     => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,H_a),Ra) ) ).

% next_present
tff(fact_147_noteq__I,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,Ra: ref_a,R: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),Ra)
     => ( ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(present_a,Ha),R)
       => aa_ref_a_o(aa_ref_a_fun_ref_a_o(noteq_a_a,Ra),R) ) ) ).

% noteq_I
tff(fact_148_effect__heapE,axiom,
    ! [F: fun_he3288789352778960325t_unit,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_heap_ref_a(F),Ha),H_a),Ra)
     => ~ ( ( H_a = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
         => ( Ra != aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(F,Ha)) ) ) ) ).

% effect_heapE
tff(fact_149_effect__heapI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,F: fun_he3288789352778960325t_unit,Ha: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( ( H_a = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
     => ( ( Ra = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_heap_ref_a(F),Ha),H_a),Ra) ) ) ).

% effect_heapI
tff(fact_150_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fun_se5672428230568430382_nat_o,Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(P2,Xa),Y2)
     => ( ( A = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2) )
       => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(P2,aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,A)),aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_151_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fun_se5639961202347202350_nat_o,Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(P2,Xa),Y2)
     => ( ( A = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(P2,aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,A)),aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_152_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fun_nat_fun_num_o,Xa: nat,Y2: num,A: product_prod_nat_num] :
      ( aa_num_o(aa_nat_fun_num_o(P2,Xa),Y2)
     => ( ( A = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2) )
       => aa_num_o(aa_nat_fun_num_o(P2,aa_Pro6261568864010571487um_nat(product_fst_nat_num,A)),aa_Pro2818900650166350121um_num(product_snd_nat_num,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_153_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fun_nat_fun_nat_o,Xa: nat,Y2: nat,A: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P2,Xa),Y2)
     => ( ( A = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2) )
       => aa_nat_o(aa_nat_fun_nat_o(P2,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_154_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fun_int_fun_int_o,Xa: int,Y2: int,A: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P2,Xa),Y2)
     => ( ( A = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2) )
       => aa_int_o(aa_int_fun_int_o(P2,aa_Pro6799258434465214121nt_int(product_fst_int_int,A)),aa_Pro6799258434465214121nt_int(product_snd_int_int,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_155_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: fun_re5167877524888533480unit_o,Xa: ref_a,Y2: heap_e7401611519738050253t_unit,A: produc7812382424243267045t_unit] :
      ( aa_hea7994211679605476167unit_o(aa_ref4509144387093619241unit_o(P2,Xa),Y2)
     => ( ( A = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Xa),Y2) )
       => aa_hea7994211679605476167unit_o(aa_ref4509144387093619241unit_o(P2,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,A)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,A)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_156_sndI,axiom,
    ! [Xa: produc1319942482725812455at_nat,Y2: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( Xa = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Y2),Z) )
     => ( aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,Xa) = Z ) ) ).

% sndI
tff(fact_157_sndI,axiom,
    ! [Xa: produc3843707927480180839at_nat,Y2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( Xa = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Y2),Z) )
     => ( aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,Xa) = Z ) ) ).

% sndI
tff(fact_158_sndI,axiom,
    ! [Xa: product_prod_nat_num,Y2: nat,Z: num] :
      ( ( Xa = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Y2),Z) )
     => ( aa_Pro2818900650166350121um_num(product_snd_nat_num,Xa) = Z ) ) ).

% sndI
tff(fact_159_sndI,axiom,
    ! [Xa: product_prod_nat_nat,Y2: nat,Z: nat] :
      ( ( Xa = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y2),Z) )
     => ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa) = Z ) ) ).

% sndI
tff(fact_160_sndI,axiom,
    ! [Xa: product_prod_int_int,Y2: int,Z: int] :
      ( ( Xa = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y2),Z) )
     => ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa) = Z ) ) ).

% sndI
tff(fact_161_sndI,axiom,
    ! [Xa: produc7812382424243267045t_unit,Y2: ref_a,Z: heap_e7401611519738050253t_unit] :
      ( ( Xa = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Y2),Z) )
     => ( aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa) = Z ) ) ).

% sndI
tff(fact_162_eq__snd__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
      ( ( B = aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,P) )
    <=> ? [A5: set_Pr4329608150637261639at_nat] : P = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A5),B) ) ).

% eq_snd_iff
tff(fact_163_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( B = aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,P) )
    <=> ? [A5: set_Pr1261947904930325089at_nat] : P = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A5),B) ) ).

% eq_snd_iff
tff(fact_164_eq__snd__iff,axiom,
    ! [B: num,P: product_prod_nat_num] :
      ( ( B = aa_Pro2818900650166350121um_num(product_snd_nat_num,P) )
    <=> ? [A5: nat] : P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A5),B) ) ).

% eq_snd_iff
tff(fact_165_eq__snd__iff,axiom,
    ! [B: nat,P: product_prod_nat_nat] :
      ( ( B = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P) )
    <=> ? [A5: nat] : P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A5),B) ) ).

% eq_snd_iff
tff(fact_166_eq__snd__iff,axiom,
    ! [B: int,P: product_prod_int_int] :
      ( ( B = aa_Pro6799258434465214121nt_int(product_snd_int_int,P) )
    <=> ? [A5: int] : P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A5),B) ) ).

% eq_snd_iff
tff(fact_167_eq__snd__iff,axiom,
    ! [B: heap_e7401611519738050253t_unit,P: produc7812382424243267045t_unit] :
      ( ( B = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P) )
    <=> ? [A5: ref_a] : P = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(A5),B) ) ).

% eq_snd_iff
tff(fact_168_fstI,axiom,
    ! [Xa: produc1319942482725812455at_nat,Y2: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( Xa = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Y2),Z) )
     => ( aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,Xa) = Y2 ) ) ).

% fstI
tff(fact_169_fstI,axiom,
    ! [Xa: produc3843707927480180839at_nat,Y2: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( Xa = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Y2),Z) )
     => ( aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,Xa) = Y2 ) ) ).

% fstI
tff(fact_170_fstI,axiom,
    ! [Xa: product_prod_nat_num,Y2: nat,Z: num] :
      ( ( Xa = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Y2),Z) )
     => ( aa_Pro6261568864010571487um_nat(product_fst_nat_num,Xa) = Y2 ) ) ).

% fstI
tff(fact_171_fstI,axiom,
    ! [Xa: product_prod_nat_nat,Y2: nat,Z: nat] :
      ( ( Xa = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y2),Z) )
     => ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa) = Y2 ) ) ).

% fstI
tff(fact_172_fstI,axiom,
    ! [Xa: product_prod_int_int,Y2: int,Z: int] :
      ( ( Xa = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y2),Z) )
     => ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa) = Y2 ) ) ).

% fstI
tff(fact_173_fstI,axiom,
    ! [Xa: produc7812382424243267045t_unit,Y2: ref_a,Z: heap_e7401611519738050253t_unit] :
      ( ( Xa = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Y2),Z) )
     => ( aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa) = Y2 ) ) ).

% fstI
tff(fact_174_eq__fst__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
      ( ( A = aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,P) )
    <=> ? [B4: set_Pr4329608150637261639at_nat] : P = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B4) ) ).

% eq_fst_iff
tff(fact_175_eq__fst__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( A = aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,P) )
    <=> ? [B4: set_Pr1261947904930325089at_nat] : P = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B4) ) ).

% eq_fst_iff
tff(fact_176_eq__fst__iff,axiom,
    ! [A: nat,P: product_prod_nat_num] :
      ( ( A = aa_Pro6261568864010571487um_nat(product_fst_nat_num,P) )
    <=> ? [B4: num] : P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B4) ) ).

% eq_fst_iff
tff(fact_177_eq__fst__iff,axiom,
    ! [A: nat,P: product_prod_nat_nat] :
      ( ( A = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P) )
    <=> ? [B4: nat] : P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B4) ) ).

% eq_fst_iff
tff(fact_178_eq__fst__iff,axiom,
    ! [A: int,P: product_prod_int_int] :
      ( ( A = aa_Pro6799258434465214121nt_int(product_fst_int_int,P) )
    <=> ? [B4: int] : P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B4) ) ).

% eq_fst_iff
tff(fact_179_eq__fst__iff,axiom,
    ! [A: ref_a,P: produc7812382424243267045t_unit] :
      ( ( A = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P) )
    <=> ? [B4: heap_e7401611519738050253t_unit] : P = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(A),B4) ) ).

% eq_fst_iff
tff(fact_180_effect__guardE,axiom,
    ! [P2: fun_he8413527710729349838unit_o,F: fun_he3288789352778960325t_unit,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_guard_ref_a(P2,F),Ha),H_a),Ra)
     => ~ ( ( H_a = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
         => ( ( Ra = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
           => ~ aa_hea7994211679605476167unit_o(P2,Ha) ) ) ) ).

% effect_guardE
tff(fact_181_effect__guardI,axiom,
    ! [P2: fun_he8413527710729349838unit_o,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,F: fun_he3288789352778960325t_unit,Ra: ref_a] :
      ( aa_hea7994211679605476167unit_o(P2,Ha)
     => ( ( H_a = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
       => ( ( Ra = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_hea1409956391221964662t_unit(F,Ha)) )
         => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_guard_ref_a(P2,F),Ha),H_a),Ra) ) ) ) ).

% effect_guardI
tff(fact_182_effect__ifI,axiom,
    ! [C: $o,T: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a,E: heap_Heap_ref_a] :
      ( ( (C)
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(T,Ha),H_a),Ra) )
     => ( ( ~ (C)
         => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(E,Ha),H_a),Ra) )
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a($ite((C),T,E),Ha),H_a),Ra) ) ) ).

% effect_ifI
tff(fact_183_effect__deterministic_I2_J,axiom,
    ! [F: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,A: ref_a,H: heap_e7401611519738050253t_unit,B: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(F,Ha),H_a),A)
     => ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(F,Ha),H),B)
       => ( H_a = H ) ) ) ).

% effect_deterministic(2)
tff(fact_184_effect__deterministic_I1_J,axiom,
    ! [F: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,A: ref_a,H: heap_e7401611519738050253t_unit,B: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(F,Ha),H_a),A)
     => ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(F,Ha),H),B)
       => ( A = B ) ) ) ).

% effect_deterministic(1)
tff(fact_185_effect__ifE,axiom,
    ! [C: $o,T: heap_Heap_ref_a,E: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a($ite((C),T,E),Ha),H_a),Ra)
     => ( ( (C)
         => ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(T,Ha),H_a),Ra) )
       => ~ ( ~ (C)
           => ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(E,Ha),H_a),Ra) ) ) ) ).

% effect_ifE
tff(fact_186_image2__eqI,axiom,
    ! [B: nat,F: fun_o_nat,Xa: $o,C: num,G: fun_o_num,A4: set_o] :
      ( ( B = aa_o_nat(F,(Xa)) )
     => ( ( C = aa_o_num(G,(Xa)) )
       => ( aa_set_o_o(member_o((Xa)),A4)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,B),C),bNF_Gr7727518618234538026at_num(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_187_image2__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,Xa: nat,C: num,G: fun_nat_num,A4: set_nat] :
      ( ( B = aa_nat_nat(F,Xa) )
     => ( ( C = aa_nat_num(G,Xa) )
       => ( aa_set_nat_o(member_nat(Xa),A4)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,B),C),bNF_Gr3875488559241464598at_num(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_188_image2__eqI,axiom,
    ! [B: nat,F: fun_int_nat,Xa: int,C: num,G: fun_int_num,A4: set_int] :
      ( ( B = aa_int_nat(F,Xa) )
     => ( ( C = aa_int_num(G,Xa) )
       => ( aa_set_int_o(member_int(Xa),A4)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,B),C),bNF_Gr4247142119478309234at_num(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_189_image2__eqI,axiom,
    ! [B: nat,F: fun_o_nat,Xa: $o,C: nat,G: fun_o_nat,A4: set_o] :
      ( ( B = aa_o_nat(F,(Xa)) )
     => ( ( C = aa_o_nat(G,(Xa)) )
       => ( aa_set_o_o(member_o((Xa)),A4)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),C)),bNF_Gr1946814795223983584at_nat(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_190_image2__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,Xa: nat,C: nat,G: fun_nat_nat,A4: set_nat] :
      ( ( B = aa_nat_nat(F,Xa) )
     => ( ( C = aa_nat_nat(G,Xa) )
       => ( aa_set_nat_o(member_nat(Xa),A4)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),C)),bNF_Gr7318156773085685964at_nat(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_191_image2__eqI,axiom,
    ! [B: nat,F: fun_int_nat,Xa: int,C: nat,G: fun_int_nat,A4: set_int] :
      ( ( B = aa_int_nat(F,Xa) )
     => ( ( C = aa_int_nat(G,Xa) )
       => ( aa_set_int_o(member_int(Xa),A4)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),C)),bNF_Gr7689810333322530600at_nat(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_192_image2__eqI,axiom,
    ! [B: int,F: fun_o_int,Xa: $o,C: int,G: fun_o_int,A4: set_o] :
      ( ( B = aa_o_int(F,(Xa)) )
     => ( ( C = aa_o_int(G,(Xa)) )
       => ( aa_set_o_o(member_o((Xa)),A4)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),C)),bNF_Gr2943678036615987864nt_int(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_193_image2__eqI,axiom,
    ! [B: int,F: fun_nat_int,Xa: nat,C: int,G: fun_nat_int,A4: set_nat] :
      ( ( B = aa_nat_int(F,Xa) )
     => ( ( C = aa_nat_int(G,Xa) )
       => ( aa_set_nat_o(member_nat(Xa),A4)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),C)),bNF_Gr8315020014477690244nt_int(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_194_image2__eqI,axiom,
    ! [B: int,F: fun_int_int,Xa: int,C: int,G: fun_int_int,A4: set_int] :
      ( ( B = aa_int_int(F,Xa) )
     => ( ( C = aa_int_int(G,Xa) )
       => ( aa_set_int_o(member_int(Xa),A4)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),C)),bNF_Gr8686673574714534880nt_int(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_195_image2__eqI,axiom,
    ! [B: nat,F: fun_Pr2602117396471556196at_nat,Xa: product_prod_nat_nat,C: num,G: fun_Pr3310361333420393902at_num,A4: set_Pr1261947904930325089at_nat] :
      ( ( B = aa_Pro4243173770044397973at_nat(F,Xa) )
     => ( ( C = aa_Pro800505556200176607at_num(G,Xa) )
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),A4)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,B),C),bNF_Gr5511461681843259283at_num(A4,F,G)) ) ) ) ).

% image2_eqI
tff(fact_196_lim__set,axiom,
    ! [Ra: ref_a,V: a,Ha: heap_e7401611519738050253t_unit] : aa_hea5848002036693495265it_nat(lim_Product_unit,aa_hea2136485429539994620t_unit(set_a(Ra,V),Ha)) = aa_hea5848002036693495265it_nat(lim_Product_unit,Ha) ).

% lim_set
tff(fact_197_prod_Osplit__sel__asm,axiom,
    ! [P2: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(P2,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_fun3538214411680374423_nat_o(P2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_198_prod_Osplit__sel__asm,axiom,
    ! [P2: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(P2,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_fun8408151826566693824at_o_o(P2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_199_prod_Osplit__sel__asm,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(P2,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_Pro4592068669816069595_int_o(P2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_200_prod_Osplit__sel__asm,axiom,
    ! [P2: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P2,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_o_o(P2,aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_201_prod_Osplit__sel__asm,axiom,
    ! [P2: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(P2,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_int_o(P2,aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_202_prod_Osplit__sel,axiom,
    ! [P2: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(P2,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_fun3538214411680374423_nat_o(P2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_203_prod_Osplit__sel,axiom,
    ! [P2: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(P2,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_fun8408151826566693824at_o_o(P2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_204_prod_Osplit__sel,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(P2,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_Pro4592068669816069595_int_o(P2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_205_prod_Osplit__sel,axiom,
    ! [P2: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P2,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_o_o(P2,aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_206_prod_Osplit__sel,axiom,
    ! [P2: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(P2,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_int_o(P2,aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_207_ssubst__Pair__rhs,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Ra),S),R2)
     => ( ( S2 = S )
       => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Ra),S2),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_208_ssubst__Pair__rhs,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Ra),S)),R2)
     => ( ( S2 = S )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Ra),S2)),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_209_ssubst__Pair__rhs,axiom,
    ! [Ra: nat,S: num,R2: set_Pr6200539531224447659at_num,S2: num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Ra),S),R2)
     => ( ( S2 = S )
       => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Ra),S2),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_210_ssubst__Pair__rhs,axiom,
    ! [Ra: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Ra),S)),R2)
     => ( ( S2 = S )
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Ra),S2)),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_211_ssubst__Pair__rhs,axiom,
    ! [Ra: int,S: int,R2: set_Pr958786334691620121nt_int,S2: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ra),S)),R2)
     => ( ( S2 = S )
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Ra),S2)),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_212_effect__raiseE,axiom,
    ! [Xa: literal,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] : ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_raise_ref_a(Xa),Ha),H_a),Ra) ).

% effect_raiseE
tff(fact_213_effect__updateE,axiom,
    ! [R: ref_a,V: a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: product_unit] :
      ( aa_Product_unit_o(heap_e2830298082929533060t_unit(update_a(R,V),Ha,H_a),Ra)
     => ( H_a = aa_hea2136485429539994620t_unit(set_a(R,V),Ha) ) ) ).

% effect_updateE
tff(fact_214_curryI,axiom,
    ! [F: fun_Pr2734351334351715892_nat_o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_Pro4518350200472837805_nat_o(F,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B))
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(produc447603880154497357_nat_o(F),A),B) ) ).

% curryI
tff(fact_215_curryI,axiom,
    ! [F: fun_Pr3059359313727237556_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_Pro5947980084102686253_nat_o(F,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B))
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(produc7820828192191229133_nat_o(F),A),B) ) ).

% curryI
tff(fact_216_curryI,axiom,
    ! [F: fun_Pr5555124106300582480_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(F,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))
     => aa_num_o(aa_nat_fun_num_o(produc156083480235303841_num_o(F),A),B) ) ).

% curryI
tff(fact_217_curryI,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(F,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
     => aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(F),A),B) ) ).

% curryI
tff(fact_218_curryI,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(F,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(F),A),B) ) ).

% curryI
tff(fact_219_case__prod__curry,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,produc4701859392678344816at_nat(F)) = F ).

% case_prod_curry
tff(fact_220_case__prod__curry,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,produc2062876521443793799_nat_o(F)) = F ).

% case_prod_curry
tff(fact_221_case__prod__curry,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,produc8249235968001453780nt_int(F)) = F ).

% case_prod_curry
tff(fact_222_case__prod__curry,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,produc175634133007206835_int_o(F)) = F ).

% case_prod_curry
tff(fact_223_case__prod__curry,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,produc1016772743285680337nt_int(F)) = F ).

% case_prod_curry
tff(fact_224_curry__case__prod,axiom,
    ! [F: fun_na1668100129766719005at_nat] : produc4701859392678344816at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F)) = F ).

% curry_case_prod
tff(fact_225_curry__case__prod,axiom,
    ! [F: fun_na5914473681949881652_nat_o] : produc2062876521443793799_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F)) = F ).

% curry_case_prod
tff(fact_226_curry__case__prod,axiom,
    ! [F: fun_in1696471567231710611nt_int] : produc8249235968001453780nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F)) = F ).

% curry_case_prod
tff(fact_227_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_o] : produc175634133007206835_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F)) = F ).

% curry_case_prod
tff(fact_228_curry__case__prod,axiom,
    ! [F: fun_int_fun_int_int] : produc1016772743285680337nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F)) = F ).

% curry_case_prod
tff(fact_229_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_na3850162164366384704_set_o,P: product_prod_nat_num] :
      ( aa_set_o_o(member_o((Z)),produc836176033315069408_set_o(C,P))
     => ~ ! [X: nat,Y: num] :
            ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) )
           => ~ aa_set_o_o(member_o((Z)),aa_num_set_o(aa_nat_fun_num_set_o(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_230_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,P: product_prod_nat_num] :
      ( aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P))
     => ~ ! [X: nat,Y: num] :
            ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_231_mem__case__prodE,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,P: product_prod_nat_num] :
      ( aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,P))
     => ~ ! [X: nat,Y: num] :
            ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) )
           => ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_232_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_na3100059926620910090_set_o,P: product_prod_nat_nat] :
      ( aa_set_o_o(member_o((Z)),aa_Pro5970036531051701619_set_o(produc59986286002894506_set_o(C),P))
     => ~ ! [X: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_set_o_o(member_o((Z)),aa_nat_set_o(aa_nat_fun_nat_set_o(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_233_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na6128946301132549280et_nat,P: product_prod_nat_nat] :
      ( aa_set_nat_o(member_nat(Z),aa_Pro9123573313595863371et_nat(produc6189476227299908564et_nat(C),P))
     => ~ ! [X: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_234_mem__case__prodE,axiom,
    ! [Z: int,C: fun_na1315397882344630524et_int,P: product_prod_nat_nat] :
      ( aa_set_int_o(member_int(Z),produc2011625207790711856et_int(C,P))
     => ~ ! [X: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_set_int_o(member_int(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_235_mem__case__prodE,axiom,
    ! [Z: $o,C: fun_in4068701459849370962_set_o,P: product_prod_int_int] :
      ( aa_set_o_o(member_o((Z)),produc4257766111578684402_set_o(C,P))
     => ~ ! [X: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_set_o_o(member_o((Z)),aa_int_set_o(aa_int_fun_int_set_o(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_236_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_in2630332840903238488et_nat,P: product_prod_int_int] :
      ( aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C,P))
     => ~ ! [X: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_237_mem__case__prodE,axiom,
    ! [Z: int,C: fun_in7040156458970095540et_int,P: product_prod_int_int] :
      ( aa_set_int_o(member_int(Z),produc73460835934605544et_int(C,P))
     => ~ ! [X: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_238_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: fun_na1206552039118883625at_nat,P: product_prod_nat_num] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc859037500352796105at_nat(C,P))
     => ~ ! [X: nat,Y: num] :
            ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) )
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_num8947610712640478729at_nat(aa_nat157436013921436562at_nat(C,X),Y)) ) ) ).

% mem_case_prodE
tff(fact_239_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X22: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X22) ).

% old.prod.case
tff(fact_240_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X22: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X22) ).

% old.prod.case
tff(fact_241_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X22: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X22) ).

% old.prod.case
tff(fact_242_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X22: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X22) ) ).

% old.prod.case
tff(fact_243_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X22: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int_int(aa_int_fun_int_int(F,X1),X22) ).

% old.prod.case
tff(fact_244_curryD,axiom,
    ! [F: fun_Pr2734351334351715892_nat_o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(produc447603880154497357_nat_o(F),A),B)
     => aa_Pro4518350200472837805_nat_o(F,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B)) ) ).

% curryD
tff(fact_245_curryD,axiom,
    ! [F: fun_Pr3059359313727237556_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(produc7820828192191229133_nat_o(F),A),B)
     => aa_Pro5947980084102686253_nat_o(F,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)) ) ).

% curryD
tff(fact_246_curryD,axiom,
    ! [F: fun_Pr5555124106300582480_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(produc156083480235303841_num_o(F),A),B)
     => aa_Pro6908818167415475913_num_o(F,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) ) ).

% curryD
tff(fact_247_curryD,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(F),A),B)
     => aa_Pro6150637604445964947_nat_o(F,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) ) ).

% curryD
tff(fact_248_curryD,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(F),A),B)
     => aa_Pro4592068669816069595_int_o(F,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% curryD
tff(fact_249_curryE,axiom,
    ! [F: fun_Pr2734351334351715892_nat_o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(produc447603880154497357_nat_o(F),A),B)
     => aa_Pro4518350200472837805_nat_o(F,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B)) ) ).

% curryE
tff(fact_250_curryE,axiom,
    ! [F: fun_Pr3059359313727237556_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(produc7820828192191229133_nat_o(F),A),B)
     => aa_Pro5947980084102686253_nat_o(F,aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)) ) ).

% curryE
tff(fact_251_curryE,axiom,
    ! [F: fun_Pr5555124106300582480_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(produc156083480235303841_num_o(F),A),B)
     => aa_Pro6908818167415475913_num_o(F,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) ) ).

% curryE
tff(fact_252_curryE,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(F),A),B)
     => aa_Pro6150637604445964947_nat_o(F,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) ) ).

% curryE
tff(fact_253_curryE,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(F),A),B)
     => aa_Pro4592068669816069595_int_o(F,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% curryE
tff(fact_254_Product__Type_OCollect__case__prodD,axiom,
    ! [Xa: produc3843707927480180839at_nat,A4: fun_se5639961202347202350_nat_o] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,A4)))
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(A4,aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,Xa)),aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_255_Product__Type_OCollect__case__prodD,axiom,
    ! [Xa: product_prod_nat_nat,A4: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,A4)))
     => aa_nat_o(aa_nat_fun_nat_o(A4,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_256_Product__Type_OCollect__case__prodD,axiom,
    ! [Xa: produc7812382424243267045t_unit,A4: fun_re5167877524888533480unit_o] :
      ( member7849944260330398460t_unit(Xa,collec3252306575900995386t_unit(produc4924168586796184916unit_o(A4)))
     => aa_hea7994211679605476167unit_o(aa_ref4509144387093619241unit_o(A4,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_257_Product__Type_OCollect__case__prodD,axiom,
    ! [Xa: product_prod_int_int,A4: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Xa),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A4)))
     => aa_int_o(aa_int_fun_int_o(A4,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_258_effect__returnI,axiom,
    ! [Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Xa: ref_a] :
      ( ( Ha = H_a )
     => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_ref97115632722995649_ref_a(heap_return_ref_a,Xa),Ha),H_a),Xa) ) ).

% effect_returnI
tff(fact_259_effect__returnE,axiom,
    ! [Xa: ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_ref97115632722995649_ref_a(heap_return_ref_a,Xa),Ha),H_a),Ra)
     => ~ ( ( Ra = Xa )
         => ( H_a != Ha ) ) ) ).

% effect_returnE
tff(fact_260_split__beta,axiom,
    ! [F: fun_na1668100129766719005at_nat,Prod: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Prod) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% split_beta
tff(fact_261_split__beta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% split_beta
tff(fact_262_split__beta,axiom,
    ! [F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_263_split__beta,axiom,
    ! [F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ) ).

% split_beta
tff(fact_264_split__beta,axiom,
    ! [F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_265_case__prod__beta,axiom,
    ! [F: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)) ).

% case_prod_beta
tff(fact_266_case__prod__beta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)) ).

% case_prod_beta
tff(fact_267_case__prod__beta,axiom,
    ! [F: fun_in1696471567231710611nt_int,P: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)) ).

% case_prod_beta
tff(fact_268_case__prod__beta,axiom,
    ! [F: fun_int_fun_int_o,P: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)) ) ).

% case_prod_beta
tff(fact_269_case__prod__beta,axiom,
    ! [F: fun_int_fun_int_int,P: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,P)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P)) ).

% case_prod_beta
tff(fact_270_internal__case__prod__def,axiom,
    produc8104577556401574439at_nat = produc4528391901023682385at_nat ).

% internal_case_prod_def
tff(fact_271_internal__case__prod__def,axiom,
    produc9189008374811844670_nat_o = produc6395135316500328296_nat_o ).

% internal_case_prod_def
tff(fact_272_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
tff(fact_273_internal__case__prod__def,axiom,
    produc8005341501107743676_int_o = produc4947309494688390418_int_o ).

% internal_case_prod_def
tff(fact_274_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
tff(fact_275_effect__updateI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,Ra: ref_a,V: a,Ha: heap_e7401611519738050253t_unit,Xa: product_unit] :
      ( ( H_a = aa_hea2136485429539994620t_unit(set_a(Ra,V),Ha) )
     => aa_Product_unit_o(heap_e2830298082929533060t_unit(update_a(Ra,V),Ha,H_a),Xa) ) ).

% effect_updateI
tff(fact_276_split__cong,axiom,
    ! [Q: product_prod_nat_nat,F: fun_na1668100129766719005at_nat,G: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] :
      ( ! [X: nat,Y: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) = Q )
         => ( aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X),Y) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(G,X),Y) ) )
     => ( ( P = Q )
       => ( aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,G),Q) ) ) ) ).

% split_cong
tff(fact_277_split__cong,axiom,
    ! [Q: product_prod_nat_nat,F: fun_na5914473681949881652_nat_o,G: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat] :
      ( ! [X: nat,Y: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) = Q )
         => ( aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X),Y) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(G,X),Y) ) )
     => ( ( P = Q )
       => ( aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,G),Q) ) ) ) ).

% split_cong
tff(fact_278_split__cong,axiom,
    ! [Q: product_prod_int_int,F: fun_in1696471567231710611nt_int,G: fun_in1696471567231710611nt_int,P: product_prod_int_int] :
      ( ! [X: int,Y: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) = Q )
         => ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X),Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(G,X),Y) ) )
     => ( ( P = Q )
       => ( aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,G),Q) ) ) ) ).

% split_cong
tff(fact_279_split__cong,axiom,
    ! [Q: product_prod_int_int,F: fun_int_fun_int_o,G: fun_int_fun_int_o,P: product_prod_int_int] :
      ( ! [X: int,Y: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) = Q )
         => ( aa_int_o(aa_int_fun_int_o(F,X),Y)
          <=> aa_int_o(aa_int_fun_int_o(G,X),Y) ) )
     => ( ( P = Q )
       => ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)
        <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,G),Q) ) ) ) ).

% split_cong
tff(fact_280_split__cong,axiom,
    ! [Q: product_prod_int_int,F: fun_int_fun_int_int,G: fun_int_fun_int_int,P: product_prod_int_int] :
      ( ! [X: int,Y: int] :
          ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) = Q )
         => ( aa_int_int(aa_int_fun_int_int(F,X),Y) = aa_int_int(aa_int_fun_int_int(G,X),Y) ) )
     => ( ( P = Q )
       => ( aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,G),Q) ) ) ) ).

% split_cong
tff(fact_281_csquare__fstOp__sndOp,axiom,
    ! [F: fun_fu8189108216148119015_int_o,P2: fun_Pr2028404973800837934_nat_o,Q2: fun_Pr4738963024209204295_int_o] : bNF_cs7194553038297458973at_int(collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(F,relcom8911704206367525917at_int(P2,Q2))),produc6408287024330202629at_nat,produc4619704898428564714at_int,bNF_fs6342258632037972510at_int(P2,Q2),bNF_sn649997494946348640at_int(P2,Q2)) ).

% csquare_fstOp_sndOp
tff(fact_282_csquare__fstOp__sndOp,axiom,
    ! [F: fun_fu2367713963901291397eger_o,P2: fun_int_fun_int_o,Q2: fun_in4891666191172872833eger_o] : bNF_cs4943907971211754724nteger(collec7215431352809750357nteger(aa_fun7359524620130742070eger_o(F,relcom967497705283160040nteger(P2,Q2))),product_snd_int_int,produc45500017171379470nteger,bNF_fs4543836202474805097nteger(P2,Q2),bNF_sn8000174127721201323nteger(P2,Q2)) ).

% csquare_fstOp_sndOp
tff(fact_283_effect__tapI,axiom,
    ! [H_a: heap_e7401611519738050253t_unit,Ha: heap_e7401611519738050253t_unit,Ra: ref_a,F: fun_he2462007625301486205_ref_a] :
      ( ( H_a = Ha )
     => ( ( Ra = aa_hea4634405385475018798_ref_a(F,Ha) )
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_tap_ref_a(F),Ha),H_a),Ra) ) ) ).

% effect_tapI
tff(fact_284_effect__tapE,axiom,
    ! [F: fun_he2462007625301486205_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_tap_ref_a(F),Ha),H_a),Ra)
     => ~ ( ( H_a = Ha )
         => ( Ra != aa_hea4634405385475018798_ref_a(F,Ha) ) ) ) ).

% effect_tapE
tff(fact_285_pair__in__swap__image,axiom,
    ! [Y2: num,Xa: nat,A4: set_Pr6200539531224447659at_num] :
      ( member6570852975090897396um_nat(product_Pair_num_nat(Y2,Xa),image_2347970371937200581um_nat(product_swap_nat_num,A4))
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2),A4) ) ).

% pair_in_swap_image
tff(fact_286_pair__in__swap__image,axiom,
    ! [Y2: set_Pr4329608150637261639at_nat,Xa: set_Pr4329608150637261639at_nat,A4: set_Pr7459493094073627847at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Y2),Xa),image_3313167189156628421at_nat(produc5396952528824381687at_nat,A4))
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2),A4) ) ).

% pair_in_swap_image
tff(fact_287_pair__in__swap__image,axiom,
    ! [Y2: set_Pr1261947904930325089at_nat,Xa: set_Pr1261947904930325089at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Y2),Xa)),aa_set4883462369968598652at_nat(image_960689587819713477at_nat(produc3946833401584108663at_nat),A4))
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),A4) ) ).

% pair_in_swap_image
tff(fact_288_pair__in__swap__image,axiom,
    ! [Y2: nat,Xa: num,A4: set_Pr3280342998896498603um_nat] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Y2),Xa),image_7762062833379733445at_num(product_swap_num_nat,A4))
    <=> member6570852975090897396um_nat(product_Pair_num_nat(Xa,Y2),A4) ) ).

% pair_in_swap_image
tff(fact_289_pair__in__swap__image,axiom,
    ! [Y2: nat,Xa: nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y2),Xa)),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(product_swap_nat_nat),A4))
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),A4) ) ).

% pair_in_swap_image
tff(fact_290_pair__in__swap__image,axiom,
    ! [Y2: int,Xa: int,A4: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y2),Xa)),image_2653370878348428101nt_int(product_swap_int_int,A4))
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),A4) ) ).

% pair_in_swap_image
tff(fact_291_snd__sndOp,axiom,
    ! [Bc: produc7812382424243267045t_unit,P2: fun_re1985804342488220176ef_a_o,Q2: fun_re5167877524888533480unit_o] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Bc) = aa_Pro8575920914306358506t_unit(comp_P8019343326919024234t_unit(produc1126150082675451753t_unit,bNF_sn8323453914685745821t_unit(P2,Q2)),Bc) ).

% snd_sndOp
tff(fact_292_comp__cong,axiom,
    ! [F: fun_o_o,G: fun_nat_o,Xa: nat,F2: fun_o_o,G2: fun_nat_o,X2: nat] :
      ( ( aa_o_o(F,aa_nat_o(G,Xa))
      <=> aa_o_o(F2,aa_nat_o(G2,X2)) )
     => ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Xa)
      <=> aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F2),G2),X2) ) ) ).

% comp_cong
tff(fact_293_comp__cong,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int,F2: fun_nat_num,G2: fun_int_nat,X2: int] :
      ( ( aa_nat_num(F,aa_int_nat(G,Xa)) = aa_nat_num(F2,aa_int_nat(G2,X2)) )
     => ( aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),Xa) = aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F2),G2),X2) ) ) ).

% comp_cong
tff(fact_294_comp__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,F2: fun_nat_nat,G2: fun_nat_nat,X2: nat] :
      ( ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = aa_nat_nat(F2,aa_nat_nat(G2,X2)) )
     => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),Xa) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F2),G2),X2) ) ) ).

% comp_cong
tff(fact_295_comp__cong,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,F2: fun_int_nat,G2: fun_int_int,X2: int] :
      ( ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = aa_int_nat(F2,aa_int_int(G2,X2)) )
     => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),Xa) = aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F2),G2),X2) ) ) ).

% comp_cong
tff(fact_296_comp__cong,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,F2: fun_nat_nat,G2: fun_nat_nat,X2: nat] :
      ( ( aa_int_nat(F,aa_int_int(G,Xa)) = aa_nat_nat(F2,aa_nat_nat(G2,X2)) )
     => ( aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),Xa) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F2),G2),X2) ) ) ).

% comp_cong
tff(fact_297_comp__cong,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,F2: fun_int_nat,G2: fun_int_int,X2: int] :
      ( ( aa_int_nat(F,aa_int_int(G,Xa)) = aa_int_nat(F2,aa_int_int(G2,X2)) )
     => ( aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),Xa) = aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F2),G2),X2) ) ) ).

% comp_cong
tff(fact_298_comp__cong,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer,F2: fun_fu4800041728187187127nteger,G2: fun_Co1981900195452921617nteger,X2: code_integer] :
      ( ( aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,Xa)) = aa_fun2509385170507305080nteger(F2,aa_Cod4161879620478718930nteger(G2,X2)) )
     => ( aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Xa) = aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F2),G2),X2) ) ) ).

% comp_cong
tff(fact_299_comp__apply__eq,axiom,
    ! [F: fun_o_o,G: fun_nat_o,Xa: nat,Ha: fun_o_o,K: fun_nat_o] :
      ( ( aa_o_o(F,aa_nat_o(G,Xa))
      <=> aa_o_o(Ha,aa_nat_o(K,Xa)) )
     => ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Xa)
      <=> aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Ha),K),Xa) ) ) ).

% comp_apply_eq
tff(fact_300_comp__apply__eq,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int,Ha: fun_nat_num,K: fun_int_nat] :
      ( ( aa_nat_num(F,aa_int_nat(G,Xa)) = aa_nat_num(Ha,aa_int_nat(K,Xa)) )
     => ( aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),Xa) = aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Ha),K),Xa) ) ) ).

% comp_apply_eq
tff(fact_301_comp__apply__eq,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,Ha: fun_nat_nat,K: fun_nat_nat] :
      ( ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = aa_nat_nat(Ha,aa_nat_nat(K,Xa)) )
     => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),Xa) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Ha),K),Xa) ) ) ).

% comp_apply_eq
tff(fact_302_comp__apply__eq,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,Ha: fun_int_nat,K: fun_int_int] :
      ( ( aa_int_nat(F,aa_int_int(G,Xa)) = aa_int_nat(Ha,aa_int_int(K,Xa)) )
     => ( aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),Xa) = aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Ha),K),Xa) ) ) ).

% comp_apply_eq
tff(fact_303_comp__apply__eq,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer,Ha: fun_fu4800041728187187127nteger,K: fun_Co1981900195452921617nteger] :
      ( ( aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,Xa)) = aa_fun2509385170507305080nteger(Ha,aa_Cod4161879620478718930nteger(K,Xa)) )
     => ( aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Xa) = aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(Ha),K),Xa) ) ) ).

% comp_apply_eq
tff(fact_304_pick__middlep,axiom,
    ! [P2: fun_Pr2028404973800837934_nat_o,Q2: fun_Pr4738963024209204295_int_o,A: product_prod_nat_nat,C: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(relcom8911704206367525917at_int(P2,Q2),A),C)
     => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(P2,A),aa_int1417691210696471063at_nat(bNF_pi770922605693484880at_int(P2,Q2,A),C))
        & aa_int_o(aa_Pro2912763917461432128_int_o(Q2,aa_int1417691210696471063at_nat(bNF_pi770922605693484880at_int(P2,Q2,A),C)),C) ) ) ).

% pick_middlep
tff(fact_305_pick__middlep,axiom,
    ! [P2: fun_int_fun_int_o,Q2: fun_in4891666191172872833eger_o,A: int,C: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(relcom967497705283160040nteger(P2,Q2),A),C)
     => ( aa_int_o(aa_int_fun_int_o(P2,A),aa_Code_integer_int(bNF_pi4515785505327359899nteger(P2,Q2,A),C))
        & aa_Code_integer_o(aa_int5993532643906289898eger_o(Q2,aa_Code_integer_int(bNF_pi4515785505327359899nteger(P2,Q2,A),C)),C) ) ) ).

% pick_middlep
tff(fact_306_sndOp__in,axiom,
    ! [Ac: produc3843707927480180839at_nat,P2: fun_se5639961202347202350_nat_o,Q2: fun_se5639961202347202350_nat_o] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Ac),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,relcom8363394528208558896at_nat(P2,Q2))))
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_Pro9111386209873968764at_nat(bNF_sn1857951156667551021at_nat(P2,Q2),Ac)),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,Q2))) ) ).

% sndOp_in
tff(fact_307_sndOp__in,axiom,
    ! [Ac: product_prod_nat_nat,P2: fun_nat_fun_nat_o,Q2: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ac),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,relcompp_nat_nat_nat(P2,Q2))))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_Pro262675715343573244at_nat(bNF_sn8916957246138178918at_nat(P2,Q2),Ac)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Q2))) ) ).

% sndOp_in
tff(fact_308_sndOp__in,axiom,
    ! [Ac: produc4196048018000912732at_int,P2: fun_Pr2028404973800837934_nat_o,Q2: fun_Pr4738963024209204295_int_o] :
      ( member8394280151737582451at_int(Ac,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,relcom8911704206367525917at_int(P2,Q2))))
     => member8394280151737582451at_int(aa_Pro4417875771598949494at_int(bNF_sn649997494946348640at_int(P2,Q2),Ac),collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,Q2))) ) ).

% sndOp_in
tff(fact_309_sndOp__in,axiom,
    ! [Ac: produc2963269191615333738nteger,P2: fun_int_fun_int_o,Q2: fun_in4891666191172872833eger_o] :
      ( member4345257455949424659nteger(Ac,collec7215431352809750357nteger(aa_fun7359524620130742070eger_o(produc540600059291021473eger_o,relcom967497705283160040nteger(P2,Q2))))
     => member4345257455949424659nteger(aa_Pro8000154941595009468nteger(bNF_sn8000174127721201323nteger(P2,Q2),Ac),collec7215431352809750357nteger(aa_fun7359524620130742070eger_o(produc540600059291021473eger_o,Q2))) ) ).

% sndOp_in
tff(fact_310_sndOp__in,axiom,
    ! [Ac: product_prod_int_int,P2: fun_int_fun_int_o,Q2: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Ac),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,relcompp_int_int_int(P2,Q2))))
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_Pro6970504127699319548nt_int(bNF_sn1062102010912252026nt_int(P2,Q2),Ac)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Q2))) ) ).

% sndOp_in
tff(fact_311_fstOp__in,axiom,
    ! [Ac: produc3843707927480180839at_nat,P2: fun_se5639961202347202350_nat_o,Q2: fun_se5639961202347202350_nat_o] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Ac),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,relcom8363394528208558896at_nat(P2,Q2))))
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_Pro9111386209873968764at_nat(bNF_fs1946732770515719151at_nat(P2,Q2),Ac)),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,P2))) ) ).

% fstOp_in
tff(fact_312_fstOp__in,axiom,
    ! [Ac: product_prod_nat_nat,P2: fun_nat_fun_nat_o,Q2: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Ac),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,relcompp_nat_nat_nat(P2,Q2))))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_Pro262675715343573244at_nat(bNF_fs6799373675401686564at_nat(P2,Q2),Ac)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,P2))) ) ).

% fstOp_in
tff(fact_313_fstOp__in,axiom,
    ! [Ac: produc4196048018000912732at_int,P2: fun_Pr2028404973800837934_nat_o,Q2: fun_Pr4738963024209204295_int_o] :
      ( member8394280151737582451at_int(Ac,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,relcom8911704206367525917at_int(P2,Q2))))
     => member8206827879206165904at_nat(aa_Pro1635230920266491085at_nat(bNF_fs6342258632037972510at_int(P2,Q2),Ac),collec7088162979684241874at_nat(produc6590410687421337004_nat_o(P2))) ) ).

% fstOp_in
tff(fact_314_fstOp__in,axiom,
    ! [Ac: produc2963269191615333738nteger,P2: fun_int_fun_int_o,Q2: fun_in4891666191172872833eger_o] :
      ( member4345257455949424659nteger(Ac,collec7215431352809750357nteger(aa_fun7359524620130742070eger_o(produc540600059291021473eger_o,relcom967497705283160040nteger(P2,Q2))))
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_Pro526324081969021963nt_int(bNF_fs4543836202474805097nteger(P2,Q2),Ac)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,P2))) ) ).

% fstOp_in
tff(fact_315_fstOp__in,axiom,
    ! [Ac: product_prod_int_int,P2: fun_int_fun_int_o,Q2: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Ac),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,relcompp_int_int_int(P2,Q2))))
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_Pro6970504127699319548nt_int(bNF_fs8167890477030535480nt_int(P2,Q2),Ac)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,P2))) ) ).

% fstOp_in
tff(fact_316_effect__bindE,axiom,
    ! [F: heap_Heap_ref_a,G: fun_re8176683087113984344_ref_a,Ha: heap_e7401611519738050253t_unit,H: heap_e7401611519738050253t_unit,R: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_b146640354529102224_ref_a(F,G),Ha),H),R)
     => ~ ! [H2: heap_e7401611519738050253t_unit,R3: ref_a] :
            ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(F,Ha),H2),R3)
           => ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_ref97115632722995649_ref_a(G,R3),H2),H),R) ) ) ).

% effect_bindE
tff(fact_317_effect__bindI,axiom,
    ! [F: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a,G: fun_re8176683087113984344_ref_a,H: heap_e7401611519738050253t_unit,R: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(F,Ha),H_a),Ra)
     => ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(aa_ref97115632722995649_ref_a(G,Ra),H_a),H),R)
       => aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(heap_b146640354529102224_ref_a(F,G),Ha),H),R) ) ) ).

% effect_bindI
tff(fact_318_fst__fstOp,axiom,
    ! [Bc: produc7812382424243267045t_unit,P2: fun_re5167877524888533480unit_o,Q2: fun_he3009834522632631726unit_o] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Bc) = aa_Pro7711270743135521984_ref_a(comp_P4044372497395451596t_unit(produc473321116828791851t_unit,bNF_fs409234558581511823t_unit(P2,Q2)),Bc) ).

% fst_fstOp
tff(fact_319_comp__apply,axiom,
    ! [F: fun_o_o,G: fun_nat_o,Xa: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Xa)
    <=> aa_o_o(F,aa_nat_o(G,Xa)) ) ).

% comp_apply
tff(fact_320_comp__apply,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int] : aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),Xa) = aa_nat_num(F,aa_int_nat(G,Xa)) ).

% comp_apply
tff(fact_321_comp__apply,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),Xa) = aa_nat_nat(F,aa_nat_nat(G,Xa)) ).

% comp_apply
tff(fact_322_comp__apply,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int] : aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),Xa) = aa_int_nat(F,aa_int_int(G,Xa)) ).

% comp_apply
tff(fact_323_comp__apply,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer] : aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Xa) = aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,Xa)) ).

% comp_apply
tff(fact_324_image__eqI,axiom,
    ! [B: $o,F: fun_o_o,Xa: $o,A4: set_o] :
      ( ( (B)
      <=> aa_o_o(F,(Xa)) )
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_set_o_o(member_o((B)),aa_set_o_set_o(image_o_o(F),A4)) ) ) ).

% image_eqI
tff(fact_325_image__eqI,axiom,
    ! [B: nat,F: fun_o_nat,Xa: $o,A4: set_o] :
      ( ( B = aa_o_nat(F,(Xa)) )
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_set_nat_o(member_nat(B),image_o_nat(F,A4)) ) ) ).

% image_eqI
tff(fact_326_image__eqI,axiom,
    ! [B: int,F: fun_o_int,Xa: $o,A4: set_o] :
      ( ( B = aa_o_int(F,(Xa)) )
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_set_int_o(member_int(B),image_o_int(F,A4)) ) ) ).

% image_eqI
tff(fact_327_image__eqI,axiom,
    ! [B: rat,F: fun_nat_rat,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_rat(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_rat_o(member_rat(B),aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% image_eqI
tff(fact_328_image__eqI,axiom,
    ! [B: $o,F: fun_nat_o,Xa: nat,A4: set_nat] :
      ( ( (B)
      <=> aa_nat_o(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_o_o(member_o((B)),image_nat_o(F,A4)) ) ) ).

% image_eqI
tff(fact_329_image__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_nat(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% image_eqI
tff(fact_330_image__eqI,axiom,
    ! [B: int,F: fun_nat_int,Xa: nat,A4: set_nat] :
      ( ( B = aa_nat_int(F,Xa) )
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% image_eqI
tff(fact_331_image__eqI,axiom,
    ! [B: $o,F: fun_int_o,Xa: int,A4: set_int] :
      ( ( (B)
      <=> aa_int_o(F,Xa) )
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_set_o_o(member_o((B)),image_int_o(F,A4)) ) ) ).

% image_eqI
tff(fact_332_image__eqI,axiom,
    ! [B: nat,F: fun_int_nat,Xa: int,A4: set_int] :
      ( ( B = aa_int_nat(F,Xa) )
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_set_nat_o(member_nat(B),aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ).

% image_eqI
tff(fact_333_image__eqI,axiom,
    ! [B: int,F: fun_int_int,Xa: int,A4: set_int] :
      ( ( B = aa_int_int(F,Xa) )
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_set_int_o(member_int(B),aa_set_int_set_int(image_int_int(F),A4)) ) ) ).

% image_eqI
tff(fact_334_Inf_OINF__image,axiom,
    ! [Inf: fun_set_o_o,G: fun_o_o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(Inf,aa_set_o_set_o(image_o_o(G),image_nat_o(F,A4)))
    <=> aa_set_o_o(Inf,image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F),A4)) ) ).

% Inf.INF_image
tff(fact_335_Inf_OINF__image,axiom,
    ! [Inf: fun_set_rat_rat,G: fun_rat_rat,F: fun_nat_rat,A4: set_nat] : aa_set_rat_rat(Inf,image_rat_rat(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(comp_rat_rat_nat(G,F)),A4)) ).

% Inf.INF_image
tff(fact_336_Inf_OINF__image,axiom,
    ! [Inf: fun_set_nat_nat,G: fun_rat_nat,F: fun_nat_rat,A4: set_nat] : aa_set_nat_nat(Inf,image_rat_nat(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(G,F)),A4)) ).

% Inf.INF_image
tff(fact_337_Inf_OINF__image,axiom,
    ! [Inf: fun_set_int_int,G: fun_rat_int,F: fun_nat_rat,A4: set_nat] : aa_set_int_int(Inf,image_rat_int(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_int_int(Inf,aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(G,F)),A4)) ).

% Inf.INF_image
tff(fact_338_Inf_OINF__image,axiom,
    ! [Inf: fun_set_rat_rat,G: fun_int_rat,F: fun_nat_int,A4: set_nat] : aa_set_rat_rat(Inf,image_int_rat(G,aa_set_nat_set_int(image_nat_int(F),A4))) = aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(comp_int_rat_nat(G,F)),A4)) ).

% Inf.INF_image
tff(fact_339_Inf_OINF__image,axiom,
    ! [Inf: fun_set_num_num,G: fun_nat_num,F: fun_int_nat,A4: set_int] : aa_set_num_num(Inf,image_nat_num(G,aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_num_num(Inf,image_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F),A4)) ).

% Inf.INF_image
tff(fact_340_Inf_OINF__image,axiom,
    ! [Inf: fun_set_rat_rat,G: fun_nat_rat,F: fun_nat_nat,A4: set_nat] : aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(comp_nat_rat_nat(G,F)),A4)) ).

% Inf.INF_image
tff(fact_341_Inf_OINF__image,axiom,
    ! [Inf: fun_set_rat_rat,G: fun_nat_rat,F: fun_int_nat,A4: set_int] : aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_rat_rat(Inf,image_int_rat(comp_nat_rat_int(G,F),A4)) ).

% Inf.INF_image
tff(fact_342_Inf_OINF__image,axiom,
    ! [Inf: fun_set_nat_nat,G: fun_nat_nat,F: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),A4)) ).

% Inf.INF_image
tff(fact_343_Inf_OINF__image,axiom,
    ! [Inf: fun_set_nat_nat,G: fun_nat_nat,F: fun_int_nat,A4: set_int] : aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_nat_nat(Inf,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),F)),A4)) ).

% Inf.INF_image
tff(fact_344_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_o_o,G: fun_o_o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(Sup,aa_set_o_set_o(image_o_o(G),image_nat_o(F,A4)))
    <=> aa_set_o_o(Sup,image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F),A4)) ) ).

% Sup.SUP_image
tff(fact_345_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_rat_rat,G: fun_rat_rat,F: fun_nat_rat,A4: set_nat] : aa_set_rat_rat(Sup,image_rat_rat(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(comp_rat_rat_nat(G,F)),A4)) ).

% Sup.SUP_image
tff(fact_346_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_nat_nat,G: fun_rat_nat,F: fun_nat_rat,A4: set_nat] : aa_set_nat_nat(Sup,image_rat_nat(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(G,F)),A4)) ).

% Sup.SUP_image
tff(fact_347_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_int_int,G: fun_rat_int,F: fun_nat_rat,A4: set_nat] : aa_set_int_int(Sup,image_rat_int(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_int_int(Sup,aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(G,F)),A4)) ).

% Sup.SUP_image
tff(fact_348_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_rat_rat,G: fun_int_rat,F: fun_nat_int,A4: set_nat] : aa_set_rat_rat(Sup,image_int_rat(G,aa_set_nat_set_int(image_nat_int(F),A4))) = aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(comp_int_rat_nat(G,F)),A4)) ).

% Sup.SUP_image
tff(fact_349_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_num_num,G: fun_nat_num,F: fun_int_nat,A4: set_int] : aa_set_num_num(Sup,image_nat_num(G,aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_num_num(Sup,image_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F),A4)) ).

% Sup.SUP_image
tff(fact_350_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_rat_rat,G: fun_nat_rat,F: fun_nat_nat,A4: set_nat] : aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(comp_nat_rat_nat(G,F)),A4)) ).

% Sup.SUP_image
tff(fact_351_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_rat_rat,G: fun_nat_rat,F: fun_int_nat,A4: set_int] : aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_rat_rat(Sup,image_int_rat(comp_nat_rat_int(G,F),A4)) ).

% Sup.SUP_image
tff(fact_352_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_nat_nat,G: fun_nat_nat,F: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),A4)) ).

% Sup.SUP_image
tff(fact_353_Sup_OSUP__image,axiom,
    ! [Sup: fun_set_nat_nat,G: fun_nat_nat,F: fun_int_nat,A4: set_int] : aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_nat_nat(Sup,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),F)),A4)) ).

% Sup.SUP_image
tff(fact_354_image__comp,axiom,
    ! [F: fun_o_o,G: fun_nat_o,Ra: set_nat] : aa_set_o_set_o(image_o_o(F),image_nat_o(G,Ra)) = image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Ra) ).

% image_comp
tff(fact_355_image__comp,axiom,
    ! [F: fun_rat_rat,G: fun_nat_rat,Ra: set_nat] : image_rat_rat(F,aa_set_nat_set_rat(image_nat_rat(G),Ra)) = aa_set_nat_set_rat(image_nat_rat(comp_rat_rat_nat(F,G)),Ra) ).

% image_comp
tff(fact_356_image__comp,axiom,
    ! [F: fun_rat_nat,G: fun_nat_rat,Ra: set_nat] : image_rat_nat(F,aa_set_nat_set_rat(image_nat_rat(G),Ra)) = aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(F,G)),Ra) ).

% image_comp
tff(fact_357_image__comp,axiom,
    ! [F: fun_rat_int,G: fun_nat_rat,Ra: set_nat] : image_rat_int(F,aa_set_nat_set_rat(image_nat_rat(G),Ra)) = aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(F,G)),Ra) ).

% image_comp
tff(fact_358_image__comp,axiom,
    ! [F: fun_int_rat,G: fun_nat_int,Ra: set_nat] : image_int_rat(F,aa_set_nat_set_int(image_nat_int(G),Ra)) = aa_set_nat_set_rat(image_nat_rat(comp_int_rat_nat(F,G)),Ra) ).

% image_comp
tff(fact_359_image__comp,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Ra: set_int] : image_nat_num(F,aa_set_int_set_nat(image_int_nat(G),Ra)) = image_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),Ra) ).

% image_comp
tff(fact_360_image__comp,axiom,
    ! [F: fun_nat_rat,G: fun_nat_nat,Ra: set_nat] : aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(image_nat_nat(G),Ra)) = aa_set_nat_set_rat(image_nat_rat(comp_nat_rat_nat(F,G)),Ra) ).

% image_comp
tff(fact_361_image__comp,axiom,
    ! [F: fun_nat_rat,G: fun_int_nat,Ra: set_int] : aa_set_nat_set_rat(image_nat_rat(F),aa_set_int_set_nat(image_int_nat(G),Ra)) = image_int_rat(comp_nat_rat_int(F,G),Ra) ).

% image_comp
tff(fact_362_image__comp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Ra: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(image_nat_nat(G),Ra)) = aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G)),Ra) ).

% image_comp
tff(fact_363_image__comp,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,Ra: set_int] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_int_set_nat(image_int_nat(G),Ra)) = aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),G)),Ra) ).

% image_comp
tff(fact_364_image__eq__imp__comp,axiom,
    ! [F: fun_nat_o,A4: set_nat,G: fun_nat_o,B5: set_nat,Ha: fun_o_o] :
      ( ( image_nat_o(F,A4) = image_nat_o(G,B5) )
     => ( image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Ha),F),A4) = image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Ha),G),B5) ) ) ).

% image_eq_imp_comp
tff(fact_365_image__eq__imp__comp,axiom,
    ! [F: fun_int_rat,A4: set_int,G: fun_nat_rat,B5: set_nat,Ha: fun_rat_nat] :
      ( ( image_int_rat(F,A4) = aa_set_nat_set_rat(image_nat_rat(G),B5) )
     => ( aa_set_int_set_nat(image_int_nat(comp_rat_nat_int(Ha,F)),A4) = aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_366_image__eq__imp__comp,axiom,
    ! [F: fun_int_rat,A4: set_int,G: fun_nat_rat,B5: set_nat,Ha: fun_rat_int] :
      ( ( image_int_rat(F,A4) = aa_set_nat_set_rat(image_nat_rat(G),B5) )
     => ( aa_set_int_set_int(image_int_int(comp_rat_int_int(Ha,F)),A4) = aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_367_image__eq__imp__comp,axiom,
    ! [F: fun_nat_rat,A4: set_nat,G: fun_int_rat,B5: set_int,Ha: fun_rat_nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = image_int_rat(G,B5) )
     => ( aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(Ha,F)),A4) = aa_set_int_set_nat(image_int_nat(comp_rat_nat_int(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_368_image__eq__imp__comp,axiom,
    ! [F: fun_nat_rat,A4: set_nat,G: fun_int_rat,B5: set_int,Ha: fun_rat_int] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = image_int_rat(G,B5) )
     => ( aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(Ha,F)),A4) = aa_set_int_set_int(image_int_int(comp_rat_int_int(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_369_image__eq__imp__comp,axiom,
    ! [F: fun_nat_rat,A4: set_nat,G: fun_nat_rat,B5: set_nat,Ha: fun_rat_rat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = aa_set_nat_set_rat(image_nat_rat(G),B5) )
     => ( aa_set_nat_set_rat(image_nat_rat(comp_rat_rat_nat(Ha,F)),A4) = aa_set_nat_set_rat(image_nat_rat(comp_rat_rat_nat(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_370_image__eq__imp__comp,axiom,
    ! [F: fun_nat_rat,A4: set_nat,G: fun_nat_rat,B5: set_nat,Ha: fun_rat_nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = aa_set_nat_set_rat(image_nat_rat(G),B5) )
     => ( aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(Ha,F)),A4) = aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_371_image__eq__imp__comp,axiom,
    ! [F: fun_nat_rat,A4: set_nat,G: fun_nat_rat,B5: set_nat,Ha: fun_rat_int] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = aa_set_nat_set_rat(image_nat_rat(G),B5) )
     => ( aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(Ha,F)),A4) = aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_372_image__eq__imp__comp,axiom,
    ! [F: fun_nat_nat,A4: set_nat,G: fun_nat_nat,B5: set_nat,Ha: fun_nat_rat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = aa_set_nat_set_nat(image_nat_nat(G),B5) )
     => ( aa_set_nat_set_rat(image_nat_rat(comp_nat_rat_nat(Ha,F)),A4) = aa_set_nat_set_rat(image_nat_rat(comp_nat_rat_nat(Ha,G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_373_image__eq__imp__comp,axiom,
    ! [F: fun_nat_nat,A4: set_nat,G: fun_nat_nat,B5: set_nat,Ha: fun_nat_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = aa_set_nat_set_nat(image_nat_nat(G),B5) )
     => ( aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Ha),F)),A4) = aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Ha),G)),B5) ) ) ).

% image_eq_imp_comp
tff(fact_374_snd__comp__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] : comp_P3428181540658402820nteger(produc6174133586879617921nteger,aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F)) = comp_C1043215000170864188nteger(F,produc6174133586879617921nteger) ).

% snd_comp_apsnd
tff(fact_375_snd__comp__apsnd,axiom,
    ! [F: fun_he2331061685413951235t_unit] : comp_P8019343326919024234t_unit(produc1126150082675451753t_unit,produc3038839249364385243_ref_a(F)) = comp_h4837689053573563144t_unit(F,produc1126150082675451753t_unit) ).

% snd_comp_apsnd
tff(fact_376_fst__comp__apfst,axiom,
    ! [F: fun_ref_a_ref_a] : comp_P4044372497395451596t_unit(produc473321116828791851t_unit,produc4508741384315284437t_unit(F)) = comp_r2143332150499412868t_unit(F,produc473321116828791851t_unit) ).

% fst_comp_apfst
tff(fact_377_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat,F: fun_se320340636488258563at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Z),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,bNF_Gr6677838525845560078at_nat(A4,F))))
     => ( aa_Pro441628658106787830at_nat(comp_s7355055891894327172at_nat(F,produc6011863565407846595at_nat),Z) = aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,Z) ) ) ).

% Collect_case_prod_Grp_eqD
tff(fact_378_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: product_prod_nat_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,bNF_Grp_nat_nat(A4,F))))
     => ( aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aa_fun2514183836563211038at_nat(comp_n2711240574518091068at_nat,F),product_fst_nat_nat),Z) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Z) ) ) ).

% Collect_case_prod_Grp_eqD
tff(fact_379_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc7812382424243267045t_unit,A4: set_ref_a,F: fun_re8475190471885019465t_unit] :
      ( member7849944260330398460t_unit(Z,collec3252306575900995386t_unit(produc4924168586796184916unit_o(bNF_Gr6510375311308651680t_unit(A4,F))))
     => ( comp_r3604920825623061426t_unit(F,produc473321116828791851t_unit,Z) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Z) ) ) ).

% Collect_case_prod_Grp_eqD
tff(fact_380_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: product_prod_int_int,A4: set_int,F: fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Z),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,bNF_Grp_int_int(A4,F))))
     => ( aa_Pro6799258434465214121nt_int(comp_i4330622015127193916nt_int(F,product_fst_int_int),Z) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Z) ) ) ).

% Collect_case_prod_Grp_eqD
tff(fact_381_Collect__split__mono__strong,axiom,
    ! [X3: set_nat,A4: set_Pr1261947904930325089at_nat,Y3: set_nat,P2: fun_nat_fun_nat_o,Q2: fun_nat_fun_nat_o] :
      ( ( X3 = image_2486076414777270412at_nat(product_fst_nat_nat,A4) )
     => ( ( Y3 = image_2486076414777270412at_nat(product_snd_nat_nat,A4) )
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),X3)
             => ! [Xa2: nat] :
                  ( aa_set_nat_o(member_nat(Xa2),Y3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P2,X),Xa2)
                   => aa_nat_o(aa_nat_fun_nat_o(Q2,X),Xa2) ) ) )
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,P2)))
           => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Q2))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_382_Collect__split__mono__strong,axiom,
    ! [X3: set_ref_a,A4: set_Pr1133966238863718811t_unit,Y3: set_he8644174446132254637t_unit,P2: fun_re5167877524888533480unit_o,Q2: fun_re5167877524888533480unit_o] :
      ( ( X3 = image_7136766188234240905_ref_a(produc473321116828791851t_unit,A4) )
     => ( ( Y3 = image_17660834435998945t_unit(produc1126150082675451753t_unit,A4) )
       => ( ! [X: ref_a] :
              ( member_ref_a(X,X3)
             => ! [Xa2: heap_e7401611519738050253t_unit] :
                  ( member242389114213480694t_unit(Xa2,Y3)
                 => ( aa_hea7994211679605476167unit_o(aa_ref4509144387093619241unit_o(P2,X),Xa2)
                   => aa_hea7994211679605476167unit_o(aa_ref4509144387093619241unit_o(Q2,X),Xa2) ) ) )
         => ( ord_le5525191956872776187t_unit(A4,collec3252306575900995386t_unit(produc4924168586796184916unit_o(P2)))
           => ord_le5525191956872776187t_unit(A4,collec3252306575900995386t_unit(produc4924168586796184916unit_o(Q2))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_383_Collect__split__mono__strong,axiom,
    ! [X3: set_int,A4: set_Pr958786334691620121nt_int,Y3: set_int,P2: fun_int_fun_int_o,Q2: fun_int_fun_int_o] :
      ( ( X3 = image_5042161079198086560nt_int(product_fst_int_int,A4) )
     => ( ( Y3 = image_5042161079198086560nt_int(product_snd_int_int,A4) )
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),X3)
             => ! [Xa2: int] :
                  ( aa_set_int_o(member_int(Xa2),Y3)
                 => ( aa_int_o(aa_int_fun_int_o(P2,X),Xa2)
                   => aa_int_o(aa_int_fun_int_o(Q2,X),Xa2) ) ) )
         => ( ord_le2843351958646193337nt_int(A4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,P2)))
           => ord_le2843351958646193337nt_int(A4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Q2))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_384_snd__comp__map__prod,axiom,
    ! [F: fun_ref_a_ref_a,G: fun_he2331061685413951235t_unit] : comp_P8019343326919024234t_unit(produc1126150082675451753t_unit,produc3456481717235892722t_unit(F,G)) = comp_h4837689053573563144t_unit(G,produc1126150082675451753t_unit) ).

% snd_comp_map_prod
tff(fact_385_fst__comp__map__prod,axiom,
    ! [F: fun_ref_a_ref_a,G: fun_he2331061685413951235t_unit] : comp_P4044372497395451596t_unit(produc473321116828791851t_unit,produc3456481717235892722t_unit(F,G)) = comp_r2143332150499412868t_unit(F,produc473321116828791851t_unit) ).

% fst_comp_map_prod
tff(fact_386_map__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_num_num,A: nat,B: num] : aa_Pro19645118074684540at_num(produc1665548884793878990um_num(F,G),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,A)),aa_num_num(G,B)) ).

% map_prod_simp
tff(fact_387_map__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_num_nat,A: nat,B: num] : aa_Pro8534773217980622642at_nat(produc5108217098638100356um_nat(F,G),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,A)),aa_num_nat(G,B)) ).

% map_prod_simp
tff(fact_388_map__prod__simp,axiom,
    ! [F: fun_nat_int,G: fun_num_int,A: nat,B: num] : aa_Pro5356275910372480490nt_int(produc5477380188365894716um_int(F,G),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,A)),aa_num_int(G,B)) ).

% map_prod_simp
tff(fact_389_map__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_num,A: nat,B: nat] : aa_Pro970919652292410950at_num(produc3535218481486409604at_num(F,G),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,A)),aa_nat_num(G,B)) ).

% map_prod_simp
tff(fact_390_map__prod__simp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A: nat,B: nat] : aa_Pro262675715343573244at_nat(produc6977886695330630970at_nat(F,G),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,A)),aa_nat_nat(G,B)) ).

% map_prod_simp
tff(fact_391_map__prod__simp,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,A: nat,B: nat] : aa_Pro6307550444590206900nt_int(produc7347049785058425330at_int(F,G),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,A)),aa_nat_int(G,B)) ).

% map_prod_simp
tff(fact_392_map__prod__simp,axiom,
    ! [F: fun_int_nat,G: fun_int_num,A: int,B: int] : aa_Pro1633873335401523598at_num(produc2224754261294513212nt_num(F,G),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_int_nat(F,A)),aa_int_num(G,B)) ).

% map_prod_simp
tff(fact_393_map__prod__simp,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A: int,B: int] : aa_Pro925629398452685892at_nat(produc5667422475138734578nt_nat(F,G),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_int_nat(F,A)),aa_int_nat(G,B)) ).

% map_prod_simp
tff(fact_394_map__prod__simp,axiom,
    ! [F: fun_int_int,G: fun_int_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(F,A)),aa_int_int(G,B)) ).

% map_prod_simp
tff(fact_395_map__prod__simp,axiom,
    ! [F: fun_se4097146774670233028at_nat,G: fun_se4805390711619070734at_num,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_Pro2794932780342069216at_num(produc3932168101365401472at_num(F,G),aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_set4749074711808745205at_nat(F,A)),aa_set1306406497964523839at_num(G,B)) ).

% map_prod_simp
tff(fact_396_fst__map__prod,axiom,
    ! [F: fun_ref_a_ref_a,G: fun_he2331061685413951235t_unit,Xa: produc7812382424243267045t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_Pro1957077887386443784t_unit(produc3456481717235892722t_unit(F,G),Xa)) = aa_ref_a_ref_a(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa)) ).

% fst_map_prod
tff(fact_397_snd__map__prod,axiom,
    ! [F: fun_ref_a_ref_a,G: fun_he2331061685413951235t_unit,Xa: produc7812382424243267045t_unit] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_Pro1957077887386443784t_unit(produc3456481717235892722t_unit(F,G),Xa)) = aa_hea2136485429539994620t_unit(G,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa)) ).

% snd_map_prod
tff(fact_398_apfst__conv,axiom,
    ! [F: fun_se6846035212209387651at_nat,Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : produc6529295065807390923at_nat(F,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_set4883462369968598652at_nat(F,Xa)),Y2) ).

% apfst_conv
tff(fact_399_apfst__conv,axiom,
    ! [F: fun_se320340636488258563at_nat,Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_Pro9111386209873968764at_nat(produc5380969780118646373at_nat(F),aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_set8868280134176360700at_nat(F,Xa)),Y2) ).

% apfst_conv
tff(fact_400_apfst__conv,axiom,
    ! [F: fun_nat_nat,Xa: nat,Y2: num] : produc6767424583952363640at_num(F,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,Xa)),Y2) ).

% apfst_conv
tff(fact_401_apfst__conv,axiom,
    ! [F: fun_nat_nat,Xa: nat,Y2: nat] : aa_Pro262675715343573244at_nat(produc986720760941809198at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,Xa)),Y2) ).

% apfst_conv
tff(fact_402_apfst__conv,axiom,
    ! [F: fun_int_int,Xa: int,Y2: int] : aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(F,Xa)),Y2) ).

% apfst_conv
tff(fact_403_apsnd__conv,axiom,
    ! [F: fun_se6846035212209387651at_nat,Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : produc8057050257526108681at_nat(F,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),aa_set4883462369968598652at_nat(F,Y2)) ).

% apsnd_conv
tff(fact_404_apsnd__conv,axiom,
    ! [F: fun_se320340636488258563at_nat,Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_Pro9111386209873968764at_nat(produc6468969725218530979at_nat(F),aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),aa_set8868280134176360700at_nat(F,Y2)) ).

% apsnd_conv
tff(fact_405_apsnd__conv,axiom,
    ! [F: fun_num_num,Xa: nat,Y2: num] : produc8389275579088931588um_nat(F,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),aa_num_num(F,Y2)) ).

% apsnd_conv
tff(fact_406_apsnd__conv,axiom,
    ! [F: fun_num_nat,Xa: nat,Y2: num] : produc1035573138926686394at_nat(F,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),aa_num_nat(F,Y2)) ).

% apsnd_conv
tff(fact_407_apsnd__conv,axiom,
    ! [F: fun_nat_num,Xa: nat,Y2: nat] : produc1225095714263496890um_nat(F,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),aa_nat_num(F,Y2)) ).

% apsnd_conv
tff(fact_408_apsnd__conv,axiom,
    ! [F: fun_nat_nat,Xa: nat,Y2: nat] : aa_Pro262675715343573244at_nat(produc3094765310956027504at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),aa_nat_nat(F,Y2)) ).

% apsnd_conv
tff(fact_409_apsnd__conv,axiom,
    ! [F: fun_int_int,Xa: int,Y2: int] : aa_Pro6970504127699319548nt_int(produc4463282112584876420nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),aa_int_int(F,Y2)) ).

% apsnd_conv
tff(fact_410_apsnd__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: code_integer,Y2: code_integer] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),produc1086072967326762835nteger(Xa,Y2)) = produc1086072967326762835nteger(Xa,aa_Cod1783974652533396728nteger(F,Y2)) ).

% apsnd_conv
tff(fact_411_fst__apfst,axiom,
    ! [F: fun_ref_a_ref_a,Xa: produc7812382424243267045t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_Pro1957077887386443784t_unit(produc4508741384315284437t_unit(F),Xa)) = aa_ref_a_ref_a(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa)) ).

% fst_apfst
tff(fact_412_snd__apfst,axiom,
    ! [F: fun_ref_a_ref_a,Xa: produc7812382424243267045t_unit] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_Pro1957077887386443784t_unit(produc4508741384315284437t_unit(F),Xa)) = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa) ).

% snd_apfst
tff(fact_413_fst__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),Xa)) = aa_Pro5245656335456409788nteger(produc8508995932063986495nteger,Xa) ).

% fst_apsnd
tff(fact_414_fst__apsnd,axiom,
    ! [F: fun_he2331061685413951235t_unit,Xa: produc7812382424243267045t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,aa_Pro1957077887386443784t_unit(produc3038839249364385243_ref_a(F),Xa)) = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa) ).

% fst_apsnd
tff(fact_415_snd__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger] : aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),Xa)) = aa_Cod1783974652533396728nteger(F,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xa)) ).

% snd_apsnd
tff(fact_416_snd__apsnd,axiom,
    ! [F: fun_he2331061685413951235t_unit,Xa: produc7812382424243267045t_unit] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,aa_Pro1957077887386443784t_unit(produc3038839249364385243_ref_a(F),Xa)) = aa_hea2136485429539994620t_unit(F,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa)) ).

% snd_apsnd
tff(fact_417_apsnd__eq__conv,axiom,
    ! [F: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger,G: fun_Co8499238673658658103nteger] :
      ( ( aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),Xa) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),Xa) )
    <=> ( aa_Cod1783974652533396728nteger(F,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xa)) = aa_Cod1783974652533396728nteger(G,aa_Pro5245656335456409788nteger(produc6174133586879617921nteger,Xa)) ) ) ).

% apsnd_eq_conv
tff(fact_418_map__prod__imageI,axiom,
    ! [A: nat,B: num,R2: set_Pr6200539531224447659at_num,F: fun_nat_nat,G: fun_num_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),R2)
     => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,A)),aa_num_num(G,B)),image_4925883905578568901at_num(produc1665548884793878990um_num(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_419_map__prod__imageI,axiom,
    ! [A: nat,B: num,R2: set_Pr6200539531224447659at_num,F: fun_nat_nat,G: fun_num_nat] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),R2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,A)),aa_num_nat(G,B))),image_4217639968629731195at_nat(produc5108217098638100356um_nat(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_420_map__prod__imageI,axiom,
    ! [A: nat,B: num,R2: set_Pr6200539531224447659at_num,F: fun_nat_int,G: fun_num_int] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),R2)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,A)),aa_num_int(G,B))),image_1039142661021589043nt_int(produc5477380188365894716um_int(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_421_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: fun_nat_nat,G: fun_nat_num] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),R2)
     => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,A)),aa_nat_num(G,B)),image_5877158439796295311at_num(produc3535218481486409604at_num(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_422_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),R2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,A)),aa_nat_nat(G,B))),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(produc6977886695330630970at_nat(F,G)),R2)) ) ).

% map_prod_imageI
tff(fact_423_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),R2)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,A)),aa_nat_int(G,B))),image_1990417195239315453nt_int(produc7347049785058425330at_int(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_424_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F: fun_int_nat,G: fun_int_num] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),R2)
     => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_int_nat(F,A)),aa_int_num(G,B)),image_6540112122905407959at_num(produc2224754261294513212nt_num(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_425_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),R2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_int_nat(F,A)),aa_int_nat(G,B))),image_5831868185956570253at_nat(produc5667422475138734578nt_nat(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_426_map__prod__imageI,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),R2)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(F,A)),aa_int_int(G,B))),image_2653370878348428101nt_int(produc6036585564866528938nt_int(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_427_map__prod__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,F: fun_se4097146774670233028at_nat,G: fun_se4805390711619070734at_num] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),R2)
     => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_set4749074711808745205at_nat(F,A)),aa_set1306406497964523839at_num(G,B)),image_8033574621161215657at_num(produc3932168101365401472at_num(F,G),R2)) ) ).

% map_prod_imageI
tff(fact_428_snd__comp__apfst,axiom,
    ! [F: fun_ref_a_ref_a] : comp_P8019343326919024234t_unit(produc1126150082675451753t_unit,produc4508741384315284437t_unit(F)) = produc1126150082675451753t_unit ).

% snd_comp_apfst
tff(fact_429_fst__comp__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] : comp_P3428181540658402820nteger(produc8508995932063986495nteger,aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F)) = produc8508995932063986495nteger ).

% fst_comp_apsnd
tff(fact_430_fst__comp__apsnd,axiom,
    ! [F: fun_he2331061685413951235t_unit] : comp_P4044372497395451596t_unit(produc473321116828791851t_unit,produc3038839249364385243_ref_a(F)) = produc473321116828791851t_unit ).

% fst_comp_apsnd
tff(fact_431_apfst__apsnd,axiom,
    ! [F: fun_re3272678581022320579at_nat,G: fun_he3825718725942840189at_nat,Xa: produc7812382424243267045t_unit] : produc2316621926130398489at_nat(F,produc5080965012319039829_ref_a(G,Xa)) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_ref1870931717416665516at_nat(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea8916874021999987062at_nat(G,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apfst_apsnd
tff(fact_432_apfst__apsnd,axiom,
    ! [F: fun_re2769948457291452253at_nat,G: fun_he7657674246925140503at_nat,Xa: produc7812382424243267045t_unit] : produc4016808307611786905at_nat(F,produc2928627021203296111_ref_a(G,Xa)) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_ref8328983138776998982at_nat(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea171790370555646736at_nat(G,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apfst_apsnd
tff(fact_433_apfst__apsnd,axiom,
    ! [F: fun_ref_a_nat,G: fun_he1741356246944716282it_num,Xa: produc7812382424243267045t_unit] : produc1704413584773205829at_num(F,produc7633249859560231688_ref_a(G,Xa)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_ref_a_nat(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea2405333822849273899it_num(G,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apfst_apsnd
tff(fact_434_apfst__apsnd,axiom,
    ! [F: fun_ref_a_nat,G: fun_he1033112309995878576it_nat,Xa: produc7812382424243267045t_unit] : produc5147081798617427195at_nat(F,produc8366372614399963582_ref_a(G,Xa)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_ref_a_nat(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea5848002036693495265it_nat(G,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apfst_apsnd
tff(fact_435_apfst__apsnd,axiom,
    ! [F: fun_ref_a_int,G: fun_he6078633327341457676it_int,Xa: produc7812382424243267045t_unit] : produc6143945040009431475nt_int(F,produc3558794870892339738_ref_a(G,Xa)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_ref_a_int(F,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea5845511566184444989it_int(G,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apfst_apsnd
tff(fact_436_apsnd__apfst,axiom,
    ! [F: fun_he3825718725942840189at_nat,G: fun_re3272678581022320579at_nat,Xa: produc7812382424243267045t_unit] : produc5990651871820846223at_nat(F,produc5988837738927868575t_unit(G,Xa)) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_ref1870931717416665516at_nat(G,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea8916874021999987062at_nat(F,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apsnd_apfst
tff(fact_437_apsnd__apfst,axiom,
    ! [F: fun_he7657674246925140503at_nat,G: fun_re2769948457291452253at_nat,Xa: produc7812382424243267045t_unit] : produc8705118745468743695at_nat(F,produc5323809043756447493t_unit(G,Xa)) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_ref8328983138776998982at_nat(G,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea171790370555646736at_nat(F,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apsnd_apfst
tff(fact_438_apsnd__apfst,axiom,
    ! [F: fun_he1741356246944716282it_num,G: fun_ref_a_nat,Xa: produc7812382424243267045t_unit] : produc3902377510303101371um_nat(F,produc6942186945380262242t_unit(G,Xa)) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_ref_a_nat(G,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea2405333822849273899it_num(F,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apsnd_apfst
tff(fact_439_apsnd__apfst,axiom,
    ! [F: fun_he1033112309995878576it_nat,G: fun_ref_a_nat,Xa: produc7812382424243267045t_unit] : produc5772047106995631985at_nat(F,produc6942186945380262242t_unit(G,Xa)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_ref_a_nat(G,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea5848002036693495265it_nat(F,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apsnd_apfst
tff(fact_440_apsnd__apfst,axiom,
    ! [F: fun_he6078633327341457676it_int,G: fun_ref_a_int,Xa: produc7812382424243267045t_unit] : produc6768910348387636265nt_int(F,produc5845103964736263686t_unit(G,Xa)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_ref_a_int(G,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xa))),aa_hea5845511566184444989it_int(F,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xa))) ).

% apsnd_apfst
tff(fact_441_apsnd__apfst__commute,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger,P: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc3195571438000033749nteger,G),P)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc3195571438000033749nteger,G),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),P)) ).

% apsnd_apfst_commute
tff(fact_442_subset__image__iff,axiom,
    ! [B5: set_rat,F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(AA),A4)
          & ( B5 = aa_set_nat_set_rat(image_nat_rat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_443_subset__image__iff,axiom,
    ! [B5: set_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_nat_set_nat(image_nat_nat(F),A4))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(AA),A4)
          & ( B5 = aa_set_nat_set_nat(image_nat_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_444_subset__image__iff,axiom,
    ! [B5: set_int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_nat_set_int(image_nat_int(F),A4))
    <=> ? [AA: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(AA),A4)
          & ( B5 = aa_set_nat_set_int(image_nat_int(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_445_subset__image__iff,axiom,
    ! [B5: set_nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_int_set_nat(image_int_nat(F),A4))
    <=> ? [AA: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(AA),A4)
          & ( B5 = aa_set_int_set_nat(image_int_nat(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_446_subset__image__iff,axiom,
    ! [B5: set_int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_int_set_int(image_int_int(F),A4))
    <=> ? [AA: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(AA),A4)
          & ( B5 = aa_set_int_set_int(image_int_int(F),AA) ) ) ) ).

% subset_image_iff
tff(fact_447_image__subset__iff,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_set_rat_o(member_rat(aa_nat_rat(F,X4)),B5) ) ) ).

% image_subset_iff
tff(fact_448_image__subset__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X4)),B5) ) ) ).

% image_subset_iff
tff(fact_449_image__subset__iff,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X4)),B5) ) ) ).

% image_subset_iff
tff(fact_450_image__subset__iff,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => aa_set_int_o(member_int(aa_nat_int(F,X4)),B5) ) ) ).

% image_subset_iff
tff(fact_451_image__subset__iff,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => aa_set_int_o(member_int(aa_int_int(F,X4)),B5) ) ) ).

% image_subset_iff
tff(fact_452_subset__imageE,axiom,
    ! [B5: set_rat,F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4))
     => ~ ! [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
           => ( B5 != aa_set_nat_set_rat(image_nat_rat(F),C2) ) ) ) ).

% subset_imageE
tff(fact_453_subset__imageE,axiom,
    ! [B5: set_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ~ ! [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
           => ( B5 != aa_set_nat_set_nat(image_nat_nat(F),C2) ) ) ) ).

% subset_imageE
tff(fact_454_subset__imageE,axiom,
    ! [B5: set_int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_nat_set_int(image_nat_int(F),A4))
     => ~ ! [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
           => ( B5 != aa_set_nat_set_int(image_nat_int(F),C2) ) ) ) ).

% subset_imageE
tff(fact_455_subset__imageE,axiom,
    ! [B5: set_nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_int_set_nat(image_int_nat(F),A4))
     => ~ ! [C2: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C2),A4)
           => ( B5 != aa_set_int_set_nat(image_int_nat(F),C2) ) ) ) ).

% subset_imageE
tff(fact_456_subset__imageE,axiom,
    ! [B5: set_int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_int_set_int(image_int_int(F),A4))
     => ~ ! [C2: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C2),A4)
           => ( B5 != aa_set_int_set_int(image_int_int(F),C2) ) ) ) ).

% subset_imageE
tff(fact_457_image__subsetI,axiom,
    ! [A4: set_o,F: fun_o_o,B5: set_o] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_o_o(member_o(aa_o_o(F,(X))),B5) )
     => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(image_o_o(F),A4)),B5) ) ).

% image_subsetI
tff(fact_458_image__subsetI,axiom,
    ! [A4: set_o,F: fun_o_nat,B5: set_nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_nat_o(member_nat(aa_o_nat(F,(X))),B5) )
     => aa_set_nat_o(ord_less_eq_set_nat(image_o_nat(F,A4)),B5) ) ).

% image_subsetI
tff(fact_459_image__subsetI,axiom,
    ! [A4: set_o,F: fun_o_int,B5: set_int] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_int_o(member_int(aa_o_int(F,(X))),B5) )
     => aa_set_int_o(ord_less_eq_set_int(image_o_int(F,A4)),B5) ) ).

% image_subsetI
tff(fact_460_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_rat,B5: set_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_rat_o(member_rat(aa_nat_rat(F,X)),B5) )
     => aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5) ) ).

% image_subsetI
tff(fact_461_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_o,B5: set_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_o_o(member_o(aa_nat_o(F,X)),B5) )
     => aa_set_o_o(ord_less_eq_set_o(image_nat_o(F,A4)),B5) ) ).

% image_subsetI
tff(fact_462_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_nat,B5: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X)),B5) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5) ) ).

% image_subsetI
tff(fact_463_image__subsetI,axiom,
    ! [A4: set_nat,F: fun_nat_int,B5: set_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_int_o(member_int(aa_nat_int(F,X)),B5) )
     => aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5) ) ).

% image_subsetI
tff(fact_464_image__subsetI,axiom,
    ! [A4: set_int,F: fun_int_o,B5: set_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_o_o(member_o(aa_int_o(F,X)),B5) )
     => aa_set_o_o(ord_less_eq_set_o(image_int_o(F,A4)),B5) ) ).

% image_subsetI
tff(fact_465_image__subsetI,axiom,
    ! [A4: set_int,F: fun_int_nat,B5: set_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X)),B5) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5) ) ).

% image_subsetI
tff(fact_466_image__subsetI,axiom,
    ! [A4: set_int,F: fun_int_int,B5: set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_int_o(member_int(aa_int_int(F,X)),B5) )
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5) ) ).

% image_subsetI
tff(fact_467_image__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),aa_set_nat_set_rat(image_nat_rat(F),B5)) ) ).

% image_mono
tff(fact_468_image__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ) ).

% image_mono
tff(fact_469_image__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),aa_set_nat_set_int(image_nat_int(F),B5)) ) ).

% image_mono
tff(fact_470_image__mono,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5)) ) ).

% image_mono
tff(fact_471_image__mono,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),aa_set_int_set_int(image_int_int(F),B5)) ) ).

% image_mono
tff(fact_472_map__prod_Ocomp,axiom,
    ! [F: fun_o_o,G: fun_o_o,Ha: fun_nat_o,I: fun_nat_o] : comp_P4785046977597218846at_nat(produc1928436952130685430_o_o_o(F,G),produc6957950696844810486_nat_o(Ha,I)) = produc6957950696844810486_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),I)) ).

% map_prod.comp
tff(fact_473_map__prod_Ocomp,axiom,
    ! [F: fun_o_o,G: fun_nat_num,Ha: fun_nat_o,I: fun_int_nat] : comp_P9057111047297513408at_int(produc3628557804203378146at_num(F,G),produc8250590009387450446nt_nat(Ha,I)) = produc4807921795543229080nt_num(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),I)) ).

% map_prod.comp
tff(fact_474_map__prod_Ocomp,axiom,
    ! [F: fun_o_o,G: fun_nat_nat,Ha: fun_nat_o,I: fun_nat_nat] : comp_P4962864564169660718at_nat(produc7071226018047599512at_nat(F,G),produc7251236297486395890at_nat(Ha,I)) = produc7251236297486395890at_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),I)) ).

% map_prod.comp
tff(fact_475_map__prod_Ocomp,axiom,
    ! [F: fun_o_o,G: fun_int_nat,Ha: fun_nat_o,I: fun_int_int] : comp_P465755969289462830at_int(produc8070579729948654068nt_nat(F,G),produc8248099538878400170nt_int(Ha,I)) = produc8250590009387450446nt_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),I)) ).

% map_prod.comp
tff(fact_476_map__prod_Ocomp,axiom,
    ! [F: fun_nat_num,G: fun_o_o,Ha: fun_int_nat,I: fun_nat_o] : comp_P5807473265676440968nt_nat(produc3264868786340801506um_o_o(F,G),produc8298419091753553490_nat_o(Ha,I)) = produc9074608839065728392_nat_o(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),I)) ).

% map_prod.comp
tff(fact_477_map__prod_Ocomp,axiom,
    ! [F: fun_nat_num,G: fun_nat_num,Ha: fun_int_nat,I: fun_int_nat] : comp_P9218946435557410242nt_int(produc1476026309457068494at_num(F,G),produc5667422475138734578nt_nat(Ha,I)) = produc165562089265172102nt_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),I)) ).

% map_prod.comp
tff(fact_478_map__prod_Ocomp,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,Ha: fun_int_nat,I: fun_nat_nat] : comp_P5124699952429557552nt_nat(produc4918694523301289860at_nat(F,G),produc4668068763237680022at_nat(Ha,I)) = produc2608876591208338912at_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),I)) ).

% map_prod.comp
tff(fact_479_map__prod_Ocomp,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Ha: fun_int_nat,I: fun_int_int] : comp_P3772045228931366376nt_int(produc5918048235202344416nt_nat(F,G),produc5664932004629684302nt_int(Ha,I)) = produc3608230303109393468nt_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),I)) ).

% map_prod.comp
tff(fact_480_map__prod_Ocomp,axiom,
    ! [F: fun_nat_nat,G: fun_o_o,Ha: fun_nat_nat,I: fun_nat_o] : comp_P8901546994122883554at_nat(produc2001155298098703512at_o_o(F,G),produc4787215299218314158_nat_o(Ha,I)) = produc4787215299218314158_nat_o(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),I)) ).

% map_prod.comp
tff(fact_481_map__prod_Ocomp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_num,Ha: fun_nat_nat,I: fun_int_nat] : comp_P6334688330072917552at_int(produc3535218481486409604at_num(F,G),produc7977240407231685526nt_nat(Ha,I)) = produc4534572193387464160nt_num(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),I)) ).

% map_prod.comp
tff(fact_482_map__prod_Ocompositionality,axiom,
    ! [F: fun_o_o,G: fun_o_o,Ha: fun_nat_o,I: fun_nat_o,Prod: product_prod_nat_nat] : aa_Pro5376729675435676284od_o_o(produc1928436952130685430_o_o_o(F,G),aa_Pro9072700582319063714od_o_o(produc6957950696844810486_nat_o(Ha,I),Prod)) = aa_Pro9072700582319063714od_o_o(produc6957950696844810486_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_483_map__prod_Ocompositionality,axiom,
    ! [F: fun_o_o,G: fun_nat_num,Ha: fun_nat_o,I: fun_int_nat,Prod: product_prod_nat_int] : aa_Pro279734039161488822_o_num(produc3628557804203378146at_num(F,G),aa_Pro5791804135702779960_o_nat(produc8250590009387450446nt_nat(Ha,I),Prod)) = aa_Pro6500048072651617666_o_num(produc4807921795543229080nt_num(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_484_map__prod_Ocompositionality,axiom,
    ! [F: fun_o_o,G: fun_nat_nat,Ha: fun_nat_o,I: fun_nat_nat,Prod: product_prod_nat_nat] : aa_Pro8794862139067426924_o_nat(produc7071226018047599512at_nat(F,G),aa_Pro8467631284203683292_o_nat(produc7251236297486395890at_nat(Ha,I),Prod)) = aa_Pro8467631284203683292_o_nat(produc7251236297486395890at_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_485_map__prod_Ocompositionality,axiom,
    ! [F: fun_o_o,G: fun_int_nat,Ha: fun_nat_o,I: fun_int_int,Prod: product_prod_nat_int] : aa_Pro6119034990566523592_o_nat(produc8070579729948654068nt_nat(F,G),aa_Pro1613953116193583252_o_int(produc8248099538878400170nt_int(Ha,I),Prod)) = aa_Pro5791804135702779960_o_nat(produc8250590009387450446nt_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_486_map__prod_Ocompositionality,axiom,
    ! [F: fun_nat_num,G: fun_o_o,Ha: fun_int_nat,I: fun_nat_o,Prod: product_prod_int_nat] : aa_Pro7407088302862950382_num_o(produc3264868786340801506um_o_o(F,G),aa_Pro4720224260291850206_nat_o(produc8298419091753553490_nat_o(Ha,I),Prod)) = aa_Pro3566207295127809812_num_o(produc9074608839065728392_nat_o(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_487_map__prod_Ocompositionality,axiom,
    ! [F: fun_nat_num,G: fun_nat_num,Ha: fun_int_nat,I: fun_int_nat,Prod: product_prod_int_int] : aa_Pro8324622092454656144um_num(produc1476026309457068494at_num(F,G),aa_Pro925629398452685892at_nat(produc5667422475138734578nt_nat(Ha,I),Prod)) = aa_Pro8987575775563768792um_num(produc165562089265172102nt_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_488_map__prod_Ocompositionality,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,Ha: fun_int_nat,I: fun_nat_nat,Prod: product_prod_int_nat] : aa_Pro7616378155505818438um_nat(produc4918694523301289860at_nat(F,G),aa_Pro2022712379096684448at_nat(produc4668068763237680022at_nat(Ha,I),Prod)) = aa_Pro153042782404153834um_nat(produc2608876591208338912at_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_489_map__prod_Ocompositionality,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Ha: fun_int_nat,I: fun_int_int,Prod: product_prod_int_int] : aa_Pro6519295174861819882um_nat(produc5918048235202344416nt_nat(F,G),aa_Pro5971150415798264992at_int(produc5664932004629684302nt_int(Ha,I),Prod)) = aa_Pro8279331838614931086um_nat(produc3608230303109393468nt_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_490_map__prod_Ocompositionality,axiom,
    ! [F: fun_nat_nat,G: fun_o_o,Ha: fun_nat_nat,I: fun_nat_o,Prod: product_prod_nat_nat] : aa_Pro8561105268026990776_nat_o(produc2001155298098703512at_o_o(F,G),aa_Pro2752793771404400514_nat_o(produc4787215299218314158_nat_o(Ha,I),Prod)) = aa_Pro2752793771404400514_nat_o(produc4787215299218314158_nat_o(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_491_map__prod_Ocompositionality,axiom,
    ! [F: fun_nat_nat,G: fun_nat_num,Ha: fun_nat_nat,I: fun_int_nat,Prod: product_prod_nat_int] : aa_Pro970919652292410950at_num(produc3535218481486409604at_num(F,G),aa_Pro8388964771554350496at_nat(produc7977240407231685526nt_nat(Ha,I),Prod)) = aa_Pro9097208708503188202at_num(produc4534572193387464160nt_num(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),I)),Prod) ).

% map_prod.compositionality
tff(fact_492_map__prod__compose,axiom,
    ! [F1: fun_o_o,F22: fun_nat_o,G1: fun_o_o,G22: fun_nat_o] : produc6957950696844810486_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F1),F22),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),G22)) = comp_P4785046977597218846at_nat(produc1928436952130685430_o_o_o(F1,G1),produc6957950696844810486_nat_o(F22,G22)) ).

% map_prod_compose
tff(fact_493_map__prod__compose,axiom,
    ! [F1: fun_o_o,F22: fun_nat_o,G1: fun_nat_num,G22: fun_int_nat] : produc4807921795543229080nt_num(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F1),F22),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),G22)) = comp_P9057111047297513408at_int(produc3628557804203378146at_num(F1,G1),produc8250590009387450446nt_nat(F22,G22)) ).

% map_prod_compose
tff(fact_494_map__prod__compose,axiom,
    ! [F1: fun_o_o,F22: fun_nat_o,G1: fun_nat_nat,G22: fun_nat_nat] : produc7251236297486395890at_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F1),F22),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G1),G22)) = comp_P4962864564169660718at_nat(produc7071226018047599512at_nat(F1,G1),produc7251236297486395890at_nat(F22,G22)) ).

% map_prod_compose
tff(fact_495_map__prod__compose,axiom,
    ! [F1: fun_o_o,F22: fun_nat_o,G1: fun_int_nat,G22: fun_int_int] : produc8250590009387450446nt_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F1),F22),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G1),G22)) = comp_P465755969289462830at_int(produc8070579729948654068nt_nat(F1,G1),produc8248099538878400170nt_int(F22,G22)) ).

% map_prod_compose
tff(fact_496_map__prod__compose,axiom,
    ! [F1: fun_nat_num,F22: fun_int_nat,G1: fun_o_o,G22: fun_nat_o] : produc9074608839065728392_nat_o(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F1),F22),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),G22)) = comp_P5807473265676440968nt_nat(produc3264868786340801506um_o_o(F1,G1),produc8298419091753553490_nat_o(F22,G22)) ).

% map_prod_compose
tff(fact_497_map__prod__compose,axiom,
    ! [F1: fun_nat_num,F22: fun_int_nat,G1: fun_nat_num,G22: fun_int_nat] : produc165562089265172102nt_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F1),F22),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),G22)) = comp_P9218946435557410242nt_int(produc1476026309457068494at_num(F1,G1),produc5667422475138734578nt_nat(F22,G22)) ).

% map_prod_compose
tff(fact_498_map__prod__compose,axiom,
    ! [F1: fun_nat_num,F22: fun_int_nat,G1: fun_nat_nat,G22: fun_nat_nat] : produc2608876591208338912at_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F1),F22),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G1),G22)) = comp_P5124699952429557552nt_nat(produc4918694523301289860at_nat(F1,G1),produc4668068763237680022at_nat(F22,G22)) ).

% map_prod_compose
tff(fact_499_map__prod__compose,axiom,
    ! [F1: fun_nat_num,F22: fun_int_nat,G1: fun_int_nat,G22: fun_int_int] : produc3608230303109393468nt_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F1),F22),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G1),G22)) = comp_P3772045228931366376nt_int(produc5918048235202344416nt_nat(F1,G1),produc5664932004629684302nt_int(F22,G22)) ).

% map_prod_compose
tff(fact_500_map__prod__compose,axiom,
    ! [F1: fun_nat_nat,F22: fun_nat_nat,G1: fun_o_o,G22: fun_nat_o] : produc4787215299218314158_nat_o(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F1),F22),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),G22)) = comp_P8901546994122883554at_nat(produc2001155298098703512at_o_o(F1,G1),produc4787215299218314158_nat_o(F22,G22)) ).

% map_prod_compose
tff(fact_501_map__prod__compose,axiom,
    ! [F1: fun_nat_nat,F22: fun_nat_nat,G1: fun_nat_num,G22: fun_int_nat] : produc4534572193387464160nt_num(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F1),F22),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),G22)) = comp_P6334688330072917552at_int(produc3535218481486409604at_num(F1,G1),produc7977240407231685526nt_nat(F22,G22)) ).

% map_prod_compose
tff(fact_502_apsnd__compose,axiom,
    ! [F: fun_Co8499238673658658103nteger,G: fun_Co8499238673658658103nteger,Xa: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,G),Xa)) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,aa_fun4435975922079263404nteger(comp_C7449957260575251196nteger(F),G)),Xa) ).

% apsnd_compose
tff(fact_503_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat,F: fun_se320340636488258563at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Z),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,bNF_Gr6677838525845560078at_nat(A4,F))))
     => member2643936169264416010at_nat(aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,Z),A4) ) ).

% Collect_case_prod_Grp_in
tff(fact_504_Collect__case__prod__Grp__in,axiom,
    ! [Z: product_prod_nat_nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,bNF_Grp_nat_nat(A4,F))))
     => aa_set_nat_o(member_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Z)),A4) ) ).

% Collect_case_prod_Grp_in
tff(fact_505_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc7812382424243267045t_unit,A4: set_ref_a,F: fun_re8475190471885019465t_unit] :
      ( member7849944260330398460t_unit(Z,collec3252306575900995386t_unit(produc4924168586796184916unit_o(bNF_Gr6510375311308651680t_unit(A4,F))))
     => member_ref_a(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Z),A4) ) ).

% Collect_case_prod_Grp_in
tff(fact_506_Collect__case__prod__Grp__in,axiom,
    ! [Z: product_prod_int_int,A4: set_int,F: fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Z),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,bNF_Grp_int_int(A4,F))))
     => aa_set_int_o(member_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Z)),A4) ) ).

% Collect_case_prod_Grp_in
tff(fact_507_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_num,F: fun_nat_nat,G: fun_num_num,R2: set_Pr6200539531224447659at_num] :
      ( member9148766508732265716at_num(C,image_4925883905578568901at_num(produc1665548884793878990um_num(F,G),R2))
     => ~ ! [X: nat,Y: num] :
            ( ( C = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,X)),aa_num_num(G,Y)) )
           => ~ member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y),R2) ) ) ).

% prod_fun_imageE
tff(fact_508_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_num,F: fun_nat_nat,G: fun_nat_num,R2: set_Pr1261947904930325089at_nat] :
      ( member9148766508732265716at_num(C,image_5877158439796295311at_num(produc3535218481486409604at_num(F,G),R2))
     => ~ ! [X: nat,Y: nat] :
            ( ( C = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,X)),aa_nat_num(G,Y)) )
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),R2) ) ) ).

% prod_fun_imageE
tff(fact_509_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_num,F: fun_int_nat,G: fun_int_num,R2: set_Pr958786334691620121nt_int] :
      ( member9148766508732265716at_num(C,image_6540112122905407959at_num(produc2224754261294513212nt_num(F,G),R2))
     => ~ ! [X: int,Y: int] :
            ( ( C = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_int_nat(F,X)),aa_int_num(G,Y)) )
           => ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),R2) ) ) ).

% prod_fun_imageE
tff(fact_510_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_nat_nat,G: fun_num_nat,R2: set_Pr6200539531224447659at_num] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),image_4217639968629731195at_nat(produc5108217098638100356um_nat(F,G),R2))
     => ~ ! [X: nat,Y: num] :
            ( ( C = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,X)),aa_num_nat(G,Y)) )
           => ~ member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y),R2) ) ) ).

% prod_fun_imageE
tff(fact_511_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_nat_nat,G: fun_nat_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(produc6977886695330630970at_nat(F,G)),R2))
     => ~ ! [X: nat,Y: nat] :
            ( ( C = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,X)),aa_nat_nat(G,Y)) )
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),R2) ) ) ).

% prod_fun_imageE
tff(fact_512_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: fun_int_nat,G: fun_int_nat,R2: set_Pr958786334691620121nt_int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),image_5831868185956570253at_nat(produc5667422475138734578nt_nat(F,G),R2))
     => ~ ! [X: int,Y: int] :
            ( ( C = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_int_nat(F,X)),aa_int_nat(G,Y)) )
           => ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),R2) ) ) ).

% prod_fun_imageE
tff(fact_513_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: fun_nat_int,G: fun_num_int,R2: set_Pr6200539531224447659at_num] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(C),image_1039142661021589043nt_int(produc5477380188365894716um_int(F,G),R2))
     => ~ ! [X: nat,Y: num] :
            ( ( C = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,X)),aa_num_int(G,Y)) )
           => ~ member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y),R2) ) ) ).

% prod_fun_imageE
tff(fact_514_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: fun_nat_int,G: fun_nat_int,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(C),image_1990417195239315453nt_int(produc7347049785058425330at_int(F,G),R2))
     => ~ ! [X: nat,Y: nat] :
            ( ( C = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,X)),aa_nat_int(G,Y)) )
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),R2) ) ) ).

% prod_fun_imageE
tff(fact_515_prod__fun__imageE,axiom,
    ! [C: product_prod_int_int,F: fun_int_int,G: fun_int_int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(C),image_2653370878348428101nt_int(produc6036585564866528938nt_int(F,G),R2))
     => ~ ! [X: int,Y: int] :
            ( ( C = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(F,X)),aa_int_int(G,Y)) )
           => ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),R2) ) ) ).

% prod_fun_imageE
tff(fact_516_prod__fun__imageE,axiom,
    ! [C: produc3843707927480180839at_nat,F: fun_na7044673412300895210at_nat,G: fun_nu8929577805884333344at_nat,R2: set_Pr6200539531224447659at_num] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),image_2708129733610033121at_nat(produc170006858110003436at_nat(F,G),R2))
     => ~ ! [X: nat,Y: num] :
            ( ( C = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_nat7062706319057040595at_nat(F,X)),aa_num8947610712640478729at_nat(G,Y)) )
           => ~ member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y),R2) ) ) ).

% prod_fun_imageE
tff(fact_517_rev__image__eqI,axiom,
    ! [Xa: $o,A4: set_o,B: $o,F: fun_o_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( ( (B)
        <=> aa_o_o(F,(Xa)) )
       => aa_set_o_o(member_o((B)),aa_set_o_set_o(image_o_o(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_518_rev__image__eqI,axiom,
    ! [Xa: $o,A4: set_o,B: nat,F: fun_o_nat] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( ( B = aa_o_nat(F,(Xa)) )
       => aa_set_nat_o(member_nat(B),image_o_nat(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_519_rev__image__eqI,axiom,
    ! [Xa: $o,A4: set_o,B: int,F: fun_o_int] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( ( B = aa_o_int(F,(Xa)) )
       => aa_set_int_o(member_int(B),image_o_int(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_520_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: rat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_rat(F,Xa) )
       => aa_set_rat_o(member_rat(B),aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_521_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: $o,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( (B)
        <=> aa_nat_o(F,Xa) )
       => aa_set_o_o(member_o((B)),image_nat_o(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_522_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_nat(F,Xa) )
       => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_523_rev__image__eqI,axiom,
    ! [Xa: nat,A4: set_nat,B: int,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( ( B = aa_nat_int(F,Xa) )
       => aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_524_rev__image__eqI,axiom,
    ! [Xa: int,A4: set_int,B: $o,F: fun_int_o] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( ( (B)
        <=> aa_int_o(F,Xa) )
       => aa_set_o_o(member_o((B)),image_int_o(F,A4)) ) ) ).

% rev_image_eqI
tff(fact_525_rev__image__eqI,axiom,
    ! [Xa: int,A4: set_int,B: nat,F: fun_int_nat] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( ( B = aa_int_nat(F,Xa) )
       => aa_set_nat_o(member_nat(B),aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_526_rev__image__eqI,axiom,
    ! [Xa: int,A4: set_int,B: int,F: fun_int_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( ( B = aa_int_int(F,Xa) )
       => aa_set_int_o(member_int(B),aa_set_int_set_int(image_int_int(F),A4)) ) ) ).

% rev_image_eqI
tff(fact_527_ball__imageD,axiom,
    ! [F: fun_nat_rat,A4: set_nat,P2: fun_rat_o] :
      ( ! [X: rat] :
          ( aa_set_rat_o(member_rat(X),aa_set_nat_set_rat(image_nat_rat(F),A4))
         => aa_rat_o(P2,X) )
     => ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_rat_o(P2,aa_nat_rat(F,X5)) ) ) ).

% ball_imageD
tff(fact_528_ball__imageD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(F),A4))
         => aa_nat_o(P2,X) )
     => ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_nat_o(P2,aa_nat_nat(F,X5)) ) ) ).

% ball_imageD
tff(fact_529_ball__imageD,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),aa_set_nat_set_int(image_nat_int(F),A4))
         => aa_int_o(P2,X) )
     => ! [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),A4)
         => aa_int_o(P2,aa_nat_int(F,X5)) ) ) ).

% ball_imageD
tff(fact_530_ball__imageD,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_set_int_set_nat(image_int_nat(F),A4))
         => aa_nat_o(P2,X) )
     => ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_nat_o(P2,aa_int_nat(F,X5)) ) ) ).

% ball_imageD
tff(fact_531_ball__imageD,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),aa_set_int_set_int(image_int_int(F),A4))
         => aa_int_o(P2,X) )
     => ! [X5: int] :
          ( aa_set_int_o(member_int(X5),A4)
         => aa_int_o(P2,aa_int_int(F,X5)) ) ) ).

% ball_imageD
tff(fact_532_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ( M = N )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),N)
           => ( aa_nat_rat(F,X) = aa_nat_rat(G,X) ) )
       => ( aa_set_nat_set_rat(image_nat_rat(F),M) = aa_set_nat_set_rat(image_nat_rat(G),N) ) ) ) ).

% image_cong
tff(fact_533_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( M = N )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),N)
           => ( aa_nat_nat(F,X) = aa_nat_nat(G,X) ) )
       => ( aa_set_nat_set_nat(image_nat_nat(F),M) = aa_set_nat_set_nat(image_nat_nat(G),N) ) ) ) ).

% image_cong
tff(fact_534_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ( M = N )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),N)
           => ( aa_nat_int(F,X) = aa_nat_int(G,X) ) )
       => ( aa_set_nat_set_int(image_nat_int(F),M) = aa_set_nat_set_int(image_nat_int(G),N) ) ) ) ).

% image_cong
tff(fact_535_image__cong,axiom,
    ! [M: set_int,N: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ( M = N )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),N)
           => ( aa_int_nat(F,X) = aa_int_nat(G,X) ) )
       => ( aa_set_int_set_nat(image_int_nat(F),M) = aa_set_int_set_nat(image_int_nat(G),N) ) ) ) ).

% image_cong
tff(fact_536_image__cong,axiom,
    ! [M: set_int,N: set_int,F: fun_int_int,G: fun_int_int] :
      ( ( M = N )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),N)
           => ( aa_int_int(F,X) = aa_int_int(G,X) ) )
       => ( aa_set_int_set_int(image_int_int(F),M) = aa_set_int_set_int(image_int_int(G),N) ) ) ) ).

% image_cong
tff(fact_537_bex__imageD,axiom,
    ! [F: fun_nat_rat,A4: set_nat,P2: fun_rat_o] :
      ( ? [X5: rat] :
          ( aa_set_rat_o(member_rat(X5),aa_set_nat_set_rat(image_nat_rat(F),A4))
          & aa_rat_o(P2,X5) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_rat_o(P2,aa_nat_rat(F,X)) ) ) ).

% bex_imageD
tff(fact_538_bex__imageD,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_nat_o] :
      ( ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_set_nat_set_nat(image_nat_nat(F),A4))
          & aa_nat_o(P2,X5) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_nat_o(P2,aa_nat_nat(F,X)) ) ) ).

% bex_imageD
tff(fact_539_bex__imageD,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_int_o] :
      ( ? [X5: int] :
          ( aa_set_int_o(member_int(X5),aa_set_nat_set_int(image_nat_int(F),A4))
          & aa_int_o(P2,X5) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
          & aa_int_o(P2,aa_nat_int(F,X)) ) ) ).

% bex_imageD
tff(fact_540_bex__imageD,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_nat_o] :
      ( ? [X5: nat] :
          ( aa_set_nat_o(member_nat(X5),aa_set_int_set_nat(image_int_nat(F),A4))
          & aa_nat_o(P2,X5) )
     => ? [X: int] :
          ( aa_set_int_o(member_int(X),A4)
          & aa_nat_o(P2,aa_int_nat(F,X)) ) ) ).

% bex_imageD
tff(fact_541_bex__imageD,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_int_o] :
      ( ? [X5: int] :
          ( aa_set_int_o(member_int(X5),aa_set_int_set_int(image_int_int(F),A4))
          & aa_int_o(P2,X5) )
     => ? [X: int] :
          ( aa_set_int_o(member_int(X),A4)
          & aa_int_o(P2,aa_int_int(F,X)) ) ) ).

% bex_imageD
tff(fact_542_image__iff,axiom,
    ! [Z: rat,F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(member_rat(Z),aa_set_nat_set_rat(image_nat_rat(F),A4))
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
          & ( Z = aa_nat_rat(F,X4) ) ) ) ).

% image_iff
tff(fact_543_image__iff,axiom,
    ! [Z: nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(Z),aa_set_nat_set_nat(image_nat_nat(F),A4))
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
          & ( Z = aa_nat_nat(F,X4) ) ) ) ).

% image_iff
tff(fact_544_image__iff,axiom,
    ! [Z: nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(member_nat(Z),aa_set_int_set_nat(image_int_nat(F),A4))
    <=> ? [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
          & ( Z = aa_int_nat(F,X4) ) ) ) ).

% image_iff
tff(fact_545_image__iff,axiom,
    ! [Z: int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(member_int(Z),aa_set_nat_set_int(image_nat_int(F),A4))
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
          & ( Z = aa_nat_int(F,X4) ) ) ) ).

% image_iff
tff(fact_546_image__iff,axiom,
    ! [Z: int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(member_int(Z),aa_set_int_set_int(image_int_int(F),A4))
    <=> ? [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
          & ( Z = aa_int_int(F,X4) ) ) ) ).

% image_iff
tff(fact_547_imageI,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => aa_set_o_o(member_o(aa_o_o(F,(Xa))),aa_set_o_set_o(image_o_o(F),A4)) ) ).

% imageI
tff(fact_548_imageI,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_nat] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => aa_set_nat_o(member_nat(aa_o_nat(F,(Xa))),image_o_nat(F,A4)) ) ).

% imageI
tff(fact_549_imageI,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_int] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => aa_set_int_o(member_int(aa_o_int(F,(Xa))),image_o_int(F,A4)) ) ).

% imageI
tff(fact_550_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_rat_o(member_rat(aa_nat_rat(F,Xa)),aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ).

% imageI
tff(fact_551_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_o_o(member_o(aa_nat_o(F,Xa)),image_nat_o(F,A4)) ) ).

% imageI
tff(fact_552_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_nat_o(member_nat(aa_nat_nat(F,Xa)),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% imageI
tff(fact_553_imageI,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set_int_o(member_int(aa_nat_int(F,Xa)),aa_set_nat_set_int(image_nat_int(F),A4)) ) ).

% imageI
tff(fact_554_imageI,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_o] :
      ( aa_set_int_o(member_int(Xa),A4)
     => aa_set_o_o(member_o(aa_int_o(F,Xa)),image_int_o(F,A4)) ) ).

% imageI
tff(fact_555_imageI,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(Xa),A4)
     => aa_set_nat_o(member_nat(aa_int_nat(F,Xa)),aa_set_int_set_nat(image_int_nat(F),A4)) ) ).

% imageI
tff(fact_556_imageI,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => aa_set_int_o(member_int(aa_int_int(F,Xa)),aa_set_int_set_int(image_int_int(F),A4)) ) ).

% imageI
tff(fact_557_Sup_OSUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_rat,D: fun_nat_rat,Sup: fun_set_rat_rat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_rat(C3,X) = aa_nat_rat(D,X) ) )
       => ( aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(C3),A4)) = aa_set_rat_rat(Sup,aa_set_nat_set_rat(image_nat_rat(D),B5)) ) ) ) ).

% Sup.SUP_cong
tff(fact_558_Sup_OSUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_nat,D: fun_nat_nat,Sup: fun_set_nat_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_nat(C3,X) = aa_nat_nat(D,X) ) )
       => ( aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(C3),A4)) = aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(D),B5)) ) ) ) ).

% Sup.SUP_cong
tff(fact_559_Sup_OSUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_int,D: fun_nat_int,Sup: fun_set_int_int] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_int(C3,X) = aa_nat_int(D,X) ) )
       => ( aa_set_int_int(Sup,aa_set_nat_set_int(image_nat_int(C3),A4)) = aa_set_int_int(Sup,aa_set_nat_set_int(image_nat_int(D),B5)) ) ) ) ).

% Sup.SUP_cong
tff(fact_560_Sup_OSUP__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_nat,D: fun_int_nat,Sup: fun_set_nat_nat] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_nat(C3,X) = aa_int_nat(D,X) ) )
       => ( aa_set_nat_nat(Sup,aa_set_int_set_nat(image_int_nat(C3),A4)) = aa_set_nat_nat(Sup,aa_set_int_set_nat(image_int_nat(D),B5)) ) ) ) ).

% Sup.SUP_cong
tff(fact_561_Sup_OSUP__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_int,D: fun_int_int,Sup: fun_set_int_int] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_int(C3,X) = aa_int_int(D,X) ) )
       => ( aa_set_int_int(Sup,aa_set_int_set_int(image_int_int(C3),A4)) = aa_set_int_int(Sup,aa_set_int_set_int(image_int_int(D),B5)) ) ) ) ).

% Sup.SUP_cong
tff(fact_562_Inf_OINF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_rat,D: fun_nat_rat,Inf: fun_set_rat_rat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_rat(C3,X) = aa_nat_rat(D,X) ) )
       => ( aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(C3),A4)) = aa_set_rat_rat(Inf,aa_set_nat_set_rat(image_nat_rat(D),B5)) ) ) ) ).

% Inf.INF_cong
tff(fact_563_Inf_OINF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_nat,D: fun_nat_nat,Inf: fun_set_nat_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_nat(C3,X) = aa_nat_nat(D,X) ) )
       => ( aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(C3),A4)) = aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(D),B5)) ) ) ) ).

% Inf.INF_cong
tff(fact_564_Inf_OINF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_int,D: fun_nat_int,Inf: fun_set_int_int] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_int(C3,X) = aa_nat_int(D,X) ) )
       => ( aa_set_int_int(Inf,aa_set_nat_set_int(image_nat_int(C3),A4)) = aa_set_int_int(Inf,aa_set_nat_set_int(image_nat_int(D),B5)) ) ) ) ).

% Inf.INF_cong
tff(fact_565_Inf_OINF__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_nat,D: fun_int_nat,Inf: fun_set_nat_nat] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_nat(C3,X) = aa_int_nat(D,X) ) )
       => ( aa_set_nat_nat(Inf,aa_set_int_set_nat(image_int_nat(C3),A4)) = aa_set_nat_nat(Inf,aa_set_int_set_nat(image_int_nat(D),B5)) ) ) ) ).

% Inf.INF_cong
tff(fact_566_Inf_OINF__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_int,D: fun_int_int,Inf: fun_set_int_int] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_int(C3,X) = aa_int_int(D,X) ) )
       => ( aa_set_int_int(Inf,aa_set_int_set_int(image_int_int(C3),A4)) = aa_set_int_int(Inf,aa_set_int_set_int(image_int_int(D),B5)) ) ) ) ).

% Inf.INF_cong
tff(fact_567_comp__eq__dest__lhs,axiom,
    ! [A: fun_o_o,B: fun_nat_o,C: fun_nat_o,V: nat] :
      ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,A),B) = C )
     => ( aa_o_o(A,aa_nat_o(B,V))
      <=> aa_nat_o(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_568_comp__eq__dest__lhs,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_int_num,V: int] :
      ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,A),B) = C )
     => ( aa_nat_num(A,aa_int_nat(B,V)) = aa_int_num(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_569_comp__eq__dest__lhs,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,V: nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,A),B) = C )
     => ( aa_nat_nat(A,aa_nat_nat(B,V)) = aa_nat_nat(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_570_comp__eq__dest__lhs,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,V: int] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,A),B) = C )
     => ( aa_int_nat(A,aa_int_int(B,V)) = aa_int_nat(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_571_comp__eq__dest__lhs,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_Co1561183069634336733nteger,V: code_integer] :
      ( ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(A),B) = C )
     => ( aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V)) = aa_Cod1978467581589928094nteger(C,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_572_comp__eq__elim,axiom,
    ! [A: fun_o_o,B: fun_nat_o,C: fun_o_o,D2: fun_nat_o] :
      ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,A),B) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,C),D2) )
     => ! [V3: nat] :
          ( aa_o_o(A,aa_nat_o(B,V3))
        <=> aa_o_o(C,aa_nat_o(D2,V3)) ) ) ).

% comp_eq_elim
tff(fact_573_comp__eq__elim,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_nat_num,D2: fun_int_nat] :
      ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,A),B) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,C),D2) )
     => ! [V3: int] : aa_nat_num(A,aa_int_nat(B,V3)) = aa_nat_num(C,aa_int_nat(D2,V3)) ) ).

% comp_eq_elim
tff(fact_574_comp__eq__elim,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,D2: fun_nat_nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,A),B) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,C),D2) )
     => ! [V3: nat] : aa_nat_nat(A,aa_nat_nat(B,V3)) = aa_nat_nat(C,aa_nat_nat(D2,V3)) ) ).

% comp_eq_elim
tff(fact_575_comp__eq__elim,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,D2: fun_int_int] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,A),B) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,C),D2) )
     => ! [V3: int] : aa_int_nat(A,aa_int_int(B,V3)) = aa_int_nat(C,aa_int_int(D2,V3)) ) ).

% comp_eq_elim
tff(fact_576_comp__eq__elim,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_fu4800041728187187127nteger,D2: fun_Co1981900195452921617nteger] :
      ( ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(A),B) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(C),D2) )
     => ! [V3: code_integer] : aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V3)) = aa_fun2509385170507305080nteger(C,aa_Cod4161879620478718930nteger(D2,V3)) ) ).

% comp_eq_elim
tff(fact_577_comp__eq__dest,axiom,
    ! [A: fun_o_o,B: fun_nat_o,C: fun_o_o,D2: fun_nat_o,V: nat] :
      ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,A),B) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,C),D2) )
     => ( aa_o_o(A,aa_nat_o(B,V))
      <=> aa_o_o(C,aa_nat_o(D2,V)) ) ) ).

% comp_eq_dest
tff(fact_578_comp__eq__dest,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_nat_num,D2: fun_int_nat,V: int] :
      ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,A),B) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,C),D2) )
     => ( aa_nat_num(A,aa_int_nat(B,V)) = aa_nat_num(C,aa_int_nat(D2,V)) ) ) ).

% comp_eq_dest
tff(fact_579_comp__eq__dest,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,D2: fun_nat_nat,V: nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,A),B) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,C),D2) )
     => ( aa_nat_nat(A,aa_nat_nat(B,V)) = aa_nat_nat(C,aa_nat_nat(D2,V)) ) ) ).

% comp_eq_dest
tff(fact_580_comp__eq__dest,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,D2: fun_int_int,V: int] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,A),B) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,C),D2) )
     => ( aa_int_nat(A,aa_int_int(B,V)) = aa_int_nat(C,aa_int_int(D2,V)) ) ) ).

% comp_eq_dest
tff(fact_581_comp__eq__dest,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_fu4800041728187187127nteger,D2: fun_Co1981900195452921617nteger,V: code_integer] :
      ( ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(A),B) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(C),D2) )
     => ( aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V)) = aa_fun2509385170507305080nteger(C,aa_Cod4161879620478718930nteger(D2,V)) ) ) ).

% comp_eq_dest
tff(fact_582_comp__assoc,axiom,
    ! [F: fun_o_o,G: fun_nat_o,Ha: fun_nat_nat] : comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),comp_nat_o_nat(G,Ha)) ).

% comp_assoc
tff(fact_583_comp__assoc,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Ha: fun_int_int] : aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G)),Ha) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha)) ).

% comp_assoc
tff(fact_584_comp__assoc,axiom,
    ! [F: fun_o_o,G: fun_o_o,Ha: fun_nat_o] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(F,G)),Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ha)) ).

% comp_assoc
tff(fact_585_comp__assoc,axiom,
    ! [F: fun_num_num,G: fun_nat_num,Ha: fun_int_nat] : aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_num_num_nat(F,G)),Ha) = aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,F),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) ).

% comp_assoc
tff(fact_586_comp__assoc,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,Ha: fun_int_nat] : aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_nat_num_nat(F,G)),Ha) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),Ha)) ).

% comp_assoc
tff(fact_587_comp__assoc,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Ha: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G)),Ha) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Ha)) ).

% comp_assoc
tff(fact_588_comp__assoc,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,Ha: fun_int_int] : aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),G)),Ha) = aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha)) ).

% comp_assoc
tff(fact_589_comp__assoc,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Ha: fun_int_int] : aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G)),Ha) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),comp_int_int_int(G,Ha)) ).

% comp_assoc
tff(fact_590_comp__assoc,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Ha: fun_Co8499238673658658103nteger] : comp_C7848664941760151970nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Ha) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),comp_C7233611567195229654nteger(G,Ha)) ).

% comp_assoc
tff(fact_591_comp__assoc,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu1763652423599918571nteger,Ha: fun_Co1981900195452921617nteger] : aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(F,G)),Ha) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),comp_f228016600672082096nteger(G,Ha)) ).

% comp_assoc
tff(fact_592_comp__def,axiom,
    ! [F: fun_o_o,G: fun_nat_o,X5: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),X5)
    <=> aa_o_o(F,aa_nat_o(G,X5)) ) ).

% comp_def
tff(fact_593_comp__def,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,X5: int] : aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),X5) = aa_nat_num(F,aa_int_nat(G,X5)) ).

% comp_def
tff(fact_594_comp__def,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,X5: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),X5) = aa_nat_nat(F,aa_nat_nat(G,X5)) ).

% comp_def
tff(fact_595_comp__def,axiom,
    ! [F: fun_int_nat,G: fun_int_int,X5: int] : aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),X5) = aa_int_nat(F,aa_int_int(G,X5)) ).

% comp_def
tff(fact_596_comp__def,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,X5: code_integer] : aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),X5) = aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,X5)) ).

% comp_def
tff(fact_597_apfst__convE,axiom,
    ! [Q: produc1319942482725812455at_nat,F: fun_se6846035212209387651at_nat,P: produc1319942482725812455at_nat] :
      ( ( Q = produc6529295065807390923at_nat(F,P) )
     => ~ ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
            ( ( P = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),Y) )
           => ( Q != aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_set4883462369968598652at_nat(F,X)),Y) ) ) ) ).

% apfst_convE
tff(fact_598_apfst__convE,axiom,
    ! [Q: produc3843707927480180839at_nat,F: fun_se320340636488258563at_nat,P: produc3843707927480180839at_nat] :
      ( ( Q = aa_Pro9111386209873968764at_nat(produc5380969780118646373at_nat(F),P) )
     => ~ ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
            ( ( P = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),Y) )
           => ( Q != aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_set8868280134176360700at_nat(F,X)),Y) ) ) ) ).

% apfst_convE
tff(fact_599_apfst__convE,axiom,
    ! [Q: product_prod_nat_num,F: fun_nat_nat,P: product_prod_nat_num] :
      ( ( Q = produc6767424583952363640at_num(F,P) )
     => ~ ! [X: nat,Y: num] :
            ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) )
           => ( Q != aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,aa_nat_nat(F,X)),Y) ) ) ) ).

% apfst_convE
tff(fact_600_apfst__convE,axiom,
    ! [Q: product_prod_nat_nat,F: fun_nat_nat,P: product_prod_nat_nat] :
      ( ( Q = aa_Pro262675715343573244at_nat(produc986720760941809198at_nat(F),P) )
     => ~ ! [X: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ( Q != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,X)),Y) ) ) ) ).

% apfst_convE
tff(fact_601_apfst__convE,axiom,
    ! [Q: product_prod_int_int,F: fun_int_int,P: product_prod_int_int] :
      ( ( Q = aa_Pro6970504127699319548nt_int(produc2355237562570658114nt_int(F),P) )
     => ~ ! [X: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ( Q != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(F,X)),Y) ) ) ) ).

% apfst_convE
tff(fact_602_prod_Omap__comp,axiom,
    ! [G1: fun_o_o,G22: fun_o_o,F1: fun_nat_o,F22: fun_nat_o,V: product_prod_nat_nat] : aa_Pro5376729675435676284od_o_o(produc1928436952130685430_o_o_o(G1,G22),aa_Pro9072700582319063714od_o_o(produc6957950696844810486_nat_o(F1,F22),V)) = aa_Pro9072700582319063714od_o_o(produc6957950696844810486_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),F1),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G22),F22)),V) ).

% prod.map_comp
tff(fact_603_prod_Omap__comp,axiom,
    ! [G1: fun_o_o,G22: fun_nat_num,F1: fun_nat_o,F22: fun_int_nat,V: product_prod_nat_int] : aa_Pro279734039161488822_o_num(produc3628557804203378146at_num(G1,G22),aa_Pro5791804135702779960_o_nat(produc8250590009387450446nt_nat(F1,F22),V)) = aa_Pro6500048072651617666_o_num(produc4807921795543229080nt_num(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),F1),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G22),F22)),V) ).

% prod.map_comp
tff(fact_604_prod_Omap__comp,axiom,
    ! [G1: fun_o_o,G22: fun_nat_nat,F1: fun_nat_o,F22: fun_nat_nat,V: product_prod_nat_nat] : aa_Pro8794862139067426924_o_nat(produc7071226018047599512at_nat(G1,G22),aa_Pro8467631284203683292_o_nat(produc7251236297486395890at_nat(F1,F22),V)) = aa_Pro8467631284203683292_o_nat(produc7251236297486395890at_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),F1),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G22),F22)),V) ).

% prod.map_comp
tff(fact_605_prod_Omap__comp,axiom,
    ! [G1: fun_o_o,G22: fun_int_nat,F1: fun_nat_o,F22: fun_int_int,V: product_prod_nat_int] : aa_Pro6119034990566523592_o_nat(produc8070579729948654068nt_nat(G1,G22),aa_Pro1613953116193583252_o_int(produc8248099538878400170nt_int(F1,F22),V)) = aa_Pro5791804135702779960_o_nat(produc8250590009387450446nt_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),F1),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G22),F22)),V) ).

% prod.map_comp
tff(fact_606_prod_Omap__comp,axiom,
    ! [G1: fun_nat_num,G22: fun_o_o,F1: fun_int_nat,F22: fun_nat_o,V: product_prod_int_nat] : aa_Pro7407088302862950382_num_o(produc3264868786340801506um_o_o(G1,G22),aa_Pro4720224260291850206_nat_o(produc8298419091753553490_nat_o(F1,F22),V)) = aa_Pro3566207295127809812_num_o(produc9074608839065728392_nat_o(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),F1),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G22),F22)),V) ).

% prod.map_comp
tff(fact_607_prod_Omap__comp,axiom,
    ! [G1: fun_nat_num,G22: fun_nat_num,F1: fun_int_nat,F22: fun_int_nat,V: product_prod_int_int] : aa_Pro8324622092454656144um_num(produc1476026309457068494at_num(G1,G22),aa_Pro925629398452685892at_nat(produc5667422475138734578nt_nat(F1,F22),V)) = aa_Pro8987575775563768792um_num(produc165562089265172102nt_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),F1),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G22),F22)),V) ).

% prod.map_comp
tff(fact_608_prod_Omap__comp,axiom,
    ! [G1: fun_nat_num,G22: fun_nat_nat,F1: fun_int_nat,F22: fun_nat_nat,V: product_prod_int_nat] : aa_Pro7616378155505818438um_nat(produc4918694523301289860at_nat(G1,G22),aa_Pro2022712379096684448at_nat(produc4668068763237680022at_nat(F1,F22),V)) = aa_Pro153042782404153834um_nat(produc2608876591208338912at_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),F1),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G22),F22)),V) ).

% prod.map_comp
tff(fact_609_prod_Omap__comp,axiom,
    ! [G1: fun_nat_num,G22: fun_int_nat,F1: fun_int_nat,F22: fun_int_int,V: product_prod_int_int] : aa_Pro6519295174861819882um_nat(produc5918048235202344416nt_nat(G1,G22),aa_Pro5971150415798264992at_int(produc5664932004629684302nt_int(F1,F22),V)) = aa_Pro8279331838614931086um_nat(produc3608230303109393468nt_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),F1),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G22),F22)),V) ).

% prod.map_comp
tff(fact_610_prod_Omap__comp,axiom,
    ! [G1: fun_nat_nat,G22: fun_o_o,F1: fun_nat_nat,F22: fun_nat_o,V: product_prod_nat_nat] : aa_Pro8561105268026990776_nat_o(produc2001155298098703512at_o_o(G1,G22),aa_Pro2752793771404400514_nat_o(produc4787215299218314158_nat_o(F1,F22),V)) = aa_Pro2752793771404400514_nat_o(produc4787215299218314158_nat_o(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G1),F1),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G22),F22)),V) ).

% prod.map_comp
tff(fact_611_prod_Omap__comp,axiom,
    ! [G1: fun_nat_nat,G22: fun_nat_num,F1: fun_nat_nat,F22: fun_int_nat,V: product_prod_nat_int] : aa_Pro970919652292410950at_num(produc3535218481486409604at_num(G1,G22),aa_Pro8388964771554350496at_nat(produc7977240407231685526nt_nat(F1,F22),V)) = aa_Pro9097208708503188202at_num(produc4534572193387464160nt_num(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G1),F1),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G22),F22)),V) ).

% prod.map_comp
tff(fact_612_subrelI,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
          ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),Y),Ra)
         => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),Y),S) )
     => ord_le5997549366648089703at_nat(Ra,S) ) ).

% subrelI
tff(fact_613_subrelI,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),Y)),Ra)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),Y)),S) )
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,Ra),S) ) ).

% subrelI
tff(fact_614_subrelI,axiom,
    ! [Ra: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
      ( ! [X: nat,Y: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y),Ra)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y),S) )
     => ord_le8085105155179020875at_num(Ra,S) ) ).

% subrelI
tff(fact_615_subrelI,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),Ra)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),S) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),S) ) ).

% subrelI
tff(fact_616_subrelI,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Y: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),Ra)
         => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),S) )
     => ord_le2843351958646193337nt_int(Ra,S) ) ).

% subrelI
tff(fact_617_all__subset__image,axiom,
    ! [F: fun_nat_rat,A4: set_nat,P2: fun_set_rat_o] :
      ( ! [B6: set_rat] :
          ( aa_set_rat_o(ord_less_eq_set_rat(B6),aa_set_nat_set_rat(image_nat_rat(F),A4))
         => aa_set_rat_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
         => aa_set_rat_o(P2,aa_set_nat_set_rat(image_nat_rat(F),B6)) ) ) ).

% all_subset_image
tff(fact_618_all__subset__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(B6),aa_set_nat_set_nat(image_nat_nat(F),A4))
         => aa_set_nat_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
         => aa_set_nat_o(P2,aa_set_nat_set_nat(image_nat_nat(F),B6)) ) ) ).

% all_subset_image
tff(fact_619_all__subset__image,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B6),aa_set_nat_set_int(image_nat_int(F),A4))
         => aa_set_int_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
         => aa_set_int_o(P2,aa_set_nat_set_int(image_nat_int(F),B6)) ) ) ).

% all_subset_image
tff(fact_620_all__subset__image,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(B6),aa_set_int_set_nat(image_int_nat(F),A4))
         => aa_set_nat_o(P2,B6) )
    <=> ! [B6: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B6),A4)
         => aa_set_nat_o(P2,aa_set_int_set_nat(image_int_nat(F),B6)) ) ) ).

% all_subset_image
tff(fact_621_all__subset__image,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B6),aa_set_int_set_int(image_int_int(F),A4))
         => aa_set_int_o(P2,B6) )
    <=> ! [B6: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(B6),A4)
         => aa_set_int_o(P2,aa_set_int_set_int(image_int_int(F),B6)) ) ) ).

% all_subset_image
tff(fact_622_rel__fun__Collect__case__prodD,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat,X3: set_Pr49084382684228242at_int,Xa: produc4196048018000912732at_int] :
      ( aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(A4,B5),F),G)
     => ( ord_le4604737605482584818at_int(X3,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,A4)))
       => ( member8394280151737582451at_int(Xa,X3)
         => aa_nat_o(aa_nat_fun_nat_o(B5,comp_P8635049891247748876at_int(F,produc4619704898428564714at_int,Xa)),comp_i5627883460353289149at_int(G,produc3064508320661873192at_int,Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_623_rel__fun__Collect__case__prodD,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int,X3: set_Pr49084382684228242at_int,Xa: produc4196048018000912732at_int] :
      ( bNF_re8869769828555196650nt_int(A4,B5,F,G)
     => ( ord_le4604737605482584818at_int(X3,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,A4)))
       => ( member8394280151737582451at_int(Xa,X3)
         => aa_fun_int_int_o(aa_fun856177901615811547_int_o(B5,comp_P9048719235609908799at_int(F,produc4619704898428564714at_int,Xa)),comp_i8088662155339470274at_int(G,produc3064508320661873192at_int,Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_624_rel__fun__Collect__case__prodD,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o,X3: set_Pr49084382684228242at_int,Xa: produc4196048018000912732at_int] :
      ( bNF_re4195927319312321177_int_o(A4,B5,F,G)
     => ( ord_le4604737605482584818at_int(X3,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,A4)))
       => ( member8394280151737582451at_int(Xa,X3)
         => aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(B5,comp_P2384775063863716054at_int(F,produc4619704898428564714at_int,Xa)),comp_i3006441595874701266at_int(G,produc3064508320661873192at_int,Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_625_rel__fun__Collect__case__prodD,axiom,
    ! [A4: fun_in4891666191172872833eger_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num,X3: set_Pr8415741480598773322nteger,Xa: produc2963269191615333738nteger] :
      ( aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(A4,B5),F),G)
     => ( ord_le2638047946463454698nteger(X3,collec7215431352809750357nteger(aa_fun7359524620130742070eger_o(produc540600059291021473eger_o,A4)))
       => ( member4345257455949424659nteger(Xa,X3)
         => aa_num_o(aa_num_fun_num_o(B5,comp_i3846818780818535807nteger(F,produc45500017171379470nteger,Xa)),comp_C8580465762518603278nteger(G,produc2153544567185597776nteger,Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_626_rel__fun__Collect__case__prodD,axiom,
    ! [A4: fun_int_fun_int_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num,X3: set_Pr958786334691620121nt_int,Xa: product_prod_int_int] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(A4,B5),F),G)
     => ( ord_le2843351958646193337nt_int(X3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A4)))
       => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Xa),X3)
         => aa_num_o(aa_num_fun_num_o(B5,aa_Pro3359080691130043031nt_num(comp_i7231038107349025742nt_int(F,product_fst_int_int),Xa)),aa_Pro3359080691130043031nt_num(comp_i7231038107349025742nt_int(G,product_snd_int_int),Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_627_convol__image__vimage2p,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,R2: fun_nat_fun_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(bNF_co8276296344278409722at_nat(aa_fun2351638177499875132at_nat(aa_fun2514183836563211038at_nat(comp_n2711240574518091068at_nat,F),product_fst_nat_nat),aa_fun2351638177499875132at_nat(aa_fun2514183836563211038at_nat(comp_n2711240574518091068at_nat,G),product_snd_nat_nat))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,bNF_vi6667456707345531737_nat_o(F,G,R2))))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R2))) ).

% convol_image_vimage2p
tff(fact_628_convol__image__vimage2p,axiom,
    ! [F: fun_ref_a_nat,G: fun_he1033112309995878576it_nat,R2: fun_nat_fun_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_4710180989675653397at_nat(bNF_co5610825919353547504at_nat(comp_r8155216869944865719t_unit(F,produc473321116828791851t_unit),comp_h8354993580257045729t_unit(G,produc1126150082675451753t_unit)),collec3252306575900995386t_unit(produc4924168586796184916unit_o(bNF_vi4623496019404053219_nat_o(F,G,R2))))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R2))) ).

% convol_image_vimage2p
tff(fact_629_convol__image__vimage2p,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,R2: fun_int_fun_int_o] : ord_le2843351958646193337nt_int(image_1990417195239315453nt_int(bNF_co49787548815638194nt_int(comp_n5749082658982224864at_nat(F,product_fst_nat_nat),comp_n5749082658982224864at_nat(G,product_snd_nat_nat)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,bNF_vi8357618154448083105_int_o(F,G,R2)))),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R2))) ).

% convol_image_vimage2p
tff(fact_630_convol__image__vimage2p,axiom,
    ! [F: fun_ref_a_int,G: fun_he6078633327341457676it_int,R2: fun_int_fun_int_o] : ord_le2843351958646193337nt_int(image_1531683682067511245nt_int(bNF_co6607689160745551784nt_int(comp_r7249815618783853587t_unit(F,produc473321116828791851t_unit),comp_h7449592329096033597t_unit(G,produc1126150082675451753t_unit)),collec3252306575900995386t_unit(produc4924168586796184916unit_o(bNF_vi7828673959370436579_int_o(F,G,R2)))),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R2))) ).

% convol_image_vimage2p
tff(fact_631_convol__image__vimage2p,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,R2: fun_nat_fun_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5831868185956570253at_nat(bNF_co2751916026326257842at_nat(comp_i1292779930663060120nt_int(F,product_fst_int_int),comp_i1292779930663060120nt_int(G,product_snd_int_int)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,bNF_vi2193923016096921761_nat_o(F,G,R2))))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R2))) ).

% convol_image_vimage2p
tff(fact_632_convol__image__vimage2p,axiom,
    ! [F: fun_int_int,G: fun_int_int,R2: fun_int_fun_int_o] : ord_le2843351958646193337nt_int(image_2653370878348428101nt_int(bNF_co3748779267718262122nt_int(comp_i4330622015127193916nt_int(F,product_fst_int_int),comp_i4330622015127193916nt_int(G,product_snd_int_int)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,bNF_vi3884084463199473129_int_o(F,G,R2)))),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R2))) ).

% convol_image_vimage2p
tff(fact_633_prod_Osize__gen__o__map,axiom,
    ! [F: fun_nat_nat,Fa: fun_nat_nat,G: fun_nat_nat,Ga: fun_nat_nat] : comp_P1567445206330693457at_nat(basic_876126793109182934at_nat(F,Fa),produc6977886695330630970at_nat(G,Ga)) = basic_876126793109182934at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Fa),Ga)) ).

% prod.size_gen_o_map
tff(fact_634_prod_Osize__gen__o__map,axiom,
    ! [F: fun_nat_nat,Fa: fun_int_nat,G: fun_nat_nat,Ga: fun_int_int] : comp_P397146161216181329at_int(basic_873636322600132658at_int(F,Fa),produc7974749936722635250nt_int(G,Ga)) = basic_873636322600132658at_int(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Fa),Ga)) ).

% prod.size_gen_o_map
tff(fact_635_prod_Osize__gen__o__map,axiom,
    ! [F: fun_int_nat,Fa: fun_nat_nat,G: fun_int_int,Ga: fun_nat_nat] : comp_P3488689526429045841nt_nat(basic_1875480505010237490nt_nat(F,Fa),produc5039722323474524658at_nat(G,Ga)) = basic_1875480505010237490nt_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Fa),Ga)) ).

% prod.size_gen_o_map
tff(fact_636_prod_Osize__gen__o__map,axiom,
    ! [F: fun_int_nat,Fa: fun_int_nat,G: fun_int_int,Ga: fun_int_int] : comp_P2318390481314533713nt_int(basic_1872990034501187214nt_int(F,Fa),produc6036585564866528938nt_int(G,Ga)) = basic_1872990034501187214nt_int(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Fa),Ga)) ).

% prod.size_gen_o_map
tff(fact_637_relChain__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,As: fun_nat_nat] :
      ( bNF_Ca968750328013420230at_nat(Ra,As)
    <=> ! [I2: nat,J: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,I2),J)),Ra)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(As,I2)),aa_nat_nat(As,J)) ) ) ).

% relChain_def
tff(fact_638_relChain__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,As: fun_int_nat] :
      ( bNF_Ca1968104039914474786nt_nat(Ra,As)
    <=> ! [I2: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J)),Ra)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(As,I2)),aa_int_nat(As,J)) ) ) ).

% relChain_def
tff(fact_639_relChain__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,As: fun_nat_int] :
      ( bNF_Ca966259857504369954at_int(Ra,As)
    <=> ! [I2: nat,J: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,I2),J)),Ra)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(As,I2)),aa_nat_int(As,J)) ) ) ).

% relChain_def
tff(fact_640_relChain__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,As: fun_int_int] :
      ( bNF_Ca1965613569405424510nt_int(Ra,As)
    <=> ! [I2: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J)),Ra)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(As,I2)),aa_int_int(As,J)) ) ) ).

% relChain_def
tff(fact_641_relChain__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,As: fun_nat_rat] :
      ( bNF_Ca333620267926924494at_rat(Ra,As)
    <=> ! [I2: nat,J: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,I2),J)),Ra)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(As,I2)),aa_nat_rat(As,J)) ) ) ).

% relChain_def
tff(fact_642_relChain__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,As: fun_int_rat] :
      ( bNF_Ca1332973979827979050nt_rat(Ra,As)
    <=> ! [I2: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J)),Ra)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(As,I2)),aa_int_rat(As,J)) ) ) ).

% relChain_def
tff(fact_643_relChain__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,As: fun_nat_Code_integer] :
      ( bNF_Ca30959104405961555nteger(Ra,As)
    <=> ! [I2: nat,J: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,I2),J)),Ra)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(As,I2)),aa_nat_Code_integer(As,J)) ) ) ).

% relChain_def
tff(fact_644_relChain__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,As: fun_int_Code_integer] :
      ( bNF_Ca402612664642806191nteger(Ra,As)
    <=> ! [I2: int,J: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J)),Ra)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(As,I2)),aa_int_Code_integer(As,J)) ) ) ).

% relChain_def
tff(fact_645_relChain__def,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,As: fun_se4097146774670233028at_nat] :
      ( bNF_Ca4949091444211332621at_nat(Ra,As)
    <=> ! [I2: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,I2),J)),Ra)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(As,I2)),aa_set4749074711808745205at_nat(As,J)) ) ) ).

% relChain_def
tff(fact_646_relChain__def,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,As: fun_se9142667792015812128at_int] :
      ( bNF_Ca4946600973702282345at_int(Ra,As)
    <=> ! [I2: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,I2),J)),Ra)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set4746584241299694929at_int(As,I2)),aa_set4746584241299694929at_int(As,J)) ) ) ).

% relChain_def
tff(fact_647_Collect__case__prod__in__rel__leI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y3)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,fun_in_rel_nat_nat(Y3)))) ) ).

% Collect_case_prod_in_rel_leI
tff(fact_648_Collect__case__prod__in__rel__leI,axiom,
    ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
      ( ord_le2843351958646193337nt_int(X3,Y3)
     => ord_le2843351958646193337nt_int(X3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,fun_in_rel_int_int(Y3)))) ) ).

% Collect_case_prod_in_rel_leI
tff(fact_649_Collect__case__prod__in__rel__leE,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,fun_in_rel_nat_nat(Y3))))
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),Y3) ) ).

% Collect_case_prod_in_rel_leE
tff(fact_650_Collect__case__prod__in__rel__leE,axiom,
    ! [X3: set_Pr958786334691620121nt_int,Y3: set_Pr958786334691620121nt_int] :
      ( ord_le2843351958646193337nt_int(X3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,fun_in_rel_int_int(Y3))))
     => ord_le2843351958646193337nt_int(X3,Y3) ) ).

% Collect_case_prod_in_rel_leE
tff(fact_651_relcompp__assoc,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,S: fun_Pr4738963024209204295_int_o,T: fun_int_fun_int_o] : relcom2949942408940504938nt_int(relcom8911704206367525917at_int(Ra,S),T) = relcom8911704206367525917at_int(Ra,relcom2949942408940504938nt_int(S,T)) ).

% relcompp_assoc
tff(fact_652_relcompp__assoc,axiom,
    ! [Ra: fun_int_fun_int_o,S: fun_in4891666191172872833eger_o,T: fun_Co2081214437782444048eger_o] : relcom8320293257703906841nteger(relcom967497705283160040nteger(Ra,S),T) = relcom967497705283160040nteger(Ra,relcom8320293257703906841nteger(S,T)) ).

% relcompp_assoc
tff(fact_653_relcompp__assoc,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,S: fun_Pr2028404973800837934_nat_o,T: fun_Pr4738963024209204295_int_o] : relcom8911704206367525917at_int(relcom282571730187242960at_nat(Ra,S),T) = relcom8911704206367525917at_int(Ra,relcom8911704206367525917at_int(S,T)) ).

% relcompp_assoc
tff(fact_654_relcompp__assoc,axiom,
    ! [Ra: fun_int_fun_int_o,S: fun_int_fun_int_o,T: fun_in4891666191172872833eger_o] : relcom967497705283160040nteger(relcompp_int_int_int(Ra,S),T) = relcom967497705283160040nteger(Ra,relcom967497705283160040nteger(S,T)) ).

% relcompp_assoc
tff(fact_655_relcompp__apply,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o,A: product_prod_nat_nat,C: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(relcom8911704206367525917at_int(R2,S3),A),C)
    <=> ? [B4: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R2,A),B4)
          & aa_int_o(aa_Pro2912763917461432128_int_o(S3,B4),C) ) ) ).

% relcompp_apply
tff(fact_656_relcompp__apply,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o,A: int,C: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(relcom967497705283160040nteger(R2,S3),A),C)
    <=> ? [B4: int] :
          ( aa_int_o(aa_int_fun_int_o(R2,A),B4)
          & aa_Code_integer_o(aa_int5993532643906289898eger_o(S3,B4),C) ) ) ).

% relcompp_apply
tff(fact_657_rel__funI,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat] :
      ( ! [X: product_prod_nat_nat,Y: int] :
          ( aa_int_o(aa_Pro2912763917461432128_int_o(A4,X),Y)
         => aa_nat_o(aa_nat_fun_nat_o(B5,aa_Pro4243173770044397973at_nat(F,X)),aa_int_nat(G,Y)) )
     => aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(A4,B5),F),G) ) ).

% rel_funI
tff(fact_658_rel__funI,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int] :
      ( ! [X: product_prod_nat_nat,Y: int] :
          ( aa_int_o(aa_Pro2912763917461432128_int_o(A4,X),Y)
         => aa_fun_int_int_o(aa_fun856177901615811547_int_o(B5,aa_Pro8023792847356852296at_nat(F,X)),aa_int_fun_int_int(G,Y)) )
     => bNF_re8869769828555196650nt_int(A4,B5,F,G) ) ).

% rel_funI
tff(fact_659_rel__funI,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o] :
      ( ! [X: product_prod_nat_nat,Y: int] :
          ( aa_int_o(aa_Pro2912763917461432128_int_o(A4,X),Y)
         => aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(B5,aa_Pro989595756709598047_nat_o(F,X)),aa_int_fun_int_o(G,Y)) )
     => bNF_re4195927319312321177_int_o(A4,B5,F,G) ) ).

% rel_funI
tff(fact_660_rel__funI,axiom,
    ! [A4: fun_in4891666191172872833eger_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num] :
      ( ! [X: int,Y: code_integer] :
          ( aa_Code_integer_o(aa_int5993532643906289898eger_o(A4,X),Y)
         => aa_num_o(aa_num_fun_num_o(B5,aa_int_num(F,X)),aa_Code_integer_num(G,Y)) )
     => aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(A4,B5),F),G) ) ).

% rel_funI
tff(fact_661_rel__funI,axiom,
    ! [A4: fun_int_fun_int_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num] :
      ( ! [X: int,Y: int] :
          ( aa_int_o(aa_int_fun_int_o(A4,X),Y)
         => aa_num_o(aa_num_fun_num_o(B5,aa_int_num(F,X)),aa_int_num(G,Y)) )
     => aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(A4,B5),F),G) ) ).

% rel_funI
tff(fact_662_fun_Omap__transfer,axiom,
    ! [Rb: fun_o_fun_o_o,Sd: fun_o_fun_o_o] : bNF_re3325573624036370925_nat_o(bNF_rel_fun_o_o_o_o(Rb,Sd),bNF_re7140302142790738653_nat_o(bNF_re4705727531993890431at_o_o(fequal_nat,Rb),bNF_re4705727531993890431at_o_o(fequal_nat,Sd)),comp_o_o_nat,comp_o_o_nat) ).

% fun.map_transfer
tff(fact_663_fun_Omap__transfer,axiom,
    ! [Rb: fun_o_fun_nat_o,Sd: fun_o_fun_nat_o] : bNF_re2134824042319328379at_nat(bNF_re4209105892566519389_o_nat(Rb,Sd),bNF_re5947481727989837021at_nat(bNF_re7523573913982466217_o_nat(fequal_nat,Rb),bNF_re7523573913982466217_o_nat(fequal_nat,Sd)),comp_o_o_nat,comp_nat_nat_nat) ).

% fun.map_transfer
tff(fact_664_fun_Omap__transfer,axiom,
    ! [Rb: fun_nat_fun_int_o,Sd: fun_num_fun_nat_o] : bNF_re7052405848218697413nt_nat(bNF_re4155804983283621959um_nat(Rb,Sd),bNF_re7620155230884812267nt_nat(bNF_re3713166177374151349at_int(fequal_int,Rb),bNF_re1845987051190671011um_nat(fequal_int,Sd)),comp_nat_num_int,comp_int_nat_int) ).

% fun.map_transfer
tff(fact_665_fun_Omap__transfer,axiom,
    ! [Rb: fun_nat_fun_o_o,Sd: fun_nat_fun_o_o] : bNF_re6900549463078501243_nat_o(bNF_re439150893885221597_nat_o(Rb,Sd),bNF_re1097902388772303837_nat_o(bNF_re1402873238235875719_nat_o(fequal_nat,Rb),bNF_re1402873238235875719_nat_o(fequal_nat,Sd)),comp_nat_nat_nat,comp_o_o_nat) ).

% fun.map_transfer
tff(fact_666_fun_Omap__transfer,axiom,
    ! [Rb: fun_nat_fun_nat_o,Sd: fun_nat_fun_nat_o] : bNF_re8005335818930388233at_nat(bNF_re5653821019739307937at_nat(Rb,Sd),bNF_re2068050147316706005at_nat(bNF_re5653821019739307937at_nat(fequal_nat,Rb),bNF_re5653821019739307937at_nat(fequal_nat,Sd)),comp_nat_nat_nat,comp_nat_nat_nat) ).

% fun.map_transfer
tff(fact_667_fun_Omap__transfer,axiom,
    ! [Rb: fun_int_fun_nat_o,Sd: fun_nat_fun_num_o] : bNF_re8542743173187387077nt_num(bNF_re9124706910656911431at_num(Rb,Sd),bNF_re4431224253247398379nt_num(bNF_re4715010359784256181nt_nat(fequal_int,Rb),bNF_re272988434038980259at_num(fequal_int,Sd)),comp_int_nat_int,comp_nat_num_int) ).

% fun.map_transfer
tff(fact_668_fun_Omap__transfer,axiom,
    ! [Rb: fun_int_fun_int_o,Sd: fun_nat_fun_nat_o] : bNF_re6308950812890564585nt_nat(bNF_re3715656647883201625at_nat(Rb,Sd),bNF_re4932164143273258237nt_nat(bNF_re4712519889275205905nt_int(fequal_int,Rb),bNF_re3715656647883201625at_nat(fequal_int,Sd)),comp_int_nat_int,comp_int_nat_int) ).

% fun.map_transfer
tff(fact_669_fun_Omap__transfer,axiom,
    ! [Rb: fun_nat_fun_nat_o,Sd: fun_num_fun_num_o] : bNF_re62826171660744097nt_num(bNF_re341483209202555957um_num(Rb,Sd),bNF_re7119215340858952409nt_num(bNF_re3715656647883201625at_nat(fequal_int,Rb),bNF_re7626690874201225453um_num(fequal_int,Sd)),comp_nat_num_int,comp_nat_num_int) ).

% fun.map_transfer
tff(fact_670_fun_Omap__transfer,axiom,
    ! [Rb: fun_num_fun_num_o,Sd: fun_num_fun_num_o] : bNF_re1151091782021442761nt_num(bNF_re4736500879113541833um_num(Rb,Sd),bNF_re2388990614730936685nt_num(bNF_re7626690874201225453um_num(fequal_int,Rb),bNF_re7626690874201225453um_num(fequal_int,Sd)),comp_num_num_int,comp_num_num_int) ).

% fun.map_transfer
tff(fact_671_fun_Omap__transfer,axiom,
    ! [Rb: fun_in4891666191172872833eger_o,Sd: fun_num_fun_num_o] : bNF_re2207678385751437555nt_num(bNF_re6718328864250387230um_num(Rb,Sd),bNF_re1332718545512871234nt_num(bNF_re982302072995117890nteger(fequal_int,Rb),bNF_re7626690874201225453um_num(fequal_int,Sd)),comp_int_num_int,comp_C2177396528461469320um_int) ).

% fun.map_transfer
tff(fact_672_rel__fun__iff__leq__vimage2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat] :
      ( aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(R2,S3),F),G)
    <=> ord_le1980079291468326055_int_o(R2,bNF_vi5493721401438417730_nat_o(F,G,S3)) ) ).

% rel_fun_iff_leq_vimage2p
tff(fact_673_rel__fun__iff__leq__vimage2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int] :
      ( bNF_re8869769828555196650nt_int(R2,S3,F,G)
    <=> ord_le1980079291468326055_int_o(R2,bNF_vi1906344552888871946_int_o(F,G,S3)) ) ).

% rel_fun_iff_leq_vimage2p
tff(fact_674_rel__fun__iff__leq__vimage2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o] :
      ( bNF_re4195927319312321177_int_o(R2,S3,F,G)
    <=> ord_le1980079291468326055_int_o(R2,bNF_vi8983165433838618593nt_o_o(F,G,S3)) ) ).

% rel_fun_iff_leq_vimage2p
tff(fact_675_rel__fun__iff__leq__vimage2p,axiom,
    ! [R2: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num] :
      ( aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(R2,S3),F),G)
    <=> ord_le6795820959886135841eger_o(R2,bNF_vi2326032003723069084_num_o(F,G,S3)) ) ).

% rel_fun_iff_leq_vimage2p
tff(fact_676_rel__fun__iff__leq__vimage2p,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(R2,S3),F),G)
    <=> ord_le77980682512674322_int_o(R2,bNF_vi2107963681178519309_num_o(F,G,S3)) ) ).

% rel_fun_iff_leq_vimage2p
tff(fact_677_rel__fun__mono_H,axiom,
    ! [Y3: fun_Pr4738963024209204295_int_o,X3: fun_Pr4738963024209204295_int_o,A4: fun_nat_fun_nat_o,B5: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat] :
      ( ! [X: product_prod_nat_nat,Y: int] :
          ( aa_int_o(aa_Pro2912763917461432128_int_o(Y3,X),Y)
         => aa_int_o(aa_Pro2912763917461432128_int_o(X3,X),Y) )
     => ( ! [X: nat,Y: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(A4,X),Y)
           => aa_nat_o(aa_nat_fun_nat_o(B5,X),Y) )
       => ( aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(X3,A4),F),G)
         => aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(Y3,B5),F),G) ) ) ) ).

% rel_fun_mono'
tff(fact_678_rel__fun__mono_H,axiom,
    ! [Y3: fun_Pr4738963024209204295_int_o,X3: fun_Pr4738963024209204295_int_o,A4: fun_fu4638078822479986034_int_o,B5: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int] :
      ( ! [X: product_prod_nat_nat,Y: int] :
          ( aa_int_o(aa_Pro2912763917461432128_int_o(Y3,X),Y)
         => aa_int_o(aa_Pro2912763917461432128_int_o(X3,X),Y) )
     => ( ! [X: fun_Pr2118491707810114563at_nat,Y: fun_int_int] :
            ( aa_fun_int_int_o(aa_fun856177901615811547_int_o(A4,X),Y)
           => aa_fun_int_int_o(aa_fun856177901615811547_int_o(B5,X),Y) )
       => ( bNF_re8869769828555196650nt_int(X3,A4,F,G)
         => bNF_re8869769828555196650nt_int(Y3,B5,F,G) ) ) ) ).

% rel_fun_mono'
tff(fact_679_rel__fun__mono_H,axiom,
    ! [Y3: fun_Pr4738963024209204295_int_o,X3: fun_Pr4738963024209204295_int_o,A4: fun_fu9090758901314268781nt_o_o,B5: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o] :
      ( ! [X: product_prod_nat_nat,Y: int] :
          ( aa_int_o(aa_Pro2912763917461432128_int_o(Y3,X),Y)
         => aa_int_o(aa_Pro2912763917461432128_int_o(X3,X),Y) )
     => ( ! [X: fun_Pr4796943543331071514_nat_o,Y: fun_int_o] :
            ( aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(A4,X),Y)
           => aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(B5,X),Y) )
       => ( bNF_re4195927319312321177_int_o(X3,A4,F,G)
         => bNF_re4195927319312321177_int_o(Y3,B5,F,G) ) ) ) ).

% rel_fun_mono'
tff(fact_680_rel__fun__mono_H,axiom,
    ! [Y3: fun_in4891666191172872833eger_o,X3: fun_in4891666191172872833eger_o,A4: fun_num_fun_num_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num] :
      ( ! [X: int,Y: code_integer] :
          ( aa_Code_integer_o(aa_int5993532643906289898eger_o(Y3,X),Y)
         => aa_Code_integer_o(aa_int5993532643906289898eger_o(X3,X),Y) )
     => ( ! [X: num,Y: num] :
            ( aa_num_o(aa_num_fun_num_o(A4,X),Y)
           => aa_num_o(aa_num_fun_num_o(B5,X),Y) )
       => ( aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(X3,A4),F),G)
         => aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(Y3,B5),F),G) ) ) ) ).

% rel_fun_mono'
tff(fact_681_rel__fun__mono_H,axiom,
    ! [Y3: fun_int_fun_int_o,X3: fun_int_fun_int_o,A4: fun_num_fun_num_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num] :
      ( ! [X: int,Y: int] :
          ( aa_int_o(aa_int_fun_int_o(Y3,X),Y)
         => aa_int_o(aa_int_fun_int_o(X3,X),Y) )
     => ( ! [X: num,Y: num] :
            ( aa_num_o(aa_num_fun_num_o(A4,X),Y)
           => aa_num_o(aa_num_fun_num_o(B5,X),Y) )
       => ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(X3,A4),F),G)
         => aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(Y3,B5),F),G) ) ) ) ).

% rel_fun_mono'
tff(fact_682_rel__fun__mono,axiom,
    ! [X3: fun_Pr4738963024209204295_int_o,A4: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat,Y3: fun_Pr4738963024209204295_int_o,B5: fun_nat_fun_nat_o] :
      ( aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(X3,A4),F),G)
     => ( ! [X: product_prod_nat_nat,Y: int] :
            ( aa_int_o(aa_Pro2912763917461432128_int_o(Y3,X),Y)
           => aa_int_o(aa_Pro2912763917461432128_int_o(X3,X),Y) )
       => ( ! [X: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(A4,X),Y)
             => aa_nat_o(aa_nat_fun_nat_o(B5,X),Y) )
         => aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(Y3,B5),F),G) ) ) ) ).

% rel_fun_mono
tff(fact_683_rel__fun__mono,axiom,
    ! [X3: fun_Pr4738963024209204295_int_o,A4: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int,Y3: fun_Pr4738963024209204295_int_o,B5: fun_fu4638078822479986034_int_o] :
      ( bNF_re8869769828555196650nt_int(X3,A4,F,G)
     => ( ! [X: product_prod_nat_nat,Y: int] :
            ( aa_int_o(aa_Pro2912763917461432128_int_o(Y3,X),Y)
           => aa_int_o(aa_Pro2912763917461432128_int_o(X3,X),Y) )
       => ( ! [X: fun_Pr2118491707810114563at_nat,Y: fun_int_int] :
              ( aa_fun_int_int_o(aa_fun856177901615811547_int_o(A4,X),Y)
             => aa_fun_int_int_o(aa_fun856177901615811547_int_o(B5,X),Y) )
         => bNF_re8869769828555196650nt_int(Y3,B5,F,G) ) ) ) ).

% rel_fun_mono
tff(fact_684_rel__fun__mono,axiom,
    ! [X3: fun_Pr4738963024209204295_int_o,A4: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o,Y3: fun_Pr4738963024209204295_int_o,B5: fun_fu9090758901314268781nt_o_o] :
      ( bNF_re4195927319312321177_int_o(X3,A4,F,G)
     => ( ! [X: product_prod_nat_nat,Y: int] :
            ( aa_int_o(aa_Pro2912763917461432128_int_o(Y3,X),Y)
           => aa_int_o(aa_Pro2912763917461432128_int_o(X3,X),Y) )
       => ( ! [X: fun_Pr4796943543331071514_nat_o,Y: fun_int_o] :
              ( aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(A4,X),Y)
             => aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(B5,X),Y) )
         => bNF_re4195927319312321177_int_o(Y3,B5,F,G) ) ) ) ).

% rel_fun_mono
tff(fact_685_rel__fun__mono,axiom,
    ! [X3: fun_in4891666191172872833eger_o,A4: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num,Y3: fun_in4891666191172872833eger_o,B5: fun_num_fun_num_o] :
      ( aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(X3,A4),F),G)
     => ( ! [X: int,Y: code_integer] :
            ( aa_Code_integer_o(aa_int5993532643906289898eger_o(Y3,X),Y)
           => aa_Code_integer_o(aa_int5993532643906289898eger_o(X3,X),Y) )
       => ( ! [X: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(A4,X),Y)
             => aa_num_o(aa_num_fun_num_o(B5,X),Y) )
         => aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(Y3,B5),F),G) ) ) ) ).

% rel_fun_mono
tff(fact_686_rel__fun__mono,axiom,
    ! [X3: fun_int_fun_int_o,A4: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num,Y3: fun_int_fun_int_o,B5: fun_num_fun_num_o] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(X3,A4),F),G)
     => ( ! [X: int,Y: int] :
            ( aa_int_o(aa_int_fun_int_o(Y3,X),Y)
           => aa_int_o(aa_int_fun_int_o(X3,X),Y) )
       => ( ! [X: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(A4,X),Y)
             => aa_num_o(aa_num_fun_num_o(B5,X),Y) )
         => aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(Y3,B5),F),G) ) ) ) ).

% rel_fun_mono
tff(fact_687_rel__funD,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat,Xa: product_prod_nat_nat,Y2: int] :
      ( aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(A4,B5),F),G)
     => ( aa_int_o(aa_Pro2912763917461432128_int_o(A4,Xa),Y2)
       => aa_nat_o(aa_nat_fun_nat_o(B5,aa_Pro4243173770044397973at_nat(F,Xa)),aa_int_nat(G,Y2)) ) ) ).

% rel_funD
tff(fact_688_rel__funD,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int,Xa: product_prod_nat_nat,Y2: int] :
      ( bNF_re8869769828555196650nt_int(A4,B5,F,G)
     => ( aa_int_o(aa_Pro2912763917461432128_int_o(A4,Xa),Y2)
       => aa_fun_int_int_o(aa_fun856177901615811547_int_o(B5,aa_Pro8023792847356852296at_nat(F,Xa)),aa_int_fun_int_int(G,Y2)) ) ) ).

% rel_funD
tff(fact_689_rel__funD,axiom,
    ! [A4: fun_Pr4738963024209204295_int_o,B5: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o,Xa: product_prod_nat_nat,Y2: int] :
      ( bNF_re4195927319312321177_int_o(A4,B5,F,G)
     => ( aa_int_o(aa_Pro2912763917461432128_int_o(A4,Xa),Y2)
       => aa_fun_int_o_o(aa_fun7250270402284233902nt_o_o(B5,aa_Pro989595756709598047_nat_o(F,Xa)),aa_int_fun_int_o(G,Y2)) ) ) ).

% rel_funD
tff(fact_690_rel__funD,axiom,
    ! [A4: fun_in4891666191172872833eger_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num,Xa: int,Y2: code_integer] :
      ( aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(A4,B5),F),G)
     => ( aa_Code_integer_o(aa_int5993532643906289898eger_o(A4,Xa),Y2)
       => aa_num_o(aa_num_fun_num_o(B5,aa_int_num(F,Xa)),aa_Code_integer_num(G,Y2)) ) ) ).

% rel_funD
tff(fact_691_rel__funD,axiom,
    ! [A4: fun_int_fun_int_o,B5: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num,Xa: int,Y2: int] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(A4,B5),F),G)
     => ( aa_int_o(aa_int_fun_int_o(A4,Xa),Y2)
       => aa_num_o(aa_num_fun_num_o(B5,aa_int_num(F,Xa)),aa_int_num(G,Y2)) ) ) ).

% rel_funD
tff(fact_692_fun_Orel__compp,axiom,
    ! [R2: fun_num_fun_num_o,S3: fun_num_fun_num_o] : bNF_re7626690874201225453um_num(fequal_int,relcompp_num_num_num(R2,S3)) = relcom8560396942509463548nt_num(bNF_re7626690874201225453um_num(fequal_int,R2),bNF_re7626690874201225453um_num(fequal_int,S3)) ).

% fun.rel_compp
tff(fact_693_relcompp__mono,axiom,
    ! [R: fun_Pr2028404973800837934_nat_o,Ra: fun_Pr2028404973800837934_nat_o,S2: fun_Pr4738963024209204295_int_o,S: fun_Pr4738963024209204295_int_o] :
      ( ord_le3650406611588909774_nat_o(R,Ra)
     => ( ord_le1980079291468326055_int_o(S2,S)
       => ord_le1980079291468326055_int_o(relcom8911704206367525917at_int(R,S2),relcom8911704206367525917at_int(Ra,S)) ) ) ).

% relcompp_mono
tff(fact_694_relcompp__mono,axiom,
    ! [R: fun_int_fun_int_o,Ra: fun_int_fun_int_o,S2: fun_in4891666191172872833eger_o,S: fun_in4891666191172872833eger_o] :
      ( ord_le77980682512674322_int_o(R,Ra)
     => ( ord_le6795820959886135841eger_o(S2,S)
       => ord_le6795820959886135841eger_o(relcom967497705283160040nteger(R,S2),relcom967497705283160040nteger(Ra,S)) ) ) ).

% relcompp_mono
tff(fact_695_in__rel__def,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(fun_in6812141508599261570at_nat(R2),Xa),Y2)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2),R2) ) ).

% in_rel_def
tff(fact_696_in__rel__def,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(fun_in8056734985859589634at_nat(R2),Xa),Y2)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),R2) ) ).

% in_rel_def
tff(fact_697_in__rel__def,axiom,
    ! [R2: set_Pr6200539531224447659at_num,Xa: nat,Y2: num] :
      ( aa_num_o(aa_nat_fun_num_o(fun_in_rel_nat_num(R2),Xa),Y2)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2),R2) ) ).

% in_rel_def
tff(fact_698_in__rel__def,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Xa: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(fun_in_rel_nat_nat(R2),Xa),Y2)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),R2) ) ).

% in_rel_def
tff(fact_699_in__rel__def,axiom,
    ! [R2: set_Pr958786334691620121nt_int,Xa: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(fun_in_rel_int_int(R2),Xa),Y2)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),R2) ) ).

% in_rel_def
tff(fact_700_in__rel__Collect__case__prod__eq,axiom,
    ! [X3: fun_nat_fun_nat_o] : fun_in_rel_nat_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,X3))) = X3 ).

% in_rel_Collect_case_prod_eq
tff(fact_701_in__rel__Collect__case__prod__eq,axiom,
    ! [X3: fun_int_fun_int_o] : fun_in_rel_int_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,X3))) = X3 ).

% in_rel_Collect_case_prod_eq
tff(fact_702_Collect__case__prod__mono,axiom,
    ! [A4: fun_nat_fun_nat_o,B5: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(A4,B5)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,A4))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,B5))) ) ).

% Collect_case_prod_mono
tff(fact_703_Collect__case__prod__mono,axiom,
    ! [A4: fun_int_fun_int_o,B5: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A4,B5)
     => ord_le2843351958646193337nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,A4)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,B5))) ) ).

% Collect_case_prod_mono
tff(fact_704_fun_Omap__comp,axiom,
    ! [G: fun_num_num,F: fun_nat_num,V: fun_int_nat] : aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,G),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),V)) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_num_num_nat(G,F)),V) ).

% fun.map_comp
tff(fact_705_fun_Omap__comp,axiom,
    ! [G: fun_nat_nat,F: fun_int_nat,V: fun_int_int] : aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),V)) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),F)),V) ).

% fun.map_comp
tff(fact_706_fun_Omap__comp,axiom,
    ! [G: fun_o_o,F: fun_nat_o,V: fun_nat_nat] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),comp_nat_o_nat(F,V)) = comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F),V) ).

% fun.map_comp
tff(fact_707_fun_Omap__comp,axiom,
    ! [G: fun_o_o,F: fun_o_o,V: fun_nat_o] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),V)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(G,F)),V) ).

% fun.map_comp
tff(fact_708_fun_Omap__comp,axiom,
    ! [G: fun_nat_num,F: fun_nat_nat,V: fun_int_nat] : aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),V)) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_nat_num_nat(G,F)),V) ).

% fun.map_comp
tff(fact_709_fun_Omap__comp,axiom,
    ! [G: fun_nat_num,F: fun_int_nat,V: fun_int_int] : aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),V)) = aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F)),V) ).

% fun.map_comp
tff(fact_710_fun_Omap__comp,axiom,
    ! [G: fun_nat_nat,F: fun_nat_nat,V: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),V)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),V) ).

% fun.map_comp
tff(fact_711_fun_Omap__comp,axiom,
    ! [G: fun_int_nat,F: fun_int_int,V: fun_int_int] : aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),comp_int_int_int(F,V)) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),V) ).

% fun.map_comp
tff(fact_712_fun_Omap__comp,axiom,
    ! [G: fun_fu4800041728187187127nteger,F: fun_Co1981900195452921617nteger,V: fun_Co8499238673658658103nteger] : aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),comp_C7233611567195229654nteger(F,V)) = comp_C7848664941760151970nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),F),V) ).

% fun.map_comp
tff(fact_713_fun_Omap__comp,axiom,
    ! [G: fun_fu4800041728187187127nteger,F: fun_fu1763652423599918571nteger,V: fun_Co1981900195452921617nteger] : aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),comp_f228016600672082096nteger(F,V)) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(G,F)),V) ).

% fun.map_comp
tff(fact_714_relcompp_Ocases,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,S: fun_Pr4738963024209204295_int_o,A1: product_prod_nat_nat,A22: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(relcom8911704206367525917at_int(Ra,S),A1),A22)
     => ~ ! [B3: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,A1),B3)
           => ~ aa_int_o(aa_Pro2912763917461432128_int_o(S,B3),A22) ) ) ).

% relcompp.cases
tff(fact_715_relcompp_Ocases,axiom,
    ! [Ra: fun_int_fun_int_o,S: fun_in4891666191172872833eger_o,A1: int,A22: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(relcom967497705283160040nteger(Ra,S),A1),A22)
     => ~ ! [B3: int] :
            ( aa_int_o(aa_int_fun_int_o(Ra,A1),B3)
           => ~ aa_Code_integer_o(aa_int5993532643906289898eger_o(S,B3),A22) ) ) ).

% relcompp.cases
tff(fact_716_relcompp_Osimps,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,S: fun_Pr4738963024209204295_int_o,A1: product_prod_nat_nat,A22: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(relcom8911704206367525917at_int(Ra,S),A1),A22)
    <=> ? [A5: product_prod_nat_nat,B4: product_prod_nat_nat,C4: int] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,A5),B4)
          & aa_int_o(aa_Pro2912763917461432128_int_o(S,B4),C4) ) ) ).

% relcompp.simps
tff(fact_717_relcompp_Osimps,axiom,
    ! [Ra: fun_int_fun_int_o,S: fun_in4891666191172872833eger_o,A1: int,A22: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(relcom967497705283160040nteger(Ra,S),A1),A22)
    <=> ? [A5: int,B4: int,C4: code_integer] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & aa_int_o(aa_int_fun_int_o(Ra,A5),B4)
          & aa_Code_integer_o(aa_int5993532643906289898eger_o(S,B4),C4) ) ) ).

% relcompp.simps
tff(fact_718_eq__OO,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o] : relcom8911704206367525917at_int(fequal9188845342289841878at_nat,R2) = R2 ).

% eq_OO
tff(fact_719_eq__OO,axiom,
    ! [R2: fun_in4891666191172872833eger_o] : relcom967497705283160040nteger(fequal_int,R2) = R2 ).

% eq_OO
tff(fact_720_relcomppE,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,S: fun_Pr4738963024209204295_int_o,A: product_prod_nat_nat,C: int] :
      ( aa_int_o(aa_Pro2912763917461432128_int_o(relcom8911704206367525917at_int(Ra,S),A),C)
     => ~ ! [B3: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,A),B3)
           => ~ aa_int_o(aa_Pro2912763917461432128_int_o(S,B3),C) ) ) ).

% relcomppE
tff(fact_721_relcomppE,axiom,
    ! [Ra: fun_int_fun_int_o,S: fun_in4891666191172872833eger_o,A: int,C: code_integer] :
      ( aa_Code_integer_o(aa_int5993532643906289898eger_o(relcom967497705283160040nteger(Ra,S),A),C)
     => ~ ! [B3: int] :
            ( aa_int_o(aa_int_fun_int_o(Ra,A),B3)
           => ~ aa_Code_integer_o(aa_int5993532643906289898eger_o(S,B3),C) ) ) ).

% relcomppE
tff(fact_722_relcomppI,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,A: product_prod_nat_nat,B: product_prod_nat_nat,S: fun_Pr4738963024209204295_int_o,C: int] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Ra,A),B)
     => ( aa_int_o(aa_Pro2912763917461432128_int_o(S,B),C)
       => aa_int_o(aa_Pro2912763917461432128_int_o(relcom8911704206367525917at_int(Ra,S),A),C) ) ) ).

% relcomppI
tff(fact_723_relcomppI,axiom,
    ! [Ra: fun_int_fun_int_o,A: int,B: int,S: fun_in4891666191172872833eger_o,C: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(Ra,A),B)
     => ( aa_Code_integer_o(aa_int5993532643906289898eger_o(S,B),C)
       => aa_Code_integer_o(aa_int5993532643906289898eger_o(relcom967497705283160040nteger(Ra,S),A),C) ) ) ).

% relcomppI
tff(fact_724_map__prod__o__convol,axiom,
    ! [H1: fun_o_o,H22: fun_o_o,F: fun_nat_o,G: fun_nat_o] : comp_P7034811116587396339_o_nat(produc1928436952130685430_o_o_o(H1,H22),bNF_convol_nat_o_o(F,G)) = bNF_convol_nat_o_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,H1),F),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,H22),G)) ).

% map_prod_o_convol
tff(fact_725_map__prod__o__convol,axiom,
    ! [H1: fun_o_o,H22: fun_nat_nat,F: fun_nat_o,G: fun_nat_nat] : comp_P1391661485614827491at_nat(produc7071226018047599512at_nat(H1,H22),bNF_convol_nat_o_nat(F,G)) = bNF_convol_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,H1),F),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,H22),G)) ).

% map_prod_o_convol
tff(fact_726_map__prod__o__convol,axiom,
    ! [H1: fun_nat_num,H22: fun_nat_num,F: fun_int_nat,G: fun_int_nat] : comp_P7387482301570536291um_int(produc1476026309457068494at_num(H1,H22),bNF_co1177303703936631735at_nat(F,G)) = bNF_co5088337930254655563um_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,H1),F),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,H22),G)) ).

% map_prod_o_convol
tff(fact_727_map__prod__o__convol,axiom,
    ! [H1: fun_nat_num,H22: fun_int_nat,F: fun_int_nat,G: fun_int_int] : comp_P8376639181999047357at_int(produc5918048235202344416nt_nat(H1,H22),bNF_co1174813233427581459at_int(F,G)) = bNF_co8531006144098876929um_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,H1),F),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,H22),G)) ).

% map_prod_o_convol
tff(fact_728_map__prod__o__convol,axiom,
    ! [H1: fun_nat_nat,H22: fun_o_o,F: fun_nat_nat,G: fun_nat_o] : comp_P5799945282029801519_o_nat(produc2001155298098703512at_o_o(H1,H22),bNF_convol_nat_nat_o(F,G)) = bNF_convol_nat_nat_o(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,H1),F),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,H22),G)) ).

% map_prod_o_convol
tff(fact_729_map__prod__o__convol,axiom,
    ! [H1: fun_nat_nat,H22: fun_nat_nat,F: fun_nat_nat,G: fun_nat_nat] : comp_P7430769850142754163at_nat(produc6977886695330630970at_nat(H1,H22),bNF_co805650143699787099at_nat(F,G)) = bNF_co805650143699787099at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,H1),F),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,H22),G)) ).

% map_prod_o_convol
tff(fact_730_map__prod__o__convol,axiom,
    ! [H1: fun_int_nat,H22: fun_nat_num,F: fun_int_int,G: fun_int_nat] : comp_P1145193044942399421um_int(produc1225400549393458656at_num(H1,H22),bNF_co2176657415837686291nt_nat(F,G)) = bNF_co6958007526947186177at_num(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,H1),F),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,H22),G)) ).

% map_prod_o_convol
tff(fact_731_map__prod__o__convol,axiom,
    ! [H1: fun_int_nat,H22: fun_int_nat,F: fun_int_int,G: fun_int_int] : comp_P2134349925370910487at_int(produc5667422475138734578nt_nat(H1,H22),bNF_co2174166945328636015nt_int(F,G)) = bNF_co1177303703936631735at_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,H1),F),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,H22),G)) ).

% map_prod_o_convol
tff(fact_732_map__prod__o__convol,axiom,
    ! [H1: fun_fu4800041728187187127nteger,H22: fun_fu4800041728187187127nteger,F: fun_Co1981900195452921617nteger,G: fun_Co1981900195452921617nteger] : comp_P2457517555449687320nteger(produc7836332429974410862nteger(H1,H22),bNF_co5315716916936888502nteger(F,G)) = bNF_co1420163402147158094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(H1),F),aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(H22),G)) ).

% map_prod_o_convol
tff(fact_733_convol__o,axiom,
    ! [F: fun_o_o,G: fun_o_o,Ha: fun_nat_o] : comp_o7658611401268416066_o_nat(bNF_convol_o_o_o(F,G),Ha) = bNF_convol_nat_o_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ha),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ha)) ).

% convol_o
tff(fact_734_convol__o,axiom,
    ! [F: fun_nat_num,G: fun_nat_num,Ha: fun_int_nat] : comp_n8531277669757933902um_int(bNF_co4716684370017810927um_num(F,G),Ha) = bNF_co5088337930254655563um_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ha),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) ).

% convol_o
tff(fact_735_convol__o,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Ha: fun_nat_nat] : comp_n8574565218330151774at_nat(bNF_co805650143699787099at_nat(F,G),Ha) = bNF_co805650143699787099at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Ha),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Ha)) ).

% convol_o
tff(fact_736_convol__o,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,Ha: fun_int_int] : comp_i1108739374719436894at_int(bNF_co1177303703936631735at_nat(F,G),Ha) = bNF_co1177303703936631735at_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),Ha),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha)) ).

% convol_o
tff(fact_737_convol__o,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu4800041728187187127nteger,Ha: fun_Co1981900195452921617nteger] : comp_f4474543515786798506nteger(bNF_co1646691892022060328nteger(F,G),Ha) = bNF_co1420163402147158094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),Ha),aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),Ha)) ).

% convol_o
tff(fact_738_vimage2p__relcompp__mono,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o,T2: fun_Pr4738963024209204295_int_o,F: fun_Pr2118491707810114563at_nat,G: fun_Pr2118491707810114563at_nat,Ha: fun_int_int] :
      ( ord_le1980079291468326055_int_o(relcom8911704206367525917at_int(R2,S3),T2)
     => ord_le1980079291468326055_int_o(relcom8911704206367525917at_int(bNF_vi763171854521451549_nat_o(F,G,R2),bNF_vi5426221348228882691_int_o(G,Ha,S3)),bNF_vi5426221348228882691_int_o(F,Ha,T2)) ) ).

% vimage2p_relcompp_mono
tff(fact_739_vimage2p__relcompp__mono,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o,T2: fun_Pr4738963024209204295_int_o,F: fun_in4514472496968388910at_nat,G: fun_in4514472496968388910at_nat,Ha: fun_Code_integer_int] :
      ( ord_le1980079291468326055_int_o(relcom8911704206367525917at_int(R2,S3),T2)
     => ord_le6795820959886135841eger_o(relcom967497705283160040nteger(bNF_vi1204582724742824157_nat_o(F,G,R2),bNF_vi2881549544236934259_int_o(G,Ha,S3)),bNF_vi2881549544236934259_int_o(F,Ha,T2)) ) ).

% vimage2p_relcompp_mono
tff(fact_740_vimage2p__relcompp__mono,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o,T2: fun_in4891666191172872833eger_o,F: fun_Pr7647638413817135296at_int,G: fun_Pr7647638413817135296at_int,Ha: fun_int_Code_integer] :
      ( ord_le6795820959886135841eger_o(relcom967497705283160040nteger(R2,S3),T2)
     => ord_le1980079291468326055_int_o(relcom8911704206367525917at_int(bNF_vi2682369418337388893_int_o(F,G,R2),bNF_vi33923956287962137eger_o(G,Ha,S3)),bNF_vi33923956287962137eger_o(F,Ha,T2)) ) ).

% vimage2p_relcompp_mono
tff(fact_741_vimage2p__relcompp__mono,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o,T2: fun_in4891666191172872833eger_o,F: fun_int_int,G: fun_int_int,Ha: fun_Co8499238673658658103nteger] :
      ( ord_le6795820959886135841eger_o(relcom967497705283160040nteger(R2,S3),T2)
     => ord_le6795820959886135841eger_o(relcom967497705283160040nteger(bNF_vi3884084463199473129_int_o(F,G,R2),bNF_vi699694178825965831eger_o(G,Ha,S3)),bNF_vi699694178825965831eger_o(F,Ha,T2)) ) ).

% vimage2p_relcompp_mono
tff(fact_742_pos__fun__distr,axiom,
    ! [R2: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o,R4: fun_Co2081214437782444048eger_o,S4: fun_num_fun_num_o] : ord_le7846159698440226291_num_o(relcom4173150990864199262er_num(bNF_re6718328864250387230um_num(R2,S3),bNF_re40212662316476751um_num(R4,S4)),bNF_re6718328864250387230um_num(relcom8320293257703906841nteger(R2,R4),relcompp_num_num_num(S3,S4))) ).

% pos_fun_distr
tff(fact_743_pos__fun__distr,axiom,
    ! [R2: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o,R4: fun_Co8353275306710069249_int_o,S4: fun_num_fun_num_o] : ord_le6143850148583763428_num_o(relcom8634939491988117549nt_num(bNF_re6718328864250387230um_num(R2,S3),bNF_re8313794964846168862um_num(R4,S4)),bNF_re7626690874201225453um_num(relcom328798225758222312er_int(R2,R4),relcompp_num_num_num(S3,S4))) ).

% pos_fun_distr
tff(fact_744_pos__fun__distr,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_num_fun_num_o,R4: fun_int_fun_int_o,S4: fun_num_fun_num_o] : ord_le6143850148583763428_num_o(relcom8560396942509463548nt_num(bNF_re7626690874201225453um_num(R2,S3),bNF_re7626690874201225453um_num(R4,S4)),bNF_re7626690874201225453um_num(relcompp_int_int_int(R2,R4),relcompp_num_num_num(S3,S4))) ).

% pos_fun_distr
tff(fact_745_pos__fun__distr,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_num_fun_num_o,R4: fun_in4891666191172872833eger_o,S4: fun_num_fun_num_o] : ord_le7846159698440226291_num_o(relcom2053567597505753645er_num(bNF_re7626690874201225453um_num(R2,S3),bNF_re6718328864250387230um_num(R4,S4)),bNF_re6718328864250387230um_num(relcom967497705283160040nteger(R2,R4),relcompp_num_num_num(S3,S4))) ).

% pos_fun_distr
tff(fact_746_pos__fun__distr,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_int_fun_int_o,R4: fun_in4891666191172872833eger_o,S4: fun_in4891666191172872833eger_o] : ord_le8507588522978985054eger_o(relcom444288326586520852nteger(bNF_re4712519889275205905nt_int(R2,S3),bNF_re3379532845092657523nteger(R4,S4)),bNF_re3379532845092657523nteger(relcom967497705283160040nteger(R2,R4),relcom967497705283160040nteger(S3,S4))) ).

% pos_fun_distr
tff(fact_747_pos__fun__distr,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_nat_fun_nat_o,R4: fun_int_fun_int_o,S4: fun_nat_fun_nat_o] : ord_le150341387783501559_nat_o(relcom3487822462411678311nt_nat(bNF_re4555766996558763186at_nat(R2,S3),bNF_re3715656647883201625at_nat(R4,S4)),bNF_re4555766996558763186at_nat(relcom2949942408940504938nt_int(R2,R4),relcompp_nat_nat_nat(S3,S4))) ).

% pos_fun_distr
tff(fact_748_pos__fun__distr,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_nat_fun_nat_o,R4: fun_Pr4738963024209204295_int_o,S4: fun_nat_fun_nat_o] : ord_le150341387783501559_nat_o(relcom8518660330154759978nt_nat(bNF_re8246922863344978751at_nat(R2,S3),bNF_re4555766996558763186at_nat(R4,S4)),bNF_re4555766996558763186at_nat(relcom8911704206367525917at_int(R2,R4),relcompp_nat_nat_nat(S3,S4))) ).

% pos_fun_distr
tff(fact_749_pos__fun__distr,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_int_fun_int_o,R4: fun_Pr4738963024209204295_int_o,S4: fun_in4891666191172872833eger_o] : ord_le9076892547966399182eger_o(relcom4123854648742082103nteger(bNF_re20414067882207223nt_int(R2,S3),bNF_re6202762543029211035nteger(R4,S4)),bNF_re6202762543029211035nteger(relcom8911704206367525917at_int(R2,R4),relcom967497705283160040nteger(S3,S4))) ).

% pos_fun_distr
tff(fact_750_pos__fun__distr,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_Pr2028404973800837934_nat_o,R4: fun_in4891666191172872833eger_o,S4: fun_Pr4738963024209204295_int_o] : ord_le2215172278096386656_int_o(relcom7261622426688989495er_int(bNF_re8766404587435608311at_nat(R2,S3),bNF_re2828929484557863493at_int(R4,S4)),bNF_re2828929484557863493at_int(relcom967497705283160040nteger(R2,R4),relcom8911704206367525917at_int(S3,S4))) ).

% pos_fun_distr
tff(fact_751_pos__fun__distr,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr2028404973800837934_nat_o,R4: fun_Pr4738963024209204295_int_o,S4: fun_Pr4738963024209204295_int_o] : ord_le6589200864103675154_int_o(relcom5467440779735672206nt_int(bNF_re2241393799969408733at_nat(R2,S3),bNF_re7400052026677387805at_int(R4,S4)),bNF_re7400052026677387805at_int(relcom8911704206367525917at_int(R2,R4),relcom8911704206367525917at_int(S3,S4))) ).

% pos_fun_distr
tff(fact_752_o__rsp_I2_J,axiom,
    ! [R1: fun_nat_fun_nat_o] : bNF_re3325573624036370925_nat_o(fequal_fun_o_o,bNF_re7140302142790738653_nat_o(bNF_re4705727531993890431at_o_o(R1,fequal_o),bNF_re4705727531993890431at_o_o(R1,fequal_o)),comp_o_o_nat,comp_o_o_nat) ).

% o_rsp(2)
tff(fact_753_o__rsp_I2_J,axiom,
    ! [R1: fun_nat_fun_nat_o] : bNF_re8005335818930388233at_nat(fequal_fun_nat_nat,bNF_re2068050147316706005at_nat(bNF_re5653821019739307937at_nat(R1,fequal_nat),bNF_re5653821019739307937at_nat(R1,fequal_nat)),comp_nat_nat_nat,comp_nat_nat_nat) ).

% o_rsp(2)
tff(fact_754_o__rsp_I2_J,axiom,
    ! [R1: fun_int_fun_int_o] : bNF_re6308950812890564585nt_nat(fequal_fun_int_nat,bNF_re4932164143273258237nt_nat(bNF_re4712519889275205905nt_int(R1,fequal_int),bNF_re3715656647883201625at_nat(R1,fequal_nat)),comp_int_nat_int,comp_int_nat_int) ).

% o_rsp(2)
tff(fact_755_o__rsp_I2_J,axiom,
    ! [R1: fun_in4891666191172872833eger_o] : bNF_re8100248535414471555er_num(fequal_fun_nat_num,bNF_re3347705605836830523er_num(bNF_re2807294637932363402at_nat(R1,fequal_nat),bNF_re6718328864250387230um_num(R1,fequal_num)),comp_nat_num_int,comp_n2103536230552569900nteger) ).

% o_rsp(2)
tff(fact_756_o__rsp_I2_J,axiom,
    ! [R1: fun_int_fun_int_o] : bNF_re62826171660744097nt_num(fequal_fun_nat_num,bNF_re7119215340858952409nt_num(bNF_re3715656647883201625at_nat(R1,fequal_nat),bNF_re7626690874201225453um_num(R1,fequal_num)),comp_nat_num_int,comp_nat_num_int) ).

% o_rsp(2)
tff(fact_757_o__rsp_I2_J,axiom,
    ! [R1: fun_in4891666191172872833eger_o] : bNF_re4528850393734421163er_num(fequal_fun_num_num,bNF_re3419773739737105871er_num(bNF_re6718328864250387230um_num(R1,fequal_num),bNF_re6718328864250387230um_num(R1,fequal_num)),comp_num_num_int,comp_n8557746072492896886nteger) ).

% o_rsp(2)
tff(fact_758_o__rsp_I2_J,axiom,
    ! [R1: fun_int_fun_int_o] : bNF_re1151091782021442761nt_num(fequal_fun_num_num,bNF_re2388990614730936685nt_num(bNF_re7626690874201225453um_num(R1,fequal_num),bNF_re7626690874201225453um_num(R1,fequal_num)),comp_num_num_int,comp_num_num_int) ).

% o_rsp(2)
tff(fact_759_o__rsp_I2_J,axiom,
    ! [R1: fun_Pr4738963024209204295_int_o] : bNF_re771439050147916165nt_nat(fequal_fun_int_nat,bNF_re8260947740043152229nt_nat(bNF_re5552630237950767466nt_int(R1,fequal_int),bNF_re4555766996558763186at_nat(R1,fequal_nat)),comp_i4471277238271202272at_nat,comp_int_nat_int) ).

% o_rsp(2)
tff(fact_760_o__rsp_I2_J,axiom,
    ! [R1: fun_Pr4738963024209204295_int_o] : bNF_re7448902332430545141nt_num(fequal_fun_nat_num,bNF_re4416501007135400113nt_num(bNF_re4555766996558763186at_nat(R1,fequal_nat),bNF_re8466801222876787014um_num(R1,fequal_num)),comp_n8649498751204056690at_nat,comp_nat_num_int) ).

% o_rsp(2)
tff(fact_761_o__rsp_I2_J,axiom,
    ! [R1: fun_Pr4738963024209204295_int_o] : bNF_re8026677594887939021nt_nat(fequal_fun_nat_nat,bNF_re6147058543053867549nt_nat(bNF_re4555766996558763186at_nat(R1,fequal_nat),bNF_re4555766996558763186at_nat(R1,fequal_nat)),comp_n2711240574518091068at_nat,comp_nat_nat_int) ).

% o_rsp(2)
tff(fact_762_o__rsp_I1_J,axiom,
    ! [R22: fun_o_fun_o_o,R32: fun_o_fun_o_o,R1: fun_nat_fun_nat_o] : bNF_re3325573624036370925_nat_o(bNF_rel_fun_o_o_o_o(R22,R32),bNF_re7140302142790738653_nat_o(bNF_re4705727531993890431at_o_o(R1,R22),bNF_re4705727531993890431at_o_o(R1,R32)),comp_o_o_nat,comp_o_o_nat) ).

% o_rsp(1)
tff(fact_763_o__rsp_I1_J,axiom,
    ! [R22: fun_o_fun_nat_o,R32: fun_o_fun_num_o,R1: fun_nat_fun_int_o] : bNF_re8356455757435110855nt_num(bNF_re766437678722298023_o_num(R22,R32),bNF_re5108275045863451431nt_num(bNF_re8210149947023016781_o_nat(R1,R22),bNF_re4767481733178795415_o_num(R1,R32)),comp_o_o_nat,comp_nat_num_int) ).

% o_rsp(1)
tff(fact_764_o__rsp_I1_J,axiom,
    ! [R22: fun_o_fun_nat_o,R32: fun_o_fun_nat_o,R1: fun_nat_fun_nat_o] : bNF_re2134824042319328379at_nat(bNF_re4209105892566519389_o_nat(R22,R32),bNF_re5947481727989837021at_nat(bNF_re7523573913982466217_o_nat(R1,R22),bNF_re7523573913982466217_o_nat(R1,R32)),comp_o_o_nat,comp_nat_nat_nat) ).

% o_rsp(1)
tff(fact_765_o__rsp_I1_J,axiom,
    ! [R22: fun_o_fun_int_o,R32: fun_o_fun_nat_o,R1: fun_nat_fun_int_o] : bNF_re1135261559000399851nt_nat(bNF_re4895681925607069953_o_nat(R22,R32),bNF_re159862433797838465nt_nat(bNF_re8207659476513966505_o_int(R1,R22),bNF_re8210149947023016781_o_nat(R1,R32)),comp_o_o_nat,comp_int_nat_int) ).

% o_rsp(1)
tff(fact_766_o__rsp_I1_J,axiom,
    ! [R22: fun_nat_fun_o_o,R32: fun_num_fun_o_o,R1: fun_int_fun_nat_o] : bNF_re1911363672597680839_nat_o(bNF_re8508505965575957011_num_o(R22,R32),bNF_re2633672112943174419_nat_o(bNF_re4914077030771115051_nat_o(R1,R22),bNF_re3760060065607074657_num_o(R1,R32)),comp_nat_num_int,comp_o_o_nat) ).

% o_rsp(1)
tff(fact_767_o__rsp_I1_J,axiom,
    ! [R22: fun_nat_fun_nat_o,R32: fun_num_fun_nat_o,R1: fun_int_fun_nat_o] : bNF_re4492593190157324629at_nat(bNF_re3784151423046777323um_nat(R22,R32),bNF_re6754414501021085911at_nat(bNF_re3344003087646356989at_nat(R1,R22),bNF_re1474333490953826375um_nat(R1,R32)),comp_nat_num_int,comp_nat_nat_nat) ).

% o_rsp(1)
tff(fact_768_o__rsp_I1_J,axiom,
    ! [R22: fun_nat_fun_int_o,R32: fun_num_fun_nat_o,R1: fun_int_fun_int_o] : bNF_re7052405848218697413nt_nat(bNF_re4155804983283621959um_nat(R22,R32),bNF_re7620155230884812267nt_nat(bNF_re3713166177374151349at_int(R1,R22),bNF_re1845987051190671011um_nat(R1,R32)),comp_nat_num_int,comp_int_nat_int) ).

% o_rsp(1)
tff(fact_769_o__rsp_I1_J,axiom,
    ! [R22: fun_nat_fun_o_o,R32: fun_nat_fun_o_o,R1: fun_nat_fun_nat_o] : bNF_re6900549463078501243_nat_o(bNF_re439150893885221597_nat_o(R22,R32),bNF_re1097902388772303837_nat_o(bNF_re1402873238235875719_nat_o(R1,R22),bNF_re1402873238235875719_nat_o(R1,R32)),comp_nat_nat_nat,comp_o_o_nat) ).

% o_rsp(1)
tff(fact_770_o__rsp_I1_J,axiom,
    ! [R22: fun_nat_fun_nat_o,R32: fun_nat_fun_num_o,R1: fun_nat_fun_int_o] : bNF_re3575568800433807701nt_num(bNF_re2211152805895086571at_num(R22,R32),bNF_re2432850987154572503nt_num(bNF_re6025474579976152573at_nat(R1,R22),bNF_re2582806366131931207at_num(R1,R32)),comp_nat_nat_nat,comp_nat_num_int) ).

% o_rsp(1)
tff(fact_771_o__rsp_I1_J,axiom,
    ! [R22: fun_nat_fun_nat_o,R32: fun_nat_fun_nat_o,R1: fun_nat_fun_nat_o] : bNF_re8005335818930388233at_nat(bNF_re5653821019739307937at_nat(R22,R32),bNF_re2068050147316706005at_nat(bNF_re5653821019739307937at_nat(R1,R22),bNF_re5653821019739307937at_nat(R1,R32)),comp_nat_nat_nat,comp_nat_nat_nat) ).

% o_rsp(1)
tff(fact_772_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_int,Ha: fun_nat_int,R12: fun_nat_int,R23: fun_nat_nat,F: fun_int_nat,L: fun_nat_nat] :
      ( ( comp_int_int_nat(G,Ha) = comp_nat_int_nat(R12,R23) )
     => ( ( comp_int_nat_nat(F,R12) = L )
       => ( comp_int_nat_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),Ha) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,L),R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_773_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_int,Ha: fun_nat_nat,R12: fun_int_int,R23: fun_nat_int,F: fun_int_nat,L: fun_int_nat] :
      ( ( comp_nat_int_nat(G,Ha) = comp_int_int_nat(R12,R23) )
     => ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),R12) = L )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_int_nat_nat(F,G)),Ha) = comp_int_nat_nat(L,R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_774_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,R12: fun_int_int,R23: fun_int_int,F: fun_int_nat,L: fun_int_nat] :
      ( ( comp_int_int_int(G,Ha) = comp_int_int_int(R12,R23) )
     => ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),R12) = L )
       => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G)),Ha) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,L),R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_775_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_o,Ha: fun_nat_nat,R12: fun_o_o,R23: fun_nat_o,F: fun_o_o,L: fun_o_o] :
      ( ( comp_nat_o_nat(G,Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,R12),R23) )
     => ( ( comp_o_o_o(F,R12) = L )
       => ( comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L),R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_776_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_o,Ha: fun_nat_nat,R12: fun_o_o,R23: fun_nat_o,F: fun_o_nat,L: fun_o_nat] :
      ( ( comp_nat_o_nat(G,Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,R12),R23) )
     => ( ( comp_o_nat_o(F,R12) = L )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(F,G)),Ha) = comp_o_nat_nat(L,R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_777_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_num,Ha: fun_int_int,R12: fun_nat_num,R23: fun_int_nat,F: fun_num_nat,L: fun_nat_nat] :
      ( ( aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,G),Ha) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,R12),R23) )
     => ( ( comp_num_nat_nat(F,R12) = L )
       => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_num_nat_int(F,G)),Ha) = aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,L),R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_778_rewriteR__comp__comp2,axiom,
    ! [G: fun_int_nat,Ha: fun_nat_int,R12: fun_nat_nat,R23: fun_nat_nat,F: fun_nat_num,L: fun_nat_num] :
      ( ( comp_int_nat_nat(G,Ha) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,R12),R23) )
     => ( ( comp_nat_num_nat(F,R12) = L )
       => ( comp_int_num_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),Ha) = comp_nat_num_nat(L,R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_779_rewriteR__comp__comp2,axiom,
    ! [G: fun_o_nat,Ha: fun_nat_o,R12: fun_nat_nat,R23: fun_nat_nat,F: fun_nat_o,L: fun_nat_o] :
      ( ( comp_o_nat_nat(G,Ha) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,R12),R23) )
     => ( ( comp_nat_o_nat(F,R12) = L )
       => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_nat_o_o(F,G)),Ha) = comp_nat_o_nat(L,R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_780_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_nat,Ha: fun_int_nat,R12: fun_int_nat,R23: fun_int_int,F: fun_nat_nat,L: fun_int_nat] :
      ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),Ha) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,R12),R23) )
     => ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),R12) = L )
       => ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G)),Ha) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,L),R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_781_rewriteR__comp__comp2,axiom,
    ! [G: fun_nat_nat,Ha: fun_int_nat,R12: fun_int_nat,R23: fun_int_int,F: fun_nat_num,L: fun_int_num] :
      ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),Ha) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,R12),R23) )
     => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),R12) = L )
       => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_nat_num_nat(F,G)),Ha) = aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,L),R23) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_782_rewriteL__comp__comp2,axiom,
    ! [F: fun_o_nat,G: fun_o_o,L1: fun_nat_nat,L2: fun_o_nat,Ha: fun_nat_o,Ra: fun_nat_nat] :
      ( ( comp_o_nat_o(F,G) = comp_nat_nat_o(L1,L2) )
     => ( ( comp_o_nat_nat(L2,Ha) = Ra )
       => ( comp_o_nat_nat(F,aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ha)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_783_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_num,G: fun_nat_num,L1: fun_nat_num,L2: fun_nat_nat,Ha: fun_int_nat,Ra: fun_int_nat] :
      ( ( comp_num_num_nat(F,G) = comp_nat_num_nat(L1,L2) )
     => ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,L2),Ha) = Ra )
       => ( aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,F),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_784_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_nat,G: fun_nat_num,L1: fun_int_nat,L2: fun_nat_int,Ha: fun_int_nat,Ra: fun_int_int] :
      ( ( comp_num_nat_nat(F,G) = comp_int_nat_nat(L1,L2) )
     => ( ( comp_nat_int_int(L2,Ha) = Ra )
       => ( comp_num_nat_int(F,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_785_rewriteL__comp__comp2,axiom,
    ! [F: fun_o_o,G: fun_o_o,L1: fun_o_o,L2: fun_o_o,Ha: fun_nat_o,Ra: fun_nat_o] :
      ( ( comp_o_o_o(F,G) = comp_o_o_o(L1,L2) )
     => ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L2),Ha) = Ra )
       => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ha)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_786_rewriteL__comp__comp2,axiom,
    ! [F: fun_nat_nat,G: fun_o_nat,L1: fun_o_nat,L2: fun_o_o,Ha: fun_nat_o,Ra: fun_nat_o] :
      ( ( comp_nat_nat_o(F,G) = comp_o_nat_o(L1,L2) )
     => ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L2),Ha) = Ra )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),comp_o_nat_nat(G,Ha)) = comp_o_nat_nat(L1,Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_787_rewriteL__comp__comp2,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,L1: fun_num_num,L2: fun_nat_num,Ha: fun_int_nat,Ra: fun_int_num] :
      ( ( comp_nat_num_nat(F,G) = comp_num_num_nat(L1,L2) )
     => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,L2),Ha) = Ra )
       => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),Ha)) = aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_788_rewriteL__comp__comp2,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,L1: fun_num_nat,L2: fun_nat_num,Ha: fun_int_nat,Ra: fun_int_num] :
      ( ( comp_int_nat_nat(F,G) = comp_num_nat_nat(L1,L2) )
     => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,L2),Ha) = Ra )
       => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),comp_nat_int_int(G,Ha)) = comp_num_nat_int(L1,Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_789_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_o,G: fun_nat_num,L1: fun_o_o,L2: fun_nat_o,Ha: fun_int_nat,Ra: fun_int_o] :
      ( ( comp_num_o_nat(F,G) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L1),L2) )
     => ( ( comp_nat_o_int(L2,Ha) = Ra )
       => ( comp_num_o_int(F,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) = comp_o_o_int(L1,Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_790_rewriteL__comp__comp2,axiom,
    ! [F: fun_nat_o,G: fun_nat_nat,L1: fun_o_o,L2: fun_nat_o,Ha: fun_nat_nat,Ra: fun_nat_o] :
      ( ( comp_nat_o_nat(F,G) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L1),L2) )
     => ( ( comp_nat_o_nat(L2,Ha) = Ra )
       => ( comp_nat_o_nat(F,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Ha)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_791_rewriteL__comp__comp2,axiom,
    ! [F: fun_num_nat,G: fun_nat_num,L1: fun_nat_nat,L2: fun_nat_nat,Ha: fun_int_nat,Ra: fun_int_nat] :
      ( ( comp_num_nat_nat(F,G) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,L1),L2) )
     => ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,L2),Ha) = Ra )
       => ( comp_num_nat_int(F,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) = aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,L1),Ra) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_792_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_o,Ha: fun_nat_nat,Ra: fun_nat_o,F: fun_o_o] :
      ( ( comp_nat_o_nat(G,Ha) = Ra )
     => ( comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_793_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_nat,Ha: fun_int_nat,Ra: fun_int_nat,F: fun_nat_num] :
      ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),Ha) = Ra )
     => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_nat_num_nat(F,G)),Ha) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_794_rewriteR__comp__comp,axiom,
    ! [G: fun_int_int,Ha: fun_int_int,Ra: fun_int_int,F: fun_int_nat] :
      ( ( comp_int_int_int(G,Ha) = Ra )
     => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G)),Ha) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_795_rewriteR__comp__comp,axiom,
    ! [G: fun_o_o,Ha: fun_nat_o,Ra: fun_nat_o,F: fun_o_o] :
      ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ha) = Ra )
     => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(F,G)),Ha) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_796_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_num,Ha: fun_int_nat,Ra: fun_int_num,F: fun_num_num] :
      ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha) = Ra )
     => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_num_num_nat(F,G)),Ha) = aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_797_rewriteR__comp__comp,axiom,
    ! [G: fun_nat_nat,Ha: fun_nat_nat,Ra: fun_nat_nat,F: fun_nat_nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Ha) = Ra )
     => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G)),Ha) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_798_rewriteR__comp__comp,axiom,
    ! [G: fun_int_nat,Ha: fun_int_int,Ra: fun_int_nat,F: fun_nat_num] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha) = Ra )
     => ( aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G)),Ha) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_799_rewriteR__comp__comp,axiom,
    ! [G: fun_int_nat,Ha: fun_int_int,Ra: fun_int_nat,F: fun_nat_nat] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha) = Ra )
     => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),G)),Ha) = aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_800_rewriteR__comp__comp,axiom,
    ! [G: fun_Co1981900195452921617nteger,Ha: fun_Co8499238673658658103nteger,Ra: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger] :
      ( ( comp_C7233611567195229654nteger(G,Ha) = Ra )
     => ( comp_C7848664941760151970nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Ha) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_801_rewriteR__comp__comp,axiom,
    ! [G: fun_fu1763652423599918571nteger,Ha: fun_Co1981900195452921617nteger,Ra: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger] :
      ( ( comp_f228016600672082096nteger(G,Ha) = Ra )
     => ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(comp_f3848972703483044950nteger(F,G)),Ha) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),Ra) ) ) ).

% rewriteR_comp_comp
tff(fact_802_rewriteL__comp__comp,axiom,
    ! [F: fun_num_num,G: fun_nat_num,L: fun_nat_num,Ha: fun_int_nat] :
      ( ( comp_num_num_nat(F,G) = L )
     => ( aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,F),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ha)) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_803_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,L: fun_int_nat,Ha: fun_int_int] :
      ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),G) = L )
     => ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha)) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_804_rewriteL__comp__comp,axiom,
    ! [F: fun_o_o,G: fun_o_o,L: fun_o_o,Ha: fun_nat_o] :
      ( ( comp_o_o_o(F,G) = L )
     => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ha)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_805_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,L: fun_nat_num,Ha: fun_int_nat] :
      ( ( comp_nat_num_nat(F,G) = L )
     => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),Ha)) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_806_rewriteL__comp__comp,axiom,
    ! [F: fun_o_o,G: fun_nat_o,L: fun_nat_o,Ha: fun_nat_nat] :
      ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G) = L )
     => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),comp_nat_o_nat(G,Ha)) = comp_nat_o_nat(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_807_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,L: fun_int_num,Ha: fun_int_int] :
      ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G) = L )
     => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ha)) = aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_808_rewriteL__comp__comp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,L: fun_nat_nat,Ha: fun_nat_nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G) = L )
     => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Ha)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_809_rewriteL__comp__comp,axiom,
    ! [F: fun_int_nat,G: fun_int_int,L: fun_int_nat,Ha: fun_int_int] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G) = L )
     => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),comp_int_int_int(G,Ha)) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_810_rewriteL__comp__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,L: fun_Co1561183069634336733nteger,Ha: fun_Co8499238673658658103nteger] :
      ( ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G) = L )
     => ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),comp_C7233611567195229654nteger(G,Ha)) = comp_C7848664941760151970nteger(L,Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_811_rewriteL__comp__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_fu1763652423599918571nteger,L: fun_fu4800041728187187127nteger,Ha: fun_Co1981900195452921617nteger] :
      ( ( comp_f3848972703483044950nteger(F,G) = L )
     => ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),comp_f228016600672082096nteger(G,Ha)) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(L),Ha) ) ) ).

% rewriteL_comp_comp
tff(fact_812_type__copy__map__cong0,axiom,
    ! [M: fun_nat_o,G: fun_nat_nat,Xa: nat,N: fun_o_o,Ha: fun_nat_o,F: fun_o_o] :
      ( ( aa_nat_o(M,aa_nat_nat(G,Xa))
      <=> aa_o_o(N,aa_nat_o(Ha,Xa)) )
     => ( aa_nat_o(comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),M),G),Xa)
      <=> aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(F,N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_813_type__copy__map__cong0,axiom,
    ! [M: fun_int_nat,G: fun_int_int,Xa: int,N: fun_nat_nat,Ha: fun_int_nat,F: fun_nat_num] :
      ( ( aa_int_nat(M,aa_int_int(G,Xa)) = aa_nat_nat(N,aa_int_nat(Ha,Xa)) )
     => ( aa_int_num(aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),M)),G),Xa) = aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_nat_num_nat(F,N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_814_type__copy__map__cong0,axiom,
    ! [M: fun_nat_nat,G: fun_int_nat,Xa: int,N: fun_int_nat,Ha: fun_int_int,F: fun_nat_nat] :
      ( ( aa_nat_nat(M,aa_int_nat(G,Xa)) = aa_int_nat(N,aa_int_int(Ha,Xa)) )
     => ( aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),M)),G),Xa) = aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_815_type__copy__map__cong0,axiom,
    ! [M: fun_int_int,G: fun_nat_int,Xa: nat,N: fun_nat_int,Ha: fun_nat_nat,F: fun_int_nat] :
      ( ( aa_int_int(M,aa_nat_int(G,Xa)) = aa_nat_int(N,aa_nat_nat(Ha,Xa)) )
     => ( aa_nat_nat(comp_int_nat_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),M),G),Xa) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_int_nat_nat(F,N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_816_type__copy__map__cong0,axiom,
    ! [M: fun_o_o,G: fun_nat_o,Xa: nat,N: fun_nat_o,Ha: fun_nat_nat,F: fun_o_o] :
      ( ( aa_o_o(M,aa_nat_o(G,Xa))
      <=> aa_nat_o(N,aa_nat_nat(Ha,Xa)) )
     => ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(F,M)),G),Xa)
      <=> aa_nat_o(comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),N),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_817_type__copy__map__cong0,axiom,
    ! [M: fun_nat_nat,G: fun_int_nat,Xa: int,N: fun_int_nat,Ha: fun_int_int,F: fun_nat_num] :
      ( ( aa_nat_nat(M,aa_int_nat(G,Xa)) = aa_int_nat(N,aa_int_int(Ha,Xa)) )
     => ( aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_nat_num_nat(F,M)),G),Xa) = aa_int_num(aa_fun1943068679254307798nt_num(aa_fun7450834664159010710nt_num(comp_int_num_int,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_818_type__copy__map__cong0,axiom,
    ! [M: fun_nat_int,G: fun_nat_nat,Xa: nat,N: fun_int_int,Ha: fun_nat_int,F: fun_int_nat] :
      ( ( aa_nat_int(M,aa_nat_nat(G,Xa)) = aa_int_int(N,aa_nat_int(Ha,Xa)) )
     => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_int_nat_nat(F,M)),G),Xa) = aa_nat_nat(comp_int_nat_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),N),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_819_type__copy__map__cong0,axiom,
    ! [M: fun_nat_nat,G: fun_nat_nat,Xa: nat,N: fun_nat_nat,Ha: fun_nat_nat,F: fun_nat_nat] :
      ( ( aa_nat_nat(M,aa_nat_nat(G,Xa)) = aa_nat_nat(N,aa_nat_nat(Ha,Xa)) )
     => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),M)),G),Xa) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_820_type__copy__map__cong0,axiom,
    ! [M: fun_int_nat,G: fun_int_int,Xa: int,N: fun_nat_nat,Ha: fun_int_nat,F: fun_nat_nat] :
      ( ( aa_int_nat(M,aa_int_int(G,Xa)) = aa_nat_nat(N,aa_int_nat(Ha,Xa)) )
     => ( aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,F),M)),G),Xa) = aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_821_type__copy__map__cong0,axiom,
    ! [M: fun_int_int,G: fun_int_int,Xa: int,N: fun_int_int,Ha: fun_int_int,F: fun_int_nat] :
      ( ( aa_int_int(M,aa_int_int(G,Xa)) = aa_int_int(N,aa_int_int(Ha,Xa)) )
     => ( aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),M)),G),Xa) = aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),N)),Ha),Xa) ) ) ).

% type_copy_map_cong0
tff(fact_822_comp__transfer,axiom,
    ! [B5: fun_o_fun_o_o,C3: fun_o_fun_o_o,A4: fun_nat_fun_nat_o] : bNF_re3325573624036370925_nat_o(bNF_rel_fun_o_o_o_o(B5,C3),bNF_re7140302142790738653_nat_o(bNF_re4705727531993890431at_o_o(A4,B5),bNF_re4705727531993890431at_o_o(A4,C3)),comp_o_o_nat,comp_o_o_nat) ).

% comp_transfer
tff(fact_823_comp__transfer,axiom,
    ! [B5: fun_o_fun_nat_o,C3: fun_o_fun_num_o,A4: fun_nat_fun_int_o] : bNF_re8356455757435110855nt_num(bNF_re766437678722298023_o_num(B5,C3),bNF_re5108275045863451431nt_num(bNF_re8210149947023016781_o_nat(A4,B5),bNF_re4767481733178795415_o_num(A4,C3)),comp_o_o_nat,comp_nat_num_int) ).

% comp_transfer
tff(fact_824_comp__transfer,axiom,
    ! [B5: fun_o_fun_nat_o,C3: fun_o_fun_nat_o,A4: fun_nat_fun_nat_o] : bNF_re2134824042319328379at_nat(bNF_re4209105892566519389_o_nat(B5,C3),bNF_re5947481727989837021at_nat(bNF_re7523573913982466217_o_nat(A4,B5),bNF_re7523573913982466217_o_nat(A4,C3)),comp_o_o_nat,comp_nat_nat_nat) ).

% comp_transfer
tff(fact_825_comp__transfer,axiom,
    ! [B5: fun_o_fun_int_o,C3: fun_o_fun_nat_o,A4: fun_nat_fun_int_o] : bNF_re1135261559000399851nt_nat(bNF_re4895681925607069953_o_nat(B5,C3),bNF_re159862433797838465nt_nat(bNF_re8207659476513966505_o_int(A4,B5),bNF_re8210149947023016781_o_nat(A4,C3)),comp_o_o_nat,comp_int_nat_int) ).

% comp_transfer
tff(fact_826_comp__transfer,axiom,
    ! [B5: fun_nat_fun_o_o,C3: fun_num_fun_o_o,A4: fun_int_fun_nat_o] : bNF_re1911363672597680839_nat_o(bNF_re8508505965575957011_num_o(B5,C3),bNF_re2633672112943174419_nat_o(bNF_re4914077030771115051_nat_o(A4,B5),bNF_re3760060065607074657_num_o(A4,C3)),comp_nat_num_int,comp_o_o_nat) ).

% comp_transfer
tff(fact_827_comp__transfer,axiom,
    ! [B5: fun_nat_fun_nat_o,C3: fun_num_fun_nat_o,A4: fun_int_fun_nat_o] : bNF_re4492593190157324629at_nat(bNF_re3784151423046777323um_nat(B5,C3),bNF_re6754414501021085911at_nat(bNF_re3344003087646356989at_nat(A4,B5),bNF_re1474333490953826375um_nat(A4,C3)),comp_nat_num_int,comp_nat_nat_nat) ).

% comp_transfer
tff(fact_828_comp__transfer,axiom,
    ! [B5: fun_nat_fun_int_o,C3: fun_num_fun_nat_o,A4: fun_int_fun_int_o] : bNF_re7052405848218697413nt_nat(bNF_re4155804983283621959um_nat(B5,C3),bNF_re7620155230884812267nt_nat(bNF_re3713166177374151349at_int(A4,B5),bNF_re1845987051190671011um_nat(A4,C3)),comp_nat_num_int,comp_int_nat_int) ).

% comp_transfer
tff(fact_829_comp__transfer,axiom,
    ! [B5: fun_nat_fun_o_o,C3: fun_nat_fun_o_o,A4: fun_nat_fun_nat_o] : bNF_re6900549463078501243_nat_o(bNF_re439150893885221597_nat_o(B5,C3),bNF_re1097902388772303837_nat_o(bNF_re1402873238235875719_nat_o(A4,B5),bNF_re1402873238235875719_nat_o(A4,C3)),comp_nat_nat_nat,comp_o_o_nat) ).

% comp_transfer
tff(fact_830_comp__transfer,axiom,
    ! [B5: fun_nat_fun_nat_o,C3: fun_nat_fun_num_o,A4: fun_nat_fun_int_o] : bNF_re3575568800433807701nt_num(bNF_re2211152805895086571at_num(B5,C3),bNF_re2432850987154572503nt_num(bNF_re6025474579976152573at_nat(A4,B5),bNF_re2582806366131931207at_num(A4,C3)),comp_nat_nat_nat,comp_nat_num_int) ).

% comp_transfer
tff(fact_831_comp__transfer,axiom,
    ! [B5: fun_nat_fun_nat_o,C3: fun_nat_fun_nat_o,A4: fun_nat_fun_nat_o] : bNF_re8005335818930388233at_nat(bNF_re5653821019739307937at_nat(B5,C3),bNF_re2068050147316706005at_nat(bNF_re5653821019739307937at_nat(A4,B5),bNF_re5653821019739307937at_nat(A4,C3)),comp_nat_nat_nat,comp_nat_nat_nat) ).

% comp_transfer
tff(fact_832_rel__fun__iff__geq__image2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_nat_fun_nat_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat] :
      ( aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(R2,S3),F),G)
    <=> ord_le768339454706049738_nat_o(bNF_Gr6955563364113978072nt_nat(F,G,R2),S3) ) ).

% rel_fun_iff_geq_image2p
tff(fact_833_rel__fun__iff__geq__image2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_fu4638078822479986034_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int] :
      ( bNF_re8869769828555196650nt_int(R2,S3,F,G)
    <=> ord_le6589200864103675154_int_o(bNF_Gr7638081787706718096nt_int(F,G,R2),S3) ) ).

% rel_fun_iff_geq_image2p
tff(fact_834_rel__fun__iff__geq__image2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,S3: fun_fu9090758901314268781nt_o_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o] :
      ( bNF_re4195927319312321177_int_o(R2,S3,F,G)
    <=> ord_le5696542305017016013nt_o_o(bNF_Gr4191080447388368243_int_o(F,G,R2),S3) ) ).

% rel_fun_iff_geq_image2p
tff(fact_835_rel__fun__iff__geq__image2p,axiom,
    ! [R2: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o,F: fun_int_num,G: fun_Code_integer_num] :
      ( aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(R2,S3),F),G)
    <=> ord_le3306795289226699318_num_o(bNF_Gr166583433173494904er_num(F,G,R2),S3) ) ).

% rel_fun_iff_geq_image2p
tff(fact_836_rel__fun__iff__geq__image2p,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_num_fun_num_o,F: fun_int_num,G: fun_int_num] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(R2,S3),F),G)
    <=> ord_le3306795289226699318_num_o(bNF_Gr7020624548710275655nt_num(F,G,R2),S3) ) ).

% rel_fun_iff_geq_image2p
tff(fact_837_map__prod__o__convol__id,axiom,
    ! [F: fun_nat_num,G: fun_nat_nat,Xa: nat] : comp_P5371577678113413053at_nat(produc4918694523301289860at_nat(F,id_nat),bNF_co805650143699787099at_nat(id_nat,G),Xa) = bNF_co8159352583862032293um_nat(comp_num_num_nat(id_num,F),G,Xa) ).

% map_prod_o_convol_id
tff(fact_838_map__prod__o__convol__id,axiom,
    ! [F: fun_o_nat,G: fun_o_nat,Xa: $o] : aa_o_P6312935399938478757at_nat(comp_P196761593691424337_nat_o(produc6587185857518356756at_nat(F,id_nat),bNF_convol_o_o_nat(id_o,G)),(Xa)) = aa_o_P6312935399938478757at_nat(bNF_convol_o_nat_nat(comp_nat_nat_o(id_nat,F),G),(Xa)) ).

% map_prod_o_convol_id
tff(fact_839_map__prod__o__convol__id,axiom,
    ! [F: fun_o_o,G: fun_o_nat,Xa: $o] : comp_P1422407636550552773_nat_o(produc7071226018047599512at_nat(F,id_nat),bNF_convol_o_o_nat(id_o,G),(Xa)) = aa_o_P3390874802229903347_o_nat(bNF_convol_o_o_nat(comp_o_o_o(id_o,F),G),(Xa)) ).

% map_prod_o_convol_id
tff(fact_840_map__prod__o__convol__id,axiom,
    ! [F: fun_o_num,G: fun_o_nat,Xa: $o] : comp_P972951341003599239_nat_o(produc4527993685489015646at_nat(F,id_nat),bNF_convol_o_o_nat(id_o,G),(Xa)) = bNF_convol_o_num_nat(comp_num_num_o(id_num,F),G,(Xa)) ).

% map_prod_o_convol_id
tff(fact_841_map__prod__o__convol__id,axiom,
    ! [F: fun_num_nat,G: fun_num_nat,Xa: num] : comp_P2539519431309506803at_num(produc4208724500416182148at_nat(F,id_nat),bNF_co6100160411832691183um_nat(id_num,G),Xa) = bNF_co7969830008525221797at_nat(comp_nat_nat_num(id_nat,F),G,Xa) ).

% map_prod_o_convol_id
tff(fact_842_map__prod__o__convol__id,axiom,
    ! [F: fun_num_o,G: fun_num_nat,Xa: num] : comp_P3192427084145116251at_num(produc8319293927732033832at_nat(F,id_nat),bNF_co6100160411832691183um_nat(id_num,G),Xa) = bNF_convol_num_o_nat(comp_o_o_num(id_o,F),G,Xa) ).

% map_prod_o_convol_id
tff(fact_843_map__prod__o__convol__id,axiom,
    ! [F: fun_num_num,G: fun_num_nat,Xa: num] : comp_P480327259280165693at_num(produc2149532328386841038at_nat(F,id_nat),bNF_co6100160411832691183um_nat(id_num,G),Xa) = aa_num2448437706225772275um_nat(bNF_co6100160411832691183um_nat(comp_num_num_num(id_num,F),G),Xa) ).

% map_prod_o_convol_id
tff(fact_844_map__prod__o__convol__id,axiom,
    ! [F: fun_nat_num,G: fun_nat_o,Xa: nat] : comp_P6576135029341976421_o_nat(produc3264868786340801506um_o_o(F,id_o),bNF_convol_nat_nat_o(id_nat,G),Xa) = bNF_convol_nat_num_o(comp_num_num_nat(id_num,F),G,Xa) ).

% map_prod_o_convol_id
tff(fact_845_map__prod__o__convol__id,axiom,
    ! [F: fun_o_nat,G: fun_o_o,Xa: $o] : comp_P1471321874221585045at_o_o(produc9141321880750005618at_o_o(F,id_o),bNF_convol_o_o_o(id_o,G),(Xa)) = bNF_convol_o_nat_o(comp_nat_nat_o(id_nat,F),G,(Xa)) ).

% map_prod_o_convol_id
tff(fact_846_map__prod__o__convol__id,axiom,
    ! [F: fun_o_o,G: fun_o_o,Xa: $o] : comp_P551574746845053877_o_o_o(produc1928436952130685430_o_o_o(F,id_o),bNF_convol_o_o_o(id_o,G),(Xa)) = aa_o_P8186867570850176203od_o_o(bNF_convol_o_o_o(comp_o_o_o(id_o,F),G),(Xa)) ).

% map_prod_o_convol_id
tff(fact_847_image__Fpow__mono,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
     => ord_le513522071413781156et_rat(aa_set3856643248234747582et_rat(image_4408659257933336347et_rat(image_nat_rat(F)),finite_Fpow_nat(A4)),finite_Fpow_rat(B5)) ) ).

% image_Fpow_mono
tff(fact_848_image__Fpow__mono,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
     => ord_le6893508408891458716et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(image_nat_nat(F)),finite_Fpow_nat(A4)),finite_Fpow_nat(B5)) ) ).

% image_Fpow_mono
tff(fact_849_image__Fpow__mono,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
     => ord_le4403425263959731960et_int(aa_set7746546440780698386et_int(image_3739036796817536367et_int(image_nat_int(F)),finite_Fpow_nat(A4)),finite_Fpow_int(B5)) ) ).

% image_Fpow_mono
tff(fact_850_image__Fpow__mono,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
     => ord_le6893508408891458716et_nat(aa_set7560802437211521810et_nat(image_4702325430467532143et_nat(image_int_nat(F)),finite_Fpow_int(A4)),finite_Fpow_nat(B5)) ) ).

% image_Fpow_mono
tff(fact_851_image__Fpow__mono,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
     => ord_le4403425263959731960et_int(aa_set5070719292279795054et_int(image_524474410958335435et_int(image_int_int(F)),finite_Fpow_int(A4)),finite_Fpow_int(B5)) ) ).

% image_Fpow_mono
tff(fact_852_convol__mem__GrpI,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,G: fun_se320340636488258563at_nat] :
      ( member2643936169264416010at_nat(Xa,A4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(bNF_co147202357524637304at_nat(id_set2334057505560298834at_nat,G,Xa)),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,bNF_Gr6677838525845560078at_nat(A4,G)))) ) ).

% convol_mem_GrpI
tff(fact_853_convol__mem__GrpI,axiom,
    ! [Xa: nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(bNF_co805650143699787099at_nat(id_nat,G),Xa)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,bNF_Grp_nat_nat(A4,G)))) ) ).

% convol_mem_GrpI
tff(fact_854_convol__mem__GrpI,axiom,
    ! [Xa: int,A4: set_int,G: fun_int_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(bNF_co2174166945328636015nt_int(id_int,G),Xa)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,bNF_Grp_int_int(A4,G)))) ) ).

% convol_mem_GrpI
tff(fact_855_image__id,axiom,
    image_o_o(id_o) = id_set_o ).

% image_id
tff(fact_856_image__id,axiom,
    image_num_num(id_num) = id_set_num ).

% image_id
tff(fact_857_image__id,axiom,
    image_nat_nat(id_nat) = id_set_nat ).

% image_id
tff(fact_858_image__id,axiom,
    image_int_int(id_int) = id_set_int ).

% image_id
tff(fact_859_fun_Omap__id,axiom,
    ! [T: fun_nat_o] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,id_o),T) = T ).

% fun.map_id
tff(fact_860_fun_Omap__id,axiom,
    ! [T: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,id_nat),T) = T ).

% fun.map_id
tff(fact_861_comp__id,axiom,
    ! [F: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),id_nat) = F ).

% comp_id
tff(fact_862_comp__id,axiom,
    ! [F: fun_int_nat] : aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),id_int) = F ).

% comp_id
tff(fact_863_id__comp,axiom,
    ! [G: fun_nat_o] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,id_o),G) = G ).

% id_comp
tff(fact_864_id__comp,axiom,
    ! [G: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,id_nat),G) = G ).

% id_comp
tff(fact_865_case__prod__Pair,axiom,
    produc5456455677105495163at_nat(produc9060074326276436823at_nat) = id_Pro6043886709289754456at_nat ).

% case_prod_Pair
tff(fact_866_case__prod__Pair,axiom,
    produc7548845685434208379at_nat(produc2922128104949294807at_nat) = id_Pro8722402698821300696at_nat ).

% case_prod_Pair
tff(fact_867_case__prod__Pair,axiom,
    produc49306077274653107at_num(product_Pair_nat_num) = id_Pro8267972854940751164at_num ).

% case_prod_Pair
tff(fact_868_case__prod__Pair,axiom,
    produc2626176000494625587at_nat(product_Pair_nat_nat) = id_Pro7559728917991913458at_nat ).

% case_prod_Pair
tff(fact_869_case__prod__Pair,axiom,
    aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,product_Pair_int_int) = id_Pro4381231610383771306nt_int ).

% case_prod_Pair
tff(fact_870_apsnd__id,axiom,
    aa_fun2509385170507305080nteger(produc6499014454317279255nteger,id_Code_integer) = id_Pro2309048762231606924nteger ).

% apsnd_id
tff(fact_871_range__fst,axiom,
    image_7136766188234240905_ref_a(produc473321116828791851t_unit,top_to2503719809363509323t_unit) = top_top_set_ref_a ).

% range_fst
tff(fact_872_range__snd,axiom,
    image_17660834435998945t_unit(produc1126150082675451753t_unit,top_to2503719809363509323t_unit) = top_to3956321331435058429t_unit ).

% range_snd
tff(fact_873_fun_Omap__id0,axiom,
    aa_fun5662400728881950724_nat_o(comp_o_o_nat,id_o) = id_fun_nat_o ).

% fun.map_id0
tff(fact_874_fun_Omap__id0,axiom,
    aa_fun6854695580826382482at_nat(comp_nat_nat_nat,id_nat) = id_fun_nat_nat ).

% fun.map_id0
tff(fact_875_surj__id,axiom,
    aa_set_o_set_o(image_o_o(id_o),top_top_set_o) = top_top_set_o ).

% surj_id
tff(fact_876_surj__id,axiom,
    aa_set_num_set_num(image_num_num(id_num),top_top_set_num) = top_top_set_num ).

% surj_id
tff(fact_877_surj__id,axiom,
    image_char_char(id_char,top_top_set_char) = top_top_set_char ).

% surj_id
tff(fact_878_surj__id,axiom,
    image_7976474329151083847st_nat(id_list_nat,top_top_set_list_nat) = top_top_set_list_nat ).

% surj_id
tff(fact_879_surj__id,axiom,
    image_rat_rat(id_rat,top_top_set_rat) = top_top_set_rat ).

% surj_id
tff(fact_880_surj__id,axiom,
    aa_set_nat_set_nat(image_nat_nat(id_nat),top_top_set_nat) = top_top_set_nat ).

% surj_id
tff(fact_881_surj__id,axiom,
    aa_set_int_set_int(image_int_int(id_int),top_top_set_int) = top_top_set_int ).

% surj_id
tff(fact_882_Grp__UNIV__idI,axiom,
    ! [Xa: $o,Y2: $o] :
      ( ( (Xa)
      <=> (Y2) )
     => aa_o_o(aa_o_fun_o_o(bNF_Grp_o_o(top_top_set_o,id_o),(Xa)),(Y2)) ) ).

% Grp_UNIV_idI
tff(fact_883_Grp__UNIV__idI,axiom,
    ! [Xa: num,Y2: num] :
      ( ( Xa = Y2 )
     => aa_num_o(aa_num_fun_num_o(bNF_Grp_num_num(top_top_set_num,id_num),Xa),Y2) ) ).

% Grp_UNIV_idI
tff(fact_884_Grp__UNIV__idI,axiom,
    ! [Xa: char,Y2: char] :
      ( ( Xa = Y2 )
     => aa_char_o(aa_char_fun_char_o(bNF_Grp_char_char(top_top_set_char,id_char),Xa),Y2) ) ).

% Grp_UNIV_idI
tff(fact_885_Grp__UNIV__idI,axiom,
    ! [Xa: list_nat,Y2: list_nat] :
      ( ( Xa = Y2 )
     => aa_list_nat_o(aa_lis557685189327716851_nat_o(bNF_Gr600725020315365264st_nat(top_top_set_list_nat,id_list_nat),Xa),Y2) ) ).

% Grp_UNIV_idI
tff(fact_886_Grp__UNIV__idI,axiom,
    ! [Xa: rat,Y2: rat] :
      ( ( Xa = Y2 )
     => aa_rat_o(aa_rat_fun_rat_o(bNF_Grp_rat_rat(top_top_set_rat,id_rat),Xa),Y2) ) ).

% Grp_UNIV_idI
tff(fact_887_Grp__UNIV__idI,axiom,
    ! [Xa: nat,Y2: nat] :
      ( ( Xa = Y2 )
     => aa_nat_o(aa_nat_fun_nat_o(bNF_Grp_nat_nat(top_top_set_nat,id_nat),Xa),Y2) ) ).

% Grp_UNIV_idI
tff(fact_888_Grp__UNIV__idI,axiom,
    ! [Xa: int,Y2: int] :
      ( ( Xa = Y2 )
     => aa_int_o(aa_int_fun_int_o(bNF_Grp_int_int(top_top_set_int,id_int),Xa),Y2) ) ).

% Grp_UNIV_idI
tff(fact_889_eq__alt,axiom,
    fequal_o = bNF_Grp_o_o(top_top_set_o,id_o) ).

% eq_alt
tff(fact_890_eq__alt,axiom,
    fequal_num = bNF_Grp_num_num(top_top_set_num,id_num) ).

% eq_alt
tff(fact_891_eq__alt,axiom,
    fequal_char = bNF_Grp_char_char(top_top_set_char,id_char) ).

% eq_alt
tff(fact_892_eq__alt,axiom,
    fequal_list_nat = bNF_Gr600725020315365264st_nat(top_top_set_list_nat,id_list_nat) ).

% eq_alt
tff(fact_893_eq__alt,axiom,
    fequal_rat = bNF_Grp_rat_rat(top_top_set_rat,id_rat) ).

% eq_alt
tff(fact_894_eq__alt,axiom,
    fequal_nat = bNF_Grp_nat_nat(top_top_set_nat,id_nat) ).

% eq_alt
tff(fact_895_eq__alt,axiom,
    fequal_int = bNF_Grp_int_int(top_top_set_int,id_int) ).

% eq_alt
tff(fact_896_Inf_OINF__id__eq,axiom,
    ! [Inf: fun_set_o_o,A4: set_o] :
      ( aa_set_o_o(Inf,aa_set_o_set_o(image_o_o(id_o),A4))
    <=> aa_set_o_o(Inf,A4) ) ).

% Inf.INF_id_eq
tff(fact_897_Inf_OINF__id__eq,axiom,
    ! [Inf: fun_set_num_num,A4: set_num] : aa_set_num_num(Inf,aa_set_num_set_num(image_num_num(id_num),A4)) = aa_set_num_num(Inf,A4) ).

% Inf.INF_id_eq
tff(fact_898_Inf_OINF__id__eq,axiom,
    ! [Inf: fun_set_nat_nat,A4: set_nat] : aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(id_nat),A4)) = aa_set_nat_nat(Inf,A4) ).

% Inf.INF_id_eq
tff(fact_899_Inf_OINF__id__eq,axiom,
    ! [Inf: fun_set_int_int,A4: set_int] : aa_set_int_int(Inf,aa_set_int_set_int(image_int_int(id_int),A4)) = aa_set_int_int(Inf,A4) ).

% Inf.INF_id_eq
tff(fact_900_Sup_OSUP__id__eq,axiom,
    ! [Sup: fun_set_o_o,A4: set_o] :
      ( aa_set_o_o(Sup,aa_set_o_set_o(image_o_o(id_o),A4))
    <=> aa_set_o_o(Sup,A4) ) ).

% Sup.SUP_id_eq
tff(fact_901_Sup_OSUP__id__eq,axiom,
    ! [Sup: fun_set_num_num,A4: set_num] : aa_set_num_num(Sup,aa_set_num_set_num(image_num_num(id_num),A4)) = aa_set_num_num(Sup,A4) ).

% Sup.SUP_id_eq
tff(fact_902_Sup_OSUP__id__eq,axiom,
    ! [Sup: fun_set_nat_nat,A4: set_nat] : aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(id_nat),A4)) = aa_set_nat_nat(Sup,A4) ).

% Sup.SUP_id_eq
tff(fact_903_Sup_OSUP__id__eq,axiom,
    ! [Sup: fun_set_int_int,A4: set_int] : aa_set_int_int(Sup,aa_set_int_set_int(image_int_int(id_int),A4)) = aa_set_int_int(Sup,A4) ).

% Sup.SUP_id_eq
tff(fact_904_comp__eq__id__dest,axiom,
    ! [A: fun_o_o,B: fun_nat_o,C: fun_nat_o,V: nat] :
      ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,A),B) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,id_o),C) )
     => ( aa_o_o(A,aa_nat_o(B,V))
      <=> aa_nat_o(C,V) ) ) ).

% comp_eq_id_dest
tff(fact_905_comp__eq__id__dest,axiom,
    ! [A: fun_nat_num,B: fun_int_nat,C: fun_int_num,V: int] :
      ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,A),B) = aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,id_num),C) )
     => ( aa_nat_num(A,aa_int_nat(B,V)) = aa_int_num(C,V) ) ) ).

% comp_eq_id_dest
tff(fact_906_comp__eq__id__dest,axiom,
    ! [A: fun_nat_nat,B: fun_nat_nat,C: fun_nat_nat,V: nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,A),B) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,id_nat),C) )
     => ( aa_nat_nat(A,aa_nat_nat(B,V)) = aa_nat_nat(C,V) ) ) ).

% comp_eq_id_dest
tff(fact_907_comp__eq__id__dest,axiom,
    ! [A: fun_int_nat,B: fun_int_int,C: fun_int_nat,V: int] :
      ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,A),B) = aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,id_nat),C) )
     => ( aa_int_nat(A,aa_int_int(B,V)) = aa_int_nat(C,V) ) ) ).

% comp_eq_id_dest
tff(fact_908_comp__eq__id__dest,axiom,
    ! [A: fun_fu4800041728187187127nteger,B: fun_Co1981900195452921617nteger,C: fun_Co1561183069634336733nteger,V: code_integer] :
      ( ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(A),B) = comp_f6855635597449841736nteger(id_fun8244602250740870226nteger,C) )
     => ( aa_fun2509385170507305080nteger(A,aa_Cod4161879620478718930nteger(B,V)) = aa_Cod1978467581589928094nteger(C,V) ) ) ).

% comp_eq_id_dest
tff(fact_909_pointfree__idE,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat] :
      ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G) = id_nat )
     => ( aa_nat_nat(F,aa_nat_nat(G,Xa)) = Xa ) ) ).

% pointfree_idE
tff(fact_910_surjD,axiom,
    ! [F: fun_char_char,Y2: char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ? [X: char] : Y2 = aa_char_char(F,X) ) ).

% surjD
tff(fact_911_surjD,axiom,
    ! [F: fun_char_rat,Y2: rat] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ? [X: char] : Y2 = aa_char_rat(F,X) ) ).

% surjD
tff(fact_912_surjD,axiom,
    ! [F: fun_char_nat,Y2: nat] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
     => ? [X: char] : Y2 = aa_char_nat(F,X) ) ).

% surjD
tff(fact_913_surjD,axiom,
    ! [F: fun_char_int,Y2: int] :
      ( ( image_char_int(F,top_top_set_char) = top_top_set_int )
     => ? [X: char] : Y2 = aa_char_int(F,X) ) ).

% surjD
tff(fact_914_surjD,axiom,
    ! [F: fun_rat_char,Y2: char] :
      ( ( image_rat_char(F,top_top_set_rat) = top_top_set_char )
     => ? [X: rat] : Y2 = aa_rat_char(F,X) ) ).

% surjD
tff(fact_915_surjD,axiom,
    ! [F: fun_rat_rat,Y2: rat] :
      ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
     => ? [X: rat] : Y2 = aa_rat_rat(F,X) ) ).

% surjD
tff(fact_916_surjD,axiom,
    ! [F: fun_rat_nat,Y2: nat] :
      ( ( image_rat_nat(F,top_top_set_rat) = top_top_set_nat )
     => ? [X: rat] : Y2 = aa_rat_nat(F,X) ) ).

% surjD
tff(fact_917_surjD,axiom,
    ! [F: fun_rat_int,Y2: int] :
      ( ( image_rat_int(F,top_top_set_rat) = top_top_set_int )
     => ? [X: rat] : Y2 = aa_rat_int(F,X) ) ).

% surjD
tff(fact_918_surjD,axiom,
    ! [F: fun_nat_char,Y2: char] :
      ( ( image_nat_char(F,top_top_set_nat) = top_top_set_char )
     => ? [X: nat] : Y2 = aa_nat_char(F,X) ) ).

% surjD
tff(fact_919_surjD,axiom,
    ! [F: fun_nat_rat,Y2: rat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = top_top_set_rat )
     => ? [X: nat] : Y2 = aa_nat_rat(F,X) ) ).

% surjD
tff(fact_920_surjE,axiom,
    ! [F: fun_char_char,Y2: char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ~ ! [X: char] : Y2 != aa_char_char(F,X) ) ).

% surjE
tff(fact_921_surjE,axiom,
    ! [F: fun_char_rat,Y2: rat] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ~ ! [X: char] : Y2 != aa_char_rat(F,X) ) ).

% surjE
tff(fact_922_surjE,axiom,
    ! [F: fun_char_nat,Y2: nat] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
     => ~ ! [X: char] : Y2 != aa_char_nat(F,X) ) ).

% surjE
tff(fact_923_surjE,axiom,
    ! [F: fun_char_int,Y2: int] :
      ( ( image_char_int(F,top_top_set_char) = top_top_set_int )
     => ~ ! [X: char] : Y2 != aa_char_int(F,X) ) ).

% surjE
tff(fact_924_surjE,axiom,
    ! [F: fun_rat_char,Y2: char] :
      ( ( image_rat_char(F,top_top_set_rat) = top_top_set_char )
     => ~ ! [X: rat] : Y2 != aa_rat_char(F,X) ) ).

% surjE
tff(fact_925_surjE,axiom,
    ! [F: fun_rat_rat,Y2: rat] :
      ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
     => ~ ! [X: rat] : Y2 != aa_rat_rat(F,X) ) ).

% surjE
tff(fact_926_surjE,axiom,
    ! [F: fun_rat_nat,Y2: nat] :
      ( ( image_rat_nat(F,top_top_set_rat) = top_top_set_nat )
     => ~ ! [X: rat] : Y2 != aa_rat_nat(F,X) ) ).

% surjE
tff(fact_927_surjE,axiom,
    ! [F: fun_rat_int,Y2: int] :
      ( ( image_rat_int(F,top_top_set_rat) = top_top_set_int )
     => ~ ! [X: rat] : Y2 != aa_rat_int(F,X) ) ).

% surjE
tff(fact_928_surjE,axiom,
    ! [F: fun_nat_char,Y2: char] :
      ( ( image_nat_char(F,top_top_set_nat) = top_top_set_char )
     => ~ ! [X: nat] : Y2 != aa_nat_char(F,X) ) ).

% surjE
tff(fact_929_surjE,axiom,
    ! [F: fun_nat_rat,Y2: rat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = top_top_set_rat )
     => ~ ! [X: nat] : Y2 != aa_nat_rat(F,X) ) ).

% surjE
tff(fact_930_surjI,axiom,
    ! [G: fun_char_char,F: fun_char_char] :
      ( ! [X: char] : aa_char_char(G,aa_char_char(F,X)) = X
     => ( image_char_char(G,top_top_set_char) = top_top_set_char ) ) ).

% surjI
tff(fact_931_surjI,axiom,
    ! [G: fun_char_rat,F: fun_rat_char] :
      ( ! [X: rat] : aa_char_rat(G,aa_rat_char(F,X)) = X
     => ( image_char_rat(G,top_top_set_char) = top_top_set_rat ) ) ).

% surjI
tff(fact_932_surjI,axiom,
    ! [G: fun_char_nat,F: fun_nat_char] :
      ( ! [X: nat] : aa_char_nat(G,aa_nat_char(F,X)) = X
     => ( image_char_nat(G,top_top_set_char) = top_top_set_nat ) ) ).

% surjI
tff(fact_933_surjI,axiom,
    ! [G: fun_char_int,F: fun_int_char] :
      ( ! [X: int] : aa_char_int(G,aa_int_char(F,X)) = X
     => ( image_char_int(G,top_top_set_char) = top_top_set_int ) ) ).

% surjI
tff(fact_934_surjI,axiom,
    ! [G: fun_rat_char,F: fun_char_rat] :
      ( ! [X: char] : aa_rat_char(G,aa_char_rat(F,X)) = X
     => ( image_rat_char(G,top_top_set_rat) = top_top_set_char ) ) ).

% surjI
tff(fact_935_surjI,axiom,
    ! [G: fun_rat_rat,F: fun_rat_rat] :
      ( ! [X: rat] : aa_rat_rat(G,aa_rat_rat(F,X)) = X
     => ( image_rat_rat(G,top_top_set_rat) = top_top_set_rat ) ) ).

% surjI
tff(fact_936_surjI,axiom,
    ! [G: fun_rat_nat,F: fun_nat_rat] :
      ( ! [X: nat] : aa_rat_nat(G,aa_nat_rat(F,X)) = X
     => ( image_rat_nat(G,top_top_set_rat) = top_top_set_nat ) ) ).

% surjI
tff(fact_937_surjI,axiom,
    ! [G: fun_rat_int,F: fun_int_rat] :
      ( ! [X: int] : aa_rat_int(G,aa_int_rat(F,X)) = X
     => ( image_rat_int(G,top_top_set_rat) = top_top_set_int ) ) ).

% surjI
tff(fact_938_surjI,axiom,
    ! [G: fun_nat_char,F: fun_char_nat] :
      ( ! [X: char] : aa_nat_char(G,aa_char_nat(F,X)) = X
     => ( image_nat_char(G,top_top_set_nat) = top_top_set_char ) ) ).

% surjI
tff(fact_939_surjI,axiom,
    ! [G: fun_nat_rat,F: fun_rat_nat] :
      ( ! [X: rat] : aa_nat_rat(G,aa_rat_nat(F,X)) = X
     => ( aa_set_nat_set_rat(image_nat_rat(G),top_top_set_nat) = top_top_set_rat ) ) ).

% surjI
tff(fact_940_rangeI,axiom,
    ! [F: fun_char_o,Xa: char] : aa_set_o_o(member_o(aa_char_o(F,Xa)),image_char_o(F,top_top_set_char)) ).

% rangeI
tff(fact_941_rangeI,axiom,
    ! [F: fun_char_nat,Xa: char] : aa_set_nat_o(member_nat(aa_char_nat(F,Xa)),image_char_nat(F,top_top_set_char)) ).

% rangeI
tff(fact_942_rangeI,axiom,
    ! [F: fun_char_int,Xa: char] : aa_set_int_o(member_int(aa_char_int(F,Xa)),image_char_int(F,top_top_set_char)) ).

% rangeI
tff(fact_943_rangeI,axiom,
    ! [F: fun_rat_o,Xa: rat] : aa_set_o_o(member_o(aa_rat_o(F,Xa)),image_rat_o(F,top_top_set_rat)) ).

% rangeI
tff(fact_944_rangeI,axiom,
    ! [F: fun_rat_nat,Xa: rat] : aa_set_nat_o(member_nat(aa_rat_nat(F,Xa)),image_rat_nat(F,top_top_set_rat)) ).

% rangeI
tff(fact_945_rangeI,axiom,
    ! [F: fun_rat_int,Xa: rat] : aa_set_int_o(member_int(aa_rat_int(F,Xa)),image_rat_int(F,top_top_set_rat)) ).

% rangeI
tff(fact_946_rangeI,axiom,
    ! [F: fun_nat_rat,Xa: nat] : aa_set_rat_o(member_rat(aa_nat_rat(F,Xa)),aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat)) ).

% rangeI
tff(fact_947_rangeI,axiom,
    ! [F: fun_nat_o,Xa: nat] : aa_set_o_o(member_o(aa_nat_o(F,Xa)),image_nat_o(F,top_top_set_nat)) ).

% rangeI
tff(fact_948_rangeI,axiom,
    ! [F: fun_nat_nat,Xa: nat] : aa_set_nat_o(member_nat(aa_nat_nat(F,Xa)),aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat)) ).

% rangeI
tff(fact_949_rangeI,axiom,
    ! [F: fun_nat_int,Xa: nat] : aa_set_int_o(member_int(aa_nat_int(F,Xa)),aa_set_nat_set_int(image_nat_int(F),top_top_set_nat)) ).

% rangeI
tff(fact_950_surj__def,axiom,
    ! [F: fun_char_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
    <=> ! [Y4: char] :
        ? [X4: char] : Y4 = aa_char_char(F,X4) ) ).

% surj_def
tff(fact_951_surj__def,axiom,
    ! [F: fun_char_rat] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
    <=> ! [Y4: rat] :
        ? [X4: char] : Y4 = aa_char_rat(F,X4) ) ).

% surj_def
tff(fact_952_surj__def,axiom,
    ! [F: fun_char_nat] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
    <=> ! [Y4: nat] :
        ? [X4: char] : Y4 = aa_char_nat(F,X4) ) ).

% surj_def
tff(fact_953_surj__def,axiom,
    ! [F: fun_char_int] :
      ( ( image_char_int(F,top_top_set_char) = top_top_set_int )
    <=> ! [Y4: int] :
        ? [X4: char] : Y4 = aa_char_int(F,X4) ) ).

% surj_def
tff(fact_954_surj__def,axiom,
    ! [F: fun_rat_char] :
      ( ( image_rat_char(F,top_top_set_rat) = top_top_set_char )
    <=> ! [Y4: char] :
        ? [X4: rat] : Y4 = aa_rat_char(F,X4) ) ).

% surj_def
tff(fact_955_surj__def,axiom,
    ! [F: fun_rat_rat] :
      ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
    <=> ! [Y4: rat] :
        ? [X4: rat] : Y4 = aa_rat_rat(F,X4) ) ).

% surj_def
tff(fact_956_surj__def,axiom,
    ! [F: fun_rat_nat] :
      ( ( image_rat_nat(F,top_top_set_rat) = top_top_set_nat )
    <=> ! [Y4: nat] :
        ? [X4: rat] : Y4 = aa_rat_nat(F,X4) ) ).

% surj_def
tff(fact_957_surj__def,axiom,
    ! [F: fun_rat_int] :
      ( ( image_rat_int(F,top_top_set_rat) = top_top_set_int )
    <=> ! [Y4: int] :
        ? [X4: rat] : Y4 = aa_rat_int(F,X4) ) ).

% surj_def
tff(fact_958_surj__def,axiom,
    ! [F: fun_nat_char] :
      ( ( image_nat_char(F,top_top_set_nat) = top_top_set_char )
    <=> ! [Y4: char] :
        ? [X4: nat] : Y4 = aa_nat_char(F,X4) ) ).

% surj_def
tff(fact_959_surj__def,axiom,
    ! [F: fun_nat_rat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = top_top_set_rat )
    <=> ! [Y4: rat] :
        ? [X4: nat] : Y4 = aa_nat_rat(F,X4) ) ).

% surj_def
tff(fact_960_range__eqI,axiom,
    ! [B: $o,F: fun_char_o,Xa: char] :
      ( ( (B)
      <=> aa_char_o(F,Xa) )
     => aa_set_o_o(member_o((B)),image_char_o(F,top_top_set_char)) ) ).

% range_eqI
tff(fact_961_range__eqI,axiom,
    ! [B: nat,F: fun_char_nat,Xa: char] :
      ( ( B = aa_char_nat(F,Xa) )
     => aa_set_nat_o(member_nat(B),image_char_nat(F,top_top_set_char)) ) ).

% range_eqI
tff(fact_962_range__eqI,axiom,
    ! [B: int,F: fun_char_int,Xa: char] :
      ( ( B = aa_char_int(F,Xa) )
     => aa_set_int_o(member_int(B),image_char_int(F,top_top_set_char)) ) ).

% range_eqI
tff(fact_963_range__eqI,axiom,
    ! [B: $o,F: fun_rat_o,Xa: rat] :
      ( ( (B)
      <=> aa_rat_o(F,Xa) )
     => aa_set_o_o(member_o((B)),image_rat_o(F,top_top_set_rat)) ) ).

% range_eqI
tff(fact_964_range__eqI,axiom,
    ! [B: nat,F: fun_rat_nat,Xa: rat] :
      ( ( B = aa_rat_nat(F,Xa) )
     => aa_set_nat_o(member_nat(B),image_rat_nat(F,top_top_set_rat)) ) ).

% range_eqI
tff(fact_965_range__eqI,axiom,
    ! [B: int,F: fun_rat_int,Xa: rat] :
      ( ( B = aa_rat_int(F,Xa) )
     => aa_set_int_o(member_int(B),image_rat_int(F,top_top_set_rat)) ) ).

% range_eqI
tff(fact_966_range__eqI,axiom,
    ! [B: rat,F: fun_nat_rat,Xa: nat] :
      ( ( B = aa_nat_rat(F,Xa) )
     => aa_set_rat_o(member_rat(B),aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat)) ) ).

% range_eqI
tff(fact_967_range__eqI,axiom,
    ! [B: $o,F: fun_nat_o,Xa: nat] :
      ( ( (B)
      <=> aa_nat_o(F,Xa) )
     => aa_set_o_o(member_o((B)),image_nat_o(F,top_top_set_nat)) ) ).

% range_eqI
tff(fact_968_range__eqI,axiom,
    ! [B: nat,F: fun_nat_nat,Xa: nat] :
      ( ( B = aa_nat_nat(F,Xa) )
     => aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat)) ) ).

% range_eqI
tff(fact_969_range__eqI,axiom,
    ! [B: int,F: fun_nat_int,Xa: nat] :
      ( ( B = aa_nat_int(F,Xa) )
     => aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),top_top_set_nat)) ) ).

% range_eqI
tff(fact_970_effect__success,axiom,
    ! [C: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit,H_a: heap_e7401611519738050253t_unit,Ra: ref_a] :
      ( aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(C,Ha),H_a),Ra)
     => aa_hea7994211679605476167unit_o(heap_success_ref_a(C),Ha) ) ).

% effect_success
tff(fact_971_success__effectE,axiom,
    ! [C: heap_Heap_ref_a,Ha: heap_e7401611519738050253t_unit] :
      ( aa_hea7994211679605476167unit_o(heap_success_ref_a(C),Ha)
     => ~ ! [R3: ref_a,H2: heap_e7401611519738050253t_unit] : ~ aa_ref_a_o(aa_hea7918659527173910083ef_a_o(heap_effect_ref_a(C,Ha),H2),R3) ) ).

% success_effectE
tff(fact_972_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_char_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_char(G,top_top_set_char) = top_top_set_char )
       => ( image_7676512070627423941r_char(produc7830753070125841726r_char(F,G),top_to1235241553633177651r_char) = top_to1235241553633177651r_char ) ) ) ).

% map_prod_surj
tff(fact_973_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_char_rat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_rat(G,top_top_set_char) = top_top_set_rat )
       => ( image_7275072250160737868ar_rat(produc5209943433214700613ar_rat(F,G),top_to1235241553633177651r_char) = top_to8437673547576850234ar_rat ) ) ) ).

% map_prod_surj
tff(fact_974_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_char_nat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_nat(G,top_top_set_char) = top_top_set_nat )
       => ( image_1559928771699358788ar_nat(produc5845073493301196349ar_nat(F,G),top_to1235241553633177651r_char) = top_to5594287848199751986ar_nat ) ) ) ).

% map_prod_surj
tff(fact_975_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_char_int] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_int(G,top_top_set_char) = top_top_set_int )
       => ( image_6605449789044937888ar_int(produc5842583022792146073ar_int(F,G),top_to1235241553633177651r_char) = top_to3104204703268025230ar_int ) ) ) ).

% map_prod_surj
tff(fact_976_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_rat_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_rat_char(G,top_top_set_rat) = top_top_set_char )
       => ( image_8036680598759533694r_char(produc5432746246116964165t_char(F,G),top_to8437673547576850234ar_rat) = top_to1235241553633177651r_char ) ) ) ).

% map_prod_surj
tff(fact_977_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_rat_rat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_rat_rat(G,top_top_set_rat) = top_top_set_rat )
       => ( image_5644637116751559941ar_rat(produc1230515432254573900at_rat(F,G),top_to8437673547576850234ar_rat) = top_to8437673547576850234ar_rat ) ) ) ).

% map_prod_surj
tff(fact_978_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_rat_nat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_rat_nat(G,top_top_set_rat) = top_top_set_nat )
       => ( image_9152865675144956669ar_nat(produc1865645492341069636at_nat(F,G),top_to8437673547576850234ar_rat) = top_to5594287848199751986ar_nat ) ) ) ).

% map_prod_surj
tff(fact_979_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_rat_int] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_rat_int(G,top_top_set_rat) = top_top_set_int )
       => ( image_4975014655635759961ar_int(produc1863155021832019360at_int(F,G),top_to8437673547576850234ar_rat) = top_to3104204703268025230ar_int ) ) ) ).

% map_prod_surj
tff(fact_980_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_nat_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_nat_char(G,top_top_set_nat) = top_top_set_char )
       => ( image_5973708714899544198r_char(produc8759894717825771837t_char(F,G),top_to5594287848199751986ar_nat) = top_to1235241553633177651r_char ) ) ) ).

% map_prod_surj
tff(fact_981_map__prod__surj,axiom,
    ! [F: fun_char_char,G: fun_nat_rat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( aa_set_nat_set_rat(image_nat_rat(G),top_top_set_nat) = top_top_set_rat )
       => ( image_2699825501378146573ar_rat(produc1547070076189439812at_rat(F,G),top_to5594287848199751986ar_nat) = top_to8437673547576850234ar_rat ) ) ) ).

% map_prod_surj
tff(fact_982_GrD1,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Fx: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,F: fun_se6846035212209387651at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Fx),bNF_Gr7888059597227825506at_nat(A4,F))
     => member6273874620674182256at_nat(Xa,A4) ) ).

% GrD1
tff(fact_983_GrD1,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Fx: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,F: fun_se320340636488258563at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Fx)),bNF_Gr1527221351080490466at_nat(A4,F))
     => member2643936169264416010at_nat(Xa,A4) ) ).

% GrD1
tff(fact_984_GrD1,axiom,
    ! [Xa: nat,Fx: num,A4: set_nat,F: fun_nat_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Fx),bNF_Gr_nat_num(A4,F))
     => aa_set_nat_o(member_nat(Xa),A4) ) ).

% GrD1
tff(fact_985_GrD1,axiom,
    ! [Xa: nat,Fx: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Fx)),bNF_Gr_nat_nat(A4,F))
     => aa_set_nat_o(member_nat(Xa),A4) ) ).

% GrD1
tff(fact_986_GrD1,axiom,
    ! [Xa: int,Fx: int,A4: set_int,F: fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Fx)),bNF_Gr_int_int(A4,F))
     => aa_set_int_o(member_int(Xa),A4) ) ).

% GrD1
tff(fact_987_GrD2,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Fx: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,F: fun_se6846035212209387651at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Fx),bNF_Gr7888059597227825506at_nat(A4,F))
     => ( aa_set4883462369968598652at_nat(F,Xa) = Fx ) ) ).

% GrD2
tff(fact_988_GrD2,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Fx: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,F: fun_se320340636488258563at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Fx)),bNF_Gr1527221351080490466at_nat(A4,F))
     => ( aa_set8868280134176360700at_nat(F,Xa) = Fx ) ) ).

% GrD2
tff(fact_989_GrD2,axiom,
    ! [Xa: nat,Fx: num,A4: set_nat,F: fun_nat_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Fx),bNF_Gr_nat_num(A4,F))
     => ( aa_nat_num(F,Xa) = Fx ) ) ).

% GrD2
tff(fact_990_GrD2,axiom,
    ! [Xa: nat,Fx: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Fx)),bNF_Gr_nat_nat(A4,F))
     => ( aa_nat_nat(F,Xa) = Fx ) ) ).

% GrD2
tff(fact_991_GrD2,axiom,
    ! [Xa: int,Fx: int,A4: set_int,F: fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Fx)),bNF_Gr_int_int(A4,F))
     => ( aa_int_int(F,Xa) = Fx ) ) ).

% GrD2
tff(fact_992_success__refI,axiom,
    ! [V: a,Ha: heap_e7401611519738050253t_unit] : aa_hea7994211679605476167unit_o(heap_success_ref_a(aa_a_Heap_Heap_ref_a(ref_a2,V)),Ha) ).

% success_refI
tff(fact_993_fun_Orel__cong,axiom,
    ! [Xa: fun_char_o,Ya: fun_char_o,Y2: fun_char_o,Xaa: fun_char_o,R2: fun_o_fun_o_o,Ra2: fun_o_fun_o_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: $o,Yb: $o] :
              ( aa_set_o_o(member_o((Z2)),image_char_o(Ya,top_top_set_char))
             => ( aa_set_o_o(member_o((Yb)),image_char_o(Xaa,top_top_set_char))
               => ( aa_o_o(aa_o_fun_o_o(R2,(Z2)),(Yb))
                <=> aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Yb)) ) ) )
         => ( bNF_re1692451286490762625ar_o_o(fequal_char,R2,Xa,Y2)
          <=> bNF_re1692451286490762625ar_o_o(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_994_fun_Orel__cong,axiom,
    ! [Xa: fun_char_o,Ya: fun_char_o,Y2: fun_char_nat,Xaa: fun_char_nat,R2: fun_o_fun_nat_o,Ra2: fun_o_fun_nat_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: $o,Yb: nat] :
              ( aa_set_o_o(member_o((Z2)),image_char_o(Ya,top_top_set_char))
             => ( aa_set_nat_o(member_nat(Yb),image_char_nat(Xaa,top_top_set_char))
               => ( aa_nat_o(aa_o_fun_nat_o(R2,(Z2)),Yb)
                <=> aa_nat_o(aa_o_fun_nat_o(Ra2,(Z2)),Yb) ) ) )
         => ( bNF_re4782998131235581479_o_nat(fequal_char,R2,Xa,Y2)
          <=> bNF_re4782998131235581479_o_nat(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_995_fun_Orel__cong,axiom,
    ! [Xa: fun_char_o,Ya: fun_char_o,Y2: fun_char_int,Xaa: fun_char_int,R2: fun_o_fun_int_o,Ra2: fun_o_fun_int_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: $o,Yb: int] :
              ( aa_set_o_o(member_o((Z2)),image_char_o(Ya,top_top_set_char))
             => ( aa_set_int_o(member_int(Yb),image_char_int(Xaa,top_top_set_char))
               => ( aa_int_o(aa_o_fun_int_o(R2,(Z2)),Yb)
                <=> aa_int_o(aa_o_fun_int_o(Ra2,(Z2)),Yb) ) ) )
         => ( bNF_re4780507660726531203_o_int(fequal_char,R2,Xa,Y2)
          <=> bNF_re4780507660726531203_o_int(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_996_fun_Orel__cong,axiom,
    ! [Xa: fun_char_nat,Ya: fun_char_nat,Y2: fun_char_o,Xaa: fun_char_o,R2: fun_nat_fun_o_o,Ra2: fun_nat_fun_o_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: nat,Yb: $o] :
              ( aa_set_nat_o(member_nat(Z2),image_char_nat(Ya,top_top_set_char))
             => ( aa_set_o_o(member_o((Yb)),image_char_o(Xaa,top_top_set_char))
               => ( aa_o_o(aa_nat_fun_o_o(R2,Z2),(Yb))
                <=> aa_o_o(aa_nat_fun_o_o(Ra2,Z2),(Yb)) ) ) )
         => ( bNF_re7885669492343766789_nat_o(fequal_char,R2,Xa,Y2)
          <=> bNF_re7885669492343766789_nat_o(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_997_fun_Orel__cong,axiom,
    ! [Xa: fun_char_nat,Ya: fun_char_nat,Y2: fun_char_nat,Xaa: fun_char_nat,R2: fun_nat_fun_nat_o,Ra2: fun_nat_fun_nat_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: nat,Yb: nat] :
              ( aa_set_nat_o(member_nat(Z2),image_char_nat(Ya,top_top_set_char))
             => ( aa_set_nat_o(member_nat(Yb),image_char_nat(Xaa,top_top_set_char))
               => ( aa_nat_o(aa_nat_fun_nat_o(R2,Z2),Yb)
                <=> aa_nat_o(aa_nat_fun_nat_o(Ra2,Z2),Yb) ) ) )
         => ( aa_fun_char_nat_o(aa_fun8433001934219309943_nat_o(bNF_re3734335887435638179at_nat(fequal_char,R2),Xa),Y2)
          <=> aa_fun_char_nat_o(aa_fun8433001934219309943_nat_o(bNF_re3734335887435638179at_nat(fequal_char,Ra2),Ya),Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_998_fun_Orel__cong,axiom,
    ! [Xa: fun_char_nat,Ya: fun_char_nat,Y2: fun_char_int,Xaa: fun_char_int,R2: fun_nat_fun_int_o,Ra2: fun_nat_fun_int_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: nat,Yb: int] :
              ( aa_set_nat_o(member_nat(Z2),image_char_nat(Ya,top_top_set_char))
             => ( aa_set_int_o(member_int(Yb),image_char_int(Xaa,top_top_set_char))
               => ( aa_int_o(aa_nat_fun_int_o(R2,Z2),Yb)
                <=> aa_int_o(aa_nat_fun_int_o(Ra2,Z2),Yb) ) ) )
         => ( bNF_re3731845416926587903at_int(fequal_char,R2,Xa,Y2)
          <=> bNF_re3731845416926587903at_int(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_999_fun_Orel__cong,axiom,
    ! [Xa: fun_char_int,Ya: fun_char_int,Y2: fun_char_o,Xaa: fun_char_o,R2: fun_int_fun_o_o,Ra2: fun_int_fun_o_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: int,Yb: $o] :
              ( aa_set_int_o(member_int(Z2),image_char_int(Ya,top_top_set_char))
             => ( aa_set_o_o(member_o((Yb)),image_char_o(Xaa,top_top_set_char))
               => ( aa_o_o(aa_int_fun_o_o(R2,Z2),(Yb))
                <=> aa_o_o(aa_int_fun_o_o(Ra2,Z2),(Yb)) ) ) )
         => ( bNF_re6064627146911078825_int_o(fequal_char,R2,Xa,Y2)
          <=> bNF_re6064627146911078825_int_o(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_1000_fun_Orel__cong,axiom,
    ! [Xa: fun_char_int,Ya: fun_char_int,Y2: fun_char_nat,Xaa: fun_char_nat,R2: fun_int_fun_nat_o,Ra2: fun_int_fun_nat_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: int,Yb: nat] :
              ( aa_set_int_o(member_int(Z2),image_char_int(Ya,top_top_set_char))
             => ( aa_set_nat_o(member_nat(Yb),image_char_nat(Xaa,top_top_set_char))
               => ( aa_nat_o(aa_int_fun_nat_o(R2,Z2),Yb)
                <=> aa_nat_o(aa_int_fun_nat_o(Ra2,Z2),Yb) ) ) )
         => ( bNF_re4733689599336692735nt_nat(fequal_char,R2,Xa,Y2)
          <=> bNF_re4733689599336692735nt_nat(fequal_char,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_1001_fun_Orel__cong,axiom,
    ! [Xa: fun_char_int,Ya: fun_char_int,Y2: fun_char_int,Xaa: fun_char_int,R2: fun_int_fun_int_o,Ra2: fun_int_fun_int_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: int,Yb: int] :
              ( aa_set_int_o(member_int(Z2),image_char_int(Ya,top_top_set_char))
             => ( aa_set_int_o(member_int(Yb),image_char_int(Xaa,top_top_set_char))
               => ( aa_int_o(aa_int_fun_int_o(R2,Z2),Yb)
                <=> aa_int_o(aa_int_fun_int_o(Ra2,Z2),Yb) ) ) )
         => ( aa_fun_char_int_o(aa_fun5090773985904865471_int_o(bNF_re4731199128827642459nt_int(fequal_char,R2),Xa),Y2)
          <=> aa_fun_char_int_o(aa_fun5090773985904865471_int_o(bNF_re4731199128827642459nt_int(fequal_char,Ra2),Ya),Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_1002_fun_Orel__cong,axiom,
    ! [Xa: fun_rat_o,Ya: fun_rat_o,Y2: fun_rat_o,Xaa: fun_rat_o,R2: fun_o_fun_o_o,Ra2: fun_o_fun_o_o] :
      ( ( Xa = Ya )
     => ( ( Y2 = Xaa )
       => ( ! [Z2: $o,Yb: $o] :
              ( aa_set_o_o(member_o((Z2)),image_rat_o(Ya,top_top_set_rat))
             => ( aa_set_o_o(member_o((Yb)),image_rat_o(Xaa,top_top_set_rat))
               => ( aa_o_o(aa_o_fun_o_o(R2,(Z2)),(Yb))
                <=> aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Yb)) ) ) )
         => ( bNF_re1865807574169326223at_o_o(fequal_rat,R2,Xa,Y2)
          <=> bNF_re1865807574169326223at_o_o(fequal_rat,Ra2,Ya,Xaa) ) ) ) ) ).

% fun.rel_cong
tff(fact_1003_fun_Orel__mono__strong,axiom,
    ! [R2: fun_o_fun_o_o,Xa: fun_char_o,Y2: fun_char_o,Ra2: fun_o_fun_o_o] :
      ( bNF_re1692451286490762625ar_o_o(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: $o,Yb: $o] :
            ( aa_set_o_o(member_o((Z2)),image_char_o(Xa,top_top_set_char))
           => ( aa_set_o_o(member_o((Yb)),image_char_o(Y2,top_top_set_char))
             => ( aa_o_o(aa_o_fun_o_o(R2,(Z2)),(Yb))
               => aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Yb)) ) ) )
       => bNF_re1692451286490762625ar_o_o(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1004_fun_Orel__mono__strong,axiom,
    ! [R2: fun_o_fun_nat_o,Xa: fun_char_o,Y2: fun_char_nat,Ra2: fun_o_fun_nat_o] :
      ( bNF_re4782998131235581479_o_nat(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: $o,Yb: nat] :
            ( aa_set_o_o(member_o((Z2)),image_char_o(Xa,top_top_set_char))
           => ( aa_set_nat_o(member_nat(Yb),image_char_nat(Y2,top_top_set_char))
             => ( aa_nat_o(aa_o_fun_nat_o(R2,(Z2)),Yb)
               => aa_nat_o(aa_o_fun_nat_o(Ra2,(Z2)),Yb) ) ) )
       => bNF_re4782998131235581479_o_nat(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1005_fun_Orel__mono__strong,axiom,
    ! [R2: fun_o_fun_int_o,Xa: fun_char_o,Y2: fun_char_int,Ra2: fun_o_fun_int_o] :
      ( bNF_re4780507660726531203_o_int(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: $o,Yb: int] :
            ( aa_set_o_o(member_o((Z2)),image_char_o(Xa,top_top_set_char))
           => ( aa_set_int_o(member_int(Yb),image_char_int(Y2,top_top_set_char))
             => ( aa_int_o(aa_o_fun_int_o(R2,(Z2)),Yb)
               => aa_int_o(aa_o_fun_int_o(Ra2,(Z2)),Yb) ) ) )
       => bNF_re4780507660726531203_o_int(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1006_fun_Orel__mono__strong,axiom,
    ! [R2: fun_nat_fun_o_o,Xa: fun_char_nat,Y2: fun_char_o,Ra2: fun_nat_fun_o_o] :
      ( bNF_re7885669492343766789_nat_o(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: nat,Yb: $o] :
            ( aa_set_nat_o(member_nat(Z2),image_char_nat(Xa,top_top_set_char))
           => ( aa_set_o_o(member_o((Yb)),image_char_o(Y2,top_top_set_char))
             => ( aa_o_o(aa_nat_fun_o_o(R2,Z2),(Yb))
               => aa_o_o(aa_nat_fun_o_o(Ra2,Z2),(Yb)) ) ) )
       => bNF_re7885669492343766789_nat_o(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1007_fun_Orel__mono__strong,axiom,
    ! [R2: fun_nat_fun_nat_o,Xa: fun_char_nat,Y2: fun_char_nat,Ra2: fun_nat_fun_nat_o] :
      ( aa_fun_char_nat_o(aa_fun8433001934219309943_nat_o(bNF_re3734335887435638179at_nat(fequal_char,R2),Xa),Y2)
     => ( ! [Z2: nat,Yb: nat] :
            ( aa_set_nat_o(member_nat(Z2),image_char_nat(Xa,top_top_set_char))
           => ( aa_set_nat_o(member_nat(Yb),image_char_nat(Y2,top_top_set_char))
             => ( aa_nat_o(aa_nat_fun_nat_o(R2,Z2),Yb)
               => aa_nat_o(aa_nat_fun_nat_o(Ra2,Z2),Yb) ) ) )
       => aa_fun_char_nat_o(aa_fun8433001934219309943_nat_o(bNF_re3734335887435638179at_nat(fequal_char,Ra2),Xa),Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1008_fun_Orel__mono__strong,axiom,
    ! [R2: fun_nat_fun_int_o,Xa: fun_char_nat,Y2: fun_char_int,Ra2: fun_nat_fun_int_o] :
      ( bNF_re3731845416926587903at_int(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: nat,Yb: int] :
            ( aa_set_nat_o(member_nat(Z2),image_char_nat(Xa,top_top_set_char))
           => ( aa_set_int_o(member_int(Yb),image_char_int(Y2,top_top_set_char))
             => ( aa_int_o(aa_nat_fun_int_o(R2,Z2),Yb)
               => aa_int_o(aa_nat_fun_int_o(Ra2,Z2),Yb) ) ) )
       => bNF_re3731845416926587903at_int(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1009_fun_Orel__mono__strong,axiom,
    ! [R2: fun_int_fun_o_o,Xa: fun_char_int,Y2: fun_char_o,Ra2: fun_int_fun_o_o] :
      ( bNF_re6064627146911078825_int_o(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: int,Yb: $o] :
            ( aa_set_int_o(member_int(Z2),image_char_int(Xa,top_top_set_char))
           => ( aa_set_o_o(member_o((Yb)),image_char_o(Y2,top_top_set_char))
             => ( aa_o_o(aa_int_fun_o_o(R2,Z2),(Yb))
               => aa_o_o(aa_int_fun_o_o(Ra2,Z2),(Yb)) ) ) )
       => bNF_re6064627146911078825_int_o(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1010_fun_Orel__mono__strong,axiom,
    ! [R2: fun_int_fun_nat_o,Xa: fun_char_int,Y2: fun_char_nat,Ra2: fun_int_fun_nat_o] :
      ( bNF_re4733689599336692735nt_nat(fequal_char,R2,Xa,Y2)
     => ( ! [Z2: int,Yb: nat] :
            ( aa_set_int_o(member_int(Z2),image_char_int(Xa,top_top_set_char))
           => ( aa_set_nat_o(member_nat(Yb),image_char_nat(Y2,top_top_set_char))
             => ( aa_nat_o(aa_int_fun_nat_o(R2,Z2),Yb)
               => aa_nat_o(aa_int_fun_nat_o(Ra2,Z2),Yb) ) ) )
       => bNF_re4733689599336692735nt_nat(fequal_char,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1011_fun_Orel__mono__strong,axiom,
    ! [R2: fun_int_fun_int_o,Xa: fun_char_int,Y2: fun_char_int,Ra2: fun_int_fun_int_o] :
      ( aa_fun_char_int_o(aa_fun5090773985904865471_int_o(bNF_re4731199128827642459nt_int(fequal_char,R2),Xa),Y2)
     => ( ! [Z2: int,Yb: int] :
            ( aa_set_int_o(member_int(Z2),image_char_int(Xa,top_top_set_char))
           => ( aa_set_int_o(member_int(Yb),image_char_int(Y2,top_top_set_char))
             => ( aa_int_o(aa_int_fun_int_o(R2,Z2),Yb)
               => aa_int_o(aa_int_fun_int_o(Ra2,Z2),Yb) ) ) )
       => aa_fun_char_int_o(aa_fun5090773985904865471_int_o(bNF_re4731199128827642459nt_int(fequal_char,Ra2),Xa),Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1012_fun_Orel__mono__strong,axiom,
    ! [R2: fun_o_fun_o_o,Xa: fun_rat_o,Y2: fun_rat_o,Ra2: fun_o_fun_o_o] :
      ( bNF_re1865807574169326223at_o_o(fequal_rat,R2,Xa,Y2)
     => ( ! [Z2: $o,Yb: $o] :
            ( aa_set_o_o(member_o((Z2)),image_rat_o(Xa,top_top_set_rat))
           => ( aa_set_o_o(member_o((Yb)),image_rat_o(Y2,top_top_set_rat))
             => ( aa_o_o(aa_o_fun_o_o(R2,(Z2)),(Yb))
               => aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Yb)) ) ) )
       => bNF_re1865807574169326223at_o_o(fequal_rat,Ra2,Xa,Y2) ) ) ).

% fun.rel_mono_strong
tff(fact_1013_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_char_o,Ra2: fun_o_fun_o_o] :
      ( ! [Z2: $o] :
          ( aa_set_o_o(member_o((Z2)),image_char_o(Xa,top_top_set_char))
         => aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Z2)) )
     => bNF_re1692451286490762625ar_o_o(fequal_char,Ra2,Xa,Xa) ) ).

% fun.rel_refl_strong
tff(fact_1014_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_char_nat,Ra2: fun_nat_fun_nat_o] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),image_char_nat(Xa,top_top_set_char))
         => aa_nat_o(aa_nat_fun_nat_o(Ra2,Z2),Z2) )
     => aa_fun_char_nat_o(aa_fun8433001934219309943_nat_o(bNF_re3734335887435638179at_nat(fequal_char,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1015_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_char_int,Ra2: fun_int_fun_int_o] :
      ( ! [Z2: int] :
          ( aa_set_int_o(member_int(Z2),image_char_int(Xa,top_top_set_char))
         => aa_int_o(aa_int_fun_int_o(Ra2,Z2),Z2) )
     => aa_fun_char_int_o(aa_fun5090773985904865471_int_o(bNF_re4731199128827642459nt_int(fequal_char,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1016_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_rat_o,Ra2: fun_o_fun_o_o] :
      ( ! [Z2: $o] :
          ( aa_set_o_o(member_o((Z2)),image_rat_o(Xa,top_top_set_rat))
         => aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Z2)) )
     => bNF_re1865807574169326223at_o_o(fequal_rat,Ra2,Xa,Xa) ) ).

% fun.rel_refl_strong
tff(fact_1017_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_rat_nat,Ra2: fun_nat_fun_nat_o] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),image_rat_nat(Xa,top_top_set_rat))
         => aa_nat_o(aa_nat_fun_nat_o(Ra2,Z2),Z2) )
     => aa_fun_rat_nat_o(aa_fun6999665265510530025_nat_o(bNF_re6557955094579809201at_nat(fequal_rat,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1018_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_rat_int,Ra2: fun_int_fun_int_o] :
      ( ! [Z2: int] :
          ( aa_set_int_o(member_int(Z2),image_rat_int(Xa,top_top_set_rat))
         => aa_int_o(aa_int_fun_int_o(Ra2,Z2),Z2) )
     => aa_fun_rat_int_o(aa_fun2499607139650510129_int_o(bNF_re7554818335971813481nt_int(fequal_rat,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1019_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_nat_rat,Ra2: fun_rat_fun_rat_o] :
      ( ! [Z2: rat] :
          ( aa_set_rat_o(member_rat(Z2),aa_set_nat_set_rat(image_nat_rat(Xa),top_top_set_nat))
         => aa_rat_o(aa_rat_fun_rat_o(Ra2,Z2),Z2) )
     => bNF_re4702136315717946289at_rat(fequal_nat,Ra2,Xa,Xa) ) ).

% fun.rel_refl_strong
tff(fact_1020_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_nat_o,Ra2: fun_o_fun_o_o] :
      ( ! [Z2: $o] :
          ( aa_set_o_o(member_o((Z2)),image_nat_o(Xa,top_top_set_nat))
         => aa_o_o(aa_o_fun_o_o(Ra2,(Z2)),(Z2)) )
     => aa_fun_nat_o_o(aa_fun4840518763754620511at_o_o(bNF_re4705727531993890431at_o_o(fequal_nat,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1021_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_nat_nat,Ra2: fun_nat_fun_nat_o] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),aa_set_nat_set_nat(image_nat_nat(Xa),top_top_set_nat))
         => aa_nat_o(aa_nat_fun_nat_o(Ra2,Z2),Z2) )
     => aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1022_fun_Orel__refl__strong,axiom,
    ! [Xa: fun_nat_int,Ra2: fun_int_fun_int_o] :
      ( ! [Z2: int] :
          ( aa_set_int_o(member_int(Z2),aa_set_nat_set_int(image_nat_int(Xa),top_top_set_nat))
         => aa_int_o(aa_int_fun_int_o(Ra2,Z2),Z2) )
     => aa_fun_nat_int_o(aa_fun7064809811658394945_int_o(bNF_re6650684261131312217nt_int(fequal_nat,Ra2),Xa),Xa) ) ).

% fun.rel_refl_strong
tff(fact_1023_range__subsetD,axiom,
    ! [F: fun_char_o,B5: set_o,I: char] :
      ( aa_set_o_o(ord_less_eq_set_o(image_char_o(F,top_top_set_char)),B5)
     => aa_set_o_o(member_o(aa_char_o(F,I)),B5) ) ).

% range_subsetD
tff(fact_1024_range__subsetD,axiom,
    ! [F: fun_char_nat,B5: set_nat,I: char] :
      ( aa_set_nat_o(ord_less_eq_set_nat(image_char_nat(F,top_top_set_char)),B5)
     => aa_set_nat_o(member_nat(aa_char_nat(F,I)),B5) ) ).

% range_subsetD
tff(fact_1025_range__subsetD,axiom,
    ! [F: fun_char_int,B5: set_int,I: char] :
      ( aa_set_int_o(ord_less_eq_set_int(image_char_int(F,top_top_set_char)),B5)
     => aa_set_int_o(member_int(aa_char_int(F,I)),B5) ) ).

% range_subsetD
tff(fact_1026_range__subsetD,axiom,
    ! [F: fun_rat_o,B5: set_o,I: rat] :
      ( aa_set_o_o(ord_less_eq_set_o(image_rat_o(F,top_top_set_rat)),B5)
     => aa_set_o_o(member_o(aa_rat_o(F,I)),B5) ) ).

% range_subsetD
tff(fact_1027_range__subsetD,axiom,
    ! [F: fun_rat_nat,B5: set_nat,I: rat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(image_rat_nat(F,top_top_set_rat)),B5)
     => aa_set_nat_o(member_nat(aa_rat_nat(F,I)),B5) ) ).

% range_subsetD
tff(fact_1028_range__subsetD,axiom,
    ! [F: fun_rat_int,B5: set_int,I: rat] :
      ( aa_set_int_o(ord_less_eq_set_int(image_rat_int(F,top_top_set_rat)),B5)
     => aa_set_int_o(member_int(aa_rat_int(F,I)),B5) ) ).

% range_subsetD
tff(fact_1029_range__subsetD,axiom,
    ! [F: fun_nat_rat,B5: set_rat,I: nat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat)),B5)
     => aa_set_rat_o(member_rat(aa_nat_rat(F,I)),B5) ) ).

% range_subsetD
tff(fact_1030_range__subsetD,axiom,
    ! [F: fun_nat_o,B5: set_o,I: nat] :
      ( aa_set_o_o(ord_less_eq_set_o(image_nat_o(F,top_top_set_nat)),B5)
     => aa_set_o_o(member_o(aa_nat_o(F,I)),B5) ) ).

% range_subsetD
tff(fact_1031_range__subsetD,axiom,
    ! [F: fun_nat_nat,B5: set_nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat)),B5)
     => aa_set_nat_o(member_nat(aa_nat_nat(F,I)),B5) ) ).

% range_subsetD
tff(fact_1032_range__subsetD,axiom,
    ! [F: fun_nat_int,B5: set_int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),top_top_set_nat)),B5)
     => aa_set_int_o(member_int(aa_nat_int(F,I)),B5) ) ).

% range_subsetD
tff(fact_1033_fun_Oset__map,axiom,
    ! [F: fun_nat_rat,V: fun_char_nat] : image_char_rat(comp_nat_rat_char(F,V),top_top_set_char) = aa_set_nat_set_rat(image_nat_rat(F),image_char_nat(V,top_top_set_char)) ).

% fun.set_map
tff(fact_1034_fun_Oset__map,axiom,
    ! [F: fun_nat_nat,V: fun_char_nat] : image_char_nat(comp_nat_nat_char(F,V),top_top_set_char) = aa_set_nat_set_nat(image_nat_nat(F),image_char_nat(V,top_top_set_char)) ).

% fun.set_map
tff(fact_1035_fun_Oset__map,axiom,
    ! [F: fun_nat_int,V: fun_char_nat] : image_char_int(comp_nat_int_char(F,V),top_top_set_char) = aa_set_nat_set_int(image_nat_int(F),image_char_nat(V,top_top_set_char)) ).

% fun.set_map
tff(fact_1036_fun_Oset__map,axiom,
    ! [F: fun_int_nat,V: fun_char_int] : image_char_nat(comp_int_nat_char(F,V),top_top_set_char) = aa_set_int_set_nat(image_int_nat(F),image_char_int(V,top_top_set_char)) ).

% fun.set_map
tff(fact_1037_fun_Oset__map,axiom,
    ! [F: fun_int_int,V: fun_char_int] : image_char_int(comp_int_int_char(F,V),top_top_set_char) = aa_set_int_set_int(image_int_int(F),image_char_int(V,top_top_set_char)) ).

% fun.set_map
tff(fact_1038_fun_Oset__map,axiom,
    ! [F: fun_nat_rat,V: fun_rat_nat] : image_rat_rat(comp_nat_rat_rat(F,V),top_top_set_rat) = aa_set_nat_set_rat(image_nat_rat(F),image_rat_nat(V,top_top_set_rat)) ).

% fun.set_map
tff(fact_1039_fun_Oset__map,axiom,
    ! [F: fun_nat_nat,V: fun_rat_nat] : image_rat_nat(comp_nat_nat_rat(F,V),top_top_set_rat) = aa_set_nat_set_nat(image_nat_nat(F),image_rat_nat(V,top_top_set_rat)) ).

% fun.set_map
tff(fact_1040_fun_Oset__map,axiom,
    ! [F: fun_nat_int,V: fun_rat_nat] : image_rat_int(comp_nat_int_rat(F,V),top_top_set_rat) = aa_set_nat_set_int(image_nat_int(F),image_rat_nat(V,top_top_set_rat)) ).

% fun.set_map
tff(fact_1041_fun_Oset__map,axiom,
    ! [F: fun_int_nat,V: fun_rat_int] : image_rat_nat(comp_int_nat_rat(F,V),top_top_set_rat) = aa_set_int_set_nat(image_int_nat(F),image_rat_int(V,top_top_set_rat)) ).

% fun.set_map
tff(fact_1042_fun_Oset__map,axiom,
    ! [F: fun_int_int,V: fun_rat_int] : image_rat_int(comp_int_int_rat(F,V),top_top_set_rat) = aa_set_int_set_int(image_int_int(F),image_rat_int(V,top_top_set_rat)) ).

% fun.set_map
tff(fact_1043_fun_Omap__cong,axiom,
    ! [Xa: fun_Co1981900195452921617nteger,Ya: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger,G: fun_fu4800041728187187127nteger] :
      ( ( Xa = Ya )
     => ( ! [Z2: fun_Co8499238673658658103nteger] :
            ( member8321684580148902862nteger(Z2,image_2716445779319712667nteger(Ya,top_to4645266643341252675nteger))
           => ( aa_fun2509385170507305080nteger(F,Z2) = aa_fun2509385170507305080nteger(G,Z2) ) )
       => ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),Xa) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_1044_fun_Omap__cong,axiom,
    ! [Xa: fun_nat_o,Ya: fun_nat_o,F: fun_o_o,G: fun_o_o] :
      ( ( Xa = Ya )
     => ( ! [Z2: $o] :
            ( aa_set_o_o(member_o((Z2)),image_nat_o(Ya,top_top_set_nat))
           => ( aa_o_o(F,(Z2))
            <=> aa_o_o(G,(Z2)) ) )
       => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Xa) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_1045_fun_Omap__cong,axiom,
    ! [Xa: fun_nat_nat,Ya: fun_nat_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xa = Ya )
     => ( ! [Z2: nat] :
            ( aa_set_nat_o(member_nat(Z2),aa_set_nat_set_nat(image_nat_nat(Ya),top_top_set_nat))
           => ( aa_nat_nat(F,Z2) = aa_nat_nat(G,Z2) ) )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Xa) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_1046_fun_Omap__cong,axiom,
    ! [Xa: fun_int_nat,Ya: fun_int_nat,F: fun_nat_num,G: fun_nat_num] :
      ( ( Xa = Ya )
     => ( ! [Z2: nat] :
            ( aa_set_nat_o(member_nat(Z2),aa_set_int_set_nat(image_int_nat(Ya),top_top_set_int))
           => ( aa_nat_num(F,Z2) = aa_nat_num(G,Z2) ) )
       => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Xa) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_1047_fun_Omap__cong,axiom,
    ! [Xa: fun_int_int,Ya: fun_int_int,F: fun_int_nat,G: fun_int_nat] :
      ( ( Xa = Ya )
     => ( ! [Z2: int] :
            ( aa_set_int_o(member_int(Z2),aa_set_int_set_int(image_int_int(Ya),top_top_set_int))
           => ( aa_int_nat(F,Z2) = aa_int_nat(G,Z2) ) )
       => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),Xa) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_1048_fun_Omap__cong0,axiom,
    ! [Xa: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger,G: fun_fu4800041728187187127nteger] :
      ( ! [Z2: fun_Co8499238673658658103nteger] :
          ( member8321684580148902862nteger(Z2,image_2716445779319712667nteger(Xa,top_to4645266643341252675nteger))
         => ( aa_fun2509385170507305080nteger(F,Z2) = aa_fun2509385170507305080nteger(G,Z2) ) )
     => ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),Xa) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),Xa) ) ) ).

% fun.map_cong0
tff(fact_1049_fun_Omap__cong0,axiom,
    ! [Xa: fun_nat_o,F: fun_o_o,G: fun_o_o] :
      ( ! [Z2: $o] :
          ( aa_set_o_o(member_o((Z2)),image_nat_o(Xa,top_top_set_nat))
         => ( aa_o_o(F,(Z2))
          <=> aa_o_o(G,(Z2)) ) )
     => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Xa) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),Xa) ) ) ).

% fun.map_cong0
tff(fact_1050_fun_Omap__cong0,axiom,
    ! [Xa: fun_nat_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),aa_set_nat_set_nat(image_nat_nat(Xa),top_top_set_nat))
         => ( aa_nat_nat(F,Z2) = aa_nat_nat(G,Z2) ) )
     => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Xa) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),Xa) ) ) ).

% fun.map_cong0
tff(fact_1051_fun_Omap__cong0,axiom,
    ! [Xa: fun_int_nat,F: fun_nat_num,G: fun_nat_num] :
      ( ! [Z2: nat] :
          ( aa_set_nat_o(member_nat(Z2),aa_set_int_set_nat(image_int_nat(Xa),top_top_set_int))
         => ( aa_nat_num(F,Z2) = aa_nat_num(G,Z2) ) )
     => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Xa) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Xa) ) ) ).

% fun.map_cong0
tff(fact_1052_fun_Omap__cong0,axiom,
    ! [Xa: fun_int_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [Z2: int] :
          ( aa_set_int_o(member_int(Z2),aa_set_int_set_int(image_int_int(Xa),top_top_set_int))
         => ( aa_int_nat(F,Z2) = aa_int_nat(G,Z2) ) )
     => ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),Xa) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),Xa) ) ) ).

% fun.map_cong0
tff(fact_1053_fun_Oinj__map__strong,axiom,
    ! [Xa: fun_Co1981900195452921617nteger,Xaa: fun_Co1981900195452921617nteger,F: fun_fu4800041728187187127nteger,Fa: fun_fu4800041728187187127nteger] :
      ( ! [Z2: fun_Co8499238673658658103nteger,Za: fun_Co8499238673658658103nteger] :
          ( member8321684580148902862nteger(Z2,image_2716445779319712667nteger(Xa,top_to4645266643341252675nteger))
         => ( member8321684580148902862nteger(Za,image_2716445779319712667nteger(Xaa,top_to4645266643341252675nteger))
           => ( ( aa_fun2509385170507305080nteger(F,Z2) = aa_fun2509385170507305080nteger(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),Xa) = aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(Fa),Xaa) )
       => ( Xa = Xaa ) ) ) ).

% fun.inj_map_strong
tff(fact_1054_fun_Oinj__map__strong,axiom,
    ! [Xa: fun_nat_o,Xaa: fun_nat_o,F: fun_o_o,Fa: fun_o_o] :
      ( ! [Z2: $o,Za: $o] :
          ( aa_set_o_o(member_o((Z2)),image_nat_o(Xa,top_top_set_nat))
         => ( aa_set_o_o(member_o((Za)),image_nat_o(Xaa,top_top_set_nat))
           => ( ( aa_o_o(F,(Z2))
              <=> aa_o_o(Fa,(Za)) )
             => ( (Z2)
              <=> (Za) ) ) ) )
     => ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),Xa) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Fa),Xaa) )
       => ( Xa = Xaa ) ) ) ).

% fun.inj_map_strong
tff(fact_1055_fun_Oinj__map__strong,axiom,
    ! [Xa: fun_nat_nat,Xaa: fun_nat_nat,F: fun_nat_nat,Fa: fun_nat_nat] :
      ( ! [Z2: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z2),aa_set_nat_set_nat(image_nat_nat(Xa),top_top_set_nat))
         => ( aa_set_nat_o(member_nat(Za),aa_set_nat_set_nat(image_nat_nat(Xaa),top_top_set_nat))
           => ( ( aa_nat_nat(F,Z2) = aa_nat_nat(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),Xa) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Fa),Xaa) )
       => ( Xa = Xaa ) ) ) ).

% fun.inj_map_strong
tff(fact_1056_fun_Oinj__map__strong,axiom,
    ! [Xa: fun_int_nat,Xaa: fun_int_nat,F: fun_nat_num,Fa: fun_nat_num] :
      ( ! [Z2: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z2),aa_set_int_set_nat(image_int_nat(Xa),top_top_set_int))
         => ( aa_set_nat_o(member_nat(Za),aa_set_int_set_nat(image_int_nat(Xaa),top_top_set_int))
           => ( ( aa_nat_num(F,Z2) = aa_nat_num(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),Xa) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Fa),Xaa) )
       => ( Xa = Xaa ) ) ) ).

% fun.inj_map_strong
tff(fact_1057_fun_Oinj__map__strong,axiom,
    ! [Xa: fun_int_int,Xaa: fun_int_int,F: fun_int_nat,Fa: fun_int_nat] :
      ( ! [Z2: int,Za: int] :
          ( aa_set_int_o(member_int(Z2),aa_set_int_set_int(image_int_int(Xa),top_top_set_int))
         => ( aa_set_int_o(member_int(Za),aa_set_int_set_int(image_int_int(Xaa),top_top_set_int))
           => ( ( aa_int_nat(F,Z2) = aa_int_nat(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),Xa) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Fa),Xaa) )
       => ( Xa = Xaa ) ) ) ).

% fun.inj_map_strong
tff(fact_1058_surj__fun__eq,axiom,
    ! [F: fun_nat_o,X3: set_nat,G1: fun_o_o,G22: fun_o_o] :
      ( ( image_nat_o(F,X3) = top_top_set_o )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),X3)
           => ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G1),F),X)
            <=> aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G22),F),X) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_1059_surj__fun__eq,axiom,
    ! [F: fun_Co1981900195452921617nteger,X3: set_Code_integer,G1: fun_fu4800041728187187127nteger,G22: fun_fu4800041728187187127nteger] :
      ( ( image_2716445779319712667nteger(F,X3) = top_to805621317456759325nteger )
     => ( ! [X: code_integer] :
            ( member_Code_integer(X,X3)
           => ( aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G1),F),X) = aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G22),F),X) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_1060_surj__fun__eq,axiom,
    ! [F: fun_nat_nat,X3: set_nat,G1: fun_nat_nat,G22: fun_nat_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F),X3) = top_top_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),X3)
           => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G1),F),X) = aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G22),F),X) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_1061_surj__fun__eq,axiom,
    ! [F: fun_int_nat,X3: set_int,G1: fun_nat_num,G22: fun_nat_num] :
      ( ( aa_set_int_set_nat(image_int_nat(F),X3) = top_top_set_nat )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),X3)
           => ( aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G1),F),X) = aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G22),F),X) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_1062_surj__fun__eq,axiom,
    ! [F: fun_int_int,X3: set_int,G1: fun_int_nat,G22: fun_int_nat] :
      ( ( aa_set_int_set_int(image_int_int(F),X3) = top_top_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),X3)
           => ( aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G1),F),X) = aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G22),F),X) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_1063_comp__surj,axiom,
    ! [F: fun_char_char,G: fun_char_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_char(G,top_top_set_char) = top_top_set_char )
       => ( image_char_char(comp_char_char_char(G,F),top_top_set_char) = top_top_set_char ) ) ) ).

% comp_surj
tff(fact_1064_comp__surj,axiom,
    ! [F: fun_char_char,G: fun_char_rat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_rat(G,top_top_set_char) = top_top_set_rat )
       => ( image_char_rat(comp_char_rat_char(G,F),top_top_set_char) = top_top_set_rat ) ) ) ).

% comp_surj
tff(fact_1065_comp__surj,axiom,
    ! [F: fun_char_char,G: fun_char_nat] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_nat(G,top_top_set_char) = top_top_set_nat )
       => ( image_char_nat(comp_char_nat_char(G,F),top_top_set_char) = top_top_set_nat ) ) ) ).

% comp_surj
tff(fact_1066_comp__surj,axiom,
    ! [F: fun_char_char,G: fun_char_int] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ( image_char_int(G,top_top_set_char) = top_top_set_int )
       => ( image_char_int(comp_char_int_char(G,F),top_top_set_char) = top_top_set_int ) ) ) ).

% comp_surj
tff(fact_1067_comp__surj,axiom,
    ! [F: fun_char_rat,G: fun_rat_char] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ( ( image_rat_char(G,top_top_set_rat) = top_top_set_char )
       => ( image_char_char(comp_rat_char_char(G,F),top_top_set_char) = top_top_set_char ) ) ) ).

% comp_surj
tff(fact_1068_comp__surj,axiom,
    ! [F: fun_char_rat,G: fun_rat_rat] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ( ( image_rat_rat(G,top_top_set_rat) = top_top_set_rat )
       => ( image_char_rat(comp_rat_rat_char(G,F),top_top_set_char) = top_top_set_rat ) ) ) ).

% comp_surj
tff(fact_1069_comp__surj,axiom,
    ! [F: fun_char_rat,G: fun_rat_nat] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ( ( image_rat_nat(G,top_top_set_rat) = top_top_set_nat )
       => ( image_char_nat(comp_rat_nat_char(G,F),top_top_set_char) = top_top_set_nat ) ) ) ).

% comp_surj
tff(fact_1070_comp__surj,axiom,
    ! [F: fun_char_rat,G: fun_rat_int] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ( ( image_rat_int(G,top_top_set_rat) = top_top_set_int )
       => ( image_char_int(comp_rat_int_char(G,F),top_top_set_char) = top_top_set_int ) ) ) ).

% comp_surj
tff(fact_1071_comp__surj,axiom,
    ! [F: fun_char_nat,G: fun_nat_char] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
     => ( ( image_nat_char(G,top_top_set_nat) = top_top_set_char )
       => ( image_char_char(comp_nat_char_char(G,F),top_top_set_char) = top_top_set_char ) ) ) ).

% comp_surj
tff(fact_1072_comp__surj,axiom,
    ! [F: fun_char_nat,G: fun_nat_rat] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
     => ( ( aa_set_nat_set_rat(image_nat_rat(G),top_top_set_nat) = top_top_set_rat )
       => ( image_char_rat(comp_nat_rat_char(G,F),top_top_set_char) = top_top_set_rat ) ) ) ).

% comp_surj
tff(fact_1073_apsnd__def,axiom,
    ! [F: fun_Co8499238673658658103nteger] : aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F) = aa_fun2509385170507305080nteger(produc1802105962441033582nteger(id_Code_integer),F) ).

% apsnd_def
tff(fact_1074_rel__fun__image2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,F: fun_Pr2602117396471556196at_nat,G: fun_int_nat] : aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(R2,bNF_Gr6955563364113978072nt_nat(F,G,R2)),F),G) ).

% rel_fun_image2p
tff(fact_1075_rel__fun__image2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,F: fun_Pr2477239776880934807at_nat,G: fun_int_fun_int_int] : bNF_re8869769828555196650nt_int(R2,bNF_Gr7638081787706718096nt_int(F,G,R2),F,G) ).

% rel_fun_image2p
tff(fact_1076_rel__fun__image2p,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,F: fun_Pr2028404973800837934_nat_o,G: fun_int_fun_int_o] : bNF_re4195927319312321177_int_o(R2,bNF_Gr4191080447388368243_int_o(F,G,R2),F,G) ).

% rel_fun_image2p
tff(fact_1077_rel__fun__image2p,axiom,
    ! [R2: fun_in4891666191172872833eger_o,F: fun_int_num,G: fun_Code_integer_num] : aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(R2,bNF_Gr166583433173494904er_num(F,G,R2)),F),G) ).

% rel_fun_image2p
tff(fact_1078_rel__fun__image2p,axiom,
    ! [R2: fun_int_fun_int_o,F: fun_int_num,G: fun_int_num] : aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(R2,bNF_Gr7020624548710275655nt_num(F,G,R2)),F),G) ).

% rel_fun_image2p
tff(fact_1079_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [S3: set_fu5818728815147633517nteger,F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,R2: set_Code_integer] :
      ( finite7620574444722732823nteger(S3,F)
     => ( ord_le2952524580166011853nteger(image_2716445779319712667nteger(G,top_to4645266643341252675nteger),S3)
       => finite917065715488314481nteger(R2,aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_1080_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [S3: set_fu5818728815147633517nteger,F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,R2: set_Code_integer] :
      ( finite2944250750932824146nteger(S3,F)
     => ( ord_le2952524580166011853nteger(image_2716445779319712667nteger(G,top_to4645266643341252675nteger),S3)
       => finite366418920864469932nteger(R2,aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_1081_image__Pow__mono,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
     => ord_le513522071413781156et_rat(aa_set3856643248234747582et_rat(image_4408659257933336347et_rat(image_nat_rat(F)),pow_nat(A4)),pow_rat(B5)) ) ).

% image_Pow_mono
tff(fact_1082_image__Pow__mono,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
     => ord_le6893508408891458716et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(image_nat_nat(F)),pow_nat(A4)),pow_nat(B5)) ) ).

% image_Pow_mono
tff(fact_1083_image__Pow__mono,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
     => ord_le4403425263959731960et_int(aa_set7746546440780698386et_int(image_3739036796817536367et_int(image_nat_int(F)),pow_nat(A4)),pow_int(B5)) ) ).

% image_Pow_mono
tff(fact_1084_image__Pow__mono,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
     => ord_le6893508408891458716et_nat(aa_set7560802437211521810et_nat(image_4702325430467532143et_nat(image_int_nat(F)),pow_int(A4)),pow_nat(B5)) ) ).

% image_Pow_mono
tff(fact_1085_image__Pow__mono,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
     => ord_le4403425263959731960et_int(aa_set5070719292279795054et_int(image_524474410958335435et_int(image_int_int(F)),pow_int(A4)),pow_int(B5)) ) ).

% image_Pow_mono
tff(fact_1086_vimage2p__Grp,axiom,
    ! [F: fun_char_int,G: fun_Code_integer_int,P2: fun_int_fun_int_o] : bNF_vi7262943168399475603_int_o(F,G,P2) = relcom5819909836510906509nteger(bNF_Grp_char_int(top_top_set_char,F),relcom967497705283160040nteger(P2,conver1462408618118490240er_int(bNF_Gr6045950513751025625er_int(top_to4645266643341252675nteger,G)))) ).

% vimage2p_Grp
tff(fact_1087_vimage2p__Grp,axiom,
    ! [F: fun_rat_int,G: fun_Code_integer_int,P2: fun_int_fun_int_o] : bNF_vi508398028314858188_int_o(F,G,P2) = relcom3409376902655092628nteger(bNF_Grp_rat_int(top_top_set_rat,F),relcom967497705283160040nteger(P2,conver1462408618118490240er_int(bNF_Gr6045950513751025625er_int(top_to4645266643341252675nteger,G)))) ).

% vimage2p_Grp
tff(fact_1088_vimage2p__Grp,axiom,
    ! [F: fun_nat_int,G: fun_Code_integer_int,P2: fun_int_fun_int_o] : bNF_vi7331807916455492308_int_o(F,G,P2) = relcom3277315637376110988nteger(bNF_Grp_nat_int(top_top_set_nat,F),relcom967497705283160040nteger(P2,conver1462408618118490240er_int(bNF_Gr6045950513751025625er_int(top_to4645266643341252675nteger,G)))) ).

% vimage2p_Grp
tff(fact_1089_vimage2p__Grp,axiom,
    ! [F: fun_int_int,G: fun_Code_integer_int,P2: fun_int_fun_int_o] : bNF_vi2125572119366604152_int_o(F,G,P2) = relcom967497705283160040nteger(bNF_Grp_int_int(top_top_set_int,F),relcom967497705283160040nteger(P2,conver1462408618118490240er_int(bNF_Gr6045950513751025625er_int(top_to4645266643341252675nteger,G)))) ).

% vimage2p_Grp
tff(fact_1090_vimage2p__Grp,axiom,
    ! [F: fun_list_nat_int,G: fun_Code_integer_int,P2: fun_int_fun_int_o] : bNF_vi467646977814820164_int_o(F,G,P2) = relcom7887202796854296604nteger(bNF_Grp_list_nat_int(top_top_set_list_nat,F),relcom967497705283160040nteger(P2,conver1462408618118490240er_int(bNF_Gr6045950513751025625er_int(top_to4645266643341252675nteger,G)))) ).

% vimage2p_Grp
tff(fact_1091_vimage2p__Grp,axiom,
    ! [F: fun_ch7922468903353698889at_nat,G: fun_in4514472496968388910at_nat,P2: fun_Pr2028404973800837934_nat_o] : bNF_vi728936465572800120_nat_o(F,G,P2) = relcom7879471502873553579at_int(bNF_Gr8982933507515605408at_nat(top_top_set_char,F),relcom8911704206367525917at_int(P2,conver1866313717161370910at_nat(bNF_Gr6591314147588017797at_nat(top_top_set_int,G)))) ).

% vimage2p_Grp
tff(fact_1092_vimage2p__Grp,axiom,
    ! [F: fun_ra4806459883200776834at_nat,G: fun_in4514472496968388910at_nat,P2: fun_Pr2028404973800837934_nat_o] : bNF_vi2997284877290766385_nat_o(F,G,P2) = relcom1211213461163567844at_int(bNF_Gr6883301533820405721at_nat(top_top_set_rat,F),relcom8911704206367525917at_int(P2,conver1866313717161370910at_nat(bNF_Gr6591314147588017797at_nat(top_top_set_int,G)))) ).

% vimage2p_Grp
tff(fact_1093_vimage2p__Grp,axiom,
    ! [F: fun_na1476630412504255114at_nat,G: fun_in4514472496968388910at_nat,P2: fun_Pr2028404973800837934_nat_o] : bNF_vi190728152058600505_nat_o(F,G,P2) = relcom8569543703982574316at_int(bNF_Gr3553472063123884001at_nat(top_top_set_nat,F),relcom8911704206367525917at_int(P2,conver1866313717161370910at_nat(bNF_Gr6591314147588017797at_nat(top_top_set_int,G)))) ).

% vimage2p_Grp
tff(fact_1094_vimage2p__Grp,axiom,
    ! [F: fun_in4514472496968388910at_nat,G: fun_in4514472496968388910at_nat,P2: fun_Pr2028404973800837934_nat_o] : bNF_vi1204582724742824157_nat_o(F,G,P2) = relcom1106208330880909712at_int(bNF_Gr6591314147588017797at_nat(top_top_set_int,F),relcom8911704206367525917at_int(P2,conver1866313717161370910at_nat(bNF_Gr6591314147588017797at_nat(top_top_set_int,G)))) ).

% vimage2p_Grp
tff(fact_1095_vimage2p__Grp,axiom,
    ! [F: fun_li4460313466401262074at_nat,G: fun_in4514472496968388910at_nat,P2: fun_Pr2028404973800837934_nat_o] : bNF_vi5202502398764838569_nat_o(F,G,P2) = relcom170568170053003356at_int(bNF_Gr4119388860389678161at_nat(top_top_set_list_nat,F),relcom8911704206367525917at_int(P2,conver1866313717161370910at_nat(bNF_Gr6591314147588017797at_nat(top_top_set_int,G)))) ).

% vimage2p_Grp
tff(fact_1096_fun_Oin__rel,axiom,
    ! [R2: fun_nat_fun_nat_o,A: fun_char_nat,B: fun_char_nat] :
      ( aa_fun_char_nat_o(aa_fun8433001934219309943_nat_o(bNF_re3734335887435638179at_nat(fequal_char,R2),A),B)
    <=> ? [Z3: fun_ch7922468903353698889at_nat] :
          ( member2657260328581288434at_nat(Z3,collec7752666796254765876at_nat(aTP_Lamm_af(R2)))
          & ( aa_fun1233241720372881579ar_nat(comp_P1491501995297412481t_char(product_fst_nat_nat),Z3) = A )
          & ( aa_fun1233241720372881579ar_nat(comp_P1491501995297412481t_char(product_snd_nat_nat),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1097_fun_Oin__rel,axiom,
    ! [R2: fun_nat_fun_nat_o,A: fun_rat_nat,B: fun_rat_nat] :
      ( aa_fun_rat_nat_o(aa_fun6999665265510530025_nat_o(bNF_re6557955094579809201at_nat(fequal_rat,R2),A),B)
    <=> ? [Z3: fun_ra4806459883200776834at_nat] :
          ( member6759628557233150763at_nat(Z3,collec4103600675842305645at_nat(aTP_Lamm_ag(R2)))
          & ( aa_fun4184522485684738539at_nat(comp_P1320480151300532104at_rat(product_fst_nat_nat),Z3) = A )
          & ( aa_fun4184522485684738539at_nat(comp_P1320480151300532104at_rat(product_snd_nat_nat),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1098_fun_Oin__rel,axiom,
    ! [R2: fun_nat_fun_nat_o,A: fun_nat_nat,B: fun_nat_nat] :
      ( aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,R2),A),B)
    <=> ? [Z3: fun_na1476630412504255114at_nat] :
          ( member3429799086536629043at_nat(Z3,collec773771205145783925at_nat(aTP_Lamm_ah(R2)))
          & ( aa_fun9030085588633777131at_nat(comp_P1955610211387027840at_nat(product_fst_nat_nat),Z3) = A )
          & ( aa_fun9030085588633777131at_nat(comp_P1955610211387027840at_nat(product_snd_nat_nat),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1099_fun_Oin__rel,axiom,
    ! [R2: fun_nat_fun_nat_o,A: fun_int_nat,B: fun_int_nat] :
      ( aa_fun_int_nat_o(aa_fun7045208173470077249_nat_o(bNF_re3715656647883201625at_nat(fequal_int,R2),A),B)
    <=> ? [Z3: fun_in4514472496968388910at_nat] :
          ( member6467641171000762839at_nat(Z3,collec3811613289609917721at_nat(aTP_Lamm_ai(R2)))
          & ( aa_fun7135526916484278507nt_nat(comp_P1953119740877977564at_int(product_fst_nat_nat),Z3) = A )
          & ( aa_fun7135526916484278507nt_nat(comp_P1953119740877977564at_int(product_snd_nat_nat),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1100_fun_Oin__rel,axiom,
    ! [R2: fun_num_fun_num_o,A: fun_int_num,B: fun_int_num] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,R2),A),B)
    <=> ? [Z3: fun_in2248087180373909186um_num] :
          ( member4201255854406283115um_num(Z3,collec1545227973015437997um_num(aTP_Lamm_aj(R2)))
          & ( aa_fun5436734615827139529nt_num(comp_P4305060815297604026um_int(product_fst_num_num),Z3) = A )
          & ( aa_fun5436734615827139529nt_num(comp_P4305060815297604026um_int(product_snd_num_num),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1101_fun_Oin__rel,axiom,
    ! [R2: fun_int_fun_int_o,A: fun_char_int,B: fun_char_int] :
      ( aa_fun_char_int_o(aa_fun5090773985904865471_int_o(bNF_re4731199128827642459nt_int(fequal_char,R2),A),B)
    <=> ? [Z3: fun_ch7619307333114993921nt_int] :
          ( member2354098758342583466nt_int(Z3,collec7449505226016060908nt_int(aTP_Lamm_ak(R2)))
          & ( aa_fun6394501117779864511ar_int(comp_P4161745170962941589t_char(product_fst_int_int),Z3) = A )
          & ( aa_fun6394501117779864511ar_int(comp_P4161745170962941589t_char(product_snd_int_int),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1102_fun_Oin__rel,axiom,
    ! [R2: fun_int_fun_int_o,A: fun_rat_int,B: fun_rat_int] :
      ( aa_fun_rat_int_o(aa_fun2499607139650510129_int_o(bNF_re7554818335971813481nt_int(fequal_rat,R2),A),B)
    <=> ? [Z3: fun_ra4503298312962071866nt_int] :
          ( member6456466986994445795nt_int(Z3,collec3800439105603600677nt_int(aTP_Lamm_al(R2)))
          & ( aa_fun7412441380874900735at_int(comp_P6018825582104210588nt_rat(product_fst_int_int),Z3) = A )
          & ( aa_fun7412441380874900735at_int(comp_P6018825582104210588nt_rat(product_snd_int_int),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1103_fun_Oin__rel,axiom,
    ! [R2: fun_int_fun_int_o,A: fun_nat_int,B: fun_nat_int] :
      ( aa_fun_nat_int_o(aa_fun7064809811658394945_int_o(bNF_re6650684261131312217nt_int(fequal_nat,R2),A),B)
    <=> ? [Z3: fun_na1173468842265550146nt_int] :
          ( member3126637516297924075nt_int(Z3,collec470609634907078957nt_int(aTP_Lamm_am(R2)))
          & ( aa_fun3034632446969163519at_int(comp_P6653955642190706324nt_nat(product_fst_int_int),Z3) = A )
          & ( aa_fun3034632446969163519at_int(comp_P6653955642190706324nt_nat(product_snd_int_int),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1104_fun_Oin__rel,axiom,
    ! [R2: fun_int_fun_int_o,A: fun_int_int,B: fun_int_int] :
      ( aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,R2),A),B)
    <=> ? [Z3: fun_in4211310926729683942nt_int] :
          ( member6164479600762057871nt_int(Z3,collec3508451719371212753nt_int(aTP_Lamm_an(R2)))
          & ( aa_fun1140073774819664895nt_int(comp_P6651465171681656048nt_int(product_fst_int_int),Z3) = A )
          & ( aa_fun1140073774819664895nt_int(comp_P6651465171681656048nt_int(product_snd_int_int),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1105_fun_Oin__rel,axiom,
    ! [R2: fun_nat_fun_nat_o,A: fun_list_nat_nat,B: fun_list_nat_nat] :
      ( aa_fun6009679834319641581_nat_o(aa_fun7689321269454855513_nat_o(bNF_re4629543205785817921at_nat(fequal_list_nat,R2),A),B)
    <=> ? [Z3: fun_li4460313466401262074at_nat] :
          ( member613779007764099619at_nat(Z3,collec5947443778231319653at_nat(aTP_Lamm_ao(R2)))
          & ( aa_fun4296578307644586859at_nat(comp_P5292260908303971472st_nat(product_fst_nat_nat),Z3) = A )
          & ( aa_fun4296578307644586859at_nat(comp_P5292260908303971472st_nat(product_snd_nat_nat),Z3) = B ) ) ) ).

% fun.in_rel
tff(fact_1106_Grp__UNIV__id,axiom,
    ! [F: fun_o_o] :
      ( ( F = id_o )
     => ( relcompp_o_o_o(conversep_o_o(bNF_Grp_o_o(top_top_set_o,F)),bNF_Grp_o_o(top_top_set_o,F)) = bNF_Grp_o_o(top_top_set_o,F) ) ) ).

% Grp_UNIV_id
tff(fact_1107_Grp__UNIV__id,axiom,
    ! [F: fun_num_num] :
      ( ( F = id_num )
     => ( relcompp_num_num_num(conversep_num_num(bNF_Grp_num_num(top_top_set_num,F)),bNF_Grp_num_num(top_top_set_num,F)) = bNF_Grp_num_num(top_top_set_num,F) ) ) ).

% Grp_UNIV_id
tff(fact_1108_Grp__UNIV__id,axiom,
    ! [F: fun_char_char] :
      ( ( F = id_char )
     => ( relcom5872928454965100966r_char(conversep_char_char(bNF_Grp_char_char(top_top_set_char,F)),bNF_Grp_char_char(top_top_set_char,F)) = bNF_Grp_char_char(top_top_set_char,F) ) ) ).

% Grp_UNIV_id
tff(fact_1109_Grp__UNIV__id,axiom,
    ! [F: fun_li7731370096534215997st_nat] :
      ( ( F = id_list_nat )
     => ( relcom8030397737370142931st_nat(conver909888920752868663st_nat(bNF_Gr600725020315365264st_nat(top_top_set_list_nat,F)),bNF_Gr600725020315365264st_nat(top_top_set_list_nat,F)) = bNF_Gr600725020315365264st_nat(top_top_set_list_nat,F) ) ) ).

% Grp_UNIV_id
tff(fact_1110_Grp__UNIV__id,axiom,
    ! [F: fun_rat_rat] :
      ( ( F = id_rat )
     => ( relcompp_rat_rat_rat(conversep_rat_rat(bNF_Grp_rat_rat(top_top_set_rat,F)),bNF_Grp_rat_rat(top_top_set_rat,F)) = bNF_Grp_rat_rat(top_top_set_rat,F) ) ) ).

% Grp_UNIV_id
tff(fact_1111_Grp__UNIV__id,axiom,
    ! [F: fun_nat_nat] :
      ( ( F = id_nat )
     => ( relcompp_nat_nat_nat(conversep_nat_nat(bNF_Grp_nat_nat(top_top_set_nat,F)),bNF_Grp_nat_nat(top_top_set_nat,F)) = bNF_Grp_nat_nat(top_top_set_nat,F) ) ) ).

% Grp_UNIV_id
tff(fact_1112_Grp__UNIV__id,axiom,
    ! [F: fun_int_int] :
      ( ( F = id_int )
     => ( relcompp_int_int_int(conversep_int_int(bNF_Grp_int_int(top_top_set_int,F)),bNF_Grp_int_int(top_top_set_int,F)) = bNF_Grp_int_int(top_top_set_int,F) ) ) ).

% Grp_UNIV_id
tff(fact_1113_strict__mono__inv,axiom,
    ! [F: fun_rat_rat,G: fun_rat_rat] :
      ( order_4774338944571509483at_rat(F)
     => ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
       => ( ! [X: rat] : aa_rat_rat(G,aa_rat_rat(F,X)) = X
         => order_4774338944571509483at_rat(G) ) ) ) ).

% strict_mono_inv
tff(fact_1114_strict__mono__inv,axiom,
    ! [F: fun_rat_nat,G: fun_nat_rat] :
      ( order_5409469004658005219at_nat(F)
     => ( ( image_rat_nat(F,top_top_set_rat) = top_top_set_nat )
       => ( ! [X: rat] : aa_nat_rat(G,aa_rat_nat(F,X)) = X
         => order_5090893588506375395at_rat(G) ) ) ) ).

% strict_mono_inv
tff(fact_1115_strict__mono__inv,axiom,
    ! [F: fun_rat_int,G: fun_int_rat] :
      ( order_5406978534148954943at_int(F)
     => ( ( image_rat_int(F,top_top_set_rat) = top_top_set_int )
       => ( ! [X: rat] : aa_int_rat(G,aa_rat_int(F,X)) = X
         => order_6090247300407429951nt_rat(G) ) ) ) ).

% strict_mono_inv
tff(fact_1116_strict__mono__inv,axiom,
    ! [F: fun_nat_rat,G: fun_rat_nat] :
      ( order_5090893588506375395at_rat(F)
     => ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = top_top_set_rat )
       => ( ! [X: nat] : aa_rat_nat(G,aa_nat_rat(F,X)) = X
         => order_5409469004658005219at_nat(G) ) ) ) ).

% strict_mono_inv
tff(fact_1117_strict__mono__inv,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( order_5726023648592871131at_nat(F)
     => ( ( aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat) = top_top_set_nat )
       => ( ! [X: nat] : aa_nat_nat(G,aa_nat_nat(F,X)) = X
         => order_5726023648592871131at_nat(G) ) ) ) ).

% strict_mono_inv
tff(fact_1118_strict__mono__inv,axiom,
    ! [F: fun_nat_int,G: fun_int_nat] :
      ( order_5723533178083820855at_int(F)
     => ( ( aa_set_nat_set_int(image_nat_int(F),top_top_set_nat) = top_top_set_int )
       => ( ! [X: nat] : aa_int_nat(G,aa_nat_int(F,X)) = X
         => order_6725377360493925687nt_nat(G) ) ) ) ).

% strict_mono_inv
tff(fact_1119_strict__mono__inv,axiom,
    ! [F: fun_int_rat,G: fun_rat_int] :
      ( order_6090247300407429951nt_rat(F)
     => ( ( image_int_rat(F,top_top_set_int) = top_top_set_rat )
       => ( ! [X: int] : aa_rat_int(G,aa_int_rat(F,X)) = X
         => order_5406978534148954943at_int(G) ) ) ) ).

% strict_mono_inv
tff(fact_1120_strict__mono__inv,axiom,
    ! [F: fun_int_nat,G: fun_nat_int] :
      ( order_6725377360493925687nt_nat(F)
     => ( ( aa_set_int_set_nat(image_int_nat(F),top_top_set_int) = top_top_set_nat )
       => ( ! [X: int] : aa_nat_int(G,aa_int_nat(F,X)) = X
         => order_5723533178083820855at_int(G) ) ) ) ).

% strict_mono_inv
tff(fact_1121_strict__mono__inv,axiom,
    ! [F: fun_int_int,G: fun_int_int] :
      ( order_6722886889984875411nt_int(F)
     => ( ( aa_set_int_set_int(image_int_int(F),top_top_set_int) = top_top_set_int )
       => ( ! [X: int] : aa_int_int(G,aa_int_int(F,X)) = X
         => order_6722886889984875411nt_int(G) ) ) ) ).

% strict_mono_inv
tff(fact_1122_Id__on__Gr,axiom,
    ! [A4: set_nat] : id_on_nat(A4) = bNF_Gr_nat_nat(A4,id_nat) ).

% Id_on_Gr
tff(fact_1123_Id__on__Gr,axiom,
    ! [A4: set_o] : id_on_o(A4) = bNF_Gr_o_o(A4,id_o) ).

% Id_on_Gr
tff(fact_1124_Id__on__Gr,axiom,
    ! [A4: set_num] : id_on_num(A4) = bNF_Gr_num_num(A4,id_num) ).

% Id_on_Gr
tff(fact_1125_type__copy__map__id0,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,M: fun_o_o] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( ( M = id_o )
       => ( comp_o_nat_nat(comp_o_nat_o(Abs,M),Rep) = id_nat ) ) ) ).

% type_copy_map_id0
tff(fact_1126_type__copy__map__id0,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,M: fun_o_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( ( M = id_o )
       => ( comp_o_o_o(comp_o_o_o(Abs,M),Rep) = id_o ) ) ) ).

% type_copy_map_id0
tff(fact_1127_type__copy__map__id0,axiom,
    ! [Rep: fun_num_o,Abs: fun_o_num,M: fun_o_o] :
      ( aa_set_o_o(type_d9070036042389994064_num_o(Rep,Abs),top_top_set_o)
     => ( ( M = id_o )
       => ( comp_o_num_num(comp_o_num_o(Abs,M),Rep) = id_num ) ) ) ).

% type_copy_map_id0
tff(fact_1128_type__copy__map__id0,axiom,
    ! [Rep: fun_nat_num,Abs: fun_num_nat,M: fun_num_num] :
      ( type_d2807825734933527320at_num(Rep,Abs,top_top_set_num)
     => ( ( M = id_num )
       => ( comp_num_nat_nat(comp_num_nat_num(Abs,M),Rep) = id_nat ) ) ) ).

% type_copy_map_id0
tff(fact_1129_type__copy__map__id0,axiom,
    ! [Rep: fun_o_num,Abs: fun_num_o,M: fun_num_num] :
      ( type_d3678713432601627782_o_num(Rep,Abs,top_top_set_num)
     => ( ( M = id_num )
       => ( comp_num_o_o(comp_num_o_num(Abs,M),Rep) = id_o ) ) ) ).

% type_copy_map_id0
tff(fact_1130_type__copy__map__id0,axiom,
    ! [Rep: fun_num_num,Abs: fun_num_num,M: fun_num_num] :
      ( type_d938156138240996706um_num(Rep,Abs,top_top_set_num)
     => ( ( M = id_num )
       => ( comp_num_num_num(comp_num_num_num(Abs,M),Rep) = id_num ) ) ) ).

% type_copy_map_id0
tff(fact_1131_type__copy__map__id0,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,M: fun_char_char] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( ( M = id_char )
       => ( comp_char_nat_nat(comp_char_nat_char(Abs,M),Rep) = id_nat ) ) ) ).

% type_copy_map_id0
tff(fact_1132_type__copy__map__id0,axiom,
    ! [Rep: fun_o_char,Abs: fun_char_o,M: fun_char_char] :
      ( type_d9076672964931703357o_char(Rep,Abs,top_top_set_char)
     => ( ( M = id_char )
       => ( comp_char_o_o(comp_char_o_char(Abs,M),Rep) = id_o ) ) ) ).

% type_copy_map_id0
tff(fact_1133_type__copy__map__id0,axiom,
    ! [Rep: fun_num_char,Abs: fun_char_num,M: fun_char_char] :
      ( type_d4396077731765706521m_char(Rep,Abs,top_top_set_char)
     => ( ( M = id_char )
       => ( comp_char_num_num(comp_char_num_char(Abs,M),Rep) = id_num ) ) ) ).

% type_copy_map_id0
tff(fact_1134_type__copy__map__id0,axiom,
    ! [Rep: fun_nat_rat,Abs: fun_rat_nat,M: fun_rat_rat] :
      ( aa_set_rat_o(type_d5615363888691252950at_rat(Rep,Abs),top_top_set_rat)
     => ( ( M = id_rat )
       => ( comp_rat_nat_nat(comp_rat_nat_rat(Abs,M),Rep) = id_nat ) ) ) ).

% type_copy_map_id0
tff(fact_1135_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( comp_char_nat_nat(Abs,Rep) = id_nat ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1136_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_o_char,Abs: fun_char_o] :
      ( type_d9076672964931703357o_char(Rep,Abs,top_top_set_char)
     => ( comp_char_o_o(Abs,Rep) = id_o ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1137_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_num_char,Abs: fun_char_num] :
      ( type_d4396077731765706521m_char(Rep,Abs,top_top_set_char)
     => ( comp_char_num_num(Abs,Rep) = id_num ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1138_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_nat_rat,Abs: fun_rat_nat] :
      ( aa_set_rat_o(type_d5615363888691252950at_rat(Rep,Abs),top_top_set_rat)
     => ( comp_rat_nat_nat(Abs,Rep) = id_nat ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1139_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_o_rat,Abs: fun_rat_o] :
      ( aa_set_rat_o(type_d6486251586359353412_o_rat(Rep,Abs),top_top_set_rat)
     => ( comp_rat_o_o(Abs,Rep) = id_o ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1140_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_num_rat,Abs: fun_rat_num] :
      ( aa_set_rat_o(type_d3745694291998722336um_rat(Rep,Abs),top_top_set_rat)
     => ( comp_rat_num_num(Abs,Rep) = id_num ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1141_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_o_nat,Abs: fun_nat_o] :
      ( aa_set_nat_o(type_d7121381646445849148_o_nat(Rep,Abs),top_top_set_nat)
     => ( comp_nat_o_o(Abs,Rep) = id_o ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1142_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_num_nat,Abs: fun_nat_num] :
      ( aa_set_nat_o(type_d4380824352085218072um_nat(Rep,Abs),top_top_set_nat)
     => ( comp_nat_num_num(Abs,Rep) = id_num ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1143_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),top_top_set_nat)
     => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Abs),Rep) = id_nat ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1144_type__copy__Abs__o__Rep,axiom,
    ! [Rep: fun_nat_int,Abs: fun_int_nat] :
      ( aa_set_int_o(type_d6248003478268698410at_int(Rep,Abs),top_top_set_int)
     => ( comp_int_nat_nat(Abs,Rep) = id_nat ) ) ).

% type_copy_Abs_o_Rep
tff(fact_1145_type__copy__Rep__o__Abs,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),top_top_set_nat)
     => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Rep),Abs) = id_nat ) ) ).

% type_copy_Rep_o_Abs
tff(fact_1146_type__copy__Rep__o__Abs,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( comp_P5156358013004227690it_o_o(Rep,Abs) = id_o ) ) ).

% type_copy_Rep_o_Abs
tff(fact_1147_case__prodI2_H,axiom,
    ! [P: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xa: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) = P )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A3),B3),Xa) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C),P),Xa) ) ).

% case_prodI2'
tff(fact_1148_image__ident,axiom,
    ! [Y3: set_nat] : aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_ap),Y3) = Y3 ).

% image_ident
tff(fact_1149_image__ident,axiom,
    ! [Y3: set_int] : aa_set_int_set_int(image_int_int(aTP_Lamm_aq),Y3) = Y3 ).

% image_ident
tff(fact_1150_split__part,axiom,
    ! [P2: $o,Q2: fun_int_fun_int_o,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_ar((P2)),Q2)),X5)
    <=> ( (P2)
        & aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Q2),X5) ) ) ).

% split_part
tff(fact_1151_Id__onI,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => member6009668390874684176at_nat(produc2335810615602371415at_nat(A,A),id_on_7869290023980668164at_nat(A4)) ) ).

% Id_onI
tff(fact_1152_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),A),id_on_2554058798563519774at_nat(A4)) ) ).

% Id_onI
tff(fact_1153_Id__onI,axiom,
    ! [A: $o,A4: set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(A)),id_on_o(A4)) ) ).

% Id_onI
tff(fact_1154_Id__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat] :
      ( member6273874620674182256at_nat(A,A4)
     => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),A),id_on_3275838214495744996at_nat(A4)) ) ).

% Id_onI
tff(fact_1155_Id__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat] :
      ( member2643936169264416010at_nat(A,A4)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),A)),id_on_6713504163309144702at_nat(A4)) ) ).

% Id_onI
tff(fact_1156_Id__onI,axiom,
    ! [A: nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),A)),id_on_nat(A4)) ) ).

% Id_onI
tff(fact_1157_Id__onI,axiom,
    ! [A: int,A4: set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),A)),id_on_int(A4)) ) ).

% Id_onI
tff(fact_1158_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_1159_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_1160_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_1161_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_1162_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_1163_case__prodI,axiom,
    ! [F: fun_se5672428230568430382_nat_o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(F,A),B)
     => aa_Pro4518350200472837805_nat_o(aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,F),aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B)) ) ).

% case_prodI
tff(fact_1164_case__prodI,axiom,
    ! [F: fun_se5639961202347202350_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(F,A),B)
     => aa_Pro5947980084102686253_nat_o(aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,F),aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)) ) ).

% case_prodI
tff(fact_1165_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,A),B)
     => aa_Pro6908818167415475913_num_o(aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,F),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) ) ).

% case_prodI
tff(fact_1166_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) ) ).

% case_prodI
tff(fact_1167_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_1168_case__prodI2,axiom,
    ! [P: produc1319942482725812455at_nat,C: fun_se5672428230568430382_nat_o] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
          ( ( P = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A3),B3) )
         => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(C,A3),B3) )
     => aa_Pro4518350200472837805_nat_o(aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,C),P) ) ).

% case_prodI2
tff(fact_1169_case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,C: fun_se5639961202347202350_nat_o] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A3),B3) )
         => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(C,A3),B3) )
     => aa_Pro5947980084102686253_nat_o(aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,C),P) ) ).

% case_prodI2
tff(fact_1170_case__prodI2,axiom,
    ! [P: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A3: nat,B3: num] :
          ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3) )
         => aa_num_o(aa_nat_fun_num_o(C,A3),B3) )
     => aa_Pro6908818167415475913_num_o(aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,C),P) ) ).

% case_prodI2
tff(fact_1171_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B3) )
     => aa_Pro6150637604445964947_nat_o(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,C),P) ) ).

% case_prodI2
tff(fact_1172_case__prodI2,axiom,
    ! [P: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(C,A3),B3) )
     => aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C),P) ) ).

% case_prodI2
tff(fact_1173_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_na3850162164366384704_set_o,A: nat,B: num] :
      ( aa_set_o_o(member_o((Z)),aa_num_set_o(aa_nat_fun_num_set_o(C,A),B))
     => aa_set_o_o(member_o((Z)),produc836176033315069408_set_o(C,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))) ) ).

% mem_case_prodI
tff(fact_1174_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,A: nat,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))) ) ).

% mem_case_prodI
tff(fact_1175_mem__case__prodI,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,A: nat,B: num] :
      ( aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))) ) ).

% mem_case_prodI
tff(fact_1176_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_na3100059926620910090_set_o,A: nat,B: nat] :
      ( aa_set_o_o(member_o((Z)),aa_nat_set_o(aa_nat_fun_nat_set_o(C,A),B))
     => aa_set_o_o(member_o((Z)),aa_Pro5970036531051701619_set_o(produc59986286002894506_set_o(C),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_1177_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na6128946301132549280et_nat,A: nat,B: nat] :
      ( aa_set_nat_o(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),aa_Pro9123573313595863371et_nat(produc6189476227299908564et_nat(C),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_1178_mem__case__prodI,axiom,
    ! [Z: int,C: fun_na1315397882344630524et_int,A: nat,B: nat] :
      ( aa_set_int_o(member_int(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc2011625207790711856et_int(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))) ) ).

% mem_case_prodI
tff(fact_1179_mem__case__prodI,axiom,
    ! [Z: $o,C: fun_in4068701459849370962_set_o,A: int,B: int] :
      ( aa_set_o_o(member_o((Z)),aa_int_set_o(aa_int_fun_int_set_o(C,A),B))
     => aa_set_o_o(member_o((Z)),produc4257766111578684402_set_o(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_1180_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_in2630332840903238488et_nat,A: int,B: int] :
      ( aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_1181_mem__case__prodI,axiom,
    ! [Z: int,C: fun_in7040156458970095540et_int,A: int,B: int] :
      ( aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,A),B))
     => aa_set_int_o(member_int(Z),produc73460835934605544et_int(C,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))) ) ).

% mem_case_prodI
tff(fact_1182_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: fun_na1206552039118883625at_nat,A: nat,B: num] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_num8947610712640478729at_nat(aa_nat157436013921436562at_nat(C,A),B))
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc859037500352796105at_nat(C,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))) ) ).

% mem_case_prodI
tff(fact_1183_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_num,Z: $o,C: fun_na3850162164366384704_set_o] :
      ( ! [A3: nat,B3: num] :
          ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3) )
         => aa_set_o_o(member_o((Z)),aa_num_set_o(aa_nat_fun_num_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o((Z)),produc836176033315069408_set_o(C,P)) ) ).

% mem_case_prodI2
tff(fact_1184_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_num,Z: nat,C: fun_na3896347130304379370et_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_1185_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_num,Z: int,C: fun_na8306170748371236422et_int] :
      ( ! [A3: nat,B3: num] :
          ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3) )
         => aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_1186_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: $o,C: fun_na3100059926620910090_set_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_o_o(member_o((Z)),aa_nat_set_o(aa_nat_fun_nat_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o((Z)),aa_Pro5970036531051701619_set_o(produc59986286002894506_set_o(C),P)) ) ).

% mem_case_prodI2
tff(fact_1187_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: nat,C: fun_na6128946301132549280et_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),aa_Pro9123573313595863371et_nat(produc6189476227299908564et_nat(C),P)) ) ).

% mem_case_prodI2
tff(fact_1188_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: int,C: fun_na1315397882344630524et_int] :
      ( ! [A3: nat,B3: nat] :
          ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B3) )
         => aa_set_int_o(member_int(Z),aa_nat_set_int(aa_nat8552214573506306493et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc2011625207790711856et_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_1189_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: $o,C: fun_in4068701459849370962_set_o] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_o_o(member_o((Z)),aa_int_set_o(aa_int_fun_int_set_o(C,A3),B3)) )
     => aa_set_o_o(member_o((Z)),produc4257766111578684402_set_o(C,P)) ) ).

% mem_case_prodI2
tff(fact_1190_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: nat,C: fun_in2630332840903238488et_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_nat_o(member_nat(Z),aa_int_set_nat(aa_int1906528167449992217et_nat(C,A3),B3)) )
     => aa_set_nat_o(member_nat(Z),produc4251311855443802252et_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_1191_mem__case__prodI2,axiom,
    ! [P: product_prod_int_int,Z: int,C: fun_in7040156458970095540et_int] :
      ( ! [A3: int,B3: int] :
          ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B3) )
         => aa_set_int_o(member_int(Z),aa_int_set_int(aa_int8639817059373041269et_int(C,A3),B3)) )
     => aa_set_int_o(member_int(Z),produc73460835934605544et_int(C,P)) ) ).

% mem_case_prodI2
tff(fact_1192_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_num,Z: product_prod_nat_nat,C: fun_na1206552039118883625at_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B3) )
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),aa_num8947610712640478729at_nat(aa_nat157436013921436562at_nat(C,A3),B3)) )
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Z),produc859037500352796105at_nat(C,P)) ) ).

% mem_case_prodI2
tff(fact_1193_case__swap,axiom,
    ! [F: fun_na1668100129766719005at_nat,P: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_as(F)),aa_Pro262675715343573244at_nat(product_swap_nat_nat,P)) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P) ).

% case_swap
tff(fact_1194_case__swap,axiom,
    ! [F: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_at(F)),aa_Pro262675715343573244at_nat(product_swap_nat_nat,P)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P) ).

% case_swap
tff(fact_1195_case__swap,axiom,
    ! [F: fun_in1696471567231710611nt_int,P: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_au(F)),aa_Pro6970504127699319548nt_int(product_swap_int_int,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P) ).

% case_swap
tff(fact_1196_case__swap,axiom,
    ! [F: fun_int_fun_int_o,P: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_av(F)),aa_Pro6970504127699319548nt_int(product_swap_int_int,P))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P) ) ).

% case_swap
tff(fact_1197_case__swap,axiom,
    ! [F: fun_int_fun_int_int,P: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_aw(F)),aa_Pro6970504127699319548nt_int(product_swap_int_int,P)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P) ).

% case_swap
tff(fact_1198_pair__imageI,axiom,
    ! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: fun_nat_fun_num_o] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),A4)
     => aa_set_o_o(member_o(aa_num_o(aa_nat_fun_num_o(F,A),B)),image_4451812852358507538_num_o(aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,F),A4)) ) ).

% pair_imageI
tff(fact_1199_pair__imageI,axiom,
    ! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: fun_nat_fun_num_nat] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),A4)
     => aa_set_nat_o(member_nat(aa_num_nat(aa_nat_fun_num_nat(F,A),B)),image_4504471508743443926um_nat(produc4973203077627929192um_nat(F),A4)) ) ).

% pair_imageI
tff(fact_1200_pair__imageI,axiom,
    ! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: fun_nat_fun_num_int] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),A4)
     => aa_set_int_o(member_int(aa_num_int(aa_nat_fun_num_int(F,A),B)),image_4501981038234393650um_int(produc4970712607118878916um_int(F),A4)) ) ).

% pair_imageI
tff(fact_1201_pair__imageI,axiom,
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A4)
     => aa_set_o_o(member_o(aa_nat_o(aa_nat_fun_nat_o(F,A),B)),image_3693632289388996572_nat_o(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,F),A4)) ) ).

% pair_imageI
tff(fact_1202_pair__imageI,axiom,
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: fun_nat_fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A4)
     => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,A),B)),image_2486076414777270412at_nat(produc6842872674320459806at_nat(F),A4)) ) ).

% pair_imageI
tff(fact_1203_pair__imageI,axiom,
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: fun_nat_fun_nat_int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),A4)
     => aa_set_int_o(member_int(aa_nat_int(aa_nat_fun_nat_int(F,A),B)),image_2483585944268220136at_int(produc6840382203811409530at_int(F),A4)) ) ).

% pair_imageI
tff(fact_1204_pair__imageI,axiom,
    ! [A: int,B: int,A4: set_Pr958786334691620121nt_int,F: fun_int_fun_int_nat] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),A4)
     => aa_set_nat_o(member_nat(aa_int_nat(aa_int_fun_int_nat(F,A),B)),image_5044651549707136836nt_nat(produc8213879946458358998nt_nat(F),A4)) ) ).

% pair_imageI
tff(fact_1205_pair__imageI,axiom,
    ! [A: int,B: int,A4: set_Pr958786334691620121nt_int,F: fun_int_fun_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),A4)
     => aa_set_o_o(member_o(aa_int_o(aa_int_fun_int_o(F,A),B)),image_2135063354759101220_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),A4)) ) ).

% pair_imageI
tff(fact_1206_pair__imageI,axiom,
    ! [A: int,B: int,A4: set_Pr958786334691620121nt_int,F: fun_int_fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),A4)
     => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(F,A),B)),image_5042161079198086560nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),A4)) ) ).

% pair_imageI
tff(fact_1207_pair__imageI,axiom,
    ! [A: nat,B: num,A4: set_Pr6200539531224447659at_num,F: fun_na3186776076167979465at_nat] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),A4)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_num4318107302918302889at_nat(aa_nat5518674451949681202at_nat(F,A),B)),image_4217639968629731195at_nat(produc8564434177180591209at_nat(F),A4)) ) ).

% pair_imageI
tff(fact_1208_Inf_OINF__identity__eq,axiom,
    ! [Inf: fun_set_nat_nat,A4: set_nat] : aa_set_nat_nat(Inf,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_ap),A4)) = aa_set_nat_nat(Inf,A4) ).

% Inf.INF_identity_eq
tff(fact_1209_Inf_OINF__identity__eq,axiom,
    ! [Inf: fun_set_int_int,A4: set_int] : aa_set_int_int(Inf,aa_set_int_set_int(image_int_int(aTP_Lamm_aq),A4)) = aa_set_int_int(Inf,A4) ).

% Inf.INF_identity_eq
tff(fact_1210_Sup_OSUP__identity__eq,axiom,
    ! [Sup: fun_set_nat_nat,A4: set_nat] : aa_set_nat_nat(Sup,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_ap),A4)) = aa_set_nat_nat(Sup,A4) ).

% Sup.SUP_identity_eq
tff(fact_1211_Sup_OSUP__identity__eq,axiom,
    ! [Sup: fun_set_int_int,A4: set_int] : aa_set_int_int(Sup,aa_set_int_set_int(image_int_int(aTP_Lamm_aq),A4)) = aa_set_int_int(Sup,A4) ).

% Sup.SUP_identity_eq
tff(fact_1212_imageE,axiom,
    ! [B: rat,F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(member_rat(B),aa_set_nat_set_rat(image_nat_rat(F),A4))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_rat(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_1213_imageE,axiom,
    ! [B: $o,F: fun_o_o,A4: set_o] :
      ( aa_set_o_o(member_o((B)),aa_set_o_set_o(image_o_o(F),A4))
     => ~ ! [X: $o] :
            ( ( (B)
            <=> aa_o_o(F,(X)) )
           => ~ aa_set_o_o(member_o((X)),A4) ) ) ).

% imageE
tff(fact_1214_imageE,axiom,
    ! [B: $o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(member_o((B)),image_nat_o(F,A4))
     => ~ ! [X: nat] :
            ( ( (B)
            <=> aa_nat_o(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_1215_imageE,axiom,
    ! [B: $o,F: fun_int_o,A4: set_int] :
      ( aa_set_o_o(member_o((B)),image_int_o(F,A4))
     => ~ ! [X: int] :
            ( ( (B)
            <=> aa_int_o(F,X) )
           => ~ aa_set_int_o(member_int(X),A4) ) ) ).

% imageE
tff(fact_1216_imageE,axiom,
    ! [B: nat,F: fun_o_nat,A4: set_o] :
      ( aa_set_nat_o(member_nat(B),image_o_nat(F,A4))
     => ~ ! [X: $o] :
            ( ( B = aa_o_nat(F,(X)) )
           => ~ aa_set_o_o(member_o((X)),A4) ) ) ).

% imageE
tff(fact_1217_imageE,axiom,
    ! [B: nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_nat(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_1218_imageE,axiom,
    ! [B: nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(member_nat(B),aa_set_int_set_nat(image_int_nat(F),A4))
     => ~ ! [X: int] :
            ( ( B = aa_int_nat(F,X) )
           => ~ aa_set_int_o(member_int(X),A4) ) ) ).

% imageE
tff(fact_1219_imageE,axiom,
    ! [B: int,F: fun_o_int,A4: set_o] :
      ( aa_set_int_o(member_int(B),image_o_int(F,A4))
     => ~ ! [X: $o] :
            ( ( B = aa_o_int(F,(X)) )
           => ~ aa_set_o_o(member_o((X)),A4) ) ) ).

% imageE
tff(fact_1220_imageE,axiom,
    ! [B: int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),A4))
     => ~ ! [X: nat] :
            ( ( B = aa_nat_int(F,X) )
           => ~ aa_set_nat_o(member_nat(X),A4) ) ) ).

% imageE
tff(fact_1221_imageE,axiom,
    ! [B: int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(member_int(B),aa_set_int_set_int(image_int_int(F),A4))
     => ~ ! [X: int] :
            ( ( B = aa_int_int(F,X) )
           => ~ aa_set_int_o(member_int(X),A4) ) ) ).

% imageE
tff(fact_1222_image__image,axiom,
    ! [F: fun_rat_rat,G: fun_nat_rat,A4: set_nat] : image_rat_rat(F,aa_set_nat_set_rat(image_nat_rat(G),A4)) = aa_set_nat_set_rat(image_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ax(F),G)),A4) ).

% image_image
tff(fact_1223_image__image,axiom,
    ! [F: fun_rat_nat,G: fun_nat_rat,A4: set_nat] : image_rat_nat(F,aa_set_nat_set_rat(image_nat_rat(G),A4)) = aa_set_nat_set_nat(image_nat_nat(aa_fun4273987348596450176at_nat(aTP_Lamm_ay(F),G)),A4) ).

% image_image
tff(fact_1224_image__image,axiom,
    ! [F: fun_rat_int,G: fun_nat_rat,A4: set_nat] : image_rat_int(F,aa_set_nat_set_rat(image_nat_rat(G),A4)) = aa_set_nat_set_int(image_nat_int(aa_fun96136329087253468at_int(aTP_Lamm_az(F),G)),A4) ).

% image_image
tff(fact_1225_image__image,axiom,
    ! [F: fun_int_rat,G: fun_nat_int,A4: set_nat] : image_int_rat(F,aa_set_nat_set_int(image_nat_int(G),A4)) = aa_set_nat_set_rat(image_nat_rat(aa_fun6234665583235904476at_rat(aTP_Lamm_ba(F),G)),A4) ).

% image_image
tff(fact_1226_image__image,axiom,
    ! [F: fun_nat_rat,G: fun_nat_nat,A4: set_nat] : aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(image_nat_nat(G),A4)) = aa_set_nat_set_rat(image_nat_rat(aa_fun2283220954820195200at_rat(aTP_Lamm_bb(F),G)),A4) ).

% image_image
tff(fact_1227_image__image,axiom,
    ! [F: fun_nat_rat,G: fun_int_nat,A4: set_int] : aa_set_nat_set_rat(image_nat_rat(F),aa_set_int_set_nat(image_int_nat(G),A4)) = image_int_rat(aa_fun2998523592351139896nt_rat(aTP_Lamm_bc(F),G),A4) ).

% image_image
tff(fact_1228_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A4: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(image_nat_nat(G),A4)) = aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_bd(F),G)),A4) ).

% image_image
tff(fact_1229_image__image,axiom,
    ! [F: fun_nat_nat,G: fun_int_nat,A4: set_int] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_int_set_nat(image_int_nat(G),A4)) = aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_be(F),G)),A4) ).

% image_image
tff(fact_1230_image__image,axiom,
    ! [F: fun_nat_int,G: fun_nat_nat,A4: set_nat] : aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(image_nat_nat(G),A4)) = aa_set_nat_set_int(image_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_bf(F),G)),A4) ).

% image_image
tff(fact_1231_image__image,axiom,
    ! [F: fun_nat_int,G: fun_int_nat,A4: set_int] : aa_set_nat_set_int(image_nat_int(F),aa_set_int_set_nat(image_int_nat(G),A4)) = aa_set_int_set_int(image_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_bg(F),G)),A4) ).

% image_image
tff(fact_1232_Compr__image__eq,axiom,
    ! [F: fun_o_o,A4: set_o,P2: fun_o_o] : aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_bh(F),A4),P2)) = aa_set_o_set_o(image_o_o(F),aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_bi(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1233_Compr__image__eq,axiom,
    ! [F: fun_int_o,A4: set_int,P2: fun_o_o] : aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aa_set2864532236940079139un_o_o(aTP_Lamm_bj(F),A4),P2)) = image_int_o(F,aa_fun_int_o_set_int(collect_int,aa_fun_o_o_fun_int_o(aa_set2419422181098893691_int_o(aTP_Lamm_bk(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1234_Compr__image__eq,axiom,
    ! [F: fun_o_int,A4: set_o,P2: fun_int_o] : aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aa_set1221527818376329535_int_o(aTP_Lamm_bl(F),A4),P2)) = image_o_int(F,aa_fun_o_o_set_o(collect_o,aa_fun_int_o_fun_o_o(aa_set6365818687085129567un_o_o(aTP_Lamm_bm(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1235_Compr__image__eq,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_int_o] : aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_bn(F),A4),P2)) = aa_set_int_set_int(image_int_int(F),aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_bo(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1236_Compr__image__eq,axiom,
    ! [F: fun_nat_rat,A4: set_nat,P2: fun_rat_o] : collect_rat(aa_fun2875410490525819644_rat_o(aa_set1331966092573250503_rat_o(aTP_Lamm_bp(F),A4),P2)) = aa_set_nat_set_rat(image_nat_rat(F),aa_fun_nat_o_set_nat(collect_nat,aa_fun6612743417842387716_nat_o(aa_set2081400587708086223_nat_o(aTP_Lamm_bq(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1237_Compr__image__eq,axiom,
    ! [F: fun_nat_o,A4: set_nat,P2: fun_o_o] : aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aa_set1656415104011699143un_o_o(aTP_Lamm_br(F),A4),P2)) = image_nat_o(F,aa_fun_nat_o_set_nat(collect_nat,aa_fun_o_o_fun_nat_o(aa_set2987053916987590195_nat_o(aTP_Lamm_bs(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1238_Compr__image__eq,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_int_o] : aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aa_set1288072335105258183_int_o(aTP_Lamm_bt(F),A4),P2)) = aa_set_nat_set_int(image_nat_int(F),aa_fun_nat_o_set_nat(collect_nat,aa_fun8064560132910005592_nat_o(aa_set3533217302775704099_nat_o(aTP_Lamm_bu(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1239_Compr__image__eq,axiom,
    ! [F: fun_o_nat,A4: set_o,P2: fun_nat_o] : aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aa_set1339306056044713023_nat_o(aTP_Lamm_bv(F),A4),P2)) = image_o_nat(F,aa_fun_o_o_set_o(collect_o,aa_fun_nat_o_fun_o_o(aa_set26024703028173499un_o_o(aTP_Lamm_bw(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1240_Compr__image__eq,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_nat_o] : aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aa_set563577997438114851_nat_o(aTP_Lamm_bx(F),A4),P2)) = aa_set_int_set_nat(image_int_nat(F),aa_fun_int_o_set_int(collect_int,aa_fun4116151057475255200_int_o(aa_set7541805066622444743_int_o(aTP_Lamm_by(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1241_Compr__image__eq,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_nat_o] : aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_bz(F),A4),P2)) = aa_set_nat_set_nat(image_nat_nat(F),aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_ca(F),A4),P2))) ).

% Compr_image_eq
tff(fact_1242_Id__on__def_H,axiom,
    ! [A4: fun_Pr4796943543331071514_nat_o] : id_on_2554058798563519774at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,A4)) = collec7088162979684241874at_nat(produc6590410687421337004_nat_o(aTP_Lamm_cb(A4))) ).

% Id_on_def'
tff(fact_1243_Id__on__def_H,axiom,
    ! [A4: fun_nat_o] : id_on_nat(aa_fun_nat_o_set_nat(collect_nat,A4)) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_cc(A4))) ).

% Id_on_def'
tff(fact_1244_Id__on__def_H,axiom,
    ! [A4: fun_Pr3238374608701176162_int_o] : id_on_8598933527810153430nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,A4)) = collec7447955092554649554nt_int(produc1676490119946744748_int_o(aTP_Lamm_cd(A4))) ).

% Id_on_def'
tff(fact_1245_Id__on__def_H,axiom,
    ! [A4: fun_int_o] : id_on_int(aa_fun_int_o_set_int(collect_int,A4)) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ce(A4))) ).

% Id_on_def'
tff(fact_1246_pred__equals__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat] :
      ( ! [X4: set_Pr4329608150637261639at_nat,Xa3: set_Pr4329608150637261639at_nat] :
          ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X4),Xa3),R2)
        <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X4),Xa3),S3) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_1247_pred__equals__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Xa3: set_Pr1261947904930325089at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X4),Xa3)),R2)
        <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X4),Xa3)),S3) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_1248_pred__equals__eq2,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X4: nat,Xa3: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X4),Xa3),R2)
        <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X4),Xa3),S3) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_1249_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Xa3: nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa3)),R2)
        <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa3)),S3) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_1250_pred__equals__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa3: int] :
          ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),R2)
        <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),S3) )
    <=> ( R2 = S3 ) ) ).

% pred_equals_eq2
tff(fact_1251_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,C),P),Z)
     => ~ ! [X: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X),Y),Z) ) ) ).

% case_prodE'
tff(fact_1252_case__prodD_H,axiom,
    ! [R2: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,R2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R2,A),B),C) ) ).

% case_prodD'
tff(fact_1253_case__prod__Pair__iden,axiom,
    ! [P: produc1319942482725812455at_nat] : aa_Pro5334082655537895548at_nat(produc5456455677105495163at_nat(produc9060074326276436823at_nat),P) = P ).

% case_prod_Pair_iden
tff(fact_1254_case__prod__Pair__iden,axiom,
    ! [P: produc3843707927480180839at_nat] : aa_Pro9111386209873968764at_nat(produc7548845685434208379at_nat(produc2922128104949294807at_nat),P) = P ).

% case_prod_Pair_iden
tff(fact_1255_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_num] : aa_Pro19645118074684540at_num(produc49306077274653107at_num(product_Pair_nat_num),P) = P ).

% case_prod_Pair_iden
tff(fact_1256_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(product_Pair_nat_nat),P) = P ).

% case_prod_Pair_iden
tff(fact_1257_case__prod__Pair__iden,axiom,
    ! [P: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,product_Pair_int_int),P) = P ).

% case_prod_Pair_iden
tff(fact_1258_subset__CollectI,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,Q2: fun_Pr3059359313727237556_nat_o,P2: fun_Pr3059359313727237556_nat_o] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,B5),A4)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B5)
           => ( aa_Pro5947980084102686253_nat_o(Q2,X)
             => aa_Pro5947980084102686253_nat_o(P2,X) ) )
       => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun2566793329686429606_nat_o(aTP_Lamm_cf(B5),Q2))),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun2566793329686429606_nat_o(aTP_Lamm_cf(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_1259_subset__CollectI,axiom,
    ! [B5: set_o,A4: set_o,Q2: fun_o_o,P2: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B5),A4)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_o(Q2,(X))
             => aa_o_o(P2,(X)) ) )
       => aa_set_o_o(ord_less_eq_set_o(aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aTP_Lamm_cg(B5),Q2))),aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aTP_Lamm_cg(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_1260_subset__CollectI,axiom,
    ! [B5: set_int,A4: set_int,Q2: fun_int_o,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_o(Q2,X)
             => aa_int_o(P2,X) ) )
       => aa_set_int_o(ord_less_eq_set_int(aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ch(B5),Q2))),aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ch(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_1261_subset__CollectI,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,Q2: fun_Pr4796943543331071514_nat_o,P2: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),A4)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B5)
           => ( aa_Pro6150637604445964947_nat_o(Q2,X)
             => aa_Pro6150637604445964947_nat_o(P2,X) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun359341089595923442_nat_o(aTP_Lamm_ci(B5),Q2))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun359341089595923442_nat_o(aTP_Lamm_ci(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_1262_subset__CollectI,axiom,
    ! [B5: set_nat,A4: set_nat,Q2: fun_nat_o,P2: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_o(Q2,X)
             => aa_nat_o(P2,X) ) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_cj(B5),Q2))),aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_cj(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_1263_subset__CollectI,axiom,
    ! [B5: set_Pr958786334691620121nt_int,A4: set_Pr958786334691620121nt_int,Q2: fun_Pr3238374608701176162_int_o,P2: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(B5,A4)
     => ( ! [X: product_prod_int_int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X),B5)
           => ( aa_Pro4592068669816069595_int_o(Q2,X)
             => aa_Pro4592068669816069595_int_o(P2,X) ) )
       => ord_le2843351958646193337nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun5349570537165229698_int_o(aTP_Lamm_ck(B5),Q2)),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun5349570537165229698_int_o(aTP_Lamm_ck(A4),P2))) ) ) ).

% subset_CollectI
tff(fact_1264_subset__Collect__iff,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P2: fun_Pr3059359313727237556_nat_o] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,B5),A4)
     => ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,B5),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun2566793329686429606_nat_o(aTP_Lamm_cf(A4),P2)))
      <=> ! [X4: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),B5)
           => aa_Pro5947980084102686253_nat_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_1265_subset__Collect__iff,axiom,
    ! [B5: set_o,A4: set_o,P2: fun_o_o] :
      ( aa_set_o_o(ord_less_eq_set_o(B5),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(B5),aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aTP_Lamm_cg(A4),P2)))
      <=> ! [X4: $o] :
            ( aa_set_o_o(member_o((X4)),B5)
           => aa_o_o(P2,(X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_1266_subset__Collect__iff,axiom,
    ! [B5: set_int,A4: set_int,P2: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ch(A4),P2)))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),B5)
           => aa_int_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_1267_subset__Collect__iff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun359341089595923442_nat_o(aTP_Lamm_ci(A4),P2)))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),B5)
           => aa_Pro6150637604445964947_nat_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_1268_subset__Collect__iff,axiom,
    ! [B5: set_nat,A4: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_cj(A4),P2)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B5)
           => aa_nat_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_1269_subset__Collect__iff,axiom,
    ! [B5: set_Pr958786334691620121nt_int,A4: set_Pr958786334691620121nt_int,P2: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(B5,A4)
     => ( ord_le2843351958646193337nt_int(B5,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun5349570537165229698_int_o(aTP_Lamm_ck(A4),P2)))
      <=> ! [X4: product_prod_int_int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X4),B5)
           => aa_Pro4592068669816069595_int_o(P2,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_1270_prop__restrict,axiom,
    ! [Xa: produc3843707927480180839at_nat,Z4: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat,P2: fun_Pr3059359313727237556_nat_o] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),Z4)
     => ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,Z4),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun2566793329686429606_nat_o(aTP_Lamm_cf(X3),P2)))
       => aa_Pro5947980084102686253_nat_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_1271_prop__restrict,axiom,
    ! [Xa: $o,Z4: set_o,X3: set_o,P2: fun_o_o] :
      ( aa_set_o_o(member_o((Xa)),Z4)
     => ( aa_set_o_o(ord_less_eq_set_o(Z4),aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aTP_Lamm_cg(X3),P2)))
       => aa_o_o(P2,(Xa)) ) ) ).

% prop_restrict
tff(fact_1272_prop__restrict,axiom,
    ! [Xa: int,Z4: set_int,X3: set_int,P2: fun_int_o] :
      ( aa_set_int_o(member_int(Xa),Z4)
     => ( aa_set_int_o(ord_less_eq_set_int(Z4),aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ch(X3),P2)))
       => aa_int_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_1273_prop__restrict,axiom,
    ! [Xa: product_prod_nat_nat,Z4: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,P2: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),Z4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Z4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun359341089595923442_nat_o(aTP_Lamm_ci(X3),P2)))
       => aa_Pro6150637604445964947_nat_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_1274_prop__restrict,axiom,
    ! [Xa: nat,Z4: set_nat,X3: set_nat,P2: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),Z4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Z4),aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_cj(X3),P2)))
       => aa_nat_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_1275_prop__restrict,axiom,
    ! [Xa: product_prod_int_int,Z4: set_Pr958786334691620121nt_int,X3: set_Pr958786334691620121nt_int,P2: fun_Pr3238374608701176162_int_o] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Xa),Z4)
     => ( ord_le2843351958646193337nt_int(Z4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun5349570537165229698_int_o(aTP_Lamm_ck(X3),P2)))
       => aa_Pro4592068669816069595_int_o(P2,Xa) ) ) ).

% prop_restrict
tff(fact_1276_Collect__restrict,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,P2: fun_Pr3059359313727237556_nat_o] : aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun2566793329686429606_nat_o(aTP_Lamm_cf(X3),P2))),X3) ).

% Collect_restrict
tff(fact_1277_Collect__restrict,axiom,
    ! [X3: set_o,P2: fun_o_o] : aa_set_o_o(ord_less_eq_set_o(aa_fun_o_o_set_o(collect_o,aa_fun_o_o_fun_o_o(aTP_Lamm_cg(X3),P2))),X3) ).

% Collect_restrict
tff(fact_1278_Collect__restrict,axiom,
    ! [X3: set_int,P2: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ch(X3),P2))),X3) ).

% Collect_restrict
tff(fact_1279_Collect__restrict,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,P2: fun_Pr4796943543331071514_nat_o] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun359341089595923442_nat_o(aTP_Lamm_ci(X3),P2))),X3) ).

% Collect_restrict
tff(fact_1280_Collect__restrict,axiom,
    ! [X3: set_nat,P2: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_cj(X3),P2))),X3) ).

% Collect_restrict
tff(fact_1281_Collect__restrict,axiom,
    ! [X3: set_Pr958786334691620121nt_int,P2: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun5349570537165229698_int_o(aTP_Lamm_ck(X3),P2)),X3) ).

% Collect_restrict
tff(fact_1282_fun_Omap__ident,axiom,
    ! [T: fun_nat_o] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aTP_Lamm_cl),T) = T ).

% fun.map_ident
tff(fact_1283_fun_Omap__ident,axiom,
    ! [T: fun_nat_nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aTP_Lamm_ap),T) = T ).

% fun.map_ident
tff(fact_1284_case__prod__app,axiom,
    ! [F: fun_na1668100129766719005at_nat,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),Xa),Y2) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_cm(F),Y2)),Xa) ).

% case_prod_app
tff(fact_1285_case__prod__app,axiom,
    ! [F: fun_na5914473681949881652_nat_o,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Xa),Y2)
    <=> aa_Pro6150637604445964947_nat_o(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_Pro1957341097241355813_nat_o(aTP_Lamm_cn(F),Y2)),Xa) ) ).

% case_prod_app
tff(fact_1286_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_co(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_1287_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7323606162024495668nt_int(aTP_Lamm_cp(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1288_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun82482788810740534_int_o(aTP_Lamm_cq(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_1289_prod_Ocase__distrib,axiom,
    ! [Ha: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun6119123525526402106nt_int(aTP_Lamm_cr(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1290_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun4545072666309863515_int_o(aTP_Lamm_cs(Ha),F)),Prod) ) ).

% prod.case_distrib
tff(fact_1291_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun7461304428735181397nt_int(aTP_Lamm_ct(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1292_prod_Ocase__distrib,axiom,
    ! [Ha: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(Ha,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4023131440832735197nt_int(aTP_Lamm_cu(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1293_prod_Ocase__distrib,axiom,
    ! [Ha: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(Ha,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun1034830265760468567nt_int(aTP_Lamm_cv(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1294_prod_Ocase__distrib,axiom,
    ! [Ha: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(Ha,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),Prod)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4152872853588681340nt_int(aTP_Lamm_cw(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1295_prod_Ocase__distrib,axiom,
    ! [Ha: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(Ha,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),Prod)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun824319262034445606_nat_o(aTP_Lamm_cx(Ha),F)),Prod) ).

% prod.case_distrib
tff(fact_1296_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_cy),Prod) ).

% prod.disc_eq_case
tff(fact_1297_image__Collect__subsetI,axiom,
    ! [P2: fun_int_o,F: fun_int_nat,B5: set_nat] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
         => aa_set_nat_o(member_nat(aa_int_nat(F,X)),B5) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),aa_fun_int_o_set_int(collect_int,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1298_image__Collect__subsetI,axiom,
    ! [P2: fun_int_o,F: fun_int_int,B5: set_int] :
      ( ! [X: int] :
          ( aa_int_o(P2,X)
         => aa_set_int_o(member_int(aa_int_int(F,X)),B5) )
     => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),aa_fun_int_o_set_int(collect_int,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1299_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_rat,B5: set_rat] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_rat_o(member_rat(aa_nat_rat(F,X)),B5) )
     => aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),aa_fun_nat_o_set_nat(collect_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1300_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_o,B5: set_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_o_o(member_o(aa_nat_o(F,X)),B5) )
     => aa_set_o_o(ord_less_eq_set_o(image_nat_o(F,aa_fun_nat_o_set_nat(collect_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1301_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_nat,B5: set_nat] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_nat_o(member_nat(aa_nat_nat(F,X)),B5) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),aa_fun_nat_o_set_nat(collect_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1302_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_nat_int,B5: set_int] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set_int_o(member_int(aa_nat_int(F,X)),B5) )
     => aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),aa_fun_nat_o_set_nat(collect_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1303_image__Collect__subsetI,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,F: fun_Pr4796943543331071514_nat_o,B5: set_o] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P2,X)
         => aa_set_o_o(member_o(aa_Pro6150637604445964947_nat_o(F,X)),B5) )
     => aa_set_o_o(ord_less_eq_set_o(image_3693632289388996572_nat_o(F,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1304_image__Collect__subsetI,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,F: fun_Pr2602117396471556196at_nat,B5: set_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P2,X)
         => aa_set_nat_o(member_nat(aa_Pro4243173770044397973at_nat(F,X)),B5) )
     => aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(F,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1305_image__Collect__subsetI,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,F: fun_Pr7647638413817135296at_int,B5: set_int] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(P2,X)
         => aa_set_int_o(member_int(aa_Pro4240683299535347697at_int(F,X)),B5) )
     => aa_set_int_o(ord_less_eq_set_int(image_2483585944268220136at_int(F,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1306_image__Collect__subsetI,axiom,
    ! [P2: fun_nat_o,F: fun_na1476630412504255114at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat] :
          ( aa_nat_o(P2,X)
         => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(F,X)),B5) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F,aa_fun_nat_o_set_nat(collect_nat,P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_1307_range__composition,axiom,
    ! [F: fun_nat_rat,G: fun_char_nat] : image_char_rat(aa_fun7940347660735252098ar_rat(aTP_Lamm_cz(F),G),top_top_set_char) = aa_set_nat_set_rat(image_nat_rat(F),image_char_nat(G,top_top_set_char)) ).

% range_composition
tff(fact_1308_range__composition,axiom,
    ! [F: fun_nat_nat,G: fun_char_nat] : image_char_nat(aa_fun2225204182273873018ar_nat(aTP_Lamm_da(F),G),top_top_set_char) = aa_set_nat_set_nat(image_nat_nat(F),image_char_nat(G,top_top_set_char)) ).

% range_composition
tff(fact_1309_range__composition,axiom,
    ! [F: fun_nat_int,G: fun_char_nat] : image_char_int(aa_fun7270725199619452118ar_int(aTP_Lamm_db(F),G),top_top_set_char) = aa_set_nat_set_int(image_nat_int(F),image_char_nat(G,top_top_set_char)) ).

% range_composition
tff(fact_1310_range__composition,axiom,
    ! [F: fun_int_nat,G: fun_char_int] : image_char_nat(aa_fun3120538043794061526ar_nat(aTP_Lamm_dc(F),G),top_top_set_char) = aa_set_int_set_nat(image_int_nat(F),image_char_int(G,top_top_set_char)) ).

% range_composition
tff(fact_1311_range__composition,axiom,
    ! [F: fun_int_int,G: fun_char_int] : image_char_int(aa_fun8166059061139640626ar_int(aTP_Lamm_dd(F),G),top_top_set_char) = aa_set_int_set_int(image_int_int(F),image_char_int(G,top_top_set_char)) ).

% range_composition
tff(fact_1312_range__composition,axiom,
    ! [F: fun_nat_rat,G: fun_rat_nat] : image_rat_rat(aa_fun6175819412704661392at_rat(aTP_Lamm_de(F),G),top_top_set_rat) = aa_set_nat_set_rat(image_nat_rat(F),image_rat_nat(G,top_top_set_rat)) ).

% range_composition
tff(fact_1313_range__composition,axiom,
    ! [F: fun_nat_nat,G: fun_rat_nat] : image_rat_nat(aa_fun460675934243282312at_nat(aTP_Lamm_df(F),G),top_top_set_rat) = aa_set_nat_set_nat(image_nat_nat(F),image_rat_nat(G,top_top_set_rat)) ).

% range_composition
tff(fact_1314_range__composition,axiom,
    ! [F: fun_nat_int,G: fun_rat_nat] : image_rat_int(aa_fun5506196951588861412at_int(aTP_Lamm_dg(F),G),top_top_set_rat) = aa_set_nat_set_int(image_nat_int(F),image_rat_nat(G,top_top_set_rat)) ).

% range_composition
tff(fact_1315_range__composition,axiom,
    ! [F: fun_int_nat,G: fun_rat_int] : image_rat_nat(aa_fun4412120562658991588at_nat(aTP_Lamm_dh(F),G),top_top_set_rat) = aa_set_int_set_nat(image_int_nat(F),image_rat_int(G,top_top_set_rat)) ).

% range_composition
tff(fact_1316_range__composition,axiom,
    ! [F: fun_int_int,G: fun_rat_int] : image_rat_int(aa_fun234269543149794880at_int(aTP_Lamm_di(F),G),top_top_set_rat) = aa_set_int_set_int(image_int_int(F),image_rat_int(G,top_top_set_rat)) ).

% range_composition
tff(fact_1317_rangeE,axiom,
    ! [B: $o,F: fun_char_o] :
      ( aa_set_o_o(member_o((B)),image_char_o(F,top_top_set_char))
     => ~ ! [X: char] :
            ( (B)
          <=> ~ aa_char_o(F,X) ) ) ).

% rangeE
tff(fact_1318_rangeE,axiom,
    ! [B: nat,F: fun_char_nat] :
      ( aa_set_nat_o(member_nat(B),image_char_nat(F,top_top_set_char))
     => ~ ! [X: char] : B != aa_char_nat(F,X) ) ).

% rangeE
tff(fact_1319_rangeE,axiom,
    ! [B: int,F: fun_char_int] :
      ( aa_set_int_o(member_int(B),image_char_int(F,top_top_set_char))
     => ~ ! [X: char] : B != aa_char_int(F,X) ) ).

% rangeE
tff(fact_1320_rangeE,axiom,
    ! [B: $o,F: fun_rat_o] :
      ( aa_set_o_o(member_o((B)),image_rat_o(F,top_top_set_rat))
     => ~ ! [X: rat] :
            ( (B)
          <=> ~ aa_rat_o(F,X) ) ) ).

% rangeE
tff(fact_1321_rangeE,axiom,
    ! [B: nat,F: fun_rat_nat] :
      ( aa_set_nat_o(member_nat(B),image_rat_nat(F,top_top_set_rat))
     => ~ ! [X: rat] : B != aa_rat_nat(F,X) ) ).

% rangeE
tff(fact_1322_rangeE,axiom,
    ! [B: int,F: fun_rat_int] :
      ( aa_set_int_o(member_int(B),image_rat_int(F,top_top_set_rat))
     => ~ ! [X: rat] : B != aa_rat_int(F,X) ) ).

% rangeE
tff(fact_1323_rangeE,axiom,
    ! [B: rat,F: fun_nat_rat] :
      ( aa_set_rat_o(member_rat(B),aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat))
     => ~ ! [X: nat] : B != aa_nat_rat(F,X) ) ).

% rangeE
tff(fact_1324_rangeE,axiom,
    ! [B: $o,F: fun_nat_o] :
      ( aa_set_o_o(member_o((B)),image_nat_o(F,top_top_set_nat))
     => ~ ! [X: nat] :
            ( (B)
          <=> ~ aa_nat_o(F,X) ) ) ).

% rangeE
tff(fact_1325_rangeE,axiom,
    ! [B: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat))
     => ~ ! [X: nat] : B != aa_nat_nat(F,X) ) ).

% rangeE
tff(fact_1326_rangeE,axiom,
    ! [B: int,F: fun_nat_int] :
      ( aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),top_top_set_nat))
     => ~ ! [X: nat] : B != aa_nat_int(F,X) ) ).

% rangeE
tff(fact_1327_fun_Orel__map_I1_J,axiom,
    ! [Sb: fun_num_fun_num_o,I: fun_nat_num,Xa: fun_int_nat,Y2: fun_int_num] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,Sb),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,I),Xa)),Y2)
    <=> aa_fun_int_num_o(aa_fun7803388736439588215_num_o(bNF_re272988434038980259at_num(fequal_int,aa_fun7501294707364756987_num_o(aTP_Lamm_dj(Sb),I)),Xa),Y2) ) ).

% fun.rel_map(1)
tff(fact_1328_fun_Orel__map_I1_J,axiom,
    ! [Sb: fun_num_fun_num_o,I: fun_num_num,Xa: fun_int_num,Y2: fun_int_num] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,Sb),aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,I),Xa)),Y2)
    <=> aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,aa_fun7420891108566521959_num_o(aTP_Lamm_dk(Sb),I)),Xa),Y2) ) ).

% fun.rel_map(1)
tff(fact_1329_fun_Orel__map_I2_J,axiom,
    ! [Sa: fun_num_fun_num_o,Xa: fun_int_num,G: fun_nat_num,Y2: fun_int_nat] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,Sa),Xa),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),Y2))
    <=> aa_fun_int_nat_o(aa_fun6964096368831925623_nat_o(bNF_re1845987051190671011um_nat(fequal_int,aa_fun8523389415946384635_nat_o(aTP_Lamm_dl(Sa),G)),Xa),Y2) ) ).

% fun.rel_map(2)
tff(fact_1330_fun_Orel__map_I2_J,axiom,
    ! [Sa: fun_num_fun_num_o,Xa: fun_int_num,G: fun_num_num,Y2: fun_int_num] :
      ( aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,Sa),Xa),aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,G),Y2))
    <=> aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,aa_fun7420891108566521959_num_o(aTP_Lamm_dm(Sa),G)),Xa),Y2) ) ).

% fun.rel_map(2)
tff(fact_1331_case__prodE2,axiom,
    ! [Q2: fun_fu8391177546919232046_nat_o,P2: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q2,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,P2),Z))
     => ~ ! [X: nat,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_fun3538214411680374423_nat_o(Q2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P2,X),Y)) ) ) ).

% case_prodE2
tff(fact_1332_case__prodE2,axiom,
    ! [Q2: fun_fu6891447142836604759at_o_o,P2: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q2,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,P2),Z))
     => ~ ! [X: nat,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_fun8408151826566693824at_o_o(Q2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P2,X),Y)) ) ) ).

% case_prodE2
tff(fact_1333_case__prodE2,axiom,
    ! [Q2: fun_Pr3238374608701176162_int_o,P2: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q2,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,P2),Z))
     => ~ ! [X: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_Pro4592068669816069595_int_o(Q2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P2,X),Y)) ) ) ).

% case_prodE2
tff(fact_1334_case__prodE2,axiom,
    ! [Q2: fun_o_o,P2: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q2,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,P2),Z))
     => ~ ! [X: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_o_o(Q2,aa_int_o(aa_int_fun_int_o(P2,X),Y)) ) ) ).

% case_prodE2
tff(fact_1335_case__prodE2,axiom,
    ! [Q2: fun_int_o,P2: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q2,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,P2),Z))
     => ~ ! [X: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(P2,X),Y)) ) ) ).

% case_prodE2
tff(fact_1336_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_dn(F)) = F ).

% case_prod_eta
tff(fact_1337_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_do(F)) = F ).

% case_prod_eta
tff(fact_1338_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_dp(F)) = F ).

% case_prod_eta
tff(fact_1339_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_dq(F)) = F ).

% case_prod_eta
tff(fact_1340_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_dr(F)) = F ).

% case_prod_eta
tff(fact_1341_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X: nat,Y: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X),Y) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y))
     => ( aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_1342_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X: nat,Y: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X),Y) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y))
     => ( aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_1343_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X: int,Y: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X),Y) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y))
     => ( aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_1344_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X: int,Y: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X),Y)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)) )
     => ( aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_1345_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X: int,Y: int] : aa_int_int(aa_int_fun_int_int(F,X),Y) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y))
     => ( aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_1346_case__prodD,axiom,
    ! [F: fun_se5672428230568430382_nat_o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( aa_Pro4518350200472837805_nat_o(aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,F),aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B))
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_1347_case__prodD,axiom,
    ! [F: fun_se5639961202347202350_nat_o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,F),aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B))
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_1348_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,F),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))
     => aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_1349_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_1350_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_1351_case__prodE,axiom,
    ! [C: fun_se5672428230568430382_nat_o,P: produc1319942482725812455at_nat] :
      ( aa_Pro4518350200472837805_nat_o(aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,C),P)
     => ~ ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
            ( ( P = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),Y) )
           => ~ aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(C,X),Y) ) ) ).

% case_prodE
tff(fact_1352_case__prodE,axiom,
    ! [C: fun_se5639961202347202350_nat_o,P: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,C),P)
     => ~ ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
            ( ( P = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),Y) )
           => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(C,X),Y) ) ) ).

% case_prodE
tff(fact_1353_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,C),P)
     => ~ ! [X: nat,Y: num] :
            ( ( P = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X),Y) ) ) ).

% case_prodE
tff(fact_1354_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,C),P)
     => ~ ! [X: nat,Y: nat] :
            ( ( P = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X),Y) ) ) ).

% case_prodE
tff(fact_1355_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,C),P)
     => ~ ! [X: int,Y: int] :
            ( ( P = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X),Y) ) ) ).

% case_prodE
tff(fact_1356_fst__def,axiom,
    ! [Prod: produc7812382424243267045t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Prod) = produc4846201513286513633_ref_a(aTP_Lamm_ds,Prod) ).

% fst_def
tff(fact_1357_fst__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_dt),Prod) ).

% fst_def
tff(fact_1358_snd__def,axiom,
    ! [Prod: produc7812382424243267045t_unit] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Prod) = produc2552614499379209865t_unit(aTP_Lamm_du,Prod) ).

% snd_def
tff(fact_1359_snd__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_dv),Prod) ).

% snd_def
tff(fact_1360_map__prod__def,axiom,
    ! [F: fun_int_int,G: fun_int_int] : produc6036585564866528938nt_int(F,G) = aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4035196674708996672nt_int(aTP_Lamm_dw(F),G)) ).

% map_prod_def
tff(fact_1361_case__prod__map__prod,axiom,
    ! [Ha: fun_na1668100129766719005at_nat,F: fun_nat_nat,G: fun_nat_nat,Xa: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,Ha),aa_Pro262675715343573244at_nat(produc6977886695330630970at_nat(F,G),Xa)) = aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aa_fun5672751633241614456at_nat(aa_fun6003775592413651218at_nat(aTP_Lamm_dx(Ha),F),G)),Xa) ).

% case_prod_map_prod
tff(fact_1362_case__prod__map__prod,axiom,
    ! [Ha: fun_na5914473681949881652_nat_o,F: fun_nat_nat,G: fun_nat_nat,Xa: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,Ha),aa_Pro262675715343573244at_nat(produc6977886695330630970at_nat(F,G),Xa)) = aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun5827507165208764047_nat_o(aa_fun83116112763158185_nat_o(aTP_Lamm_dy(Ha),F),G)),Xa) ).

% case_prod_map_prod
tff(fact_1363_case__prod__map__prod,axiom,
    ! [Ha: fun_in1696471567231710611nt_int,F: fun_int_int,G: fun_int_int,Xa: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,Ha),aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),Xa)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4035196674708996672nt_int(aa_fun2536542299119233284nt_int(aTP_Lamm_dz(Ha),F),G)),Xa) ).

% case_prod_map_prod
tff(fact_1364_case__prod__map__prod,axiom,
    ! [Ha: fun_int_fun_int_o,F: fun_int_int,G: fun_int_int,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Ha),aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),Xa))
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun7896090258416335903_int_o(aa_fun8092296977475930211_int_o(aTP_Lamm_ea(Ha),F),G)),Xa) ) ).

% case_prod_map_prod
tff(fact_1365_case__prod__map__prod,axiom,
    ! [Ha: fun_int_fun_int_int,F: fun_int_int,G: fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,Ha),aa_Pro6970504127699319548nt_int(produc6036585564866528938nt_int(F,G),Xa)) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun5911324870756489105nt_int(aa_fun8505558176664197859nt_int(aTP_Lamm_eb(Ha),F),G)),Xa) ).

% case_prod_map_prod
tff(fact_1366_converse__relcompp,axiom,
    ! [Ra: fun_in2375804543815556931_nat_o,S: fun_Pr2028404973800837934_nat_o] : conver1866313717161370910at_nat(relcom716700242784427165at_nat(Ra,S)) = relcom8911704206367525917at_int(conver6965824823119931701at_nat(S),conver1866313717161370910at_nat(Ra)) ).

% converse_relcompp
tff(fact_1367_converse__relcompp,axiom,
    ! [Ra: fun_Co8353275306710069249_int_o,S: fun_int_fun_int_o] : conver1462408618118490240er_int(relcom3789353511612409832nt_int(Ra,S)) = relcom967497705283160040nteger(conversep_int_int(S),conver1462408618118490240er_int(Ra)) ).

% converse_relcompp
tff(fact_1368_converse__relcompp,axiom,
    ! [Ra: fun_Pr2028404973800837934_nat_o,S: fun_Pr4738963024209204295_int_o] : conver4689305806000247544at_int(relcom8911704206367525917at_int(Ra,S)) = relcom716700242784427165at_nat(conver4689305806000247544at_int(S),conver6965824823119931701at_nat(Ra)) ).

% converse_relcompp
tff(fact_1369_converse__relcompp,axiom,
    ! [Ra: fun_int_fun_int_o,S: fun_in4891666191172872833eger_o] : conver2101108097643427968nteger(relcom967497705283160040nteger(Ra,S)) = relcom3789353511612409832nt_int(conver2101108097643427968nteger(S),conversep_int_int(Ra)) ).

% converse_relcompp
tff(fact_1370_flip__pred,axiom,
    ! [A4: set_Pr7459493094073627847at_nat,R2: fun_se5672428230568430382_nat_o] :
      ( ord_le5997549366648089703at_nat(A4,aa_fun4050081229871760341at_nat(collec7872792499061297490at_nat,aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,conver9095369822871743413at_nat(R2))))
     => ord_le5997549366648089703at_nat(image_3313167189156628421at_nat(produc5456455677105495163at_nat(aTP_Lamm_ec),A4),aa_fun4050081229871760341at_nat(collec7872792499061297490at_nat,aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,R2))) ) ).

% flip_pred
tff(fact_1371_flip__pred,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,R2: fun_se5639961202347202350_nat_o] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,conver5243090411617913141at_nat(R2))))
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,aa_set4883462369968598652at_nat(image_960689587819713477at_nat(produc7548845685434208379at_nat(aTP_Lamm_ed)),A4)),aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,R2))) ) ).

% flip_pred
tff(fact_1372_flip__pred,axiom,
    ! [A4: set_Pr3280342998896498603um_nat,R2: fun_nat_fun_num_o] :
      ( ord_le5164908622851071819um_nat(A4,collec1522684865789554998um_nat(produc6857965554392702720_nat_o(conversep_nat_num(R2))))
     => ord_le8085105155179020875at_num(image_7762062833379733445at_num(produc5219324331026901427at_num(aTP_Lamm_ee),A4),aa_fun6502660398956743453at_num(collec4100598399430923318at_num,aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,R2))) ) ).

% flip_pred
tff(fact_1373_flip__pred,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R2: fun_nat_fun_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,conversep_nat_nat(R2))))
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(produc2626176000494625587at_nat(aTP_Lamm_ef)),A4)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R2))) ) ).

% flip_pred
tff(fact_1374_flip__pred,axiom,
    ! [A4: set_Pr958786334691620121nt_int,R2: fun_int_fun_int_o] :
      ( ord_le2843351958646193337nt_int(A4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,conversep_int_int(R2))))
     => ord_le2843351958646193337nt_int(image_2653370878348428101nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_eg),A4),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R2))) ) ).

% flip_pred
tff(fact_1375_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,Xa: produc3843707927480180839at_nat,S3: fun_o_7855627885593154802at_nat,Y2: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),aa_Pro2875264725485350454at_nat(comp_o4913564575355137019t_unit(S3,Rep),aa_o_Product_unit(Abs,(Y2))))
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),aa_o_s4772616771316612843at_nat(S3,(Y2))) ) ) ).

% type_copy_wit
tff(fact_1376_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,Xa: product_prod_nat_nat,S3: fun_o_4630291819367765772at_nat,Y2: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),aa_Pro5062749757247690704at_nat(comp_o2254869825208929t_unit(S3,Rep),aa_o_Product_unit(Abs,(Y2))))
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),aa_o_s3322794332960822277at_nat(S3,(Y2))) ) ) ).

% type_copy_wit
tff(fact_1377_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,Xa: $o,S3: fun_o_set_o,Y2: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(member_o((Xa)),aa_Pro536174410435844839_set_o(comp_o6895741221744486154t_unit(S3,Rep),aa_o_Product_unit(Abs,(Y2))))
       => aa_set_o_o(member_o((Xa)),aa_o_set_o(S3,(Y2))) ) ) ).

% type_copy_wit
tff(fact_1378_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,Xa: nat,S3: fun_o_set_nat,Y2: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_nat_o(member_nat(Xa),aa_Pro6064029160526250199et_nat(comp_o9190160103024335696t_unit(S3,Rep),aa_o_Product_unit(Abs,(Y2))))
       => aa_set_nat_o(member_nat(Xa),aa_o_set_nat(S3,(Y2))) ) ) ).

% type_copy_wit
tff(fact_1379_type__copy__wit,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,Xa: int,S3: fun_o_set_int,Y2: $o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( aa_set_int_o(member_int(Xa),aa_Pro1886178141017053491et_int(comp_o1521574261646630388t_unit(S3,Rep),aa_o_Product_unit(Abs,(Y2))))
       => aa_set_int_o(member_int(Xa),aa_o_set_int(S3,(Y2))) ) ) ).

% type_copy_wit
tff(fact_1380_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,F: fun_nat_nat,P2: fun_nat_o,Ha: fun_nat_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( bNF_vi6818137723729380987_o_o_o(F,Rep,bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(F),P2)),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),Ha),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1381_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,F: fun_int_int,P2: fun_int_o,Ha: fun_int_char] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( bNF_vi9149698133064535136char_o(F,Rep,bNF_Grp_int_char(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ei(F),P2)),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_char_nat_int(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1382_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_o_char,Abs: fun_char_o,F: fun_nat_o,P2: fun_o_o,Ha: fun_o_char] :
      ( type_d9076672964931703357o_char(Rep,Abs,top_top_set_char)
     => ( bNF_vi8816436626763797690char_o(F,Rep,bNF_Grp_o_char(aa_fun_o_o_set_o(collect_o,P2),Ha)) = bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,aa_fun_o_o_fun_nat_o(aTP_Lamm_ej(F),P2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_char_o_o(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1383_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_num_char,Abs: fun_char_num,F: fun_int_nat,P2: fun_nat_o,Ha: fun_nat_char] :
      ( type_d4396077731765706521m_char(Rep,Abs,top_top_set_char)
     => ( bNF_vi1300075849944819442char_o(F,Rep,bNF_Grp_nat_char(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_int_num(aa_fun_int_o_set_int(collect_int,aa_fun4116151057475255200_int_o(aTP_Lamm_ek(F),P2)),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_char_num_nat(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1384_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,F: fun_nat_nat,P2: fun_nat_o,Ha: fun_nat_char] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( bNF_vi8943400373106913048char_o(F,Rep,bNF_Grp_nat_char(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_nat_nat(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(F),P2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_char_nat_nat(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1385_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_nat_rat,Abs: fun_rat_nat,F: fun_int_int,P2: fun_int_o,Ha: fun_int_rat] :
      ( aa_set_rat_o(type_d5615363888691252950at_rat(Rep,Abs),top_top_set_rat)
     => ( bNF_vi1281217848275042457_rat_o(F,Rep,bNF_Grp_int_rat(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ei(F),P2)),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_rat_nat_int(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1386_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_o_rat,Abs: fun_rat_o,F: fun_nat_o,P2: fun_o_o,Ha: fun_o_rat] :
      ( aa_set_rat_o(type_d6486251586359353412_o_rat(Rep,Abs),top_top_set_rat)
     => ( bNF_vi3868162310295888627_rat_o(F,Rep,bNF_Grp_o_rat(aa_fun_o_o_set_o(collect_o,P2),Ha)) = bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,aa_fun_o_o_fun_nat_o(aTP_Lamm_ej(F),P2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_rat_o_o(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1387_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_num_rat,Abs: fun_rat_num,F: fun_int_nat,P2: fun_nat_o,Ha: fun_nat_rat] :
      ( aa_set_rat_o(type_d3745694291998722336um_rat(Rep,Abs),top_top_set_rat)
     => ( bNF_vi7769575839906753835_rat_o(F,Rep,bNF_Grp_nat_rat(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_int_num(aa_fun_int_o_set_int(collect_int,aa_fun4116151057475255200_int_o(aTP_Lamm_ek(F),P2)),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_rat_num_nat(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1388_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_nat_rat,Abs: fun_rat_nat,F: fun_nat_nat,P2: fun_nat_o,Ha: fun_nat_rat] :
      ( aa_set_rat_o(type_d5615363888691252950at_rat(Rep,Abs),top_top_set_rat)
     => ( bNF_vi2930123780028963665_rat_o(F,Rep,bNF_Grp_nat_rat(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_nat_nat(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(F),P2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_rat_nat_nat(Abs,Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1389_type__copy__vimage2p__Grp__Rep,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat,F: fun_int_int,P2: fun_int_o,Ha: fun_int_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),top_top_set_nat)
     => ( bNF_vi5018550775591610529_nat_o(F,Rep,bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ei(F),P2)),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,Abs),Ha)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_1390_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_nat_num,Abs: fun_num_nat,G: fun_nat_int,P2: fun_int_o,Ha: fun_int_nat] :
      ( type_d2807825734933527320at_num(Rep,Abs,top_top_set_num)
     => ( bNF_vi1731478210338170163_nat_o(G,Abs,bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Grp_nat_num(aa_fun_nat_o_set_nat(collect_nat,aa_fun8064560132910005592_nat_o(aTP_Lamm_el(G),P2)),comp_int_num_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Rep),Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1391_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,G: fun_nat_nat,P2: fun_nat_o,Ha: fun_nat_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( bNF_vi6818137723729380987_o_o_o(G,Abs,bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(G),P2)),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Rep),Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1392_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_int_nat,Abs: fun_nat_int,G: fun_int_int,P2: fun_int_o,Ha: fun_int_int] :
      ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep,Abs),top_top_set_nat)
     => ( bNF_vi3197508430158922565_int_o(G,Abs,bNF_Grp_int_int(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,aa_fun6243517787477317628_int_o(aTP_Lamm_ei(G),P2)),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Rep),Ha)),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1393_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_int_nat,Abs: fun_nat_int,G: fun_nat_int,P2: fun_int_o,Ha: fun_int_int] :
      ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep,Abs),top_top_set_nat)
     => ( bNF_vi8357618154448083105_int_o(G,Abs,bNF_Grp_int_int(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Grp_nat_nat(aa_fun_nat_o_set_nat(collect_nat,aa_fun8064560132910005592_nat_o(aTP_Lamm_el(G),P2)),comp_int_nat_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Rep),Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1394_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat,G: fun_nat_nat,P2: fun_nat_o,Ha: fun_nat_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),top_top_set_nat)
     => ( bNF_vi6667456707345531737_nat_o(G,Abs,bNF_Grp_nat_nat(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_nat_nat(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(G),P2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Rep),Ha)),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1395_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,G: fun_nat_o,P2: fun_o_o,Ha: fun_o_Product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( bNF_vi5856767400800367702unit_o(G,Abs,bNF_Gr395642070819736771t_unit(aa_fun_o_o_set_o(collect_o,P2),Ha)) = bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,aa_fun_o_o_fun_nat_o(aTP_Lamm_ej(G),P2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_P5156358013004227690it_o_o(Rep,Ha)),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1396_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,G: fun_nat_nat,P2: fun_nat_o,Ha: fun_nat_Product_unit] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),top_top_set_o)
     => ( bNF_vi1443435967014148806unit_o(G,Abs,bNF_Gr1993012052113314029t_unit(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(G),P2)),comp_nat_o_nat(comp_P4985893458130756478_o_nat(Rep,Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1397_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_nat_num,Abs: fun_num_nat,G: fun_Pr7647638413817135296at_int,P2: fun_int_o,Ha: fun_int_nat] :
      ( type_d2807825734933527320at_num(Rep,Abs,top_top_set_num)
     => ( bNF_vi9094538908245281400_nat_o(G,Abs,bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Gr5974128493091723341at_num(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun3324632943032829859_nat_o(aTP_Lamm_em(G),P2)),comp_i1186163378102392086at_nat(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Rep),Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1398_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_nat_num,Abs: fun_num_nat,G: fun_Pr982841511892225912nt_int,P2: fun_int_o,Ha: fun_int_nat] :
      ( type_d2807825734933527320at_num(Rep,Abs,top_top_set_num)
     => ( bNF_vi1633389952966819264_nat_o(G,Abs,bNF_Grp_int_nat(aa_fun_int_o_set_int(collect_int,P2),Ha)) = bNF_Gr8532703628021589765nt_num(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun1766064008402934507_int_o(aTP_Lamm_en(G),P2)),comp_i7231038107349025742nt_int(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Rep),Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1399_type__copy__vimage2p__Grp__Abs,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,G: fun_Pr2602117396471556196at_nat,P2: fun_nat_o,Ha: fun_nat_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( bNF_vi9025507403824386624_o_o_o(G,Abs,bNF_Grp_nat_o(aa_fun_nat_o_set_nat(collect_nat,P2),Ha)) = bNF_Gr8859634586176591525_nat_o(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun3836547582011492095_nat_o(aTP_Lamm_eo(G),P2)),comp_n1662990198670838940at_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Rep),Ha),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_1400_exE__realizer,axiom,
    ! [P2: fun_nat_fun_nat_o,P: product_prod_nat_nat,Q2: fun_fu8391177546919232046_nat_o,F: fun_na1668100129766719005at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P))
     => ( ! [X: nat,Y: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P2,Y),X)
           => aa_fun3538214411680374423_nat_o(Q2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X),Y)) )
       => aa_fun3538214411680374423_nat_o(Q2,aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),P)) ) ) ).

% exE_realizer
tff(fact_1401_exE__realizer,axiom,
    ! [P2: fun_nat_fun_nat_o,P: product_prod_nat_nat,Q2: fun_fu6891447142836604759at_o_o,F: fun_na5914473681949881652_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(P2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P))
     => ( ! [X: nat,Y: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P2,Y),X)
           => aa_fun8408151826566693824at_o_o(Q2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X),Y)) )
       => aa_fun8408151826566693824at_o_o(Q2,aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),P)) ) ) ).

% exE_realizer
tff(fact_1402_exE__realizer,axiom,
    ! [P2: fun_int_fun_int_o,P: product_prod_int_int,Q2: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int] :
      ( aa_int_o(aa_int_fun_int_o(P2,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P))
     => ( ! [X: int,Y: int] :
            ( aa_int_o(aa_int_fun_int_o(P2,Y),X)
           => aa_Pro4592068669816069595_int_o(Q2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X),Y)) )
       => aa_Pro4592068669816069595_int_o(Q2,aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),P)) ) ) ).

% exE_realizer
tff(fact_1403_exE__realizer,axiom,
    ! [P2: fun_int_fun_int_o,P: product_prod_int_int,Q2: fun_o_o,F: fun_int_fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(P2,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P))
     => ( ! [X: int,Y: int] :
            ( aa_int_o(aa_int_fun_int_o(P2,Y),X)
           => aa_o_o(Q2,aa_int_o(aa_int_fun_int_o(F,X),Y)) )
       => aa_o_o(Q2,aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),P)) ) ) ).

% exE_realizer
tff(fact_1404_exE__realizer,axiom,
    ! [P2: fun_int_fun_int_o,P: product_prod_int_int,Q2: fun_int_o,F: fun_int_fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P2,aa_Pro6799258434465214121nt_int(product_snd_int_int,P)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P))
     => ( ! [X: int,Y: int] :
            ( aa_int_o(aa_int_fun_int_o(P2,Y),X)
           => aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(F,X),Y)) )
       => aa_int_o(Q2,aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),P)) ) ) ).

% exE_realizer
tff(fact_1405_case__prod__beta_H,axiom,
    ! [F: fun_na1668100129766719005at_nat,X5: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),X5) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X5)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X5)) ).

% case_prod_beta'
tff(fact_1406_case__prod__beta_H,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X5: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),X5) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X5)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X5)) ).

% case_prod_beta'
tff(fact_1407_case__prod__beta_H,axiom,
    ! [F: fun_in1696471567231710611nt_int,X5: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),X5) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X5)) ).

% case_prod_beta'
tff(fact_1408_case__prod__beta_H,axiom,
    ! [F: fun_int_fun_int_o,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),X5)
    <=> aa_int_o(aa_int_fun_int_o(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X5)) ) ).

% case_prod_beta'
tff(fact_1409_case__prod__beta_H,axiom,
    ! [F: fun_int_fun_int_int,X5: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),X5) = aa_int_int(aa_int_fun_int_int(F,aa_Pro6799258434465214121nt_int(product_fst_int_int,X5)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X5)) ).

% case_prod_beta'
tff(fact_1410_case__prod__unfold,axiom,
    ! [X5: fun_na1668100129766719005at_nat,Xa4: product_prod_nat_nat] : aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,X5),Xa4) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(X5,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa4)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa4)) ).

% case_prod_unfold
tff(fact_1411_case__prod__unfold,axiom,
    ! [X5: fun_na5914473681949881652_nat_o,Xa4: product_prod_nat_nat] : aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,X5),Xa4) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(X5,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa4)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa4)) ).

% case_prod_unfold
tff(fact_1412_case__prod__unfold,axiom,
    ! [X5: fun_in1696471567231710611nt_int,Xa4: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,X5),Xa4) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(X5,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa4)) ).

% case_prod_unfold
tff(fact_1413_case__prod__unfold,axiom,
    ! [X5: fun_int_fun_int_o,Xa4: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,X5),Xa4)
    <=> aa_int_o(aa_int_fun_int_o(X5,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa4)) ) ).

% case_prod_unfold
tff(fact_1414_case__prod__unfold,axiom,
    ! [X5: fun_int_fun_int_int,Xa4: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,X5),Xa4) = aa_int_int(aa_int_fun_int_int(X5,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa4)) ).

% case_prod_unfold
tff(fact_1415_pred__subset__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat] :
      ( ord_le6814556925368771278_nat_o(aa_set4399942904220837859_nat_o(aTP_Lamm_ep,R2),aa_set4399942904220837859_nat_o(aTP_Lamm_ep,S3))
    <=> ord_le5997549366648089703at_nat(R2,S3) ) ).

% pred_subset_eq2
tff(fact_1416_pred__subset__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ord_le5606610691101069006_nat_o(aa_set5605188905385065315_nat_o(aTP_Lamm_eq,R2),aa_set5605188905385065315_nat_o(aTP_Lamm_eq,S3))
    <=> aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,R2),S3) ) ).

% pred_subset_eq2
tff(fact_1417_pred__subset__eq2,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aa_set108017733112466993_num_o(aTP_Lamm_er,R2),aa_set108017733112466993_num_o(aTP_Lamm_er,S3))
    <=> ord_le8085105155179020875at_num(R2,S3) ) ).

% pred_subset_eq2
tff(fact_1418_pred__subset__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aa_set2634951030311766213_nat_o(aTP_Lamm_es,R2),aa_set2634951030311766213_nat_o(aTP_Lamm_es,S3))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,R2),S3) ) ).

% pred_subset_eq2
tff(fact_1419_pred__subset__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aa_set742265667465749333_int_o(aTP_Lamm_et,R2),aa_set742265667465749333_int_o(aTP_Lamm_et,S3))
    <=> ord_le2843351958646193337nt_int(R2,S3) ) ).

% pred_subset_eq2
tff(fact_1420_comp__fun__commute__on__def,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,F: fun_Pr7278371387837724055at_nat] :
      ( finite8619596206551965927at_nat(S3,F)
    <=> ! [X4: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),S3)
         => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y4),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Y4)),aa_Pro8793452203445305672at_nat(F,X4)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,X4)),aa_Pro8793452203445305672at_nat(F,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_1421_comp__fun__commute__on__def,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: fun_Pr7570940996112156337at_nat] :
      ( finite2110315540964234241at_nat(S3,F)
    <=> ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),S3)
         => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y4),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Y4)),aa_Pro4462963215392351842at_nat(F,X4)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,X4)),aa_Pro4462963215392351842at_nat(F,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_1422_comp__fun__commute__on__def,axiom,
    ! [S3: set_o,F: fun_o_fun_nat_nat] :
      ( finite4703585013658510744_o_nat(S3,F)
    <=> ! [X4: $o,Y4: $o] :
          ( aa_set_o_o(member_o((X4)),S3)
         => ( aa_set_o_o(member_o((Y4)),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Y4))),aa_o_fun_nat_nat(F,(X4))) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(X4))),aa_o_fun_nat_nat(F,(Y4))) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_1423_comp__fun__commute__on__def,axiom,
    ! [S3: set_nat,F: fun_nat_fun_nat_nat] :
      ( finite3582905537739598962at_nat(S3,F)
    <=> ! [X4: nat,Y4: nat] :
          ( aa_set_nat_o(member_nat(X4),S3)
         => ( aa_set_nat_o(member_nat(Y4),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Y4)),aa_nat_fun_nat_nat(F,X4)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,X4)),aa_nat_fun_nat_nat(F,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_1424_comp__fun__commute__on__def,axiom,
    ! [S3: set_int,F: fun_int_fun_nat_nat] :
      ( finite4582259249640653518nt_nat(S3,F)
    <=> ! [X4: int,Y4: int] :
          ( aa_set_int_o(member_int(X4),S3)
         => ( aa_set_int_o(member_int(Y4),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Y4)),aa_int_fun_nat_nat(F,X4)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,X4)),aa_int_fun_nat_nat(F,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_1425_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,F: fun_Pr7278371387837724055at_nat,Xa: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
      ( finite8619596206551965927at_nat(S3,F)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),S3)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Y2)),aa_Pro8793452203445305672at_nat(F,Xa)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Xa)),aa_Pro8793452203445305672at_nat(F,Y2)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_1426_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: fun_Pr7570940996112156337at_nat,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( finite2110315540964234241at_nat(S3,F)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),S3)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Y2)),aa_Pro4462963215392351842at_nat(F,Xa)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Xa)),aa_Pro4462963215392351842at_nat(F,Y2)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_1427_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [S3: set_o,F: fun_o_fun_nat_nat,Xa: $o,Y2: $o] :
      ( finite4703585013658510744_o_nat(S3,F)
     => ( aa_set_o_o(member_o((Xa)),S3)
       => ( aa_set_o_o(member_o((Y2)),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Y2))),aa_o_fun_nat_nat(F,(Xa))) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Xa))),aa_o_fun_nat_nat(F,(Y2))) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_1428_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [S3: set_nat,F: fun_nat_fun_nat_nat,Xa: nat,Y2: nat] :
      ( finite3582905537739598962at_nat(S3,F)
     => ( aa_set_nat_o(member_nat(Xa),S3)
       => ( aa_set_nat_o(member_nat(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Y2)),aa_nat_fun_nat_nat(F,Xa)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Xa)),aa_nat_fun_nat_nat(F,Y2)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_1429_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [S3: set_int,F: fun_int_fun_nat_nat,Xa: int,Y2: int] :
      ( finite4582259249640653518nt_nat(S3,F)
     => ( aa_set_int_o(member_int(Xa),S3)
       => ( aa_set_int_o(member_int(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Y2)),aa_int_fun_nat_nat(F,Xa)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Xa)),aa_int_fun_nat_nat(F,Y2)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_1430_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,F: fun_Pr2206598640620379197un_o_o,Xa: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat,G: fun_nat_o] :
      ( finite2818816984019561153_nat_o(S3,F)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),S3)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y2),S3)
         => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro8665576620106447086un_o_o(F,Y2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro8665576620106447086un_o_o(F,Xa)),G)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro8665576620106447086un_o_o(F,Xa)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro8665576620106447086un_o_o(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1431_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,F: fun_Pr7278371387837724055at_nat,Xa: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat,G: fun_nat_nat] :
      ( finite8619596206551965927at_nat(S3,F)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),S3)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Y2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Xa)),G)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Xa)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1432_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: fun_Pr4929039660471435863un_o_o,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat,G: fun_nat_o] :
      ( finite1496024664758132007_nat_o(S3,F)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),S3)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),S3)
         => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro7088692872332817160un_o_o(F,Y2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro7088692872332817160un_o_o(F,Xa)),G)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro7088692872332817160un_o_o(F,Xa)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_Pro7088692872332817160un_o_o(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1433_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: fun_Pr7570940996112156337at_nat,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat,G: fun_nat_nat] :
      ( finite2110315540964234241at_nat(S3,F)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),S3)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Y2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Xa)),G)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Xa)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1434_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_o,F: fun_o_fun_o_o,Xa: $o,Y2: $o,G: fun_nat_o] :
      ( finite7905178416053733328on_o_o(S3,F)
     => ( aa_set_o_o(member_o((Xa)),S3)
       => ( aa_set_o_o(member_o((Y2)),S3)
         => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_o_fun_o_o(F,(Y2))),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_o_fun_o_o(F,(Xa))),G)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_o_fun_o_o(F,(Xa))),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_o_fun_o_o(F,(Y2))),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1435_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_o,F: fun_o_fun_nat_nat,Xa: $o,Y2: $o,G: fun_nat_nat] :
      ( finite4703585013658510744_o_nat(S3,F)
     => ( aa_set_o_o(member_o((Xa)),S3)
       => ( aa_set_o_o(member_o((Y2)),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Y2))),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Xa))),G)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Xa))),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Y2))),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1436_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_nat,F: fun_nat_fun_o_o,Xa: nat,Y2: nat,G: fun_nat_o] :
      ( finite7806256374766696054_nat_o(S3,F)
     => ( aa_set_nat_o(member_nat(Xa),S3)
       => ( aa_set_nat_o(member_nat(Y2),S3)
         => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_nat_fun_o_o(F,Y2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_nat_fun_o_o(F,Xa)),G)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_nat_fun_o_o(F,Xa)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_nat_fun_o_o(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1437_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_nat,F: fun_nat_fun_nat_nat,Xa: nat,Y2: nat,G: fun_nat_nat] :
      ( finite3582905537739598962at_nat(S3,F)
     => ( aa_set_nat_o(member_nat(Xa),S3)
       => ( aa_set_nat_o(member_nat(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Y2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Xa)),G)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Xa)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1438_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_int,F: fun_int_fun_o_o,Xa: int,Y2: int,G: fun_nat_o] :
      ( finite5985214029334008090_int_o(S3,F)
     => ( aa_set_int_o(member_int(Xa),S3)
       => ( aa_set_int_o(member_int(Y2),S3)
         => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_int_fun_o_o(F,Y2)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_int_fun_o_o(F,Xa)),G)) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_int_fun_o_o(F,Xa)),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_int_fun_o_o(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1439_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [S3: set_int,F: fun_int_fun_nat_nat,Xa: int,Y2: int,G: fun_nat_nat] :
      ( finite4582259249640653518nt_nat(S3,F)
     => ( aa_set_int_o(member_int(Xa),S3)
       => ( aa_set_int_o(member_int(Y2),S3)
         => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Y2)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Xa)),G)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Xa)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Y2)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_1440_comp__fun__commute__on_Ointro,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,F: fun_Pr7278371387837724055at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),S3)
         => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Y)),aa_Pro8793452203445305672at_nat(F,X)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,X)),aa_Pro8793452203445305672at_nat(F,Y)) ) ) )
     => finite8619596206551965927at_nat(S3,F) ) ).

% comp_fun_commute_on.intro
tff(fact_1441_comp__fun__commute__on_Ointro,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: fun_Pr7570940996112156337at_nat] :
      ( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),S3)
         => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Y)),aa_Pro4462963215392351842at_nat(F,X)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,X)),aa_Pro4462963215392351842at_nat(F,Y)) ) ) )
     => finite2110315540964234241at_nat(S3,F) ) ).

% comp_fun_commute_on.intro
tff(fact_1442_comp__fun__commute__on_Ointro,axiom,
    ! [S3: set_o,F: fun_o_fun_nat_nat] :
      ( ! [X: $o,Y: $o] :
          ( aa_set_o_o(member_o((X)),S3)
         => ( aa_set_o_o(member_o((Y)),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Y))),aa_o_fun_nat_nat(F,(X))) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(X))),aa_o_fun_nat_nat(F,(Y))) ) ) )
     => finite4703585013658510744_o_nat(S3,F) ) ).

% comp_fun_commute_on.intro
tff(fact_1443_comp__fun__commute__on_Ointro,axiom,
    ! [S3: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [X: nat,Y: nat] :
          ( aa_set_nat_o(member_nat(X),S3)
         => ( aa_set_nat_o(member_nat(Y),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Y)),aa_nat_fun_nat_nat(F,X)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,X)),aa_nat_fun_nat_nat(F,Y)) ) ) )
     => finite3582905537739598962at_nat(S3,F) ) ).

% comp_fun_commute_on.intro
tff(fact_1444_comp__fun__commute__on_Ointro,axiom,
    ! [S3: set_int,F: fun_int_fun_nat_nat] :
      ( ! [X: int,Y: int] :
          ( aa_set_int_o(member_int(X),S3)
         => ( aa_set_int_o(member_int(Y),S3)
           => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Y)),aa_int_fun_nat_nat(F,X)) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,X)),aa_int_fun_nat_nat(F,Y)) ) ) )
     => finite4582259249640653518nt_nat(S3,F) ) ).

% comp_fun_commute_on.intro
tff(fact_1445_case__prod__o__map__prod,axiom,
    ! [F: fun_na1668100129766719005at_nat,G1: fun_nat_nat,G22: fun_nat_nat] : comp_P1264855332011520990at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,F),produc6977886695330630970at_nat(G1,G22)) = aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aa_fun5672751633241614456at_nat(aa_fun6003775592413651218at_nat(aTP_Lamm_dx(F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_1446_case__prod__o__map__prod,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G1: fun_nat_nat,G22: fun_nat_nat] : comp_P1901131479461194247at_nat(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,F),produc6977886695330630970at_nat(G1,G22)) = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aa_fun5827507165208764047_nat_o(aa_fun83116112763158185_nat_o(aTP_Lamm_dy(F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_1447_case__prod__o__map__prod,axiom,
    ! [F: fun_in1696471567231710611nt_int,G1: fun_int_int,G22: fun_int_int] : comp_P2788356708182965334nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,F),produc6036585564866528938nt_int(G1,G22)) = aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_fun4035196674708996672nt_int(aa_fun2536542299119233284nt_int(aTP_Lamm_dz(F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_1448_case__prod__o__map__prod,axiom,
    ! [F: fun_int_fun_int_o,G1: fun_int_int,G22: fun_int_int] : comp_P9034173210191959223nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,F),produc6036585564866528938nt_int(G1,G22)) = aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun7896090258416335903_int_o(aa_fun8092296977475930211_int_o(aTP_Lamm_ea(F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_1449_case__prod__o__map__prod,axiom,
    ! [F: fun_int_fun_int_int,G1: fun_int_int,G22: fun_int_int] : comp_P5356232565778667509nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,F),produc6036585564866528938nt_int(G1,G22)) = aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aa_fun5911324870756489105nt_int(aa_fun8505558176664197859nt_int(aTP_Lamm_eb(F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_1450_vimage2p__relcompp__converse,axiom,
    ! [Rep: fun_Pr2118491707810114563at_nat,Abs: fun_Pr2118491707810114563at_nat,F: fun_Pr2118491707810114563at_nat,G: fun_int_int,R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o] :
      ( aa_set3702741264012733747_nat_o(type_d281131942595027436at_nat(Rep,Abs),top_to4669805908274784177at_nat)
     => ( bNF_vi5426221348228882691_int_o(F,G,relcom8911704206367525917at_int(conver6965824823119931701at_nat(R2),S3)) = relcom8911704206367525917at_int(conver6965824823119931701at_nat(bNF_vi763171854521451549_nat_o(Rep,F,R2)),bNF_vi5426221348228882691_int_o(Rep,G,S3)) ) ) ).

% vimage2p_relcompp_converse
tff(fact_1451_vimage2p__relcompp__converse,axiom,
    ! [Rep: fun_in4514472496968388910at_nat,Abs: fun_Pr7647638413817135296at_int,F: fun_in4514472496968388910at_nat,G: fun_Code_integer_int,R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o] :
      ( aa_set3702741264012733747_nat_o(type_d7732240953928457511at_nat(Rep,Abs),top_to4669805908274784177at_nat)
     => ( bNF_vi2881549544236934259_int_o(F,G,relcom8911704206367525917at_int(conver6965824823119931701at_nat(R2),S3)) = relcom967497705283160040nteger(conversep_int_int(bNF_vi1204582724742824157_nat_o(Rep,F,R2)),bNF_vi2881549544236934259_int_o(Rep,G,S3)) ) ) ).

% vimage2p_relcompp_converse
tff(fact_1452_vimage2p__relcompp__converse,axiom,
    ! [Rep: fun_Pr7647638413817135296at_int,Abs: fun_in4514472496968388910at_nat,F: fun_Pr7647638413817135296at_int,G: fun_int_Code_integer,R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o] :
      ( aa_set_int_o(type_d1331861005912558337at_int(Rep,Abs),top_top_set_int)
     => ( bNF_vi33923956287962137eger_o(F,G,relcom967497705283160040nteger(conversep_int_int(R2),S3)) = relcom8911704206367525917at_int(conver6965824823119931701at_nat(bNF_vi2682369418337388893_int_o(Rep,F,R2)),bNF_vi33923956287962137eger_o(Rep,G,S3)) ) ) ).

% vimage2p_relcompp_converse
tff(fact_1453_vimage2p__relcompp__converse,axiom,
    ! [Rep: fun_int_int,Abs: fun_int_int,F: fun_int_int,G: fun_Co8499238673658658103nteger,R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o] :
      ( aa_set_int_o(type_d7247357190169752966nt_int(Rep,Abs),top_top_set_int)
     => ( bNF_vi699694178825965831eger_o(F,G,relcom967497705283160040nteger(conversep_int_int(R2),S3)) = relcom967497705283160040nteger(conversep_int_int(bNF_vi3884084463199473129_int_o(Rep,F,R2)),bNF_vi699694178825965831eger_o(Rep,G,S3)) ) ) ).

% vimage2p_relcompp_converse
tff(fact_1454_Id__on__iff,axiom,
    ! [Xa: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( member6009668390874684176at_nat(produc2335810615602371415at_nat(Xa,Y2),id_on_7869290023980668164at_nat(A4))
    <=> ( ( Xa = Y2 )
        & aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),A4) ) ) ).

% Id_on_iff
tff(fact_1455_Id__on__iff,axiom,
    ! [Xa: product_prod_nat_nat,Y2: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),id_on_2554058798563519774at_nat(A4))
    <=> ( ( Xa = Y2 )
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),A4) ) ) ).

% Id_on_iff
tff(fact_1456_Id__on__iff,axiom,
    ! [Xa: $o,Y2: $o,A4: set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa)),(Y2)),id_on_o(A4))
    <=> ( ( (Xa)
        <=> (Y2) )
        & aa_set_o_o(member_o((Xa)),A4) ) ) ).

% Id_on_iff
tff(fact_1457_Id__on__iff,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2),id_on_3275838214495744996at_nat(A4))
    <=> ( ( Xa = Y2 )
        & member6273874620674182256at_nat(Xa,A4) ) ) ).

% Id_on_iff
tff(fact_1458_Id__on__iff,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),id_on_6713504163309144702at_nat(A4))
    <=> ( ( Xa = Y2 )
        & member2643936169264416010at_nat(Xa,A4) ) ) ).

% Id_on_iff
tff(fact_1459_Id__on__iff,axiom,
    ! [Xa: nat,Y2: nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),id_on_nat(A4))
    <=> ( ( Xa = Y2 )
        & aa_set_nat_o(member_nat(Xa),A4) ) ) ).

% Id_on_iff
tff(fact_1460_Id__on__iff,axiom,
    ! [Xa: int,Y2: int,A4: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),id_on_int(A4))
    <=> ( ( Xa = Y2 )
        & aa_set_int_o(member_int(Xa),A4) ) ) ).

% Id_on_iff
tff(fact_1461_Id__on__eqI,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( A = B )
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
       => member6009668390874684176at_nat(produc2335810615602371415at_nat(A,B),id_on_7869290023980668164at_nat(A4)) ) ) ).

% Id_on_eqI
tff(fact_1462_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( A = B )
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),id_on_2554058798563519774at_nat(A4)) ) ) ).

% Id_on_eqI
tff(fact_1463_Id__on__eqI,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( (A)
      <=> (B) )
     => ( aa_set_o_o(member_o((A)),A4)
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),id_on_o(A4)) ) ) ).

% Id_on_eqI
tff(fact_1464_Id__on__eqI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat] :
      ( ( A = B )
     => ( member6273874620674182256at_nat(A,A4)
       => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),id_on_3275838214495744996at_nat(A4)) ) ) ).

% Id_on_eqI
tff(fact_1465_Id__on__eqI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat] :
      ( ( A = B )
     => ( member2643936169264416010at_nat(A,A4)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),id_on_6713504163309144702at_nat(A4)) ) ) ).

% Id_on_eqI
tff(fact_1466_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),id_on_nat(A4)) ) ) ).

% Id_on_eqI
tff(fact_1467_Id__on__eqI,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( A = B )
     => ( aa_set_int_o(member_int(A),A4)
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),id_on_int(A4)) ) ) ).

% Id_on_eqI
tff(fact_1468_Id__onE,axiom,
    ! [C: produc1906917888121415911at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( member6009668390874684176at_nat(C,id_on_7869290023980668164at_nat(A4))
     => ~ ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => ( C != produc2335810615602371415at_nat(X,X) ) ) ) ).

% Id_onE
tff(fact_1469_Id__onE,axiom,
    ! [C: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( member8206827879206165904at_nat(C,id_on_2554058798563519774at_nat(A4))
     => ~ ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => ( C != aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X),X) ) ) ) ).

% Id_onE
tff(fact_1470_Id__onE,axiom,
    ! [C: product_prod_o_o,A4: set_o] :
      ( member7466972457876170832od_o_o(C,id_on_o(A4))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( C != aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(X)) ) ) ) ).

% Id_onE
tff(fact_1471_Id__onE,axiom,
    ! [C: produc1319942482725812455at_nat,A4: set_se5479201105039950375at_nat] :
      ( member1466754251312161552at_nat(C,id_on_3275838214495744996at_nat(A4))
     => ~ ! [X: set_Pr4329608150637261639at_nat] :
            ( member6273874620674182256at_nat(X,A4)
           => ( C != aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),X) ) ) ) ).

% Id_onE
tff(fact_1472_Id__onE,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(C),id_on_6713504163309144702at_nat(A4))
     => ~ ! [X: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(X,A4)
           => ( C != aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),X) ) ) ) ).

% Id_onE
tff(fact_1473_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),id_on_nat(A4))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ( C != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),X) ) ) ) ).

% Id_onE
tff(fact_1474_Id__onE,axiom,
    ! [C: product_prod_int_int,A4: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(C),id_on_int(A4))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( C != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),X) ) ) ) ).

% Id_onE
tff(fact_1475_predicate2__transferD,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o,P2: fun_nat_fun_nat_o,Q2: fun_nat_fun_nat_o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( bNF_re1448341924010609215_nat_o(R1,bNF_re4705727531993890431at_o_o(R22,fequal_o),P2,Q2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),B5)
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R1)))
           => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R22)))
             => ( aa_nat_o(aa_nat_fun_nat_o(P2,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,B))
              <=> aa_nat_o(aa_nat_fun_nat_o(Q2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1476_predicate2__transferD,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_int_fun_int_o,P2: fun_nat_fun_int_o,Q2: fun_nat_fun_int_o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
      ( bNF_re1754666308579983679_int_o(R1,bNF_re5089333283451836215nt_o_o(R22,fequal_o),P2,Q2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
       => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(B),B5)
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R1)))
           => ( ord_le2843351958646193337nt_int(B5,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R22)))
             => ( aa_int_o(aa_nat_fun_int_o(P2,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A)),aa_Pro6799258434465214121nt_int(product_fst_int_int,B))
              <=> aa_int_o(aa_nat_fun_int_o(Q2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A)),aa_Pro6799258434465214121nt_int(product_snd_int_int,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1477_predicate2__transferD,axiom,
    ! [R1: fun_int_fun_int_o,R22: fun_nat_fun_nat_o,P2: fun_int_fun_nat_o,Q2: fun_int_fun_nat_o,A: product_prod_int_int,A4: set_Pr958786334691620121nt_int,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( bNF_re8347998229424194295_nat_o(R1,bNF_re4705727531993890431at_o_o(R22,fequal_o),P2,Q2)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(A),A4)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),B5)
         => ( ord_le2843351958646193337nt_int(A4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R1)))
           => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R22)))
             => ( aa_nat_o(aa_int_fun_nat_o(P2,aa_Pro6799258434465214121nt_int(product_fst_int_int,A)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,B))
              <=> aa_nat_o(aa_int_fun_nat_o(Q2,aa_Pro6799258434465214121nt_int(product_snd_int_int,A)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1478_predicate2__transferD,axiom,
    ! [R1: fun_int_fun_int_o,R22: fun_int_fun_int_o,P2: fun_int_fun_int_o,Q2: fun_int_fun_int_o,A: product_prod_int_int,A4: set_Pr958786334691620121nt_int,B: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
      ( bNF_re8654322613993568759_int_o(R1,bNF_re5089333283451836215nt_o_o(R22,fequal_o),P2,Q2)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(A),A4)
       => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(B),B5)
         => ( ord_le2843351958646193337nt_int(A4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R1)))
           => ( ord_le2843351958646193337nt_int(B5,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R22)))
             => ( aa_int_o(aa_int_fun_int_o(P2,aa_Pro6799258434465214121nt_int(product_fst_int_int,A)),aa_Pro6799258434465214121nt_int(product_fst_int_int,B))
              <=> aa_int_o(aa_int_fun_int_o(Q2,aa_Pro6799258434465214121nt_int(product_snd_int_int,A)),aa_Pro6799258434465214121nt_int(product_snd_int_int,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1479_predicate2__transferD,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_re5167877524888533480unit_o,P2: fun_nat_fun_ref_a_o,Q2: fun_na4670164693588179099unit_o,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: produc7812382424243267045t_unit,B5: set_Pr1133966238863718811t_unit] :
      ( bNF_re8330881586764134179unit_o(R1,bNF_re1850668159662942673it_o_o(R22,fequal_o),P2,Q2)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
       => ( member7849944260330398460t_unit(B,B5)
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R1)))
           => ( ord_le5525191956872776187t_unit(B5,collec3252306575900995386t_unit(produc4924168586796184916unit_o(R22)))
             => ( aa_ref_a_o(aa_nat_fun_ref_a_o(P2,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,A)),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,B))
              <=> aa_hea7994211679605476167unit_o(aa_nat3704611491654317276unit_o(Q2,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,A)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1480_predicate2__transferD,axiom,
    ! [R1: fun_re5167877524888533480unit_o,R22: fun_nat_fun_nat_o,P2: fun_ref_a_fun_nat_o,Q2: fun_he4135464443191253079_nat_o,A: produc7812382424243267045t_unit,A4: set_Pr1133966238863718811t_unit,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( bNF_re5614068985792486801_nat_o(R1,bNF_re4705727531993890431at_o_o(R22,fequal_o),P2,Q2)
     => ( member7849944260330398460t_unit(A,A4)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),B5)
         => ( ord_le5525191956872776187t_unit(A4,collec3252306575900995386t_unit(produc4924168586796184916unit_o(R1)))
           => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R22)))
             => ( aa_nat_o(aa_ref_a_fun_nat_o(P2,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,A)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,B))
              <=> aa_nat_o(aa_hea9023553427033036112_nat_o(Q2,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,A)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1481_predicate2__transferD,axiom,
    ! [R1: fun_re5167877524888533480unit_o,R22: fun_int_fun_int_o,P2: fun_ref_a_fun_int_o,Q2: fun_he1890319475520807163_int_o,A: produc7812382424243267045t_unit,A4: set_Pr1133966238863718811t_unit,B: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
      ( bNF_re5920393370361861265_int_o(R1,bNF_re5089333283451836215nt_o_o(R22,fequal_o),P2,Q2)
     => ( member7849944260330398460t_unit(A,A4)
       => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(B),B5)
         => ( ord_le5525191956872776187t_unit(A4,collec3252306575900995386t_unit(produc4924168586796184916unit_o(R1)))
           => ( ord_le2843351958646193337nt_int(B5,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R22)))
             => ( aa_int_o(aa_ref_a_fun_int_o(P2,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,A)),aa_Pro6799258434465214121nt_int(product_fst_int_int,B))
              <=> aa_int_o(aa_hea7202511081600348148_int_o(Q2,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,A)),aa_Pro6799258434465214121nt_int(product_snd_int_int,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1482_predicate2__transferD,axiom,
    ! [R1: fun_int_fun_int_o,R22: fun_re5167877524888533480unit_o,P2: fun_int_fun_ref_a_o,Q2: fun_in3144112885846485751unit_o,A: product_prod_int_int,A4: set_Pr958786334691620121nt_int,B: produc7812382424243267045t_unit,B5: set_Pr1133966238863718811t_unit] :
      ( bNF_re6262881856644161643unit_o(R1,bNF_re1850668159662942673it_o_o(R22,fequal_o),P2,Q2)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(A),A4)
       => ( member7849944260330398460t_unit(B,B5)
         => ( ord_le2843351958646193337nt_int(A4,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R1)))
           => ( ord_le5525191956872776187t_unit(B5,collec3252306575900995386t_unit(produc4924168586796184916unit_o(R22)))
             => ( aa_ref_a_o(aa_int_fun_ref_a_o(P2,aa_Pro6799258434465214121nt_int(product_fst_int_int,A)),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,B))
              <=> aa_hea7994211679605476167unit_o(aa_int2178559683912623928unit_o(Q2,aa_Pro6799258434465214121nt_int(product_snd_int_int,A)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1483_predicate2__transferD,axiom,
    ! [R1: fun_Pr4738963024209204295_int_o,R22: fun_Pr4738963024209204295_int_o,P2: fun_Pr2028404973800837934_nat_o,Q2: fun_int_fun_int_o,A: produc4196048018000912732at_int,A4: set_Pr49084382684228242at_int,B: produc4196048018000912732at_int,B5: set_Pr49084382684228242at_int] :
      ( bNF_re4195927319312321177_int_o(R1,bNF_re6644619430987730960nt_o_o(R22,fequal_o),P2,Q2)
     => ( member8394280151737582451at_int(A,A4)
       => ( member8394280151737582451at_int(B,B5)
         => ( ord_le4604737605482584818at_int(A4,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,R1)))
           => ( ord_le4604737605482584818at_int(B5,collec2851339944984316337at_int(aa_fun5928862226927483600_int_o(produc6937227050501860053_int_o,R22)))
             => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(P2,aa_Pro8788825115640893031at_nat(produc4619704898428564714at_int,A)),aa_Pro8788825115640893031at_nat(produc4619704898428564714at_int,B))
              <=> aa_int_o(aa_int_fun_int_o(Q2,aa_Pro8273224204977640518nt_int(produc3064508320661873192at_int,A)),aa_Pro8273224204977640518nt_int(produc3064508320661873192at_int,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1484_predicate2__transferD,axiom,
    ! [R1: fun_re5167877524888533480unit_o,R22: fun_re5167877524888533480unit_o,P2: fun_re1985804342488220176ef_a_o,Q2: fun_he3009834522632631726unit_o,A: produc7812382424243267045t_unit,A4: set_Pr1133966238863718811t_unit,B: produc7812382424243267045t_unit,B5: set_Pr1133966238863718811t_unit] :
      ( bNF_re8225015301715176273unit_o(R1,bNF_re1850668159662942673it_o_o(R22,fequal_o),P2,Q2)
     => ( member7849944260330398460t_unit(A,A4)
       => ( member7849944260330398460t_unit(B,B5)
         => ( ord_le5525191956872776187t_unit(A4,collec3252306575900995386t_unit(produc4924168586796184916unit_o(R1)))
           => ( ord_le5525191956872776187t_unit(B5,collec3252306575900995386t_unit(produc4924168586796184916unit_o(R22)))
             => ( aa_ref_a_o(aa_ref_a_fun_ref_a_o(P2,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,A)),aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,B))
              <=> aa_hea7994211679605476167unit_o(aa_hea3972832001280910815unit_o(Q2,aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,A)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,B)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_1485_fst__diag__id,axiom,
    ! [Z: $o] :
      ( aa_o_o(comp_P2137996956111167876_o_o_o(product_fst_o_o,aTP_Lamm_eu),(Z))
    <=> aa_o_o(id_o,(Z)) ) ).

% fst_diag_id
tff(fact_1486_fst__diag__id,axiom,
    ! [Z: num] : aa_num_num(comp_P864883071962432936um_num(product_fst_num_num,aTP_Lamm_ev),Z) = aa_num_num(id_num,Z) ).

% fst_diag_id
tff(fact_1487_fst__diag__id,axiom,
    ! [Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(comp_P8235817900121468292at_nat(produc4486455499028916547at_nat,aTP_Lamm_ew),Z) = aa_set4883462369968598652at_nat(id_set7849613212452405432at_nat,Z) ).

% fst_diag_id
tff(fact_1488_fst__diag__id,axiom,
    ! [Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(comp_P4528037747288561284at_nat(produc6011863565407846595at_nat,aTP_Lamm_ex),Z) = aa_set8868280134176360700at_nat(id_set2334057505560298834at_nat,Z) ).

% fst_diag_id
tff(fact_1489_fst__diag__id,axiom,
    ! [Z: nat] : aa_nat_nat(aa_fun9030085588633777131at_nat(comp_P1955610211387027840at_nat(product_fst_nat_nat),aTP_Lamm_ey),Z) = aa_nat_nat(id_nat,Z) ).

% fst_diag_id
tff(fact_1490_fst__diag__id,axiom,
    ! [Z: int] : aa_int_int(aa_fun1140073774819664895nt_int(comp_P6651465171681656048nt_int(product_fst_int_int),aTP_Lamm_ez),Z) = aa_int_int(id_int,Z) ).

% fst_diag_id
tff(fact_1491_snd__diag__id,axiom,
    ! [Z: $o] :
      ( aa_o_o(comp_P2137996956111167876_o_o_o(product_snd_o_o,aTP_Lamm_eu),(Z))
    <=> aa_o_o(id_o,(Z)) ) ).

% snd_diag_id
tff(fact_1492_snd__diag__id,axiom,
    ! [Z: num] : aa_num_num(comp_P864883071962432936um_num(product_snd_num_num,aTP_Lamm_ev),Z) = aa_num_num(id_num,Z) ).

% snd_diag_id
tff(fact_1493_snd__diag__id,axiom,
    ! [Z: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(comp_P8235817900121468292at_nat(produc826222791318859909at_nat,aTP_Lamm_ew),Z) = aa_set4883462369968598652at_nat(id_set7849613212452405432at_nat,Z) ).

% snd_diag_id
tff(fact_1494_snd__diag__id,axiom,
    ! [Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(comp_P4528037747288561284at_nat(produc2937545259423399429at_nat,aTP_Lamm_ex),Z) = aa_set8868280134176360700at_nat(id_set2334057505560298834at_nat,Z) ).

% snd_diag_id
tff(fact_1495_snd__diag__id,axiom,
    ! [Z: nat] : aa_nat_nat(aa_fun9030085588633777131at_nat(comp_P1955610211387027840at_nat(product_snd_nat_nat),aTP_Lamm_ey),Z) = aa_nat_nat(id_nat,Z) ).

% snd_diag_id
tff(fact_1496_snd__diag__id,axiom,
    ! [Z: int] : aa_int_int(aa_fun1140073774819664895nt_int(comp_P6651465171681656048nt_int(product_snd_int_int),aTP_Lamm_ez),Z) = aa_int_int(id_int,Z) ).

% snd_diag_id
tff(fact_1497_case__prod__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: produc3346328345660746739nteger] : produc4281629032016041417nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Xa) = aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(F,aa_Cod4161879620478718930nteger(G,produc1861665112190112313nteger(Xa))),produc7955729907432593271nteger(Xa)) ).

% case_prod_comp
tff(fact_1498_lookup__def,axiom,
    ! [Ra: ref_a] : lookup_a(Ra) = heap_tap_a(aTP_Lamm_fa(Ra)) ).

% lookup_def
tff(fact_1499_fst__snd__flip,axiom,
    ! [Xy: produc1319942482725812455at_nat] : aa_Pro6157208134662059484at_nat(produc4486455499028916547at_nat,Xy) = comp_P3728553842126764068at_nat(produc826222791318859909at_nat,produc5456455677105495163at_nat(aTP_Lamm_ec),Xy) ).

% fst_snd_flip
tff(fact_1500_fst__snd__flip,axiom,
    ! [Xy: produc3843707927480180839at_nat] : aa_Pro441628658106787830at_nat(produc6011863565407846595at_nat,Xy) = aa_Pro441628658106787830at_nat(comp_P4974406650942564362at_nat(produc2937545259423399429at_nat,produc7548845685434208379at_nat(aTP_Lamm_ed)),Xy) ).

% fst_snd_flip
tff(fact_1501_fst__snd__flip,axiom,
    ! [Xy: product_prod_num_nat] : product_fst_num_nat(Xy) = comp_P5051845089628979463um_nat(product_snd_nat_num,produc5219324331026901427at_num(aTP_Lamm_ee),Xy) ).

% fst_snd_flip
tff(fact_1502_fst__snd__flip,axiom,
    ! [Xy: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xy) = aa_Pro4243173770044397973at_nat(comp_P1567445206330693457at_nat(product_snd_nat_nat,produc2626176000494625587at_nat(aTP_Lamm_ef)),Xy) ).

% fst_snd_flip
tff(fact_1503_fst__snd__flip,axiom,
    ! [Xy: produc7812382424243267045t_unit] : aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Xy) = comp_P1287623958910499840t_unit(produc6369869760173755749_ref_a,produc2058451029528183389_ref_a(aTP_Lamm_fb),Xy) ).

% fst_snd_flip
tff(fact_1504_fst__snd__flip,axiom,
    ! [Xy: produc1799199577659733785_ref_a] : aa_Pro7100128628477207734t_unit(produc5717040794327095847_ref_a,Xy) = comp_P2006160480335490974_ref_a(produc1126150082675451753t_unit,produc4426278547552487717t_unit(aTP_Lamm_fc),Xy) ).

% fst_snd_flip
tff(fact_1505_fst__snd__flip,axiom,
    ! [Xy: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Xy) = aa_Pro6799258434465214121nt_int(comp_P5356232565778667509nt_int(product_snd_int_int,aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_eg)),Xy) ).

% fst_snd_flip
tff(fact_1506_snd__fst__flip,axiom,
    ! [Xy: produc1319942482725812455at_nat] : aa_Pro6157208134662059484at_nat(produc826222791318859909at_nat,Xy) = comp_P3728553842126764068at_nat(produc4486455499028916547at_nat,produc5456455677105495163at_nat(aTP_Lamm_ec),Xy) ).

% snd_fst_flip
tff(fact_1507_snd__fst__flip,axiom,
    ! [Xy: produc3843707927480180839at_nat] : aa_Pro441628658106787830at_nat(produc2937545259423399429at_nat,Xy) = aa_Pro441628658106787830at_nat(comp_P4974406650942564362at_nat(produc6011863565407846595at_nat,produc7548845685434208379at_nat(aTP_Lamm_ed)),Xy) ).

% snd_fst_flip
tff(fact_1508_snd__fst__flip,axiom,
    ! [Xy: product_prod_num_nat] : product_snd_num_nat(Xy) = comp_P8336958949797789649um_nat(product_fst_nat_num,produc5219324331026901427at_num(aTP_Lamm_ee),Xy) ).

% snd_fst_flip
tff(fact_1509_snd__fst__flip,axiom,
    ! [Xy: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xy) = aa_Pro4243173770044397973at_nat(comp_P1567445206330693457at_nat(product_fst_nat_nat,produc2626176000494625587at_nat(aTP_Lamm_ef)),Xy) ).

% snd_fst_flip
tff(fact_1510_snd__fst__flip,axiom,
    ! [Xy: produc1799199577659733785_ref_a] : aa_Pro7573530150736202740_ref_a(produc6369869760173755749_ref_a,Xy) = comp_P7254561687666694144_ref_a(produc473321116828791851t_unit,produc4426278547552487717t_unit(aTP_Lamm_fc),Xy) ).

% snd_fst_flip
tff(fact_1511_snd__fst__flip,axiom,
    ! [Xy: produc7812382424243267045t_unit] : aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Xy) = comp_P3239167524078694454t_unit(produc5717040794327095847_ref_a,produc2058451029528183389_ref_a(aTP_Lamm_fb),Xy) ).

% snd_fst_flip
tff(fact_1512_snd__fst__flip,axiom,
    ! [Xy: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Xy) = aa_Pro6799258434465214121nt_int(comp_P5356232565778667509nt_int(product_fst_int_int,aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_eg)),Xy) ).

% snd_fst_flip
tff(fact_1513_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,F: fun_Pr7278371387837724055at_nat,Xa: produc3843707927480180839at_nat] :
      ( finite352855791390415404at_nat(S3,F)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),S3)
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro8793452203445305672at_nat(F,Xa)),aa_Pro8793452203445305672at_nat(F,Xa)) = aa_Pro8793452203445305672at_nat(F,Xa) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_1514_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: fun_Pr7570940996112156337at_nat,Xa: product_prod_nat_nat] :
      ( finite5150387867909132486at_nat(S3,F)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),S3)
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_Pro4462963215392351842at_nat(F,Xa)),aa_Pro4462963215392351842at_nat(F,Xa)) = aa_Pro4462963215392351842at_nat(F,Xa) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_1515_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [S3: set_o,F: fun_o_fun_nat_nat,Xa: $o] :
      ( finite2426395113773319005_o_nat(S3,F)
     => ( aa_set_o_o(member_o((Xa)),S3)
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_o_fun_nat_nat(F,(Xa))),aa_o_fun_nat_nat(F,(Xa))) = aa_o_fun_nat_nat(F,(Xa)) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_1516_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [S3: set_nat,F: fun_nat_fun_nat_nat,Xa: nat] :
      ( finite7982400111564556781at_nat(S3,F)
     => ( aa_set_nat_o(member_nat(Xa),S3)
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(F,Xa)),aa_nat_fun_nat_nat(F,Xa)) = aa_nat_fun_nat_nat(F,Xa) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_1517_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [S3: set_int,F: fun_int_fun_nat_nat,Xa: int] :
      ( finite8981753823465611337nt_nat(S3,F)
     => ( aa_set_int_o(member_int(Xa),S3)
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_int_fun_nat_nat(F,Xa)),aa_int_fun_nat_nat(F,Xa)) = aa_int_fun_nat_nat(F,Xa) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_1518_OO__Grp__cong,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: fun_Pr2118491707810114563at_nat,G: fun_Pr7647638413817135296at_int] :
      ( ( A4 = B5 )
     => ( relcom8911704206367525917at_int(conver6965824823119931701at_nat(bNF_Gr7275288873855648526at_nat(A4,F)),bNF_Gr190934199572118623at_int(A4,G)) = relcom8911704206367525917at_int(conver6965824823119931701at_nat(bNF_Gr7275288873855648526at_nat(B5,F)),bNF_Gr190934199572118623at_int(B5,G)) ) ) ).

% OO_Grp_cong
tff(fact_1519_OO__Grp__cong,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_int,G: fun_int_Code_integer] :
      ( ( A4 = B5 )
     => ( relcom967497705283160040nteger(conversep_int_int(bNF_Grp_int_int(A4,F)),bNF_Gr6684649993275963353nteger(A4,G)) = relcom967497705283160040nteger(conversep_int_int(bNF_Grp_int_int(B5,F)),bNF_Gr6684649993275963353nteger(B5,G)) ) ) ).

% OO_Grp_cong
tff(fact_1520_fst__diag__fst,axiom,
    comp_P9134237804721252988t_unit(produc6942380573175061695_ref_a,comp_r6154141770783548544t_unit(aTP_Lamm_fd,produc473321116828791851t_unit)) = produc473321116828791851t_unit ).

% fst_diag_fst
tff(fact_1521_snd__diag__snd,axiom,
    comp_P8237986404257982958t_unit(produc1692618790031806469t_unit,comp_h5940001671827863714t_unit(aTP_Lamm_fe,produc1126150082675451753t_unit)) = produc1126150082675451753t_unit ).

% snd_diag_snd
tff(fact_1522_type__copy__map__comp0,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,M: fun_nat_nat,M1: fun_o_nat,M2: fun_nat_o,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( ( M = comp_o_nat_nat(M1,M2) )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),M)),G) = comp_o_nat_nat(comp_o_nat_o(comp_nat_nat_o(F,M1),Rep),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1523_type__copy__map__comp0,axiom,
    ! [Rep: fun_int_o,Abs: fun_o_int,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_nat,G: fun_int_nat] :
      ( aa_set_o_o(type_d8403010662121346494_int_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,comp_o_nat_nat(F,M)),G) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_o_nat_int(comp_o_nat_o(F,M1),Rep)),comp_nat_int_int(comp_o_int_nat(Abs,M2),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1524_type__copy__map__comp0,axiom,
    ! [Rep: fun_num_o,Abs: fun_o_num,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_o,G: fun_int_nat] :
      ( aa_set_o_o(type_d9070036042389994064_num_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( comp_nat_o_int(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),M),G) = comp_num_o_int(comp_o_o_num(comp_o_o_o(F,M1),Rep),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs,M2)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1525_type__copy__map__comp0,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_o,G: fun_nat_nat] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),M),G) = comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(F,M1)),Rep),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(Abs,M2)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1526_type__copy__map__comp0,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_o,G: fun_nat_nat] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),M),G) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(comp_o_o_o(F,M1),Rep)),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1527_type__copy__map__comp0,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_num,G: fun_int_nat] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(F,M)),G) = comp_o_num_int(comp_o_num_o(comp_o_num_o(F,M1),Rep),comp_nat_o_int(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1528_type__copy__map__comp0,axiom,
    ! [Rep: fun_num_o,Abs: fun_o_num,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_num,G: fun_int_nat] :
      ( aa_set_o_o(type_d9070036042389994064_num_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(F,M)),G) = aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,comp_o_num_num(comp_o_num_o(F,M1),Rep)),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs,M2)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1529_type__copy__map__comp0,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_num,G: fun_int_nat] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(F,M)),G) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(comp_o_num_o(F,M1),Rep)),aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,comp_o_nat_nat(Abs,M2)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1530_type__copy__map__comp0,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_nat,G: fun_nat_nat] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(F,M)),G) = comp_o_nat_nat(comp_o_nat_o(comp_o_nat_o(F,M1),Rep),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1531_type__copy__map__comp0,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o,F: fun_o_nat,G: fun_nat_nat] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( ( M = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) )
       => ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(F,M)),G) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(comp_o_nat_o(F,M1),Rep)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(Abs,M2)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_1532_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_int_o,Abs: fun_o_int,Rep2: fun_nat_o,Abs2: fun_o_nat,Rep3: fun_int_nat,Abs3: fun_nat_int,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_d8403010662121346494_int_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,comp_o_nat_nat(Abs2,M)),Rep3) = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_o_nat_int(comp_o_nat_o(Abs2,M1),Rep)),comp_nat_int_int(comp_o_int_nat(Abs,M2),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1533_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_num_o,Abs: fun_o_num,Rep2: fun_o_o,Abs2: fun_o_o,Rep3: fun_int_nat,Abs3: fun_nat_int,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_d9070036042389994064_num_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_definition_o_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( comp_nat_o_int(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs2),M),Rep3) = comp_num_o_int(comp_o_o_num(comp_o_o_o(Abs2,M1),Rep),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs,M2)),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1534_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,Rep2: fun_o_o,Abs2: fun_o_o,Rep3: fun_nat_nat,Abs3: fun_nat_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_definition_o_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs2),M),Rep3) = comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(Abs2,M1)),Rep),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(Abs,M2)),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1535_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,Rep2: fun_o_o,Abs2: fun_o_o,Rep3: fun_nat_nat,Abs3: fun_nat_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_definition_o_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs2),M),Rep3) = aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_o_o_o(comp_o_o_o(Abs2,M1),Rep)),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1536_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,Rep2: fun_num_o,Abs2: fun_o_num,Rep3: fun_int_nat,Abs3: fun_nat_int,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_d9070036042389994064_num_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs2,M)),Rep3) = comp_o_num_int(comp_o_num_o(comp_o_num_o(Abs2,M1),Rep),comp_nat_o_int(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1537_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_num_o,Abs: fun_o_num,Rep2: fun_num_o,Abs2: fun_o_num,Rep3: fun_int_nat,Abs3: fun_nat_int,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_d9070036042389994064_num_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_d9070036042389994064_num_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs2,M)),Rep3) = aa_fun4776194614826069188nt_num(aa_fun8775978671473068146nt_num(comp_num_num_int,comp_o_num_num(comp_o_num_o(Abs2,M1),Rep)),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs,M2)),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1538_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,Rep2: fun_num_o,Abs2: fun_o_num,Rep3: fun_int_nat,Abs3: fun_nat_int,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_d9070036042389994064_num_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(Abs2,M)),Rep3) = aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_o_num_nat(comp_o_num_o(Abs2,M1),Rep)),aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,comp_o_nat_nat(Abs,M2)),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1539_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_o_o,Abs: fun_o_o,Rep2: fun_nat_o,Abs2: fun_o_nat,Rep3: fun_nat_nat,Abs3: fun_nat_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_definition_o_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(Abs2,M)),Rep3) = comp_o_nat_nat(comp_o_nat_o(comp_o_nat_o(Abs2,M1),Rep),comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs),M2),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1540_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,Rep2: fun_nat_o,Abs2: fun_o_nat,Rep3: fun_nat_nat,Abs3: fun_nat_nat,M: fun_nat_o,M1: fun_o_o,M2: fun_nat_o] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep2,Abs2),top_top_set_o)
       => ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep3,Abs3),top_top_set_nat)
         => ( ( aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(Abs2,M)),Rep3) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(comp_o_nat_o(Abs2,M1),Rep)),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_o_nat_nat(Abs,M2)),Rep3)) )
           => ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,M1),M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1541_type__copy__map__comp0__undo,axiom,
    ! [Rep: fun_nat_o,Abs: fun_o_nat,Rep2: fun_o_char,Abs2: fun_char_o,Rep3: fun_nat_o,Abs3: fun_o_nat,M: fun_o_char,M1: fun_o_char,M2: fun_o_o] :
      ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep,Abs),top_top_set_o)
     => ( type_d9076672964931703357o_char(Rep2,Abs2,top_top_set_char)
       => ( aa_set_o_o(type_d1000680970699258650_nat_o(Rep3,Abs3),top_top_set_o)
         => ( ( aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_char_o_o(Abs2,M)),Rep3) = comp_nat_o_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_char_o_o(Abs2,M1)),Rep),comp_o_nat_nat(comp_o_nat_o(Abs,M2),Rep3)) )
           => ( comp_o_char_o(M1,M2) = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_1542_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_nat_fun_nat_o] : bNF_re3734335887435638179at_nat(fequal_char,R2) = relcom7935476141563037016ar_nat(conver220575755619908018ar_nat(bNF_Gr4589399631816671513ar_nat(collec7752666796254765876at_nat(aTP_Lamm_af(R2)),comp_P1491501995297412481t_char(product_fst_nat_nat))),bNF_Gr4589399631816671513ar_nat(collec7752666796254765876at_nat(aTP_Lamm_af(R2)),comp_P1491501995297412481t_char(product_snd_nat_nat))) ).

% fun.rel_compp_Grp
tff(fact_1543_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_nat_fun_nat_o] : bNF_re6557955094579809201at_nat(fequal_rat,R2) = relcom4095231159914400721at_nat(conver5141842042509244658at_nat(bNF_Gr8977141012135759449at_nat(collec4103600675842305645at_nat(aTP_Lamm_ag(R2)),comp_P1320480151300532104at_rat(product_fst_nat_nat))),bNF_Gr8977141012135759449at_nat(collec4103600675842305645at_nat(aTP_Lamm_ag(R2)),comp_P1320480151300532104at_rat(product_snd_nat_nat))) ).

% fun.rel_compp_Grp
tff(fact_1544_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_nat_fun_nat_o] : bNF_re5653821019739307937at_nat(fequal_nat,R2) = relcom3117122015492867545at_nat(conver764033108603507442at_nat(bNF_Gr4599332078230022233at_nat(collec773771205145783925at_nat(aTP_Lamm_ah(R2)),comp_P1955610211387027840at_nat(product_fst_nat_nat))),bNF_Gr4599332078230022233at_nat(collec773771205145783925at_nat(aTP_Lamm_ah(R2)),comp_P1955610211387027840at_nat(product_snd_nat_nat))) ).

% fun.rel_compp_Grp
tff(fact_1545_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_nat_fun_nat_o] : bNF_re3715656647883201625at_nat(fequal_int,R2) = relcom2062132017421480317nt_nat(conver8092846473308784626nt_nat(bNF_Gr2704773406080523609nt_nat(collec3811613289609917721at_nat(aTP_Lamm_ai(R2)),comp_P1953119740877977564at_int(product_fst_nat_nat))),bNF_Gr2704773406080523609nt_nat(collec3811613289609917721at_nat(aTP_Lamm_ai(R2)),comp_P1953119740877977564at_int(product_snd_nat_nat))) ).

% fun.rel_compp_Grp
tff(fact_1546_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_num_fun_num_o] : bNF_re7626690874201225453um_num(fequal_int,R2) = relcom6669847303400716945nt_num(conver6394054172651645648nt_num(bNF_Gr1005981105423384631nt_num(collec1545227973015437997um_num(aTP_Lamm_aj(R2)),comp_P4305060815297604026um_int(product_fst_num_num))),bNF_Gr1005981105423384631nt_num(collec1545227973015437997um_num(aTP_Lamm_aj(R2)),comp_P4305060815297604026um_int(product_snd_num_num))) ).

% fun.rel_compp_Grp
tff(fact_1547_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_int_fun_int_o] : bNF_re4731199128827642459nt_int(fequal_char,R2) = relcom5716382434130703632ar_int(conver5381835153026890950ar_int(bNF_Gr527286992368878637ar_int(collec7449505226016060908nt_int(aTP_Lamm_ak(R2)),comp_P4161745170962941589t_char(product_fst_int_int))),bNF_Gr527286992368878637ar_int(collec7449505226016060908nt_int(aTP_Lamm_ak(R2)),comp_P4161745170962941589t_char(product_snd_int_int))) ).

% fun.rel_compp_Grp
tff(fact_1548_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_int_fun_int_o] : bNF_re7554818335971813481nt_int(fequal_rat,R2) = relcom4534186879828360073at_int(conver8369760937699406854at_int(bNF_Gr2981687870471145837at_int(collec3800439105603600677nt_int(aTP_Lamm_al(R2)),comp_P6018825582104210588nt_rat(product_fst_int_int))),bNF_Gr2981687870471145837at_int(collec3800439105603600677nt_int(aTP_Lamm_al(R2)),comp_P6018825582104210588nt_rat(product_snd_int_int))) ).

% fun.rel_compp_Grp
tff(fact_1549_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_int_fun_int_o] : bNF_re6650684261131312217nt_int(fequal_nat,R2) = relcom3556077735406826897at_int(conver3991952003793669638at_int(bNF_Gr7827250973420184429at_int(collec470609634907078957nt_int(aTP_Lamm_am(R2)),comp_P6653955642190706324nt_nat(product_fst_int_int))),bNF_Gr7827250973420184429at_int(collec470609634907078957nt_int(aTP_Lamm_am(R2)),comp_P6653955642190706324nt_nat(product_snd_int_int))) ).

% fun.rel_compp_Grp
tff(fact_1550_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_int_fun_int_o] : bNF_re4712519889275205905nt_int(fequal_int,R2) = relcom2501087737335439669nt_int(conver2097393331644171014nt_int(bNF_Gr5932692301270685805nt_int(collec3508451719371212753nt_int(aTP_Lamm_an(R2)),comp_P6651465171681656048nt_int(product_fst_int_int))),bNF_Gr5932692301270685805nt_int(collec3508451719371212753nt_int(aTP_Lamm_an(R2)),comp_P6651465171681656048nt_int(product_snd_int_int))) ).

% fun.rel_compp_Grp
tff(fact_1551_fun_Orel__compp__Grp,axiom,
    ! [R2: fun_nat_fun_nat_o] : bNF_re4629543205785817921at_nat(fequal_list_nat,R2) = relcom8850562456331176777at_nat(conver1823943793964841842at_nat(bNF_Gr5947979870622126553at_nat(collec5947443778231319653at_nat(aTP_Lamm_ao(R2)),comp_P5292260908303971472st_nat(product_fst_nat_nat))),bNF_Gr5947979870622126553at_nat(collec5947443778231319653at_nat(aTP_Lamm_ao(R2)),comp_P5292260908303971472st_nat(product_snd_nat_nat))) ).

% fun.rel_compp_Grp
tff(fact_1552_image__Pow__surj,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = B5 )
     => ( aa_set3856643248234747582et_rat(image_4408659257933336347et_rat(image_nat_rat(F)),pow_nat(A4)) = pow_rat(B5) ) ) ).

% image_Pow_surj
tff(fact_1553_image__Pow__surj,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = B5 )
     => ( aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(image_nat_nat(F)),pow_nat(A4)) = pow_nat(B5) ) ) ).

% image_Pow_surj
tff(fact_1554_image__Pow__surj,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( ( aa_set_nat_set_int(image_nat_int(F),A4) = B5 )
     => ( aa_set7746546440780698386et_int(image_3739036796817536367et_int(image_nat_int(F)),pow_nat(A4)) = pow_int(B5) ) ) ).

% image_Pow_surj
tff(fact_1555_image__Pow__surj,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( ( aa_set_int_set_nat(image_int_nat(F),A4) = B5 )
     => ( aa_set7560802437211521810et_nat(image_4702325430467532143et_nat(image_int_nat(F)),pow_int(A4)) = pow_nat(B5) ) ) ).

% image_Pow_surj
tff(fact_1556_image__Pow__surj,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( ( aa_set_int_set_int(image_int_int(F),A4) = B5 )
     => ( aa_set5070719292279795054et_int(image_524474410958335435et_int(image_int_int(F)),pow_int(A4)) = pow_int(B5) ) ) ).

% image_Pow_surj
tff(fact_1557_fst__diag__snd,axiom,
    comp_P8237986404257982958t_unit(produc2762368996035445955t_unit,comp_h5940001671827863714t_unit(aTP_Lamm_fe,produc1126150082675451753t_unit)) = produc1126150082675451753t_unit ).

% fst_diag_snd
tff(fact_1558_snd__diag__fst,axiom,
    comp_P9134237804721252988t_unit(produc8185483580310833921_ref_a,comp_r6154141770783548544t_unit(aTP_Lamm_fd,produc473321116828791851t_unit)) = produc473321116828791851t_unit ).

% snd_diag_fst
tff(fact_1559_fun_Orel__Grp,axiom,
    ! [A4: set_fu5818728815147633517nteger,F: fun_fu4800041728187187127nteger] : bNF_re832114662117571029nteger(fequal_Code_integer,bNF_Gr5906040164712753802nteger(A4,F)) = bNF_Gr1900252008802339518nteger(collec2197094084277357286nteger(aTP_Lamm_ff(A4)),comp_f6739083541385432444nteger(F)) ).

% fun.rel_Grp
tff(fact_1560_fun_Orel__Grp,axiom,
    ! [A4: set_o,F: fun_o_o] : bNF_re4705727531993890431at_o_o(fequal_nat,bNF_Grp_o_o(A4,F)) = bNF_Gr368687572618510094_nat_o(collect_fun_nat_o(aTP_Lamm_fg(A4)),aa_fun5662400728881950724_nat_o(comp_o_o_nat,F)) ).

% fun.rel_Grp
tff(fact_1561_fun_Orel__Grp,axiom,
    ! [A4: set_nat,F: fun_nat_nat] : bNF_re5653821019739307937at_nat(fequal_nat,bNF_Grp_nat_nat(A4,F)) = bNF_Gr6503367905095020170at_nat(collect_fun_nat_nat(aTP_Lamm_fh(A4)),aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F)) ).

% fun.rel_Grp
tff(fact_1562_fun_Orel__Grp,axiom,
    ! [A4: set_nat,F: fun_nat_num] : bNF_re272988434038980259at_num(fequal_int,bNF_Grp_nat_num(A4,F)) = bNF_Gr7926914479574802572nt_num(collect_fun_int_nat(aTP_Lamm_fi(A4)),aa_fun6849364718390456542nt_num(comp_nat_num_int,F)) ).

% fun.rel_Grp
tff(fact_1563_fun_Orel__Grp,axiom,
    ! [A4: set_int,F: fun_int_nat] : bNF_re4715010359784256181nt_nat(fequal_int,bNF_Grp_int_nat(A4,F)) = bNF_Gr1946743134186898334nt_nat(collect_fun_int_int(aTP_Lamm_fj(A4)),aa_fun135663990063072770nt_nat(comp_int_nat_int,F)) ).

% fun.rel_Grp
tff(fact_1564_fun_Orel__Grp,axiom,
    ! [A4: set_num,F: fun_num_num] : bNF_re7626690874201225453um_num(fequal_int,bNF_Grp_num_num(A4,F)) = bNF_Gr5488113006707497430nt_num(collect_fun_int_num(aTP_Lamm_fk(A4)),aa_fun8775978671473068146nt_num(comp_num_num_int,F)) ).

% fun.rel_Grp
tff(fact_1565_Grp__fst__snd,axiom,
    ! [R2: fun_nat_fun_nat_o] : relcom8572034174491624592at_nat(conver4691796276509297820at_nat(bNF_Gr193424670081168899at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R2)),product_fst_nat_nat)),bNF_Gr193424670081168899at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,R2)),product_snd_nat_nat)) = R2 ).

% Grp_fst_snd
tff(fact_1566_Grp__fst__snd,axiom,
    ! [R2: fun_re5167877524888533480unit_o] : relcom3136146828236442456t_unit(conver4368408235330554745_ref_a(bNF_Gr5964920315653434322_ref_a(collec3252306575900995386t_unit(produc4924168586796184916unit_o(R2)),produc473321116828791851t_unit)),bNF_Gr5237703264515865432t_unit(collec3252306575900995386t_unit(produc4924168586796184916unit_o(R2)),produc1126150082675451753t_unit)) = R2 ).

% Grp_fst_snd
tff(fact_1567_Grp__fst__snd,axiom,
    ! [R2: fun_int_fun_int_o] : relcom3664783465810776136nt_int(conver7247880940930113968nt_int(bNF_Gr2749509334501985047nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R2)),product_fst_int_int)),bNF_Gr2749509334501985047nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,R2)),product_snd_int_int)) = R2 ).

% Grp_fst_snd
tff(fact_1568_type__copy__set__map0,axiom,
    ! [Rep: fun_o_char,Abs: fun_char_o,S3: fun_char_set_rat,M: fun_o_char,F: fun_nat_rat,S4: fun_o_set_nat,G: fun_nat_o] :
      ( type_d9076672964931703357o_char(Rep,Abs,top_top_set_char)
     => ( ( comp_char_set_rat_o(S3,M) = comp_s8507027185284273247_rat_o(image_nat_rat(F),S4) )
       => ( comp_o_set_rat_nat(comp_char_set_rat_o(S3,Rep),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_char_o_o(Abs,M)),G)) = comp_s8255929034757298889at_nat(image_nat_rat(F),comp_o_set_nat_nat(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1569_type__copy__set__map0,axiom,
    ! [Rep: fun_num_char,Abs: fun_char_num,S3: fun_char_set_rat,M: fun_nat_char,F: fun_nat_rat,S4: fun_nat_set_nat,G: fun_int_nat] :
      ( type_d4396077731765706521m_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c5738158305436223508at_nat(S3,M) = comp_s8255929034757298889at_nat(image_nat_rat(F),S4) )
       => ( comp_num_set_rat_int(comp_c2295490091592002142at_num(S3,Rep),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_char_num_nat(Abs,M)),G)) = comp_s8253438564248248613at_int(image_nat_rat(F),comp_nat_set_nat_int(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1570_type__copy__set__map0,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,S3: fun_char_set_rat,M: fun_nat_char,F: fun_nat_rat,S4: fun_nat_set_nat,G: fun_nat_nat] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c5738158305436223508at_nat(S3,M) = comp_s8255929034757298889at_nat(image_nat_rat(F),S4) )
       => ( comp_nat_set_rat_nat(comp_c5738158305436223508at_nat(S3,Rep),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_char_nat_nat(Abs,M)),G)) = comp_s8255929034757298889at_nat(image_nat_rat(F),comp_nat_set_nat_nat(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1571_type__copy__set__map0,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,S3: fun_char_set_rat,M: fun_int_char,F: fun_nat_rat,S4: fun_int_set_nat,G: fun_int_int] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c5735667834927173232at_int(S3,M) = comp_s8253438564248248613at_int(image_nat_rat(F),S4) )
       => ( comp_nat_set_rat_int(comp_c5738158305436223508at_nat(S3,Rep),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_char_nat_int(Abs,M)),G)) = comp_s8253438564248248613at_int(image_nat_rat(F),comp_int_set_nat_int(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1572_type__copy__set__map0,axiom,
    ! [Rep: fun_o_char,Abs: fun_char_o,S3: fun_char_set_nat,M: fun_o_char,F: fun_nat_nat,S4: fun_o_set_nat,G: fun_nat_o] :
      ( type_d9076672964931703357o_char(Rep,Abs,top_top_set_char)
     => ( ( comp_char_set_nat_o(S3,M) = comp_s33089643564333159_nat_o(image_nat_nat(F),S4) )
       => ( comp_o_set_nat_nat(comp_char_set_nat_o(S3,Rep),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_char_o_o(Abs,M)),G)) = comp_s3433241188411525313at_nat(image_nat_nat(F),comp_o_set_nat_nat(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1573_type__copy__set__map0,axiom,
    ! [Rep: fun_num_char,Abs: fun_char_num,S3: fun_char_set_nat,M: fun_nat_char,F: fun_nat_nat,S4: fun_nat_set_nat,G: fun_int_nat] :
      ( type_d4396077731765706521m_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c915470459090449932at_nat(S3,M) = comp_s3433241188411525313at_nat(image_nat_nat(F),S4) )
       => ( comp_num_set_nat_int(comp_c6696174282101004374at_num(S3,Rep),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_char_num_nat(Abs,M)),G)) = comp_s3430750717902475037at_int(image_nat_nat(F),comp_nat_set_nat_int(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1574_type__copy__set__map0,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,S3: fun_char_set_nat,M: fun_nat_char,F: fun_nat_nat,S4: fun_nat_set_nat,G: fun_nat_nat] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c915470459090449932at_nat(S3,M) = comp_s3433241188411525313at_nat(image_nat_nat(F),S4) )
       => ( comp_nat_set_nat_nat(comp_c915470459090449932at_nat(S3,Rep),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,comp_char_nat_nat(Abs,M)),G)) = comp_s3433241188411525313at_nat(image_nat_nat(F),comp_nat_set_nat_nat(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1575_type__copy__set__map0,axiom,
    ! [Rep: fun_nat_char,Abs: fun_char_nat,S3: fun_char_set_nat,M: fun_int_char,F: fun_nat_nat,S4: fun_int_set_nat,G: fun_int_int] :
      ( type_d5945782884131516111t_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c912979988581399656at_int(S3,M) = comp_s3430750717902475037at_int(image_nat_nat(F),S4) )
       => ( comp_nat_set_nat_int(comp_c915470459090449932at_nat(S3,Rep),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,comp_char_nat_int(Abs,M)),G)) = comp_s3430750717902475037at_int(image_nat_nat(F),comp_int_set_nat_int(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1576_type__copy__set__map0,axiom,
    ! [Rep: fun_o_char,Abs: fun_char_o,S3: fun_char_set_int,M: fun_o_char,F: fun_nat_int,S4: fun_o_set_nat,G: fun_nat_o] :
      ( type_d9076672964931703357o_char(Rep,Abs,top_top_set_char)
     => ( ( comp_char_set_int_o(S3,M) = comp_s7011316712748663051_int_o(image_nat_int(F),S4) )
       => ( comp_o_set_int_nat(comp_char_set_int_o(S3,Rep),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,comp_char_o_o(Abs,M)),G)) = comp_s5620162763104547101nt_nat(image_nat_int(F),comp_o_set_nat_nat(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1577_type__copy__set__map0,axiom,
    ! [Rep: fun_num_char,Abs: fun_char_num,S3: fun_char_set_int,M: fun_nat_char,F: fun_nat_int,S4: fun_nat_set_nat,G: fun_int_nat] :
      ( type_d4396077731765706521m_char(Rep,Abs,top_top_set_char)
     => ( ( comp_c3102392033783471720nt_nat(S3,M) = comp_s5620162763104547101nt_nat(image_nat_int(F),S4) )
       => ( comp_num_set_int_int(comp_c8883095856794026162nt_num(S3,Rep),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,comp_char_num_nat(Abs,M)),G)) = comp_s5617672292595496825nt_int(image_nat_int(F),comp_nat_set_nat_int(S4,G)) ) ) ) ).

% type_copy_set_map0
tff(fact_1578_type__definition_ORep__range,axiom,
    ! [Rep: fun_nat_rat,Abs: fun_rat_nat,A4: set_rat] :
      ( aa_set_rat_o(type_d5615363888691252950at_rat(Rep,Abs),A4)
     => ( aa_set_nat_set_rat(image_nat_rat(Rep),top_top_set_nat) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_1579_type__definition_ORep__range,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),A4)
     => ( aa_set_nat_set_nat(image_nat_nat(Rep),top_top_set_nat) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_1580_type__definition_ORep__range,axiom,
    ! [Rep: fun_nat_int,Abs: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(type_d6248003478268698410at_int(Rep,Abs),A4)
     => ( aa_set_nat_set_int(image_nat_int(Rep),top_top_set_nat) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_1581_type__definition_ORep__range,axiom,
    ! [Rep: fun_int_nat,Abs: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep,Abs),A4)
     => ( aa_set_int_set_nat(image_int_nat(Rep),top_top_set_int) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_1582_type__definition_ORep__range,axiom,
    ! [Rep: fun_int_int,Abs: fun_int_int,A4: set_int] :
      ( aa_set_int_o(type_d7247357190169752966nt_int(Rep,Abs),A4)
     => ( aa_set_int_set_int(image_int_int(Rep),top_top_set_int) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_1583_type__definition_ORep__range,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A4: set_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A4)
     => ( image_Product_unit_o(Rep,top_to1996260823553986621t_unit) = A4 ) ) ).

% type_definition.Rep_range
tff(fact_1584_type__definition_OAbs__image,axiom,
    ! [Rep: fun_rat_nat,Abs: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(type_d5933939304842882774at_nat(Rep,Abs),A4)
     => ( aa_set_nat_set_rat(image_nat_rat(Abs),A4) = top_top_set_rat ) ) ).

% type_definition.Abs_image
tff(fact_1585_type__definition_OAbs__image,axiom,
    ! [Rep: fun_nat_nat,Abs: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(type_d6250493948777748686at_nat(Rep,Abs),A4)
     => ( aa_set_nat_set_nat(image_nat_nat(Abs),A4) = top_top_set_nat ) ) ).

% type_definition.Abs_image
tff(fact_1586_type__definition_OAbs__image,axiom,
    ! [Rep: fun_nat_int,Abs: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(type_d6248003478268698410at_int(Rep,Abs),A4)
     => ( aa_set_int_set_nat(image_int_nat(Abs),A4) = top_top_set_nat ) ) ).

% type_definition.Abs_image
tff(fact_1587_type__definition_OAbs__image,axiom,
    ! [Rep: fun_int_nat,Abs: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(type_d7249847660678803242nt_nat(Rep,Abs),A4)
     => ( aa_set_nat_set_int(image_nat_int(Abs),A4) = top_top_set_int ) ) ).

% type_definition.Abs_image
tff(fact_1588_type__definition_OAbs__image,axiom,
    ! [Rep: fun_int_int,Abs: fun_int_int,A4: set_int] :
      ( aa_set_int_o(type_d7247357190169752966nt_int(Rep,Abs),A4)
     => ( aa_set_int_set_int(image_int_int(Abs),A4) = top_top_set_int ) ) ).

% type_definition.Abs_image
tff(fact_1589_type__definition_OAbs__image,axiom,
    ! [Rep: fun_Product_unit_o,Abs: fun_o_Product_unit,A4: set_o] :
      ( aa_set_o_o(type_d6188575255521822967unit_o(Rep,Abs),A4)
     => ( image_o_Product_unit(Abs,A4) = top_to1996260823553986621t_unit ) ) ).

% type_definition.Abs_image
tff(fact_1590_Quotient__alt__def5,axiom,
    ! [R2: fun_char_fun_char_o,Abs: fun_char_char,Rep: fun_char_char,T2: fun_char_fun_char_o] :
      ( quotient_char_char(R2,Abs,Rep,T2)
    <=> ( ord_le3895236127104983496char_o(T2,bNF_Grp_char_char(top_top_set_char,Abs))
        & ord_le3895236127104983496char_o(bNF_Grp_char_char(top_top_set_char,Rep),conversep_char_char(T2))
        & ( R2 = relcom5872928454965100966r_char(T2,conversep_char_char(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1591_Quotient__alt__def5,axiom,
    ! [R2: fun_char_fun_char_o,Abs: fun_char_rat,Rep: fun_rat_char,T2: fun_char_fun_rat_o] :
      ( quotient_char_rat(R2,Abs,Rep,T2)
    <=> ( ord_le3450618396672030593_rat_o(T2,bNF_Grp_char_rat(top_top_set_char,Abs))
        & ord_le4131652544857200641char_o(bNF_Grp_rat_char(top_top_set_rat,Rep),conversep_char_rat(T2))
        & ( R2 = relcom3428143170894750637t_char(T2,conversep_char_rat(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1592_Quotient__alt__def5,axiom,
    ! [R2: fun_char_fun_char_o,Abs: fun_char_nat,Rep: fun_nat_char,T2: fun_char_fun_nat_o] :
      ( quotient_char_nat(R2,Abs,Rep,T2)
    <=> ( ord_le4200052891806866313_nat_o(T2,bNF_Grp_char_nat(top_top_set_char,Abs))
        & ord_le8697944750422312969char_o(bNF_Grp_nat_char(top_top_set_nat,Rep),conversep_char_nat(T2))
        & ( R2 = relcom6755291642603558309t_char(T2,conversep_char_nat(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1593_Quotient__alt__def5,axiom,
    ! [R2: fun_char_fun_char_o,Abs: fun_char_int,Rep: fun_int_char,T2: fun_char_fun_int_o] :
      ( quotient_char_int(R2,Abs,Rep,T2)
    <=> ( ord_le1954907924136420397_int_o(T2,bNF_Grp_char_int(top_top_set_char,Abs))
        & ord_le1351501346584897197char_o(bNF_Grp_int_char(top_top_set_int,Rep),conversep_char_int(T2))
        & ( R2 = relcom750623199913152513t_char(T2,conversep_char_int(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1594_Quotient__alt__def5,axiom,
    ! [R2: fun_rat_fun_rat_o,Abs: fun_rat_char,Rep: fun_char_rat,T2: fun_rat_fun_char_o] :
      ( quotient_rat_char(R2,Abs,Rep,T2)
    <=> ( ord_le4131652544857200641char_o(T2,bNF_Grp_rat_char(top_top_set_rat,Abs))
        & ord_le3450618396672030593_rat_o(bNF_Grp_char_rat(top_top_set_char,Rep),conversep_rat_char(T2))
        & ( R2 = relcom1505247221262953652ar_rat(T2,conversep_rat_char(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1595_Quotient__alt__def5,axiom,
    ! [R2: fun_rat_fun_rat_o,Abs: fun_rat_rat,Rep: fun_rat_rat,T2: fun_rat_fun_rat_o] :
      ( quotient_rat_rat(R2,Abs,Rep,T2)
    <=> ( ord_le7525112548594501306_rat_o(T2,bNF_Grp_rat_rat(top_top_set_rat,Abs))
        & ord_le7525112548594501306_rat_o(bNF_Grp_rat_rat(top_top_set_rat,Rep),conversep_rat_rat(T2))
        & ( R2 = relcompp_rat_rat_rat(T2,conversep_rat_rat(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1596_Quotient__alt__def5,axiom,
    ! [R2: fun_rat_fun_rat_o,Abs: fun_rat_nat,Rep: fun_nat_rat,T2: fun_rat_fun_nat_o] :
      ( quotient_rat_nat(R2,Abs,Rep,T2)
    <=> ( ord_le8274547043729337026_nat_o(T2,bNF_Grp_rat_nat(top_top_set_rat,Abs))
        & ord_le18904959571214018_rat_o(bNF_Grp_nat_rat(top_top_set_nat,Rep),conversep_rat_nat(T2))
        & ( R2 = relcompp_rat_nat_rat(T2,conversep_rat_nat(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1597_Quotient__alt__def5,axiom,
    ! [R2: fun_rat_fun_rat_o,Abs: fun_rat_int,Rep: fun_int_rat,T2: fun_rat_fun_int_o] :
      ( quotient_rat_int(R2,Abs,Rep,T2)
    <=> ( ord_le6029402076058891110_int_o(T2,bNF_Grp_rat_int(top_top_set_rat,Abs))
        & ord_le1573691155048284518_rat_o(bNF_Grp_int_rat(top_top_set_int,Rep),conversep_rat_int(T2))
        & ( R2 = relcompp_rat_int_rat(T2,conversep_rat_int(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1598_Quotient__alt__def5,axiom,
    ! [R2: fun_nat_fun_nat_o,Abs: fun_nat_char,Rep: fun_char_nat,T2: fun_nat_fun_char_o] :
      ( quotient_nat_char(R2,Abs,Rep,T2)
    <=> ( ord_le8697944750422312969char_o(T2,bNF_Grp_nat_char(top_top_set_nat,Abs))
        & ord_le4200052891806866313_nat_o(bNF_Grp_char_nat(top_top_set_char,Rep),conversep_nat_char(T2))
        & ( R2 = relcom8112501251453913252ar_nat(T2,conversep_nat_char(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1599_Quotient__alt__def5,axiom,
    ! [R2: fun_nat_fun_nat_o,Abs: fun_nat_rat,Rep: fun_rat_nat,T2: fun_nat_fun_rat_o] :
      ( quotient_nat_rat(R2,Abs,Rep,T2)
    <=> ( ord_le18904959571214018_rat_o(T2,bNF_Grp_nat_rat(top_top_set_nat,Abs))
        & ord_le8274547043729337026_nat_o(bNF_Grp_rat_nat(top_top_set_rat,Rep),conversep_nat_rat(T2))
        & ( R2 = relcompp_nat_rat_nat(T2,conversep_nat_rat(T2)) ) ) ) ).

% Quotient_alt_def5
tff(fact_1600_K__record__comp,axiom,
    ! [C: $o,F: fun_nat_o,X5: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,aa_o_fun_o_o(aTP_Lamm_fl,(C))),F),X5)
    <=> (C) ) ).

% K_record_comp
tff(fact_1601_K__record__comp,axiom,
    ! [C: num,F: fun_int_nat,X5: int] : aa_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,aTP_Lamm_fm(C)),F),X5) = C ).

% K_record_comp
tff(fact_1602_K__record__comp,axiom,
    ! [C: nat,F: fun_nat_nat,X5: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(aTP_Lamm_fn,C)),F),X5) = C ).

% K_record_comp
tff(fact_1603_K__record__comp,axiom,
    ! [C: nat,F: fun_int_int,X5: int] : aa_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,aTP_Lamm_fo(C)),F),X5) = C ).

% K_record_comp
tff(fact_1604_K__record__comp,axiom,
    ! [C: fun_Pr5876161306168176259nteger,F: fun_Co1981900195452921617nteger,X5: code_integer] : aa_Cod1978467581589928094nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(aTP_Lamm_fp(C)),F),X5) = C ).

% K_record_comp
tff(fact_1605_image__paired__Times,axiom,
    ! [F: fun_nat_rat,G: fun_int_nat,A4: set_nat,B5: set_int] : image_3755276878268593137at_nat(produc5347463441023893471at_nat(aa_fun8497543784391190088at_nat(aTP_Lamm_fq(F),G)),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc140472662868866674at_nat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set5726841466071138683et_nat(aTP_Lamm_fs(G),B5)) ).

% image_paired_Times
tff(fact_1606_image__paired__Times,axiom,
    ! [F: fun_nat_rat,G: fun_int_int,A4: set_nat,B5: set_int] : image_8800797895614172237at_int(produc1169612421514696763at_int(aa_fun6462367883577042760at_int(aTP_Lamm_ft(F),G)),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc137982192359816398at_int(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set3236758321139411927et_int(aTP_Lamm_fu(G),B5)) ).

% image_paired_Times
tff(fact_1607_image__paired__Times,axiom,
    ! [F: fun_nat_nat,G: fun_int_int,A4: set_nat,B5: set_int] : image_9117352539549038149at_int(produc1486167065449562675at_int(aa_fun566144318431074624at_int(aTP_Lamm_fv(F),G)),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc454536836294682310at_int(aa_set_nat_set_nat(image_nat_nat(F),A4),aa_set2464685511577892303et_int(aTP_Lamm_fw(G),B5)) ).

% image_paired_Times
tff(fact_1608_image__paired__Times,axiom,
    ! [F: fun_nat_int,G: fun_int_nat,A4: set_nat,B5: set_int] : image_5071185234104513605nt_nat(produc6663371796859813939nt_nat(aa_fun5820023813409591964nt_nat(aTP_Lamm_fx(F),G)),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc1456381018704787142nt_nat(aa_set_nat_set_int(image_nat_int(F),A4),aa_set5326422216746463695et_nat(aTP_Lamm_fy(G),B5)) ).

% image_paired_Times
tff(fact_1609_image__paired__Times,axiom,
    ! [F: fun_int_nat,G: fun_nat_rat,A4: set_int,B5: set_nat] : image_3420722608207172081at_rat(produc877984105854340063at_rat(aa_fun1750146759014209948at_rat(aTP_Lamm_fz(F),G)),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc9045269283572012658at_rat(aa_set_int_set_nat(image_int_nat(F),A4),aa_set3405691356855822623et_rat(aTP_Lamm_gb(G),B5)) ).

% image_paired_Times
tff(fact_1610_image__paired__Times,axiom,
    ! [F: fun_int_nat,G: fun_nat_int,A4: set_int,B5: set_nat] : image_2751100147091372101at_int(produc208361644738540083at_int(aa_fun2564171078181162140at_int(aTP_Lamm_gc(F),G)),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc454536836294682310at_int(aa_set_int_set_nat(image_int_nat(F),A4),aa_set7295594549401773427et_int(aTP_Lamm_gd(G),B5)) ).

% image_paired_Times
tff(fact_1611_image__paired__Times,axiom,
    ! [F: fun_int_nat,G: fun_int_int,A4: set_int,B5: set_int] : image_1654017166447373545at_int(produc3246203729202673879at_int(aa_fun816493080544626660at_int(aTP_Lamm_ge(F),G)),produc1453890548195736866nt_int(A4,aTP_Lamm_gf(B5))) = produc454536836294682310at_int(aa_set_int_set_nat(image_int_nat(F),A4),aa_set2464685511577892303et_int(aTP_Lamm_fw(G),B5)) ).

% image_paired_Times
tff(fact_1612_image__paired__Times,axiom,
    ! [F: fun_int_int,G: fun_nat_rat,A4: set_int,B5: set_nat] : image_4420076320108226637nt_rat(produc1877337817755394619nt_rat(aa_fun4968850353178579960nt_rat(aTP_Lamm_gg(F),G)),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc821250958618291406nt_rat(aa_set_int_set_int(image_int_int(F),A4),aa_set3777344917092667259et_rat(aTP_Lamm_gh(G),B5)) ).

% image_paired_Times
tff(fact_1613_image__paired__Times,axiom,
    ! [F: fun_int_int,G: fun_nat_nat,A4: set_int,B5: set_nat] : image_7928304878501623365nt_nat(produc5385566376148791347nt_nat(aa_fun7818050573159679480nt_nat(aTP_Lamm_gi(F),G)),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc1456381018704787142nt_nat(aa_set_int_set_int(image_int_int(F),A4),aa_set933959217715569011et_nat(aTP_Lamm_gj(G),B5)) ).

% image_paired_Times
tff(fact_1614_image__paired__Times,axiom,
    ! [F: fun_int_int,G: fun_int_nat,A4: set_int,B5: set_int] : image_6831221897857624809nt_nat(produc8423408460612925143nt_nat(aa_fun6070372575523144000nt_nat(aTP_Lamm_gk(F),G)),produc1453890548195736866nt_int(A4,aTP_Lamm_gf(B5))) = produc1456381018704787142nt_nat(aa_set_int_set_int(image_int_int(F),A4),aa_set5326422216746463695et_nat(aTP_Lamm_fy(G),B5)) ).

% image_paired_Times
tff(fact_1615_Gr__incl,axiom,
    ! [A4: set_nat,F: fun_nat_rat,B5: set_rat] :
      ( ord_le5989899228261996553at_rat(bNF_Gr_nat_rat(A4,F),produc9045269283572012658at_rat(A4,aTP_Lamm_gl(B5)))
    <=> aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5) ) ).

% Gr_incl
tff(fact_1616_Gr__incl,axiom,
    ! [A4: set_nat,F: fun_nat_int,B5: set_int] :
      ( ord_le656430383953171549at_int(bNF_Gr_nat_int(A4,F),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5)))
    <=> aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5) ) ).

% Gr_incl
tff(fact_1617_Gr__incl,axiom,
    ! [A4: set_int,F: fun_int_nat,B5: set_nat] :
      ( ord_le5333435103577920093nt_nat(bNF_Gr_int_nat(A4,F),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5)))
    <=> aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5) ) ).

% Gr_incl
tff(fact_1618_Gr__incl,axiom,
    ! [A4: set_int,F: fun_int_int,B5: set_int] :
      ( ord_le2843351958646193337nt_int(bNF_Gr_int_int(A4,F),produc1453890548195736866nt_int(A4,aTP_Lamm_gf(B5)))
    <=> aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5) ) ).

% Gr_incl
tff(fact_1619_Gr__incl,axiom,
    ! [A4: set_nat,F: fun_nat_nat,B5: set_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,bNF_Gr_nat_nat(A4,F)),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5)))
    <=> aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5) ) ).

% Gr_incl
tff(fact_1620_subset__fst__snd,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),produc457027306803732586at_nat(image_2486076414777270412at_nat(product_fst_nat_nat,A4),aTP_Lamm_gn(A4))) ).

% subset_fst_snd
tff(fact_1621_subset__fst__snd,axiom,
    ! [A4: set_Pr1133966238863718811t_unit] : ord_le5525191956872776187t_unit(A4,produc3946373962484743846t_unit(image_7136766188234240905_ref_a(produc473321116828791851t_unit,A4),aTP_Lamm_go(A4))) ).

% subset_fst_snd
tff(fact_1622_inv__image__def,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat,F: fun_na5670658592214072528at_nat] : inv_im2903009620893676101at_nat(Ra,F) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_fun1198561302286688022_nat_o(aTP_Lamm_gp(Ra),F))) ).

% inv_image_def
tff(fact_1623_inv__image__def,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,F: fun_na7044673412300895210at_nat] : inv_im4437241714082187487at_nat(Ra,F) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_fun2628974918839883004_nat_o(aTP_Lamm_gq(Ra),F))) ).

% inv_image_def
tff(fact_1624_inv__image__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,F: fun_nat_nat] : inv_image_nat_nat(Ra,F) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_fun6824225949033397647_nat_o(aTP_Lamm_gr(Ra),F))) ).

% inv_image_def
tff(fact_1625_inv__image__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,F: fun_nat_int] : inv_image_int_nat(Ra,F) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_fun4569312790843823667_nat_o(aTP_Lamm_gs(Ra),F))) ).

% inv_image_def
tff(fact_1626_inv__image__def,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat,F: fun_in2433280179646926196at_nat] : inv_im2900519150384625825at_int(Ra,F) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6626869587057797306_int_o(aTP_Lamm_gt(Ra),F))) ).

% inv_image_def
tff(fact_1627_inv__image__def,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,F: fun_in8804710076054006414at_nat] : inv_im4434751243573137211at_int(Ra,F) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun7652824614168344736_int_o(aTP_Lamm_gu(Ra),F))) ).

% inv_image_def
tff(fact_1628_inv__image__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,F: fun_int_nat] : inv_image_nat_int(Ra,F) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun927631379751134075_int_o(aTP_Lamm_gv(Ra),F))) ).

% inv_image_def
tff(fact_1629_inv__image__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,F: fun_int_int] : inv_image_int_int(Ra,F) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun7896090258416335903_int_o(aTP_Lamm_gw(Ra),F))) ).

% inv_image_def
tff(fact_1630_The__case__prod,axiom,
    ! [P2: fun_re5167877524888533480unit_o] : the_Pr5012129486825234846t_unit(produc4924168586796184916unit_o(P2)) = the_Pr5012129486825234846t_unit(aTP_Lamm_gx(P2)) ).

% The_case_prod
tff(fact_1631_The__case__prod,axiom,
    ! [P2: fun_int_fun_int_o] : the_Pr4378521158711661632nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,P2)) = the_Pr4378521158711661632nt_int(aTP_Lamm_gy(P2)) ).

% The_case_prod
tff(fact_1632_Quotient__composition__ge__eq,axiom,
    ! [T2: fun_in2375804543815556931_nat_o,R2: fun_Pr2028404973800837934_nat_o] :
      ( left_t8261381459436684205at_nat(T2)
     => ( ord_le3650406611588909774_nat_o(fequal9188845342289841878at_nat,R2)
       => ord_le77980682512674322_int_o(fequal_int,relcom1106208330880909712at_int(T2,relcom8911704206367525917at_int(R2,conver1866313717161370910at_nat(T2)))) ) ) ).

% Quotient_composition_ge_eq
tff(fact_1633_Quotient__composition__ge__eq,axiom,
    ! [T2: fun_Co8353275306710069249_int_o,R2: fun_int_fun_int_o] :
      ( left_t1735413524555719857er_int(T2)
     => ( ord_le77980682512674322_int_o(fequal_int,R2)
       => ord_le6737357399843569584eger_o(fequal_Code_integer,relcom692387321444912665nteger(T2,relcom967497705283160040nteger(R2,conver1462408618118490240er_int(T2)))) ) ) ).

% Quotient_composition_ge_eq
tff(fact_1634_Quotient__composition__le__eq,axiom,
    ! [T2: fun_in2375804543815556931_nat_o,R2: fun_Pr2028404973800837934_nat_o] :
      ( left_u1182257597767569412at_nat(T2)
     => ( ord_le3650406611588909774_nat_o(R2,fequal9188845342289841878at_nat)
       => ord_le77980682512674322_int_o(relcom1106208330880909712at_int(T2,relcom8911704206367525917at_int(R2,conver1866313717161370910at_nat(T2))),fequal_int) ) ) ).

% Quotient_composition_le_eq
tff(fact_1635_Quotient__composition__le__eq,axiom,
    ! [T2: fun_Co8353275306710069249_int_o,R2: fun_int_fun_int_o] :
      ( left_u6156896403773009178er_int(T2)
     => ( ord_le77980682512674322_int_o(R2,fequal_int)
       => ord_le6737357399843569584eger_o(relcom692387321444912665nteger(T2,relcom967497705283160040nteger(R2,conver1462408618118490240er_int(T2))),fequal_Code_integer) ) ) ).

% Quotient_composition_le_eq
tff(fact_1636_mem__Sigma__iff,axiom,
    ! [A: $o,B: $o,A4: set_o,B5: fun_o_set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),product_Sigma_o_o(A4,B5))
    <=> ( aa_set_o_o(member_o((A)),A4)
        & aa_set_o_o(member_o((B)),aa_o_set_o(B5,(A))) ) ) ).

% mem_Sigma_iff
tff(fact_1637_mem__Sigma__iff,axiom,
    ! [A: $o,B: nat,A4: set_o,B5: fun_o_set_nat] :
      ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A)),B),product_Sigma_o_nat(A4,B5))
    <=> ( aa_set_o_o(member_o((A)),A4)
        & aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(A))) ) ) ).

% mem_Sigma_iff
tff(fact_1638_mem__Sigma__iff,axiom,
    ! [A: $o,B: int,A4: set_o,B5: fun_o_set_int] :
      ( member7847949116333733898_o_int(product_Pair_o_int((A),B),product_Sigma_o_int(A4,B5))
    <=> ( aa_set_o_o(member_o((A)),A4)
        & aa_set_int_o(member_int(B),aa_o_set_int(B5,(A))) ) ) ).

% mem_Sigma_iff
tff(fact_1639_mem__Sigma__iff,axiom,
    ! [A: nat,B: $o,A4: set_nat,B5: fun_nat_set_o] :
      ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A),(B)),product_Sigma_nat_o(A4,B5))
    <=> ( aa_set_nat_o(member_nat(A),A4)
        & aa_set_o_o(member_o((B)),aa_nat_set_o(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1640_mem__Sigma__iff,axiom,
    ! [A: nat,B: int,A4: set_nat,B5: fun_nat_set_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(A,B),produc454536836294682310at_int(A4,B5))
    <=> ( aa_set_nat_o(member_nat(A),A4)
        & aa_set_int_o(member_int(B),aa_nat_set_int(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1641_mem__Sigma__iff,axiom,
    ! [A: int,B: $o,A4: set_int,B5: fun_int_set_o] :
      ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A),(B)),product_Sigma_int_o(A4,B5))
    <=> ( aa_set_int_o(member_int(A),A4)
        & aa_set_o_o(member_o((B)),aa_int_set_o(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1642_mem__Sigma__iff,axiom,
    ! [A: int,B: nat,A4: set_int,B5: fun_int_set_nat] :
      ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A),B),produc1456381018704787142nt_nat(A4,B5))
    <=> ( aa_set_int_o(member_int(A),A4)
        & aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1643_mem__Sigma__iff,axiom,
    ! [A: nat,B: num,A4: set_nat,B5: fun_nat_set_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),produc6237731129814287028at_num(A4,B5))
    <=> ( aa_set_nat_o(member_nat(A),A4)
        & member_num(B,aa_nat_set_num(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1644_mem__Sigma__iff,axiom,
    ! [A: int,B: int,A4: set_int,B5: fun_int_set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),produc1453890548195736866nt_int(A4,B5))
    <=> ( aa_set_int_o(member_int(A),A4)
        & aa_set_int_o(member_int(B),aa_int_set_int(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1645_mem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),produc457027306803732586at_nat(A4,B5))
    <=> ( aa_set_nat_o(member_nat(A),A4)
        & aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A)) ) ) ).

% mem_Sigma_iff
tff(fact_1646_SigmaI,axiom,
    ! [A: $o,A4: set_o,B: $o,B5: fun_o_set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_o_o(member_o((B)),aa_o_set_o(B5,(A)))
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),product_Sigma_o_o(A4,B5)) ) ) ).

% SigmaI
tff(fact_1647_SigmaI,axiom,
    ! [A: $o,A4: set_o,B: nat,B5: fun_o_set_nat] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(A)))
       => member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A)),B),product_Sigma_o_nat(A4,B5)) ) ) ).

% SigmaI
tff(fact_1648_SigmaI,axiom,
    ! [A: $o,A4: set_o,B: int,B5: fun_o_set_int] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_int_o(member_int(B),aa_o_set_int(B5,(A)))
       => member7847949116333733898_o_int(product_Pair_o_int((A),B),product_Sigma_o_int(A4,B5)) ) ) ).

% SigmaI
tff(fact_1649_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: $o,B5: fun_nat_set_o] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_o_o(member_o((B)),aa_nat_set_o(B5,A))
       => member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A),(B)),product_Sigma_nat_o(A4,B5)) ) ) ).

% SigmaI
tff(fact_1650_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: int,B5: fun_nat_set_int] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_int_o(member_int(B),aa_nat_set_int(B5,A))
       => member4262671552274231302at_int(product_Pair_nat_int(A,B),produc454536836294682310at_int(A4,B5)) ) ) ).

% SigmaI
tff(fact_1651_SigmaI,axiom,
    ! [A: int,A4: set_int,B: $o,B5: fun_int_set_o] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_o_o(member_o((B)),aa_int_set_o(B5,A))
       => member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A),(B)),product_Sigma_int_o(A4,B5)) ) ) ).

% SigmaI
tff(fact_1652_SigmaI,axiom,
    ! [A: int,A4: set_int,B: nat,B5: fun_int_set_nat] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,A))
       => member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A),B),produc1456381018704787142nt_nat(A4,B5)) ) ) ).

% SigmaI
tff(fact_1653_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: num,B5: fun_nat_set_num] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( member_num(B,aa_nat_set_num(B5,A))
       => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),produc6237731129814287028at_num(A4,B5)) ) ) ).

% SigmaI
tff(fact_1654_SigmaI,axiom,
    ! [A: int,A4: set_int,B: int,B5: fun_int_set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_int_o(member_int(B),aa_int_set_int(B5,A))
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),produc1453890548195736866nt_int(A4,B5)) ) ) ).

% SigmaI
tff(fact_1655_SigmaI,axiom,
    ! [A: nat,A4: set_nat,B: nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A))
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),produc457027306803732586at_nat(A4,B5)) ) ) ).

% SigmaI
tff(fact_1656_in__inv__image,axiom,
    ! [Xa: nat,Y2: nat,Ra: set_Pr1261947904930325089at_nat,F: fun_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),inv_image_nat_nat(Ra,F))
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(F,Xa)),aa_nat_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1657_in__inv__image,axiom,
    ! [Xa: nat,Y2: nat,Ra: set_Pr958786334691620121nt_int,F: fun_nat_int] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),inv_image_int_nat(Ra,F))
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(F,Xa)),aa_nat_int(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1658_in__inv__image,axiom,
    ! [Xa: int,Y2: int,Ra: set_Pr1261947904930325089at_nat,F: fun_int_nat] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),inv_image_nat_int(Ra,F))
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_int_nat(F,Xa)),aa_int_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1659_in__inv__image,axiom,
    ! [Xa: int,Y2: int,Ra: set_Pr958786334691620121nt_int,F: fun_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),inv_image_int_int(Ra,F))
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(F,Xa)),aa_int_int(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1660_in__inv__image,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Ra: set_Pr1261947904930325089at_nat,F: fun_se4097146774670233028at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),inv_im6750873321330482877at_nat(Ra,F))
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_set4749074711808745205at_nat(F,Xa)),aa_set4749074711808745205at_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1661_in__inv__image,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Ra: set_Pr958786334691620121nt_int,F: fun_se9142667792015812128at_int] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),inv_im8510909985083594081at_nat(Ra,F))
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_set4746584241299694929at_int(F,Xa)),aa_set4746584241299694929at_int(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1662_in__inv__image,axiom,
    ! [Xa: nat,Y2: nat,Ra: set_Pr4329608150637261639at_nat,F: fun_na7044673412300895210at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),inv_im4437241714082187487at_nat(Ra,F))
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_nat7062706319057040595at_nat(F,Xa)),aa_nat7062706319057040595at_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1663_in__inv__image,axiom,
    ! [Xa: int,Y2: int,Ra: set_Pr4329608150637261639at_nat,F: fun_in8804710076054006414at_nat] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),inv_im4434751243573137211at_int(Ra,F))
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_int8822742982810151799at_nat(F,Xa)),aa_int8822742982810151799at_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1664_in__inv__image,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Ra: set_Pr4329608150637261639at_nat,F: fun_se320340636488258563at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),inv_im7023619800666061458at_nat(Ra,F))
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_set8868280134176360700at_nat(F,Xa)),aa_set8868280134176360700at_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1665_in__inv__image,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat,Ra: set_Pr1261947904930325089at_nat,F: fun_se4674745386781188650at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2),inv_im2354225701593503139at_nat(Ra,F))
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_set168828743186997083at_nat(F,Xa)),aa_set168828743186997083at_nat(F,Y2))),Ra) ) ).

% in_inv_image
tff(fact_1666_Collect__case__prod,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_Pr4796943543331071514_nat_o] : collec7088162979684241874at_nat(produc6590410687421337004_nat_o(aa_fun140556562124455766_nat_o(aTP_Lamm_gz(P2),Q2))) = produc2761391749766926216at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2),aTP_Lamm_ha(Q2)) ).

% Collect_case_prod
tff(fact_1667_Collect__case__prod,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_nat_o] : collec7029190964493513045at_nat(produc8758269395934548017_nat_o(aa_fun1047528069869899400_nat_o(aTP_Lamm_hb(P2),Q2))) = produc7672662199629908489at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2),aTP_Lamm_hc(Q2)) ).

% Collect_case_prod
tff(fact_1668_Collect__case__prod,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_Pr3238374608701176162_int_o] : collec6785001409445536906nt_int(produc5031841752791441652_int_o(aa_fun483471867002406230_int_o(aTP_Lamm_hd(P2),Q2))) = produc8806266479013559872nt_int(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2),aTP_Lamm_he(Q2)) ).

% Collect_case_prod
tff(fact_1669_Collect__case__prod,axiom,
    ! [P2: fun_nat_o,Q2: fun_Pr4796943543331071514_nat_o] : collec5903703980526211963at_nat(produc5864757623865647827_nat_o(aa_fun6835597611960591551_nat_o(aTP_Lamm_hf(P2),Q2))) = produc1809337555817847783at_nat(aa_fun_nat_o_set_nat(collect_nat,P2),aTP_Lamm_hg(Q2)) ).

% Collect_case_prod
tff(fact_1670_Collect__case__prod,axiom,
    ! [P2: fun_nat_o,Q2: fun_Pr3238374608701176162_int_o] : collec5600542410287506995nt_int(produc4306188689235752475_int_o(aa_fun5166854214830555983_int_o(aTP_Lamm_hh(P2),Q2))) = produc7854212285064481439nt_int(aa_fun_nat_o_set_nat(collect_nat,P2),aTP_Lamm_hi(Q2)) ).

% Collect_case_prod
tff(fact_1671_Collect__case__prod,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_Pr4796943543331071514_nat_o] : collec7751116662793354522at_nat(produc3235059054576640100_nat_o(aa_fun4662527895831814670_nat_o(aTP_Lamm_hj(P2),Q2))) = produc3424345432876038864at_nat(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2),aTP_Lamm_hk(Q2)) ).

% Collect_case_prod
tff(fact_1672_Collect__case__prod,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_nat_o] : collec364394062568603661nt_nat(produc5441891513209881465_nat_o(aa_fun2349983811102306768_nat_o(aTP_Lamm_hl(P2),Q2))) = produc1007865297704999105nt_nat(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2),aTP_Lamm_hm(Q2)) ).

% Collect_case_prod
tff(fact_1673_Collect__case__prod,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_Pr3238374608701176162_int_o] : collec7447955092554649554nt_int(produc1676490119946744748_int_o(aa_fun5005443200709765134_int_o(aTP_Lamm_hn(P2),Q2))) = produc245848125267896712nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2),aTP_Lamm_ho(Q2)) ).

% Collect_case_prod
tff(fact_1674_Collect__case__prod,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_o] : aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_fun5001253801653566085_nat_o(aTP_Lamm_hp(P2),Q2))) = produc457027306803732586at_nat(aa_fun_nat_o_set_nat(collect_nat,P2),aTP_Lamm_hq(Q2)) ).

% Collect_case_prod
tff(fact_1675_Collect__case__prod,axiom,
    ! [P2: fun_int_o,Q2: fun_int_o] : aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6736451341614545449_int_o(aTP_Lamm_hr(P2),Q2))) = produc1453890548195736866nt_int(aa_fun_int_o_set_int(collect_int,P2),aTP_Lamm_hs(Q2)) ).

% Collect_case_prod
tff(fact_1676_UNIV__Times__UNIV,axiom,
    produc2936396906406871148r_char(top_top_set_char,aTP_Lamm_ht) = top_to1235241553633177651r_char ).

% UNIV_Times_UNIV
tff(fact_1677_UNIV__Times__UNIV,axiom,
    produc1204236661673008499ar_rat(top_top_set_char,aTP_Lamm_hu) = top_to8437673547576850234ar_rat ).

% UNIV_Times_UNIV
tff(fact_1678_UNIV__Times__UNIV,axiom,
    produc1839366721759504235ar_nat(top_top_set_char,aTP_Lamm_hv) = top_to5594287848199751986ar_nat ).

% UNIV_Times_UNIV
tff(fact_1679_UNIV__Times__UNIV,axiom,
    produc1836876251250453959ar_int(top_top_set_char,aTP_Lamm_hw) = top_to3104204703268025230ar_int ).

% UNIV_Times_UNIV
tff(fact_1680_UNIV__Times__UNIV,axiom,
    produc1427039474575272051t_char(top_top_set_rat,aTP_Lamm_hx) = top_to3221206985699244474t_char ).

% UNIV_Times_UNIV
tff(fact_1681_UNIV__Times__UNIV,axiom,
    produc8728714639637146746at_rat(top_top_set_rat,aTP_Lamm_hy) = top_to3112507417142880193at_rat ).

% UNIV_Times_UNIV
tff(fact_1682_UNIV__Times__UNIV,axiom,
    produc140472662868866674at_nat(top_top_set_rat,aTP_Lamm_hz) = top_to269121717765781945at_nat ).

% UNIV_Times_UNIV
tff(fact_1683_UNIV__Times__UNIV,axiom,
    produc137982192359816398at_int(top_top_set_rat,aTP_Lamm_ia) = top_to7002410609688830997at_int ).

% UNIV_Times_UNIV
tff(fact_1684_UNIV__Times__UNIV,axiom,
    produc4754187946284079723t_char(top_top_set_nat,aTP_Lamm_ib) = top_to7107842681480589234t_char ).

% UNIV_Times_UNIV
tff(fact_1685_UNIV__Times__UNIV,axiom,
    produc9045269283572012658at_rat(top_top_set_nat,aTP_Lamm_ic) = top_to7513191607651882425at_rat ).

% UNIV_Times_UNIV
tff(fact_1686_The__split__eq,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : the_Pr4877703158196064750at_nat(aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,aa_set2950039546221647971_nat_o(aTP_Lamm_id(Xa),Y2))) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2) ).

% The_split_eq
tff(fact_1687_The__split__eq,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : the_Pr4604535484834969198at_nat(aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,aa_set8959908230745118793_nat_o(aTP_Lamm_ie(Xa),Y2))) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2) ).

% The_split_eq
tff(fact_1688_The__split__eq,axiom,
    ! [Xa: nat,Y2: num] : the_Pr8265262403268641490at_num(aa_fun4529110265455811444_num_o(produc4927758841916487424_num_o,aa_num3446304307616776968_num_o(aTP_Lamm_if(Xa),Y2))) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2) ).

% The_split_eq
tff(fact_1689_The__split__eq,axiom,
    ! [Xa: nat,Y2: nat] : the_Pr7557018466319803784at_nat(aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_nat8314798964848609180_nat_o(aTP_Lamm_ig(Xa),Y2))) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2) ).

% The_split_eq
tff(fact_1690_The__split__eq,axiom,
    ! [Xa: int,Y2: int] : the_Pr4378521158711661632nt_int(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_ih(Xa),Y2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2) ).

% The_split_eq
tff(fact_1691_Times__eq__cancel2,axiom,
    ! [Xa: nat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),C3)
     => ( ( produc457027306803732586at_nat(A4,aTP_Lamm_gm(C3)) = produc457027306803732586at_nat(B5,aTP_Lamm_gm(C3)) )
      <=> ( A4 = B5 ) ) ) ).

% Times_eq_cancel2
tff(fact_1692_Sigma__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_set_nat,D: fun_nat_set_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_set_nat(C3,X) = aa_nat_set_nat(D,X) ) )
       => ( produc457027306803732586at_nat(A4,C3) = produc457027306803732586at_nat(B5,D) ) ) ) ).

% Sigma_cong
tff(fact_1693_SigmaE2,axiom,
    ! [A: $o,B: $o,A4: set_o,B5: fun_o_set_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),product_Sigma_o_o(A4,B5))
     => ~ ( aa_set_o_o(member_o((A)),A4)
         => ~ aa_set_o_o(member_o((B)),aa_o_set_o(B5,(A))) ) ) ).

% SigmaE2
tff(fact_1694_SigmaE2,axiom,
    ! [A: $o,B: nat,A4: set_o,B5: fun_o_set_nat] :
      ( member2802428098988154798_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A)),B),product_Sigma_o_nat(A4,B5))
     => ~ ( aa_set_o_o(member_o((A)),A4)
         => ~ aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(A))) ) ) ).

% SigmaE2
tff(fact_1695_SigmaE2,axiom,
    ! [A: $o,B: int,A4: set_o,B5: fun_o_set_int] :
      ( member7847949116333733898_o_int(product_Pair_o_int((A),B),product_Sigma_o_int(A4,B5))
     => ~ ( aa_set_o_o(member_o((A)),A4)
         => ~ aa_set_int_o(member_int(B),aa_o_set_int(B5,(A))) ) ) ).

% SigmaE2
tff(fact_1696_SigmaE2,axiom,
    ! [A: nat,B: $o,A4: set_nat,B5: fun_nat_set_o] :
      ( member6310962623043647828_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A),(B)),product_Sigma_nat_o(A4,B5))
     => ~ ( aa_set_nat_o(member_nat(A),A4)
         => ~ aa_set_o_o(member_o((B)),aa_nat_set_o(B5,A)) ) ) ).

% SigmaE2
tff(fact_1697_SigmaE2,axiom,
    ! [A: nat,B: int,A4: set_nat,B5: fun_nat_set_int] :
      ( member4262671552274231302at_int(product_Pair_nat_int(A,B),produc454536836294682310at_int(A4,B5))
     => ~ ( aa_set_nat_o(member_nat(A),A4)
         => ~ aa_set_int_o(member_int(B),aa_nat_set_int(B5,A)) ) ) ).

% SigmaE2
tff(fact_1698_SigmaE2,axiom,
    ! [A: int,B: $o,A4: set_int,B5: fun_int_set_o] :
      ( member4489920277610959864_int_o(aa_o_P5078366980852708413_int_o(product_Pair_int_o(A),(B)),product_Sigma_int_o(A4,B5))
     => ~ ( aa_set_int_o(member_int(A),A4)
         => ~ aa_set_o_o(member_o((B)),aa_int_set_o(B5,A)) ) ) ).

% SigmaE2
tff(fact_1699_SigmaE2,axiom,
    ! [A: int,B: nat,A4: set_int,B5: fun_int_set_nat] :
      ( member216504246829706758nt_nat(aa_nat8602574874988167631nt_nat(product_Pair_int_nat(A),B),produc1456381018704787142nt_nat(A4,B5))
     => ~ ( aa_set_int_o(member_int(A),A4)
         => ~ aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,A)) ) ) ).

% SigmaE2
tff(fact_1700_SigmaE2,axiom,
    ! [A: nat,B: num,A4: set_nat,B5: fun_nat_set_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),produc6237731129814287028at_num(A4,B5))
     => ~ ( aa_set_nat_o(member_nat(A),A4)
         => ~ member_num(B,aa_nat_set_num(B5,A)) ) ) ).

% SigmaE2
tff(fact_1701_SigmaE2,axiom,
    ! [A: int,B: int,A4: set_int,B5: fun_int_set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),produc1453890548195736866nt_int(A4,B5))
     => ~ ( aa_set_int_o(member_int(A),A4)
         => ~ aa_set_int_o(member_int(B),aa_int_set_int(B5,A)) ) ) ).

% SigmaE2
tff(fact_1702_SigmaE2,axiom,
    ! [A: nat,B: nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),produc457027306803732586at_nat(A4,B5))
     => ~ ( aa_set_nat_o(member_nat(A),A4)
         => ~ aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A)) ) ) ).

% SigmaE2
tff(fact_1703_SigmaD2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,B5: fun_se3430100612660269923at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),produc6258361011560675592at_nat(A4,B5))
     => member6273874620674182256at_nat(B,aa_set7518756442812098908at_nat(B5,A)) ) ).

% SigmaD2
tff(fact_1704_SigmaD2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,B5: fun_se953073389798102371at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),produc7726050920695093128at_nat(A4,B5))
     => member2643936169264416010at_nat(B,aa_set6242432332095302236at_nat(B5,A)) ) ).

% SigmaD2
tff(fact_1705_SigmaD2,axiom,
    ! [A: nat,B: num,A4: set_nat,B5: fun_nat_set_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),produc6237731129814287028at_num(A4,B5))
     => member_num(B,aa_nat_set_num(B5,A)) ) ).

% SigmaD2
tff(fact_1706_SigmaD2,axiom,
    ! [A: int,B: int,A4: set_int,B5: fun_int_set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),produc1453890548195736866nt_int(A4,B5))
     => aa_set_int_o(member_int(B),aa_int_set_int(B5,A)) ) ).

% SigmaD2
tff(fact_1707_SigmaD2,axiom,
    ! [A: nat,B: nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),produc457027306803732586at_nat(A4,B5))
     => aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A)) ) ).

% SigmaD2
tff(fact_1708_SigmaD1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_se5479201105039950375at_nat,B5: fun_se3430100612660269923at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),produc6258361011560675592at_nat(A4,B5))
     => member6273874620674182256at_nat(A,A4) ) ).

% SigmaD1
tff(fact_1709_SigmaD1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_se7855581050983116737at_nat,B5: fun_se953073389798102371at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),produc7726050920695093128at_nat(A4,B5))
     => member2643936169264416010at_nat(A,A4) ) ).

% SigmaD1
tff(fact_1710_SigmaD1,axiom,
    ! [A: nat,B: num,A4: set_nat,B5: fun_nat_set_num] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),produc6237731129814287028at_num(A4,B5))
     => aa_set_nat_o(member_nat(A),A4) ) ).

% SigmaD1
tff(fact_1711_SigmaD1,axiom,
    ! [A: int,B: int,A4: set_int,B5: fun_int_set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),produc1453890548195736866nt_int(A4,B5))
     => aa_set_int_o(member_int(A),A4) ) ).

% SigmaD1
tff(fact_1712_SigmaD1,axiom,
    ! [A: nat,B: nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),produc457027306803732586at_nat(A4,B5))
     => aa_set_nat_o(member_nat(A),A4) ) ).

% SigmaD1
tff(fact_1713_SigmaE,axiom,
    ! [C: product_prod_o_o,A4: set_o,B5: fun_o_set_o] :
      ( member7466972457876170832od_o_o(C,product_Sigma_o_o(A4,B5))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ! [Y: $o] :
                ( aa_set_o_o(member_o((Y)),aa_o_set_o(B5,(X)))
               => ( C != aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)) ) ) ) ) ).

% SigmaE
tff(fact_1714_SigmaE,axiom,
    ! [C: product_prod_o_nat,A4: set_o,B5: fun_o_set_nat] :
      ( member2802428098988154798_o_nat(C,product_Sigma_o_nat(A4,B5))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ! [Y: nat] :
                ( aa_set_nat_o(member_nat(Y),aa_o_set_nat(B5,(X)))
               => ( C != aa_nat139401504800077733_o_nat(product_Pair_o_nat((X)),Y) ) ) ) ) ).

% SigmaE
tff(fact_1715_SigmaE,axiom,
    ! [C: product_prod_o_int,A4: set_o,B5: fun_o_set_int] :
      ( member7847949116333733898_o_int(C,product_Sigma_o_int(A4,B5))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ! [Y: int] :
                ( aa_set_int_o(member_int(Y),aa_o_set_int(B5,(X)))
               => ( C != product_Pair_o_int((X),Y) ) ) ) ) ).

% SigmaE
tff(fact_1716_SigmaE,axiom,
    ! [C: product_prod_nat_o,A4: set_nat,B5: fun_nat_set_o] :
      ( member6310962623043647828_nat_o(C,product_Sigma_nat_o(A4,B5))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ! [Y: $o] :
                ( aa_set_o_o(member_o((Y)),aa_nat_set_o(B5,X))
               => ( C != aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(X),(Y)) ) ) ) ) ).

% SigmaE
tff(fact_1717_SigmaE,axiom,
    ! [C: product_prod_nat_int,A4: set_nat,B5: fun_nat_set_int] :
      ( member4262671552274231302at_int(C,produc454536836294682310at_int(A4,B5))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ! [Y: int] :
                ( aa_set_int_o(member_int(Y),aa_nat_set_int(B5,X))
               => ( C != product_Pair_nat_int(X,Y) ) ) ) ) ).

% SigmaE
tff(fact_1718_SigmaE,axiom,
    ! [C: product_prod_int_o,A4: set_int,B5: fun_int_set_o] :
      ( member4489920277610959864_int_o(C,product_Sigma_int_o(A4,B5))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ! [Y: $o] :
                ( aa_set_o_o(member_o((Y)),aa_int_set_o(B5,X))
               => ( C != aa_o_P5078366980852708413_int_o(product_Pair_int_o(X),(Y)) ) ) ) ) ).

% SigmaE
tff(fact_1719_SigmaE,axiom,
    ! [C: product_prod_int_nat,A4: set_int,B5: fun_int_set_nat] :
      ( member216504246829706758nt_nat(C,produc1456381018704787142nt_nat(A4,B5))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ! [Y: nat] :
                ( aa_set_nat_o(member_nat(Y),aa_int_set_nat(B5,X))
               => ( C != aa_nat8602574874988167631nt_nat(product_Pair_int_nat(X),Y) ) ) ) ) ).

% SigmaE
tff(fact_1720_SigmaE,axiom,
    ! [C: product_prod_nat_num,A4: set_nat,B5: fun_nat_set_num] :
      ( member9148766508732265716at_num(C,produc6237731129814287028at_num(A4,B5))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ! [Y: num] :
                ( member_num(Y,aa_nat_set_num(B5,X))
               => ( C != aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y) ) ) ) ) ).

% SigmaE
tff(fact_1721_SigmaE,axiom,
    ! [C: product_prod_int_int,A4: set_int,B5: fun_int_set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(C),produc1453890548195736866nt_int(A4,B5))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ! [Y: int] :
                ( aa_set_int_o(member_int(Y),aa_int_set_int(B5,X))
               => ( C != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y) ) ) ) ) ).

% SigmaE
tff(fact_1722_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(C),produc457027306803732586at_nat(A4,B5))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ! [Y: nat] :
                ( aa_set_nat_o(member_nat(Y),aa_nat_set_nat(B5,X))
               => ( C != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y) ) ) ) ) ).

% SigmaE
tff(fact_1723_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_Pr2028404973800837934_nat_o] : collec7088162979684241874at_nat(produc6590410687421337004_nat_o(aa_fun443525351149406012_nat_o(aTP_Lamm_ii(P2),Q2))) = produc2761391749766926216at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2),aTP_Lamm_ij(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1724_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_Pr6984107991879650211_nat_o] : collec7029190964493513045at_nat(produc8758269395934548017_nat_o(aa_fun5503210535175347332_nat_o(aTP_Lamm_ik(P2),Q2))) = produc7672662199629908489at_nat(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2),aTP_Lamm_il(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1725_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,Q2: fun_Pr4496145827618669174_int_o] : collec6785001409445536906nt_int(produc5031841752791441652_int_o(aa_fun5031764673572090172_int_o(aTP_Lamm_im(P2),Q2))) = produc8806266479013559872nt_int(aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2),aTP_Lamm_in(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1726_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_nat_o,Q2: fun_na3970208099681795815_nat_o] : collec5903703980526211963at_nat(produc5864757623865647827_nat_o(aa_fun2376078316387139852_nat_o(aTP_Lamm_io(P2),Q2))) = produc1809337555817847783at_nat(aa_fun_nat_o_set_nat(collect_nat,P2),aTP_Lamm_ip(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1727_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_nat_o,Q2: fun_na6437948953499627055_int_o] : collec5600542410287506995nt_int(produc4306188689235752475_int_o(aa_fun6971596671186205596_int_o(aTP_Lamm_iq(P2),Q2))) = produc7854212285064481439nt_int(aa_fun_nat_o_set_nat(collect_nat,P2),aTP_Lamm_ir(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1728_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_Pr6550376307508196838_nat_o] : collec7751116662793354522at_nat(produc3235059054576640100_nat_o(aa_fun4736202344062180668_nat_o(aTP_Lamm_is(P2),Q2))) = produc3424345432876038864at_nat(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2),aTP_Lamm_it(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1729_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_Pr8286563733112057579_nat_o] : collec364394062568603661nt_nat(produc5441891513209881465_nat_o(aa_fun8858419945204301588_nat_o(aTP_Lamm_iu(P2),Q2))) = produc1007865297704999105nt_nat(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2),aTP_Lamm_iv(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1730_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,Q2: fun_Pr9018117161326028078_int_o] : collec7447955092554649554nt_int(produc1676490119946744748_int_o(aa_fun101069629630089020_int_o(aTP_Lamm_iw(P2),Q2))) = produc245848125267896712nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2),aTP_Lamm_ix(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1731_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_nat_o,Q2: fun_nat_fun_nat_o] : aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_fun6699960423209655484_nat_o(aTP_Lamm_iy(P2),Q2))) = produc457027306803732586at_nat(aa_fun_nat_o_set_nat(collect_nat,P2),aTP_Lamm_iz(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1732_Collect__case__prod__Sigma,axiom,
    ! [P2: fun_int_o,Q2: fun_int_fun_int_o] : aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_fun6133342665515662012_int_o(aTP_Lamm_ja(P2),Q2))) = produc1453890548195736866nt_int(aa_fun_int_o_set_int(collect_int,P2),aTP_Lamm_jb(Q2)) ).

% Collect_case_prod_Sigma
tff(fact_1733_Sigma__mono,axiom,
    ! [A4: set_nat,C3: set_nat,B5: fun_nat_set_nat,D: fun_nat_set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(B5,X)),aa_nat_set_nat(D,X)) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,produc457027306803732586at_nat(A4,B5)),produc457027306803732586at_nat(C3,D)) ) ) ).

% Sigma_mono
tff(fact_1734_Times__subset__cancel2,axiom,
    ! [Xa: nat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),C3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,produc457027306803732586at_nat(A4,aTP_Lamm_gm(C3))),produc457027306803732586at_nat(B5,aTP_Lamm_gm(C3)))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B5) ) ) ).

% Times_subset_cancel2
tff(fact_1735_mem__Times__iff,axiom,
    ! [Xa: product_prod_o_o,A4: set_o,B5: set_o] :
      ( member7466972457876170832od_o_o(Xa,product_Sigma_o_o(A4,aTP_Lamm_jc(B5)))
    <=> ( aa_set_o_o(member_o(aa_Pro2802791181778974573_o_o_o(product_fst_o_o,Xa)),A4)
        & aa_set_o_o(member_o(aa_Pro2802791181778974573_o_o_o(product_snd_o_o,Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1736_mem__Times__iff,axiom,
    ! [Xa: product_prod_o_nat,A4: set_o,B5: set_nat] :
      ( member2802428098988154798_o_nat(Xa,product_Sigma_o_nat(A4,aTP_Lamm_jd(B5)))
    <=> ( aa_set_o_o(member_o(aa_Pro5092300983057929219_nat_o(product_fst_o_nat,Xa)),A4)
        & aa_set_nat_o(member_nat(aa_Pro7345537496369368101at_nat(product_snd_o_nat,Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1737_mem__Times__iff,axiom,
    ! [Xa: product_prod_o_int,A4: set_o,B5: set_int] :
      ( member7847949116333733898_o_int(Xa,product_Sigma_o_int(A4,aTP_Lamm_je(B5)))
    <=> ( aa_set_o_o(member_o(product_fst_o_int(Xa)),A4)
        & aa_set_int_o(member_int(product_snd_o_int(Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1738_mem__Times__iff,axiom,
    ! [Xa: product_prod_nat_o,A4: set_nat,B5: set_o] :
      ( member6310962623043647828_nat_o(Xa,product_Sigma_nat_o(A4,aTP_Lamm_jf(B5)))
    <=> ( aa_set_nat_o(member_nat(aa_Pro1863436886664842699_o_nat(product_fst_nat_o,Xa)),A4)
        & aa_set_o_o(member_o(aa_Pro4464983059765771485at_o_o(product_snd_nat_o,Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1739_mem__Times__iff,axiom,
    ! [Xa: product_prod_nat_int,A4: set_nat,B5: set_int] :
      ( member4262671552274231302at_int(Xa,produc454536836294682310at_int(A4,aTP_Lamm_fr(B5)))
    <=> ( aa_set_nat_o(member_nat(product_fst_nat_int(Xa)),A4)
        & aa_set_int_o(member_int(product_snd_nat_int(Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1740_mem__Times__iff,axiom,
    ! [Xa: product_prod_int_o,A4: set_int,B5: set_o] :
      ( member4489920277610959864_int_o(Xa,product_Sigma_int_o(A4,aTP_Lamm_jg(B5)))
    <=> ( aa_set_int_o(member_int(aa_Pro2547522449196342987_o_int(product_fst_int_o,Xa)),A4)
        & aa_set_o_o(member_o(product_snd_int_o(Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1741_mem__Times__iff,axiom,
    ! [Xa: product_prod_int_nat,A4: set_int,B5: set_nat] :
      ( member216504246829706758nt_nat(Xa,produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5)))
    <=> ( aa_set_int_o(member_int(aa_Pro4612336859772192333at_int(product_fst_int_nat,Xa)),A4)
        & aa_set_nat_o(member_nat(product_snd_int_nat(Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1742_mem__Times__iff,axiom,
    ! [Xa: product_prod_int_int,A4: set_int,B5: set_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Xa),produc1453890548195736866nt_int(A4,aTP_Lamm_gf(B5)))
    <=> ( aa_set_int_o(member_int(aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),A4)
        & aa_set_int_o(member_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1743_mem__Times__iff,axiom,
    ! [Xa: product_prod_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5)))
    <=> ( aa_set_nat_o(member_nat(aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),A4)
        & aa_set_nat_o(member_nat(aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1744_mem__Times__iff,axiom,
    ! [Xa: produc652263959640188542_nat_o,A4: set_Pr1261947904930325089at_nat,B5: set_o] :
      ( member2200362271989506215_nat_o(Xa,produc203973861852599583_nat_o(A4,aTP_Lamm_jh(B5)))
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(produc6435950281062462810_nat_o(Xa)),A4)
        & aa_set_o_o(member_o(produc668916169454083228_nat_o(Xa)),B5) ) ) ).

% mem_Times_iff
tff(fact_1745_product__swap,axiom,
    ! [A4: set_nat,B5: set_nat] : aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(product_swap_nat_nat),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))) = produc457027306803732586at_nat(B5,aTP_Lamm_gm(A4)) ).

% product_swap
tff(fact_1746_Quotient__compose,axiom,
    ! [R1: fun_o_fun_o_o,Abs1: fun_o_o,Rep1: fun_o_o,T1: fun_o_fun_o_o,R22: fun_o_fun_o_o,Abs22: fun_o_nat,Rep22: fun_nat_o,T22: fun_o_fun_nat_o] :
      ( quotient_o_o(R1,Abs1,Rep1,T1)
     => ( quotient_o_nat(R22,Abs22,Rep22,T22)
       => quotient_o_nat(relcompp_o_o_o(T1,relcompp_o_o_o(R22,conversep_o_o(T1))),comp_o_nat_o(Abs22,Abs1),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Rep1),Rep22),relcompp_o_o_nat(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1747_Quotient__compose,axiom,
    ! [R1: fun_num_fun_num_o,Abs1: fun_num_nat,Rep1: fun_nat_num,T1: fun_num_fun_nat_o,R22: fun_nat_fun_nat_o,Abs22: fun_nat_int,Rep22: fun_int_nat,T22: fun_nat_fun_int_o] :
      ( quotient_num_nat(R1,Abs1,Rep1,T1)
     => ( quotient_nat_int(R22,Abs22,Rep22,T22)
       => quotient_num_int(relcompp_num_nat_num(T1,relcompp_nat_nat_num(R22,conversep_num_nat(T1))),comp_nat_int_num(Abs22,Abs1),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Rep1),Rep22),relcompp_num_nat_int(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1748_Quotient__compose,axiom,
    ! [R1: fun_nat_fun_nat_o,Abs1: fun_nat_int,Rep1: fun_int_nat,T1: fun_nat_fun_int_o,R22: fun_int_fun_int_o,Abs22: fun_int_int,Rep22: fun_int_int,T22: fun_int_fun_int_o] :
      ( quotient_nat_int(R1,Abs1,Rep1,T1)
     => ( quotient_int_int(R22,Abs22,Rep22,T22)
       => quotient_nat_int(relcompp_nat_int_nat(T1,relcompp_int_int_nat(R22,conversep_nat_int(T1))),comp_int_int_nat(Abs22,Abs1),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Rep1),Rep22),relcompp_nat_int_int(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1749_Quotient__compose,axiom,
    ! [R1: fun_nat_fun_nat_o,Abs1: fun_nat_o,Rep1: fun_o_nat,T1: fun_nat_fun_o_o,R22: fun_o_fun_o_o,Abs22: fun_o_o,Rep22: fun_o_o,T22: fun_o_fun_o_o] :
      ( quotient_nat_o(R1,Abs1,Rep1,T1)
     => ( quotient_o_o(R22,Abs22,Rep22,T22)
       => quotient_nat_o(relcompp_nat_o_nat(T1,relcompp_o_o_nat(R22,conversep_nat_o(T1))),aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,Abs22),Abs1),comp_o_nat_o(Rep1,Rep22),relcompp_nat_o_o(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1750_Quotient__compose,axiom,
    ! [R1: fun_int_fun_int_o,Abs1: fun_int_nat,Rep1: fun_nat_int,T1: fun_int_fun_nat_o,R22: fun_nat_fun_nat_o,Abs22: fun_nat_num,Rep22: fun_num_nat,T22: fun_nat_fun_num_o] :
      ( quotient_int_nat(R1,Abs1,Rep1,T1)
     => ( quotient_nat_num(R22,Abs22,Rep22,T22)
       => quotient_int_num(relcompp_int_nat_int(T1,relcompp_nat_nat_int(R22,conversep_int_nat(T1))),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,Abs22),Abs1),comp_nat_int_num(Rep1,Rep22),relcompp_int_nat_num(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1751_Quotient__compose,axiom,
    ! [R1: fun_nat_fun_nat_o,Abs1: fun_nat_nat,Rep1: fun_nat_nat,T1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o,Abs22: fun_nat_nat,Rep22: fun_nat_nat,T22: fun_nat_fun_nat_o] :
      ( quotient_nat_nat(R1,Abs1,Rep1,T1)
     => ( quotient_nat_nat(R22,Abs22,Rep22,T22)
       => quotient_nat_nat(relcompp_nat_nat_nat(T1,relcompp_nat_nat_nat(R22,conversep_nat_nat(T1))),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Abs22),Abs1),aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,Rep1),Rep22),relcompp_nat_nat_nat(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1752_Quotient__compose,axiom,
    ! [R1: fun_int_fun_int_o,Abs1: fun_int_int,Rep1: fun_int_int,T1: fun_int_fun_int_o,R22: fun_int_fun_int_o,Abs22: fun_int_nat,Rep22: fun_nat_int,T22: fun_int_fun_nat_o] :
      ( quotient_int_int(R1,Abs1,Rep1,T1)
     => ( quotient_int_nat(R22,Abs22,Rep22,T22)
       => quotient_int_nat(relcompp_int_int_int(T1,relcompp_int_int_int(R22,conversep_int_int(T1))),aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,Abs22),Abs1),comp_int_int_nat(Rep1,Rep22),relcompp_int_int_nat(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1753_Quotient__compose,axiom,
    ! [R1: fun_int_fun_int_o,Abs1: fun_int_int,Rep1: fun_int_int,T1: fun_int_fun_int_o,R22: fun_int_fun_int_o,Abs22: fun_int_Code_integer,Rep22: fun_Code_integer_int,T22: fun_in4891666191172872833eger_o] :
      ( quotient_int_int(R1,Abs1,Rep1,T1)
     => ( quotie2291698863146621601nteger(R22,Abs22,Rep22,T22)
       => quotie2291698863146621601nteger(relcompp_int_int_int(T1,relcompp_int_int_int(R22,conversep_int_int(T1))),comp_i1585864551200866970er_int(Abs22,Abs1),comp_i2224564030725804698nteger(Rep1,Rep22),relcom967497705283160040nteger(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1754_Quotient__compose,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,Abs1: fun_Pr2118491707810114563at_nat,Rep1: fun_Pr2118491707810114563at_nat,T1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o,Abs22: fun_Pr7647638413817135296at_int,Rep22: fun_in4514472496968388910at_nat,T22: fun_Pr4738963024209204295_int_o] :
      ( quotie4166942682936818134at_nat(R1,Abs1,Rep1,T1)
     => ( quotie1194848508323700631at_int(R22,Abs22,Rep22,T22)
       => quotie1194848508323700631at_int(relcom282571730187242960at_nat(T1,relcom282571730187242960at_nat(R22,conver6965824823119931701at_nat(T1))),comp_P4605287290794827253at_nat(Abs22,Abs1),comp_P7428279379633703887at_int(Rep1,Rep22),relcom8911704206367525917at_int(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1755_Quotient__compose,axiom,
    ! [R1: fun_fu7883101333820356676eger_o,Abs1: fun_fu7265242236430298039nteger,Rep1: fun_fu4800041728187187127nteger,T1: fun_fu7956894566369240976eger_o,R22: fun_fu4788845708739168348eger_o,Abs22: fun_fu5689865433563736081nteger,Rep22: fun_Co1981900195452921617nteger,T22: fun_fu855086506144988854eger_o] :
      ( quotie2859788831201447250nteger(R1,Abs1,Rep1,T1)
     => ( quotie2414878292700455852nteger(R22,Abs22,Rep22,T22)
       => quotie3207804130827357816nteger(relcom5548617121453999728nteger(T1,relcom2709991456463183012nteger(R22,conver45789976744698033nteger(T1))),comp_f1982395660488109180nteger(Abs22,Abs1),aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(Rep1),Rep22),relcom2426062844565531082nteger(T1,T22)) ) ) ).

% Quotient_compose
tff(fact_1756_swap__product,axiom,
    ! [A4: set_se5479201105039950375at_nat,B5: set_se5479201105039950375at_nat] : image_3313167189156628421at_nat(produc5456455677105495163at_nat(aTP_Lamm_ec),produc6258361011560675592at_nat(A4,aTP_Lamm_ji(B5))) = produc6258361011560675592at_nat(B5,aTP_Lamm_ji(A4)) ).

% swap_product
tff(fact_1757_swap__product,axiom,
    ! [A4: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] : aa_set4883462369968598652at_nat(image_960689587819713477at_nat(produc7548845685434208379at_nat(aTP_Lamm_ed)),produc7726050920695093128at_nat(A4,aTP_Lamm_jj(B5))) = produc7726050920695093128at_nat(B5,aTP_Lamm_jj(A4)) ).

% swap_product
tff(fact_1758_swap__product,axiom,
    ! [A4: set_num,B5: set_nat] : image_7762062833379733445at_num(produc5219324331026901427at_num(aTP_Lamm_ee),produc7810729746965977780um_nat(A4,aTP_Lamm_jk(B5))) = produc6237731129814287028at_num(B5,aTP_Lamm_jl(A4)) ).

% swap_product
tff(fact_1759_swap__product,axiom,
    ! [A4: set_nat,B5: set_nat] : aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(produc2626176000494625587at_nat(aTP_Lamm_ef)),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))) = produc457027306803732586at_nat(B5,aTP_Lamm_gm(A4)) ).

% swap_product
tff(fact_1760_swap__product,axiom,
    ! [A4: set_int,B5: set_int] : image_2653370878348428101nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_eg),produc1453890548195736866nt_int(A4,aTP_Lamm_gf(B5))) = produc1453890548195736866nt_int(B5,aTP_Lamm_gf(A4)) ).

% swap_product
tff(fact_1761_map__prod__surj__on,axiom,
    ! [F: fun_nat_rat,A4: set_nat,A6: set_rat,G: fun_int_nat,B5: set_int,B7: set_nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = A6 )
     => ( ( aa_set_int_set_nat(image_int_nat(G),B5) = B7 )
       => ( image_3755276878268593137at_nat(produc8749313216793205150nt_nat(F,G),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc140472662868866674at_nat(A6,aTP_Lamm_jm(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1762_map__prod__surj__on,axiom,
    ! [F: fun_nat_rat,A4: set_nat,A6: set_rat,G: fun_int_int,B5: set_int,B7: set_int] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = A6 )
     => ( ( aa_set_int_set_int(image_int_int(G),B5) = B7 )
       => ( image_8800797895614172237at_int(produc8746822746284154874nt_int(F,G),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc137982192359816398at_int(A6,aTP_Lamm_jn(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1763_map__prod__surj__on,axiom,
    ! [F: fun_nat_nat,A4: set_nat,A6: set_nat,G: fun_int_int,B5: set_int,B7: set_int] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = A6 )
     => ( ( aa_set_int_set_int(image_int_int(G),B5) = B7 )
       => ( image_9117352539549038149at_int(produc7974749936722635250nt_int(F,G),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc454536836294682310at_int(A6,aTP_Lamm_fr(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1764_map__prod__surj__on,axiom,
    ! [F: fun_nat_int,A4: set_nat,A6: set_int,G: fun_int_nat,B5: set_int,B7: set_nat] :
      ( ( aa_set_nat_set_int(image_nat_int(F),A4) = A6 )
     => ( ( aa_set_int_set_nat(image_int_nat(G),B5) = B7 )
       => ( image_5071185234104513605nt_nat(produc8348893967468530162nt_nat(F,G),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc1456381018704787142nt_nat(A6,aTP_Lamm_ga(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1765_map__prod__surj__on,axiom,
    ! [F: fun_nat_int,A4: set_nat,A6: set_int,G: fun_int_int,B5: set_int,B7: set_int] :
      ( ( aa_set_nat_set_int(image_nat_int(F),A4) = A6 )
     => ( ( aa_set_int_set_int(image_int_int(G),B5) = B7 )
       => ( image_893334214595316897nt_int(produc8346403496959479886nt_int(F,G),produc454536836294682310at_int(A4,aTP_Lamm_fr(B5))) = produc1453890548195736866nt_int(A6,aTP_Lamm_gf(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1766_map__prod__surj__on,axiom,
    ! [F: fun_int_nat,A4: set_int,A6: set_nat,G: fun_nat_rat,B5: set_nat,B7: set_rat] :
      ( ( aa_set_int_set_nat(image_int_nat(F),A4) = A6 )
     => ( ( aa_set_nat_set_rat(image_nat_rat(G),B5) = B7 )
       => ( image_3420722608207172081at_rat(produc4032938703151184286at_rat(F,G),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc9045269283572012658at_rat(A6,aTP_Lamm_gl(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1767_map__prod__surj__on,axiom,
    ! [F: fun_int_nat,A4: set_int,A6: set_nat,G: fun_nat_int,B5: set_nat,B7: set_int] :
      ( ( aa_set_int_set_nat(image_int_nat(F),A4) = A6 )
     => ( ( aa_set_nat_set_int(image_nat_int(G),B5) = B7 )
       => ( image_2751100147091372101at_int(produc4665578292728629746at_int(F,G),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc454536836294682310at_int(A6,aTP_Lamm_fr(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1768_map__prod__surj__on,axiom,
    ! [F: fun_int_nat,A4: set_int,A6: set_nat,G: fun_int_int,B5: set_int,B7: set_int] :
      ( ( aa_set_int_set_nat(image_int_nat(F),A4) = A6 )
     => ( ( aa_set_int_set_int(image_int_int(G),B5) = B7 )
       => ( image_1654017166447373545at_int(produc5664932004629684302nt_int(F,G),produc1453890548195736866nt_int(A4,aTP_Lamm_gf(B5))) = produc454536836294682310at_int(A6,aTP_Lamm_fr(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1769_map__prod__surj__on,axiom,
    ! [F: fun_int_int,A4: set_int,A6: set_int,G: fun_nat_rat,B5: set_nat,B7: set_rat] :
      ( ( aa_set_int_set_int(image_int_int(F),A4) = A6 )
     => ( ( aa_set_nat_set_rat(image_nat_rat(G),B5) = B7 )
       => ( image_4420076320108226637nt_rat(produc4404592263388028922at_rat(F,G),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc821250958618291406nt_rat(A6,aTP_Lamm_jo(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1770_map__prod__surj__on,axiom,
    ! [F: fun_int_int,A4: set_int,A6: set_int,G: fun_nat_nat,B5: set_nat,B7: set_nat] :
      ( ( aa_set_int_set_int(image_int_int(F),A4) = A6 )
     => ( ( aa_set_nat_set_nat(image_nat_nat(G),B5) = B7 )
       => ( image_7928304878501623365nt_nat(produc5039722323474524658at_nat(F,G),produc1456381018704787142nt_nat(A4,aTP_Lamm_ga(B5))) = produc1456381018704787142nt_nat(A6,aTP_Lamm_ga(B7)) ) ) ) ).

% map_prod_surj_on
tff(fact_1771_left__total__OO,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o] :
      ( left_t5646938096895998950at_nat(R2)
     => ( left_t1861001511420785031at_int(S3)
       => left_t1861001511420785031at_int(relcom8911704206367525917at_int(R2,S3)) ) ) ).

% left_total_OO
tff(fact_1772_left__total__OO,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o] :
      ( left_total_int_int(R2)
     => ( left_t2374113004080657585nteger(S3)
       => left_t2374113004080657585nteger(relcom967497705283160040nteger(R2,S3)) ) ) ).

% left_total_OO
tff(fact_1773_left__unique__OO,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o] :
      ( left_u7296345956935681743at_nat(R2)
     => ( left_u4005249686606446046at_int(S3)
       => left_u4005249686606446046at_int(relcom8911704206367525917at_int(R2,S3)) ) ) ).

% left_unique_OO
tff(fact_1774_left__unique__OO,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o] :
      ( left_unique_int_int(R2)
     => ( left_u6795595883297946906nteger(S3)
       => left_u6795595883297946906nteger(relcom967497705283160040nteger(R2,S3)) ) ) ).

% left_unique_OO
tff(fact_1775_Product__Type_Oproduct__def,axiom,
    ! [A4: set_nat,B5: set_nat] : produc929334515565554804at_nat(A4,B5) = produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5)) ).

% Product_Type.product_def
tff(fact_1776_member__product,axiom,
    ! [Xa: produc3843707927480180839at_nat,A4: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),produc3241716407206042770at_nat(A4,B5))
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),produc7726050920695093128at_nat(A4,aTP_Lamm_jj(B5))) ) ).

% member_product
tff(fact_1777_member__product,axiom,
    ! [Xa: product_prod_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),produc929334515565554804at_nat(A4,B5))
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))) ) ).

% member_product
tff(fact_1778_snd__image__times,axiom,
    ! [A4: set_Product_unit,B5: set_Product_unit] :
      image_262127665112854123t_unit(produc6786084904866402309t_unit,produc6121607364029222536t_unit(A4,aTP_Lamm_jp(B5))) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,B5) ).

% snd_image_times
tff(fact_1779_snd__image__times,axiom,
    ! [A4: set_Product_unit,B5: set_o] :
      image_1801747192096393899it_o_o(produc7461459235754980752unit_o,produc4996980317674386579unit_o(A4,aTP_Lamm_jq(B5))) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_o,B5) ).

% snd_image_times
tff(fact_1780_snd__image__times,axiom,
    ! [A4: set_Product_unit,B5: set_nat] :
      image_6725568594985674851at_nat(produc996414825743149016it_nat,produc6875377002296012693it_nat(A4,aTP_Lamm_jr(B5))) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,B5) ).

% snd_image_times
tff(fact_1781_snd__image__times,axiom,
    ! [A4: set_o,B5: set_Product_unit] :
      image_4419268941131634678t_unit(produc2875488738488844090t_unit,produc411009820408249917t_unit(A4,aTP_Lamm_js(B5))) = $ite(A4 = bot_bot_set_o,bot_bo3957492148770167129t_unit,B5) ).

% snd_image_times
tff(fact_1782_snd__image__times,axiom,
    ! [A4: set_o,B5: set_o] :
      image_7896445794123959606_o_o_o(product_snd_o_o,product_Sigma_o_o(A4,aTP_Lamm_jc(B5))) = $ite(A4 = bot_bot_set_o,bot_bot_set_o,B5) ).

% snd_image_times
tff(fact_1783_snd__image__times,axiom,
    ! [A4: set_o,B5: set_nat] :
      image_4888532181312399726at_nat(product_snd_o_nat,product_Sigma_o_nat(A4,aTP_Lamm_jd(B5))) = $ite(A4 = bot_bot_set_o,bot_bot_set_nat,B5) ).

% snd_image_times
tff(fact_1784_snd__image__times,axiom,
    ! [A4: set_nat,B5: set_Product_unit] :
      image_5511522805742971924t_unit(produc8850949007409916470t_unit,produc5506539147108004339t_unit(A4,aTP_Lamm_jt(B5))) = $ite(A4 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,B5) ).

% snd_image_times
tff(fact_1785_snd__image__times,axiom,
    ! [A4: set_nat,B5: set_o] :
      image_7478392465989118804at_o_o(product_snd_nat_o,product_Sigma_nat_o(A4,aTP_Lamm_jf(B5))) = $ite(A4 = bot_bot_set_nat,bot_bot_set_o,B5) ).

% snd_image_times
tff(fact_1786_snd__image__times,axiom,
    ! [A4: set_nat,B5: set_nat] :
      image_2486076414777270412at_nat(product_snd_nat_nat,produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))) = $ite(A4 = bot_bot_set_nat,bot_bot_set_nat,B5) ).

% snd_image_times
tff(fact_1787_snd__image__times,axiom,
    ! [A4: set_ref_a,B5: set_he8644174446132254637t_unit] :
      image_17660834435998945t_unit(produc1126150082675451753t_unit,produc3946373962484743846t_unit(A4,aTP_Lamm_ju(B5))) = $ite(A4 = bot_bot_set_ref_a,bot_bo3686245532977126937t_unit,B5) ).

% snd_image_times
tff(fact_1788_fst__image__times,axiom,
    ! [A4: set_Product_unit,B5: set_Product_unit] :
      image_262127665112854123t_unit(produc6963936691090040003t_unit,produc6121607364029222536t_unit(A4,aTP_Lamm_jp(B5))) = $ite(B5 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,A4) ).

% fst_image_times
tff(fact_1789_fst__image__times,axiom,
    ! [A4: set_o,B5: set_Product_unit] :
      image_8742399529945250561unit_o(produc5219655352205527288t_unit,produc411009820408249917t_unit(A4,aTP_Lamm_js(B5))) = $ite(B5 = bot_bo3957492148770167129t_unit,bot_bot_set_o,A4) ).

% fst_image_times
tff(fact_1790_fst__image__times,axiom,
    ! [A4: set_nat,B5: set_Product_unit] :
      image_3884712837920099081it_nat(produc3800628931346975736t_unit,produc5506539147108004339t_unit(A4,aTP_Lamm_jt(B5))) = $ite(B5 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,A4) ).

% fst_image_times
tff(fact_1791_fst__image__times,axiom,
    ! [A4: set_Product_unit,B5: set_o] :
      image_4246157487713968288t_unit(produc582253812616888142unit_o,produc4996980317674386579unit_o(A4,aTP_Lamm_jq(B5))) = $ite(B5 = bot_bot_set_o,bot_bo3957492148770167129t_unit,A4) ).

% fst_image_times
tff(fact_1792_fst__image__times,axiom,
    ! [A4: set_o,B5: set_o] :
      image_7896445794123959606_o_o_o(product_fst_o_o,product_Sigma_o_o(A4,aTP_Lamm_jc(B5))) = $ite(B5 = bot_bot_set_o,bot_bot_set_o,A4) ).

% fst_image_times
tff(fact_1793_fst__image__times,axiom,
    ! [A4: set_nat,B5: set_o] :
      image_8629803608462650132_o_nat(product_fst_nat_o,product_Sigma_nat_o(A4,aTP_Lamm_jf(B5))) = $ite(B5 = bot_bot_set_o,bot_bot_set_nat,A4) ).

% fst_image_times
tff(fact_1794_fst__image__times,axiom,
    ! [A4: set_Product_unit,B5: set_nat] :
      image_7474191388529947450t_unit(produc5169466786534984090it_nat,produc6875377002296012693it_nat(A4,aTP_Lamm_jr(B5))) = $ite(B5 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,A4) ).

% fst_image_times
tff(fact_1795_fst__image__times,axiom,
    ! [A4: set_o,B5: set_nat] :
      image_8105710389281276538_nat_o(product_fst_o_nat,product_Sigma_o_nat(A4,aTP_Lamm_jd(B5))) = $ite(B5 = bot_bot_set_nat,bot_bot_set_o,A4) ).

% fst_image_times
tff(fact_1796_fst__image__times,axiom,
    ! [A4: set_nat,B5: set_nat] :
      image_2486076414777270412at_nat(product_fst_nat_nat,produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))) = $ite(B5 = bot_bot_set_nat,bot_bot_set_nat,A4) ).

% fst_image_times
tff(fact_1797_fst__image__times,axiom,
    ! [A4: set_ref_a,B5: set_he8644174446132254637t_unit] :
      image_7136766188234240905_ref_a(produc473321116828791851t_unit,produc3946373962484743846t_unit(A4,aTP_Lamm_ju(B5))) = $ite(B5 = bot_bo3686245532977126937t_unit,bot_bot_set_ref_a,A4) ).

% fst_image_times
tff(fact_1798_image__is__empty,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( ( aa_set_int_set_int(image_int_int(F),A4) = bot_bot_set_int )
    <=> ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_1799_image__is__empty,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = bot_bot_set_rat )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_1800_image__is__empty,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( ( aa_set_nat_set_int(image_nat_int(F),A4) = bot_bot_set_int )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_1801_image__is__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A4: set_Product_unit] :
      ( ( image_405062704495631173t_unit(F,A4) = bot_bo3957492148770167129t_unit )
    <=> ( A4 = bot_bo3957492148770167129t_unit ) ) ).

% image_is_empty
tff(fact_1802_image__is__empty,axiom,
    ! [F: fun_o_Product_unit,A4: set_o] :
      ( ( image_o_Product_unit(F,A4) = bot_bo3957492148770167129t_unit )
    <=> ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_1803_image__is__empty,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( ( image_8730104196221521654t_unit(F,A4) = bot_bo3957492148770167129t_unit )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_1804_image__is__empty,axiom,
    ! [F: fun_Product_unit_o,A4: set_Product_unit] :
      ( ( image_Product_unit_o(F,A4) = bot_bot_set_o )
    <=> ( A4 = bot_bo3957492148770167129t_unit ) ) ).

% image_is_empty
tff(fact_1805_image__is__empty,axiom,
    ! [F: fun_o_o,A4: set_o] :
      ( ( aa_set_o_set_o(image_o_o(F),A4) = bot_bot_set_o )
    <=> ( A4 = bot_bot_set_o ) ) ).

% image_is_empty
tff(fact_1806_image__is__empty,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( ( image_nat_o(F,A4) = bot_bot_set_o )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% image_is_empty
tff(fact_1807_image__is__empty,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( ( aa_set_int_set_nat(image_int_nat(F),A4) = bot_bot_set_nat )
    <=> ( A4 = bot_bot_set_int ) ) ).

% image_is_empty
tff(fact_1808_empty__is__image,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(image_int_int(F),A4) )
    <=> ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_1809_empty__is__image,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( ( bot_bot_set_rat = aa_set_nat_set_rat(image_nat_rat(F),A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_1810_empty__is__image,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( ( bot_bot_set_int = aa_set_nat_set_int(image_nat_int(F),A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_1811_empty__is__image,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A4: set_Product_unit] :
      ( ( bot_bo3957492148770167129t_unit = image_405062704495631173t_unit(F,A4) )
    <=> ( A4 = bot_bo3957492148770167129t_unit ) ) ).

% empty_is_image
tff(fact_1812_empty__is__image,axiom,
    ! [F: fun_o_Product_unit,A4: set_o] :
      ( ( bot_bo3957492148770167129t_unit = image_o_Product_unit(F,A4) )
    <=> ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_1813_empty__is__image,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( ( bot_bo3957492148770167129t_unit = image_8730104196221521654t_unit(F,A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_1814_empty__is__image,axiom,
    ! [F: fun_Product_unit_o,A4: set_Product_unit] :
      ( ( bot_bot_set_o = image_Product_unit_o(F,A4) )
    <=> ( A4 = bot_bo3957492148770167129t_unit ) ) ).

% empty_is_image
tff(fact_1815_empty__is__image,axiom,
    ! [F: fun_o_o,A4: set_o] :
      ( ( bot_bot_set_o = aa_set_o_set_o(image_o_o(F),A4) )
    <=> ( A4 = bot_bot_set_o ) ) ).

% empty_is_image
tff(fact_1816_empty__is__image,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( ( bot_bot_set_o = image_nat_o(F,A4) )
    <=> ( A4 = bot_bot_set_nat ) ) ).

% empty_is_image
tff(fact_1817_empty__is__image,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( ( bot_bot_set_nat = aa_set_int_set_nat(image_int_nat(F),A4) )
    <=> ( A4 = bot_bot_set_int ) ) ).

% empty_is_image
tff(fact_1818_image__empty,axiom,
    ! [F: fun_int_int] : aa_set_int_set_int(image_int_int(F),bot_bot_set_int) = bot_bot_set_int ).

% image_empty
tff(fact_1819_image__empty,axiom,
    ! [F: fun_int_nat] : aa_set_int_set_nat(image_int_nat(F),bot_bot_set_int) = bot_bot_set_nat ).

% image_empty
tff(fact_1820_image__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit] : image_405062704495631173t_unit(F,bot_bo3957492148770167129t_unit) = bot_bo3957492148770167129t_unit ).

% image_empty
tff(fact_1821_image__empty,axiom,
    ! [F: fun_Product_unit_o] : image_Product_unit_o(F,bot_bo3957492148770167129t_unit) = bot_bot_set_o ).

% image_empty
tff(fact_1822_image__empty,axiom,
    ! [F: fun_Product_unit_nat] : image_875570014554754200it_nat(F,bot_bo3957492148770167129t_unit) = bot_bot_set_nat ).

% image_empty
tff(fact_1823_image__empty,axiom,
    ! [F: fun_o_Product_unit] : image_o_Product_unit(F,bot_bot_set_o) = bot_bo3957492148770167129t_unit ).

% image_empty
tff(fact_1824_image__empty,axiom,
    ! [F: fun_o_o] : aa_set_o_set_o(image_o_o(F),bot_bot_set_o) = bot_bot_set_o ).

% image_empty
tff(fact_1825_image__empty,axiom,
    ! [F: fun_o_nat] : image_o_nat(F,bot_bot_set_o) = bot_bot_set_nat ).

% image_empty
tff(fact_1826_image__empty,axiom,
    ! [F: fun_nat_rat] : aa_set_nat_set_rat(image_nat_rat(F),bot_bot_set_nat) = bot_bot_set_rat ).

% image_empty
tff(fact_1827_image__empty,axiom,
    ! [F: fun_nat_int] : aa_set_nat_set_int(image_nat_int(F),bot_bot_set_nat) = bot_bot_set_int ).

% image_empty
tff(fact_1828_Sigma__empty1,axiom,
    ! [B5: fun_se953073389798102371at_nat] : produc7726050920695093128at_nat(bot_bo3083307316010499117at_nat,B5) = bot_bo228742789529271731at_nat ).

% Sigma_empty1
tff(fact_1829_Sigma__empty1,axiom,
    ! [B5: fun_nat_set_nat] : produc457027306803732586at_nat(bot_bot_set_nat,B5) = bot_bo2099793752762293965at_nat ).

% Sigma_empty1
tff(fact_1830_Sigma__empty2,axiom,
    ! [A4: set_se7855581050983116737at_nat] : produc7726050920695093128at_nat(A4,aTP_Lamm_jv) = bot_bo228742789529271731at_nat ).

% Sigma_empty2
tff(fact_1831_Sigma__empty2,axiom,
    ! [A4: set_nat] : produc457027306803732586at_nat(A4,aTP_Lamm_jw) = bot_bo2099793752762293965at_nat ).

% Sigma_empty2
tff(fact_1832_Times__empty,axiom,
    ! [A4: set_Product_unit,B5: set_Product_unit] :
      ( ( produc6121607364029222536t_unit(A4,aTP_Lamm_jp(B5)) = bot_bo1654022762305692595t_unit )
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( B5 = bot_bo3957492148770167129t_unit ) ) ) ).

% Times_empty
tff(fact_1833_Times__empty,axiom,
    ! [A4: set_Product_unit,B5: set_o] :
      ( ( produc4996980317674386579unit_o(A4,aTP_Lamm_jq(B5)) = bot_bo3195362268743713342unit_o )
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
tff(fact_1834_Times__empty,axiom,
    ! [A4: set_Product_unit,B5: set_nat] :
      ( ( produc6875377002296012693it_nat(A4,aTP_Lamm_jr(B5)) = bot_bo2472676621365177558it_nat )
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
tff(fact_1835_Times__empty,axiom,
    ! [A4: set_o,B5: set_Product_unit] :
      ( ( produc411009820408249917t_unit(A4,aTP_Lamm_js(B5)) = bot_bo4402188677660116584t_unit )
    <=> ( ( A4 = bot_bot_set_o )
        | ( B5 = bot_bo3957492148770167129t_unit ) ) ) ).

% Times_empty
tff(fact_1836_Times__empty,axiom,
    ! [A4: set_o,B5: set_o] :
      ( ( product_Sigma_o_o(A4,aTP_Lamm_jc(B5)) = bot_bo7073875226086086771od_o_o )
    <=> ( ( A4 = bot_bot_set_o )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
tff(fact_1837_Times__empty,axiom,
    ! [A4: set_o,B5: set_nat] :
      ( ( product_Sigma_o_nat(A4,aTP_Lamm_jd(B5)) = bot_bo66526543525301729_o_nat )
    <=> ( ( A4 = bot_bot_set_o )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
tff(fact_1838_Times__empty,axiom,
    ! [A4: set_nat,B5: set_Product_unit] :
      ( ( produc5506539147108004339t_unit(A4,aTP_Lamm_jt(B5)) = bot_bo5043309098482578556t_unit )
    <=> ( ( A4 = bot_bot_set_nat )
        | ( B5 = bot_bo3957492148770167129t_unit ) ) ) ).

% Times_empty
tff(fact_1839_Times__empty,axiom,
    ! [A4: set_nat,B5: set_o] :
      ( ( product_Sigma_nat_o(A4,aTP_Lamm_jf(B5)) = bot_bo1114129665703605383_nat_o )
    <=> ( ( A4 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
tff(fact_1840_Times__empty,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5)) = bot_bo2099793752762293965at_nat )
    <=> ( ( A4 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
tff(fact_1841_Times__empty,axiom,
    ! [A4: set_Product_unit,B5: set_Pr1261947904930325089at_nat] :
      ( ( produc3904733337175673596at_nat(A4,aTP_Lamm_jx(B5)) = bot_bo6659885642593197735at_nat )
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
tff(fact_1842_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,aTP_Lamm_jy((P2)))) = $ite((P2),top_to6833984726390702231at_nat,bot_bo228742789529271731at_nat) ).

% Collect_const_case_prod
tff(fact_1843_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_jz((P2)))) = $ite((P2),top_to4669805908274784177at_nat,bot_bo2099793752762293965at_nat) ).

% Collect_const_case_prod
tff(fact_1844_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ka((P2)))) = $ite((P2),top_to4366644338036079209nt_int,bot_bo1796632182523588997nt_int) ).

% Collect_const_case_prod
tff(fact_1845_Sigma__empty__iff,axiom,
    ! [I3: set_se7855581050983116737at_nat,X3: fun_se953073389798102371at_nat] :
      ( ( produc7726050920695093128at_nat(I3,X3) = bot_bo228742789529271731at_nat )
    <=> ! [X4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X4,I3)
         => ( aa_set6242432332095302236at_nat(X3,X4) = bot_bo3083307316010499117at_nat ) ) ) ).

% Sigma_empty_iff
tff(fact_1846_Sigma__empty__iff,axiom,
    ! [I3: set_nat,X3: fun_nat_set_nat] :
      ( ( produc457027306803732586at_nat(I3,X3) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),I3)
         => ( aa_nat_set_nat(X3,X4) = bot_bot_set_nat ) ) ) ).

% Sigma_empty_iff
tff(fact_1847_subset__emptyI,axiom,
    ! [A4: set_int] :
      ( ! [X: int] : ~ aa_set_int_o(member_int(X),A4)
     => aa_set_int_o(ord_less_eq_set_int(A4),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1848_subset__emptyI,axiom,
    ! [A4: set_Product_unit] :
      ( ! [X: product_unit] : ~ member_Product_unit(X,A4)
     => ord_le3507040750410214029t_unit(A4,bot_bo3957492148770167129t_unit) ) ).

% subset_emptyI
tff(fact_1849_subset__emptyI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ! [X: produc3843707927480180839at_nat] : ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),bot_bo228742789529271731at_nat) ) ).

% subset_emptyI
tff(fact_1850_subset__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [X: product_prod_nat_nat] : ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_1851_subset__emptyI,axiom,
    ! [A4: set_o] :
      ( ! [X: $o] : ~ aa_set_o_o(member_o((X)),A4)
     => aa_set_o_o(ord_less_eq_set_o(A4),bot_bot_set_o) ) ).

% subset_emptyI
tff(fact_1852_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X: nat] : ~ aa_set_nat_o(member_nat(X),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(A4),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1853_less__by__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( A4 = bot_bo228742789529271731at_nat )
     => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),B5) ) ).

% less_by_empty
tff(fact_1854_less__by__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B5) ) ).

% less_by_empty
tff(fact_1855_times__eq__iff,axiom,
    ! [A4: set_Product_unit,B5: set_Product_unit,C3: set_Product_unit,D: set_Product_unit] :
      ( ( produc6121607364029222536t_unit(A4,aTP_Lamm_jp(B5)) = produc6121607364029222536t_unit(C3,aTP_Lamm_jp(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bo3957492148770167129t_unit )
            | ( B5 = bot_bo3957492148770167129t_unit ) )
          & ( ( C3 = bot_bo3957492148770167129t_unit )
            | ( D = bot_bo3957492148770167129t_unit ) ) ) ) ) ).

% times_eq_iff
tff(fact_1856_times__eq__iff,axiom,
    ! [A4: set_Product_unit,B5: set_o,C3: set_Product_unit,D: set_o] :
      ( ( produc4996980317674386579unit_o(A4,aTP_Lamm_jq(B5)) = produc4996980317674386579unit_o(C3,aTP_Lamm_jq(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bo3957492148770167129t_unit )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C3 = bot_bo3957492148770167129t_unit )
            | ( D = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
tff(fact_1857_times__eq__iff,axiom,
    ! [A4: set_Product_unit,B5: set_nat,C3: set_Product_unit,D: set_nat] :
      ( ( produc6875377002296012693it_nat(A4,aTP_Lamm_jr(B5)) = produc6875377002296012693it_nat(C3,aTP_Lamm_jr(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bo3957492148770167129t_unit )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C3 = bot_bo3957492148770167129t_unit )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
tff(fact_1858_times__eq__iff,axiom,
    ! [A4: set_o,B5: set_Product_unit,C3: set_o,D: set_Product_unit] :
      ( ( produc411009820408249917t_unit(A4,aTP_Lamm_js(B5)) = produc411009820408249917t_unit(C3,aTP_Lamm_js(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bot_set_o )
            | ( B5 = bot_bo3957492148770167129t_unit ) )
          & ( ( C3 = bot_bot_set_o )
            | ( D = bot_bo3957492148770167129t_unit ) ) ) ) ) ).

% times_eq_iff
tff(fact_1859_times__eq__iff,axiom,
    ! [A4: set_o,B5: set_o,C3: set_o,D: set_o] :
      ( ( product_Sigma_o_o(A4,aTP_Lamm_jc(B5)) = product_Sigma_o_o(C3,aTP_Lamm_jc(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bot_set_o )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C3 = bot_bot_set_o )
            | ( D = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
tff(fact_1860_times__eq__iff,axiom,
    ! [A4: set_o,B5: set_nat,C3: set_o,D: set_nat] :
      ( ( product_Sigma_o_nat(A4,aTP_Lamm_jd(B5)) = product_Sigma_o_nat(C3,aTP_Lamm_jd(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bot_set_o )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C3 = bot_bot_set_o )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
tff(fact_1861_times__eq__iff,axiom,
    ! [A4: set_nat,B5: set_Product_unit,C3: set_nat,D: set_Product_unit] :
      ( ( produc5506539147108004339t_unit(A4,aTP_Lamm_jt(B5)) = produc5506539147108004339t_unit(C3,aTP_Lamm_jt(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bot_set_nat )
            | ( B5 = bot_bo3957492148770167129t_unit ) )
          & ( ( C3 = bot_bot_set_nat )
            | ( D = bot_bo3957492148770167129t_unit ) ) ) ) ) ).

% times_eq_iff
tff(fact_1862_times__eq__iff,axiom,
    ! [A4: set_nat,B5: set_o,C3: set_nat,D: set_o] :
      ( ( product_Sigma_nat_o(A4,aTP_Lamm_jf(B5)) = product_Sigma_nat_o(C3,aTP_Lamm_jf(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bot_set_nat )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C3 = bot_bot_set_nat )
            | ( D = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
tff(fact_1863_times__eq__iff,axiom,
    ! [A4: set_nat,B5: set_nat,C3: set_nat,D: set_nat] :
      ( ( produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5)) = produc457027306803732586at_nat(C3,aTP_Lamm_gm(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bot_set_nat )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C3 = bot_bot_set_nat )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
tff(fact_1864_times__eq__iff,axiom,
    ! [A4: set_Product_unit,B5: set_Pr1261947904930325089at_nat,C3: set_Product_unit,D: set_Pr1261947904930325089at_nat] :
      ( ( produc3904733337175673596at_nat(A4,aTP_Lamm_jx(B5)) = produc3904733337175673596at_nat(C3,aTP_Lamm_jx(D)) )
    <=> ( ( ( A4 = C3 )
          & ( B5 = D ) )
        | ( ( ( A4 = bot_bo3957492148770167129t_unit )
            | ( B5 = bot_bo2099793752762293965at_nat ) )
          & ( ( C3 = bot_bo3957492148770167129t_unit )
            | ( D = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
tff(fact_1865_times__subset__iff,axiom,
    ! [A4: set_Product_unit,C3: set_Product_unit,B5: set_Product_unit,D: set_Product_unit] :
      ( ord_le930853871177014503t_unit(produc6121607364029222536t_unit(A4,aTP_Lamm_jp(C3)),produc6121607364029222536t_unit(B5,aTP_Lamm_jp(D)))
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( C3 = bot_bo3957492148770167129t_unit )
        | ( ord_le3507040750410214029t_unit(A4,B5)
          & ord_le3507040750410214029t_unit(C3,D) ) ) ) ).

% times_subset_iff
tff(fact_1866_times__subset__iff,axiom,
    ! [A4: set_Product_unit,C3: set_o,B5: set_Product_unit,D: set_o] :
      ( ord_le8534741403851954546unit_o(produc4996980317674386579unit_o(A4,aTP_Lamm_jq(C3)),produc4996980317674386579unit_o(B5,aTP_Lamm_jq(D)))
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( C3 = bot_bot_set_o )
        | ( ord_le3507040750410214029t_unit(A4,B5)
          & aa_set_o_o(ord_less_eq_set_o(C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1867_times__subset__iff,axiom,
    ! [A4: set_Product_unit,C3: set_nat,B5: set_Product_unit,D: set_nat] :
      ( ord_le8121013740822022818it_nat(produc6875377002296012693it_nat(A4,aTP_Lamm_jr(C3)),produc6875377002296012693it_nat(B5,aTP_Lamm_jr(D)))
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( C3 = bot_bot_set_nat )
        | ( ord_le3507040750410214029t_unit(A4,B5)
          & aa_set_nat_o(ord_less_eq_set_nat(C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1868_times__subset__iff,axiom,
    ! [A4: set_o,C3: set_Product_unit,B5: set_o,D: set_Product_unit] :
      ( ord_le518195775913581980t_unit(produc411009820408249917t_unit(A4,aTP_Lamm_js(C3)),produc411009820408249917t_unit(B5,aTP_Lamm_js(D)))
    <=> ( ( A4 = bot_bot_set_o )
        | ( C3 = bot_bo3957492148770167129t_unit )
        | ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
          & ord_le3507040750410214029t_unit(C3,D) ) ) ) ).

% times_subset_iff
tff(fact_1869_times__subset__iff,axiom,
    ! [A4: set_o,C3: set_o,B5: set_o,D: set_o] :
      ( ord_le192195778054999463od_o_o(product_Sigma_o_o(A4,aTP_Lamm_jc(C3)),product_Sigma_o_o(B5,aTP_Lamm_jc(D)))
    <=> ( ( A4 = bot_bot_set_o )
        | ( C3 = bot_bot_set_o )
        | ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
          & aa_set_o_o(ord_less_eq_set_o(C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1870_times__subset__iff,axiom,
    ! [A4: set_o,C3: set_nat,B5: set_o,D: set_nat] :
      ( ord_le6472378536984071597_o_nat(product_Sigma_o_nat(A4,aTP_Lamm_jd(C3)),product_Sigma_o_nat(B5,aTP_Lamm_jd(D)))
    <=> ( ( A4 = bot_bot_set_o )
        | ( C3 = bot_bot_set_nat )
        | ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
          & aa_set_nat_o(ord_less_eq_set_nat(C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1871_times__subset__iff,axiom,
    ! [A4: set_nat,C3: set_Product_unit,B5: set_nat,D: set_Product_unit] :
      ( ord_le1468274181084648008t_unit(produc5506539147108004339t_unit(A4,aTP_Lamm_jt(C3)),produc5506539147108004339t_unit(B5,aTP_Lamm_jt(D)))
    <=> ( ( A4 = bot_bot_set_nat )
        | ( C3 = bot_bo3957492148770167129t_unit )
        | ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
          & ord_le3507040750410214029t_unit(C3,D) ) ) ) ).

% times_subset_iff
tff(fact_1872_times__subset__iff,axiom,
    ! [A4: set_nat,C3: set_o,B5: set_nat,D: set_o] :
      ( ord_le7519981659162375251_nat_o(product_Sigma_nat_o(A4,aTP_Lamm_jf(C3)),product_Sigma_nat_o(B5,aTP_Lamm_jf(D)))
    <=> ( ( A4 = bot_bot_set_nat )
        | ( C3 = bot_bot_set_o )
        | ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
          & aa_set_o_o(ord_less_eq_set_o(C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1873_times__subset__iff,axiom,
    ! [A4: set_nat,C3: set_nat,B5: set_nat,D: set_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,produc457027306803732586at_nat(A4,aTP_Lamm_gm(C3))),produc457027306803732586at_nat(B5,aTP_Lamm_gm(D)))
    <=> ( ( A4 = bot_bot_set_nat )
        | ( C3 = bot_bot_set_nat )
        | ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
          & aa_set_nat_o(ord_less_eq_set_nat(C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1874_times__subset__iff,axiom,
    ! [A4: set_Product_unit,C3: set_Pr1261947904930325089at_nat,B5: set_Product_unit,D: set_Pr1261947904930325089at_nat] :
      ( ord_le8681960759543468507at_nat(produc3904733337175673596at_nat(A4,aTP_Lamm_jx(C3)),produc3904733337175673596at_nat(B5,aTP_Lamm_jx(D)))
    <=> ( ( A4 = bot_bo3957492148770167129t_unit )
        | ( C3 = bot_bo2099793752762293965at_nat )
        | ( ord_le3507040750410214029t_unit(A4,B5)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C3),D) ) ) ) ).

% times_subset_iff
tff(fact_1875_fst__image__Sigma,axiom,
    ! [A4: set_o,B5: fun_o_1833192751255691672t_unit] : image_8742399529945250561unit_o(produc5219655352205527288t_unit,produc411009820408249917t_unit(A4,B5)) = aa_fun_o_o_set_o(collect_o,aa_fun67325214945128089un_o_o(aTP_Lamm_kb(A4),B5)) ).

% fst_image_Sigma
tff(fact_1876_fst__image__Sigma,axiom,
    ! [A4: set_int,B5: fun_in4731230917697954842t_unit] : image_7716461468295075337it_int(produc917050878549155996t_unit,produc2622961094310184599t_unit(A4,B5)) = aa_fun_int_o_set_int(collect_int,aa_fun1317521766530534759_int_o(aTP_Lamm_kc(A4),B5)) ).

% fst_image_Sigma
tff(fact_1877_fst__image__Sigma,axiom,
    ! [A4: set_nat,B5: fun_na896991816813928310t_unit] : image_3884712837920099081it_nat(produc3800628931346975736t_unit,produc5506539147108004339t_unit(A4,B5)) = aa_fun_nat_o_set_nat(collect_nat,aa_fun4664615919704916071_nat_o(aTP_Lamm_kd(A4),B5)) ).

% fst_image_Sigma
tff(fact_1878_fst__image__Sigma,axiom,
    ! [A4: set_o,B5: fun_o_set_o] : image_7896445794123959606_o_o_o(product_fst_o_o,product_Sigma_o_o(A4,B5)) = aa_fun_o_o_set_o(collect_o,aa_fun4619127644298054564un_o_o(aTP_Lamm_ke(A4),B5)) ).

% fst_image_Sigma
tff(fact_1879_fst__image__Sigma,axiom,
    ! [A4: set_int,B5: fun_int_set_o] : image_90517134139374612_o_int(product_fst_int_o,product_Sigma_int_o(A4,B5)) = aa_fun_int_o_set_int(collect_int,aa_fun4291261932532047132_int_o(aTP_Lamm_kf(A4),B5)) ).

% fst_image_Sigma
tff(fact_1880_fst__image__Sigma,axiom,
    ! [A4: set_nat,B5: fun_nat_set_o] : image_8629803608462650132_o_nat(product_fst_nat_o,product_Sigma_nat_o(A4,B5)) = aa_fun_nat_o_set_nat(collect_nat,aa_fun4904187145036355100_nat_o(aTP_Lamm_kg(A4),B5)) ).

% fst_image_Sigma
tff(fact_1881_fst__image__Sigma,axiom,
    ! [A4: set_o,B5: fun_o_set_nat] : image_8105710389281276538_nat_o(product_fst_o_nat,product_Sigma_o_nat(A4,B5)) = aa_fun_o_o_set_o(collect_o,aa_fun8423158570077026744un_o_o(aTP_Lamm_kh(A4),B5)) ).

% fst_image_Sigma
tff(fact_1882_fst__image__Sigma,axiom,
    ! [A4: set_int,B5: fun_int_set_nat] : image_2855239504505064772at_int(product_fst_int_nat,produc1456381018704787142nt_nat(A4,B5)) = aa_fun_int_o_set_int(collect_int,aa_fun1965424933913381656_int_o(aTP_Lamm_ki(A4),B5)) ).

% fst_image_Sigma
tff(fact_1883_fst__image__Sigma,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat] : image_2486076414777270412at_nat(product_fst_nat_nat,produc457027306803732586at_nat(A4,B5)) = aa_fun_nat_o_set_nat(collect_nat,aa_fun2108954047564320208_nat_o(aTP_Lamm_kj(A4),B5)) ).

% fst_image_Sigma
tff(fact_1884_fst__image__Sigma,axiom,
    ! [A4: set_ref_a,B5: fun_re9091073251222685225t_unit] : image_7136766188234240905_ref_a(produc473321116828791851t_unit,produc3946373962484743846t_unit(A4,B5)) = collect_ref_a(aa_fun7429455494379205479ef_a_o(aTP_Lamm_kk(A4),B5)) ).

% fst_image_Sigma
tff(fact_1885_rp__inv__image__rp,axiom,
    ! [P2: produc1319942482725812455at_nat,F: fun_se320340636488258563at_nat] :
      ( aa_Pro4518350200472837805_nat_o(fun_re2478310338295953701at_nat,P2)
     => aa_Pro4518350200472837805_nat_o(fun_re2478310338295953701at_nat,fun_rp8445993449185644698at_nat(P2,F)) ) ).

% rp_inv_image_rp
tff(fact_1886_the__elem__image__unique,axiom,
    ! [A4: set_int,F: fun_int_nat,Xa: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [Y: int] :
            ( aa_set_int_o(member_int(Y),A4)
           => ( aa_int_nat(F,Y) = aa_int_nat(F,Xa) ) )
       => ( aa_set_nat_nat(the_elem_nat,aa_set_int_set_nat(image_int_nat(F),A4)) = aa_int_nat(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_1887_the__elem__image__unique,axiom,
    ! [A4: set_int,F: fun_int_int,Xa: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [Y: int] :
            ( aa_set_int_o(member_int(Y),A4)
           => ( aa_int_int(F,Y) = aa_int_int(F,Xa) ) )
       => ( aa_set_int_int(the_elem_int,aa_set_int_set_int(image_int_int(F),A4)) = aa_int_int(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_1888_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: fun_nat_rat,Xa: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y: nat] :
            ( aa_set_nat_o(member_nat(Y),A4)
           => ( aa_nat_rat(F,Y) = aa_nat_rat(F,Xa) ) )
       => ( aa_set_rat_rat(the_elem_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) = aa_nat_rat(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_1889_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: fun_nat_nat,Xa: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y: nat] :
            ( aa_set_nat_o(member_nat(Y),A4)
           => ( aa_nat_nat(F,Y) = aa_nat_nat(F,Xa) ) )
       => ( aa_set_nat_nat(the_elem_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_nat_nat(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_1890_the__elem__image__unique,axiom,
    ! [A4: set_nat,F: fun_nat_int,Xa: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [Y: nat] :
            ( aa_set_nat_o(member_nat(Y),A4)
           => ( aa_nat_int(F,Y) = aa_nat_int(F,Xa) ) )
       => ( aa_set_int_int(the_elem_int,aa_set_nat_set_int(image_nat_int(F),A4)) = aa_nat_int(F,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_1891_snd__image__Sigma,axiom,
    ! [A4: set_ref_a,B5: fun_re9091073251222685225t_unit] : image_17660834435998945t_unit(produc1126150082675451753t_unit,produc3946373962484743846t_unit(A4,B5)) = comple6989579708249578522t_unit(image_5570573867351727369t_unit(B5,A4)) ).

% snd_image_Sigma
tff(fact_1892_snd__image__Sigma,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat] : image_2486076414777270412at_nat(product_snd_nat_nat,produc457027306803732586at_nat(A4,B5)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4)) ).

% snd_image_Sigma
tff(fact_1893_range__constant,axiom,
    ! [Xa: product_unit] : image_4337914935673063477t_unit(aTP_Lamm_kl(Xa),top_top_set_char) = aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit) ).

% range_constant
tff(fact_1894_range__constant,axiom,
    ! [Xa: $o] : image_char_o(aTP_Lamm_km((Xa)),top_top_set_char) = aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o) ).

% range_constant
tff(fact_1895_range__constant,axiom,
    ! [Xa: nat] : image_char_nat(aTP_Lamm_kn(Xa),top_top_set_char) = aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat) ).

% range_constant
tff(fact_1896_range__constant,axiom,
    ! [Xa: product_unit] : image_2327579789247107822t_unit(aTP_Lamm_ko(Xa),top_top_set_rat) = aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit) ).

% range_constant
tff(fact_1897_range__constant,axiom,
    ! [Xa: $o] : image_rat_o(aTP_Lamm_kp((Xa)),top_top_set_rat) = aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o) ).

% range_constant
tff(fact_1898_range__constant,axiom,
    ! [Xa: nat] : image_rat_nat(aTP_Lamm_kq(Xa),top_top_set_rat) = aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat) ).

% range_constant
tff(fact_1899_range__constant,axiom,
    ! [Xa: rat] : aa_set_nat_set_rat(image_nat_rat(aTP_Lamm_kr(Xa)),top_top_set_nat) = insert_rat(Xa,bot_bot_set_rat) ).

% range_constant
tff(fact_1900_range__constant,axiom,
    ! [Xa: int] : aa_set_nat_set_int(image_nat_int(aTP_Lamm_ks(Xa)),top_top_set_nat) = aa_set_int_set_int(insert_int(Xa),bot_bot_set_int) ).

% range_constant
tff(fact_1901_range__constant,axiom,
    ! [Xa: product_unit] : image_8730104196221521654t_unit(aTP_Lamm_kt(Xa),top_top_set_nat) = aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit) ).

% range_constant
tff(fact_1902_range__constant,axiom,
    ! [Xa: $o] : image_nat_o(aa_o_fun_nat_o(aTP_Lamm_ku,(Xa)),top_top_set_nat) = aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o) ).

% range_constant
tff(fact_1903_INT__constant,axiom,
    ! [C: set_char,A4: set_Product_unit] :
      comple7965528767983648192t_char(image_5325685772115642575t_char(aTP_Lamm_kv(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_char,C) ).

% INT_constant
tff(fact_1904_INT__constant,axiom,
    ! [C: set_char,A4: set_o] :
      comple7965528767983648192t_char(image_o_set_char(aTP_Lamm_kw(C),A4)) = $ite(A4 = bot_bot_set_o,top_top_set_char,C) ).

% INT_constant
tff(fact_1905_INT__constant,axiom,
    ! [C: set_char,A4: set_nat] :
      comple7965528767983648192t_char(image_nat_set_char(aTP_Lamm_kx(C),A4)) = $ite(A4 = bot_bot_set_nat,top_top_set_char,C) ).

% INT_constant
tff(fact_1906_INT__constant,axiom,
    ! [C: set_rat,A4: set_Product_unit] :
      comple4298007329820168263et_rat(image_4665621433434589782et_rat(aTP_Lamm_ky(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_rat,C) ).

% INT_constant
tff(fact_1907_INT__constant,axiom,
    ! [C: set_rat,A4: set_o] :
      comple4298007329820168263et_rat(image_o_set_rat(aTP_Lamm_kz(C),A4)) = $ite(A4 = bot_bot_set_o,top_top_set_rat,C) ).

% INT_constant
tff(fact_1908_INT__constant,axiom,
    ! [C: set_rat,A4: set_nat] :
      comple4298007329820168263et_rat(image_nat_set_rat(aTP_Lamm_gl(C),A4)) = $ite(A4 = bot_bot_set_nat,top_top_set_rat,C) ).

% INT_constant
tff(fact_1909_INT__constant,axiom,
    ! [C: set_int,A4: set_Product_unit] :
      comple3628384868704368283et_int(image_3995998972318789802et_int(aTP_Lamm_la(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_int,C) ).

% INT_constant
tff(fact_1910_INT__constant,axiom,
    ! [C: set_int,A4: set_o] :
      comple3628384868704368283et_int(image_o_set_int(aTP_Lamm_je(C),A4)) = $ite(A4 = bot_bot_set_o,top_top_set_int,C) ).

% INT_constant
tff(fact_1911_INT__constant,axiom,
    ! [C: set_int,A4: set_nat] :
      comple3628384868704368283et_int(image_nat_set_int(aTP_Lamm_fr(C),A4)) = $ite(A4 = bot_bot_set_nat,top_top_set_int,C) ).

% INT_constant
tff(fact_1912_INT__constant,axiom,
    ! [C: set_nat,A4: set_Product_unit] :
      comple7806235888213564991et_nat(image_8173849991827986510et_nat(aTP_Lamm_jr(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_nat,C) ).

% INT_constant
tff(fact_1913_neg__fun__distr1,axiom,
    ! [R2: fun_in4891666191172872833eger_o,R4: fun_Co2081214437782444048eger_o,S3: fun_num_fun_num_o,S4: fun_num_fun_num_o] :
      ( left_u6795595883297946906nteger(R2)
     => ( right_7281159819619925802nteger(R2)
       => ( right_5866438497372881682nteger(R4)
         => ( left_t2884206969071330nteger(R4)
           => ord_le7846159698440226291_num_o(bNF_re6718328864250387230um_num(relcom8320293257703906841nteger(R2,R4),relcompp_num_num_num(S3,S4)),relcom4173150990864199262er_num(bNF_re6718328864250387230um_num(R2,S3),bNF_re40212662316476751um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1914_neg__fun__distr1,axiom,
    ! [R2: fun_in4891666191172872833eger_o,R4: fun_Co8353275306710069249_int_o,S3: fun_num_fun_num_o,S4: fun_num_fun_num_o] :
      ( left_u6795595883297946906nteger(R2)
     => ( right_7281159819619925802nteger(R2)
       => ( right_7836862732553076961er_int(R4)
         => ( left_t1735413524555719857er_int(R4)
           => ord_le6143850148583763428_num_o(bNF_re7626690874201225453um_num(relcom328798225758222312er_int(R2,R4),relcompp_num_num_num(S3,S4)),relcom8634939491988117549nt_num(bNF_re6718328864250387230um_num(R2,S3),bNF_re8313794964846168862um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1915_neg__fun__distr1,axiom,
    ! [R2: fun_int_fun_int_o,R4: fun_int_fun_int_o,S3: fun_num_fun_num_o,S4: fun_num_fun_num_o] :
      ( left_unique_int_int(R2)
     => ( right_total_int_int(R2)
       => ( right_unique_int_int(R4)
         => ( left_total_int_int(R4)
           => ord_le6143850148583763428_num_o(bNF_re7626690874201225453um_num(relcompp_int_int_int(R2,R4),relcompp_num_num_num(S3,S4)),relcom8560396942509463548nt_num(bNF_re7626690874201225453um_num(R2,S3),bNF_re7626690874201225453um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1916_neg__fun__distr1,axiom,
    ! [R2: fun_int_fun_int_o,R4: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o,S4: fun_num_fun_num_o] :
      ( left_unique_int_int(R2)
     => ( right_total_int_int(R2)
       => ( right_8475562212078014689nteger(R4)
         => ( left_t2374113004080657585nteger(R4)
           => ord_le7846159698440226291_num_o(bNF_re6718328864250387230um_num(relcom967497705283160040nteger(R2,R4),relcompp_num_num_num(S3,S4)),relcom2053567597505753645er_num(bNF_re7626690874201225453um_num(R2,S3),bNF_re6718328864250387230um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1917_neg__fun__distr1,axiom,
    ! [R2: fun_int_fun_int_o,R4: fun_in4891666191172872833eger_o,S3: fun_int_fun_int_o,S4: fun_in4891666191172872833eger_o] :
      ( left_unique_int_int(R2)
     => ( right_total_int_int(R2)
       => ( right_8475562212078014689nteger(R4)
         => ( left_t2374113004080657585nteger(R4)
           => ord_le8507588522978985054eger_o(bNF_re3379532845092657523nteger(relcom967497705283160040nteger(R2,R4),relcom967497705283160040nteger(S3,S4)),relcom444288326586520852nteger(bNF_re4712519889275205905nt_int(R2,S3),bNF_re3379532845092657523nteger(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1918_neg__fun__distr1,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o,R4: fun_int_fun_int_o,S3: fun_nat_fun_nat_o,S4: fun_nat_fun_nat_o] :
      ( left_u4005249686606446046at_int(R2)
     => ( right_1157779049177982926at_int(R2)
       => ( right_unique_int_int(R4)
         => ( left_total_int_int(R4)
           => ord_le150341387783501559_nat_o(bNF_re4555766996558763186at_nat(relcom2949942408940504938nt_int(R2,R4),relcompp_nat_nat_nat(S3,S4)),relcom3487822462411678311nt_nat(bNF_re4555766996558763186at_nat(R2,S3),bNF_re3715656647883201625at_nat(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1919_neg__fun__distr1,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,R4: fun_Pr4738963024209204295_int_o,S3: fun_nat_fun_nat_o,S4: fun_nat_fun_nat_o] :
      ( left_u7296345956935681743at_nat(R2)
     => ( right_4094035864313884895at_nat(R2)
       => ( right_8621877368619751767at_int(R4)
         => ( left_t1861001511420785031at_int(R4)
           => ord_le150341387783501559_nat_o(bNF_re4555766996558763186at_nat(relcom8911704206367525917at_int(R2,R4),relcompp_nat_nat_nat(S3,S4)),relcom8518660330154759978nt_nat(bNF_re8246922863344978751at_nat(R2,S3),bNF_re4555766996558763186at_nat(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1920_neg__fun__distr1,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,R4: fun_Pr4738963024209204295_int_o,S3: fun_int_fun_int_o,S4: fun_in4891666191172872833eger_o] :
      ( left_u7296345956935681743at_nat(R2)
     => ( right_4094035864313884895at_nat(R2)
       => ( right_8621877368619751767at_int(R4)
         => ( left_t1861001511420785031at_int(R4)
           => ord_le9076892547966399182eger_o(bNF_re6202762543029211035nteger(relcom8911704206367525917at_int(R2,R4),relcom967497705283160040nteger(S3,S4)),relcom4123854648742082103nteger(bNF_re20414067882207223nt_int(R2,S3),bNF_re6202762543029211035nteger(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1921_neg__fun__distr1,axiom,
    ! [R2: fun_int_fun_int_o,R4: fun_in4891666191172872833eger_o,S3: fun_Pr2028404973800837934_nat_o,S4: fun_Pr4738963024209204295_int_o] :
      ( left_unique_int_int(R2)
     => ( right_total_int_int(R2)
       => ( right_8475562212078014689nteger(R4)
         => ( left_t2374113004080657585nteger(R4)
           => ord_le2215172278096386656_int_o(bNF_re2828929484557863493at_int(relcom967497705283160040nteger(R2,R4),relcom8911704206367525917at_int(S3,S4)),relcom7261622426688989495er_int(bNF_re8766404587435608311at_nat(R2,S3),bNF_re2828929484557863493at_int(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1922_neg__fun__distr1,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,R4: fun_Pr4738963024209204295_int_o,S3: fun_Pr2028404973800837934_nat_o,S4: fun_Pr4738963024209204295_int_o] :
      ( left_u7296345956935681743at_nat(R2)
     => ( right_4094035864313884895at_nat(R2)
       => ( right_8621877368619751767at_int(R4)
         => ( left_t1861001511420785031at_int(R4)
           => ord_le6589200864103675154_int_o(bNF_re7400052026677387805at_int(relcom8911704206367525917at_int(R2,R4),relcom8911704206367525917at_int(S3,S4)),relcom5467440779735672206nt_int(bNF_re2241393799969408733at_nat(R2,S3),bNF_re7400052026677387805at_int(R4,S4))) ) ) ) ) ).

% neg_fun_distr1
tff(fact_1923_neg__fun__distr2,axiom,
    ! [R4: fun_Co2081214437782444048eger_o,S4: fun_num_fun_num_o,R2: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o] :
      ( right_5866438497372881682nteger(R4)
     => ( left_t2884206969071330nteger(R4)
       => ( left_unique_num_num(S4)
         => ( right_total_num_num(S4)
           => ord_le7846159698440226291_num_o(bNF_re6718328864250387230um_num(relcom8320293257703906841nteger(R2,R4),relcompp_num_num_num(S3,S4)),relcom4173150990864199262er_num(bNF_re6718328864250387230um_num(R2,S3),bNF_re40212662316476751um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1924_neg__fun__distr2,axiom,
    ! [R4: fun_Co8353275306710069249_int_o,S4: fun_num_fun_num_o,R2: fun_in4891666191172872833eger_o,S3: fun_num_fun_num_o] :
      ( right_7836862732553076961er_int(R4)
     => ( left_t1735413524555719857er_int(R4)
       => ( left_unique_num_num(S4)
         => ( right_total_num_num(S4)
           => ord_le6143850148583763428_num_o(bNF_re7626690874201225453um_num(relcom328798225758222312er_int(R2,R4),relcompp_num_num_num(S3,S4)),relcom8634939491988117549nt_num(bNF_re6718328864250387230um_num(R2,S3),bNF_re8313794964846168862um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1925_neg__fun__distr2,axiom,
    ! [R4: fun_int_fun_int_o,S4: fun_num_fun_num_o,R2: fun_int_fun_int_o,S3: fun_num_fun_num_o] :
      ( right_unique_int_int(R4)
     => ( left_total_int_int(R4)
       => ( left_unique_num_num(S4)
         => ( right_total_num_num(S4)
           => ord_le6143850148583763428_num_o(bNF_re7626690874201225453um_num(relcompp_int_int_int(R2,R4),relcompp_num_num_num(S3,S4)),relcom8560396942509463548nt_num(bNF_re7626690874201225453um_num(R2,S3),bNF_re7626690874201225453um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1926_neg__fun__distr2,axiom,
    ! [R4: fun_in4891666191172872833eger_o,S4: fun_num_fun_num_o,R2: fun_int_fun_int_o,S3: fun_num_fun_num_o] :
      ( right_8475562212078014689nteger(R4)
     => ( left_t2374113004080657585nteger(R4)
       => ( left_unique_num_num(S4)
         => ( right_total_num_num(S4)
           => ord_le7846159698440226291_num_o(bNF_re6718328864250387230um_num(relcom967497705283160040nteger(R2,R4),relcompp_num_num_num(S3,S4)),relcom2053567597505753645er_num(bNF_re7626690874201225453um_num(R2,S3),bNF_re6718328864250387230um_num(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1927_neg__fun__distr2,axiom,
    ! [R4: fun_in4891666191172872833eger_o,S4: fun_in4891666191172872833eger_o,R2: fun_int_fun_int_o,S3: fun_int_fun_int_o] :
      ( right_8475562212078014689nteger(R4)
     => ( left_t2374113004080657585nteger(R4)
       => ( left_u6795595883297946906nteger(S4)
         => ( right_7281159819619925802nteger(S4)
           => ord_le8507588522978985054eger_o(bNF_re3379532845092657523nteger(relcom967497705283160040nteger(R2,R4),relcom967497705283160040nteger(S3,S4)),relcom444288326586520852nteger(bNF_re4712519889275205905nt_int(R2,S3),bNF_re3379532845092657523nteger(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1928_neg__fun__distr2,axiom,
    ! [R4: fun_int_fun_int_o,S4: fun_nat_fun_nat_o,R2: fun_Pr4738963024209204295_int_o,S3: fun_nat_fun_nat_o] :
      ( right_unique_int_int(R4)
     => ( left_total_int_int(R4)
       => ( left_unique_nat_nat(S4)
         => ( right_total_nat_nat(S4)
           => ord_le150341387783501559_nat_o(bNF_re4555766996558763186at_nat(relcom2949942408940504938nt_int(R2,R4),relcompp_nat_nat_nat(S3,S4)),relcom3487822462411678311nt_nat(bNF_re4555766996558763186at_nat(R2,S3),bNF_re3715656647883201625at_nat(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1929_neg__fun__distr2,axiom,
    ! [R4: fun_Pr4738963024209204295_int_o,S4: fun_nat_fun_nat_o,R2: fun_Pr2028404973800837934_nat_o,S3: fun_nat_fun_nat_o] :
      ( right_8621877368619751767at_int(R4)
     => ( left_t1861001511420785031at_int(R4)
       => ( left_unique_nat_nat(S4)
         => ( right_total_nat_nat(S4)
           => ord_le150341387783501559_nat_o(bNF_re4555766996558763186at_nat(relcom8911704206367525917at_int(R2,R4),relcompp_nat_nat_nat(S3,S4)),relcom8518660330154759978nt_nat(bNF_re8246922863344978751at_nat(R2,S3),bNF_re4555766996558763186at_nat(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1930_neg__fun__distr2,axiom,
    ! [R4: fun_Pr4738963024209204295_int_o,S4: fun_in4891666191172872833eger_o,R2: fun_Pr2028404973800837934_nat_o,S3: fun_int_fun_int_o] :
      ( right_8621877368619751767at_int(R4)
     => ( left_t1861001511420785031at_int(R4)
       => ( left_u6795595883297946906nteger(S4)
         => ( right_7281159819619925802nteger(S4)
           => ord_le9076892547966399182eger_o(bNF_re6202762543029211035nteger(relcom8911704206367525917at_int(R2,R4),relcom967497705283160040nteger(S3,S4)),relcom4123854648742082103nteger(bNF_re20414067882207223nt_int(R2,S3),bNF_re6202762543029211035nteger(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1931_neg__fun__distr2,axiom,
    ! [R4: fun_in4891666191172872833eger_o,S4: fun_Pr4738963024209204295_int_o,R2: fun_int_fun_int_o,S3: fun_Pr2028404973800837934_nat_o] :
      ( right_8475562212078014689nteger(R4)
     => ( left_t2374113004080657585nteger(R4)
       => ( left_u4005249686606446046at_int(S4)
         => ( right_1157779049177982926at_int(S4)
           => ord_le2215172278096386656_int_o(bNF_re2828929484557863493at_int(relcom967497705283160040nteger(R2,R4),relcom8911704206367525917at_int(S3,S4)),relcom7261622426688989495er_int(bNF_re8766404587435608311at_nat(R2,S3),bNF_re2828929484557863493at_int(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1932_neg__fun__distr2,axiom,
    ! [R4: fun_Pr4738963024209204295_int_o,S4: fun_Pr4738963024209204295_int_o,R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr2028404973800837934_nat_o] :
      ( right_8621877368619751767at_int(R4)
     => ( left_t1861001511420785031at_int(R4)
       => ( left_u4005249686606446046at_int(S4)
         => ( right_1157779049177982926at_int(S4)
           => ord_le6589200864103675154_int_o(bNF_re7400052026677387805at_int(relcom8911704206367525917at_int(R2,R4),relcom8911704206367525917at_int(S3,S4)),relcom5467440779735672206nt_int(bNF_re2241393799969408733at_nat(R2,S3),bNF_re7400052026677387805at_int(R4,S4))) ) ) ) ) ).

% neg_fun_distr2
tff(fact_1933_refl__onI,axiom,
    ! [Ra: set_Pr5564308138774400199at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ord_le3091953645718357095at_nat(Ra,produc4441300235185154312at_nat(A4,aTP_Lamm_lb(A4)))
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => member6009668390874684176at_nat(produc2335810615602371415at_nat(X,X),Ra) )
       => refl_o7446464703481863270at_nat(A4,Ra) ) ) ).

% refl_onI
tff(fact_1934_refl__onI,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ord_le3000389064537975527at_nat(Ra,produc2761391749766926216at_nat(A4,aTP_Lamm_lc(A4)))
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X),X),Ra) )
       => refl_o7122697186039111552at_nat(A4,Ra) ) ) ).

% refl_onI
tff(fact_1935_refl__onI,axiom,
    ! [Ra: set_Product_prod_o_o,A4: set_o] :
      ( ord_le192195778054999463od_o_o(Ra,product_Sigma_o_o(A4,aTP_Lamm_jc(A4)))
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(X)),Ra) )
       => refl_on_o(A4,Ra) ) ) ).

% refl_onI
tff(fact_1936_refl__onI,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat,A4: set_se5479201105039950375at_nat] :
      ( ord_le5997549366648089703at_nat(Ra,produc6258361011560675592at_nat(A4,aTP_Lamm_ji(A4)))
     => ( ! [X: set_Pr4329608150637261639at_nat] :
            ( member6273874620674182256at_nat(X,A4)
           => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),X),Ra) )
       => refl_o6272981002401706822at_nat(A4,Ra) ) ) ).

% refl_onI
tff(fact_1937_refl__onI,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,A4: set_se7855581050983116737at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,Ra),produc7726050920695093128at_nat(A4,aTP_Lamm_jj(A4)))
     => ( ! [X: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(X,A4)
           => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),X)),Ra) )
       => aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(A4),Ra) ) ) ).

% refl_onI
tff(fact_1938_refl__onI,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,A4: set_int] :
      ( ord_le2843351958646193337nt_int(Ra,produc1453890548195736866nt_int(A4,aTP_Lamm_gf(A4)))
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),X)),Ra) )
       => refl_on_int(A4,Ra) ) ) ).

% refl_onI
tff(fact_1939_refl__onI,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),produc457027306803732586at_nat(A4,aTP_Lamm_gm(A4)))
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),X)),Ra) )
       => aa_set3702741264012733747_nat_o(refl_on_nat(A4),Ra) ) ) ).

% refl_onI
tff(fact_1940_refl__on__def,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat] :
      ( refl_o6272981002401706822at_nat(A4,Ra)
    <=> ( ord_le5997549366648089703at_nat(Ra,produc6258361011560675592at_nat(A4,aTP_Lamm_ji(A4)))
        & ! [X4: set_Pr4329608150637261639at_nat] :
            ( member6273874620674182256at_nat(X4,A4)
           => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X4),X4),Ra) ) ) ) ).

% refl_on_def
tff(fact_1941_refl__on__def,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(A4),Ra)
    <=> ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,Ra),produc7726050920695093128at_nat(A4,aTP_Lamm_jj(A4)))
        & ! [X4: set_Pr1261947904930325089at_nat] :
            ( member2643936169264416010at_nat(X4,A4)
           => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X4),X4)),Ra) ) ) ) ).

% refl_on_def
tff(fact_1942_refl__on__def,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int] :
      ( refl_on_int(A4,Ra)
    <=> ( ord_le2843351958646193337nt_int(Ra,produc1453890548195736866nt_int(A4,aTP_Lamm_gf(A4)))
        & ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A4)
           => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),X4)),Ra) ) ) ) ).

% refl_on_def
tff(fact_1943_refl__on__def,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(refl_on_nat(A4),Ra)
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),produc457027306803732586at_nat(A4,aTP_Lamm_gm(A4)))
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),X4)),Ra) ) ) ) ).

% refl_on_def
tff(fact_1944_image__insert,axiom,
    ! [F: fun_int_int,A: int,B5: set_int] : aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(insert_int(A),B5)) = aa_set_int_set_int(insert_int(aa_int_int(F,A)),aa_set_int_set_int(image_int_int(F),B5)) ).

% image_insert
tff(fact_1945_image__insert,axiom,
    ! [F: fun_int_nat,A: int,B5: set_int] : aa_set_int_set_nat(image_int_nat(F),aa_set_int_set_int(insert_int(A),B5)) = aa_set_nat_set_nat(insert_nat(aa_int_nat(F,A)),aa_set_int_set_nat(image_int_nat(F),B5)) ).

% image_insert
tff(fact_1946_image__insert,axiom,
    ! [F: fun_nat_rat,A: nat,B5: set_nat] : aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(insert_nat(A),B5)) = insert_rat(aa_nat_rat(F,A),aa_set_nat_set_rat(image_nat_rat(F),B5)) ).

% image_insert
tff(fact_1947_image__insert,axiom,
    ! [F: fun_nat_int,A: nat,B5: set_nat] : aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(insert_nat(A),B5)) = aa_set_int_set_int(insert_int(aa_nat_int(F,A)),aa_set_nat_set_int(image_nat_int(F),B5)) ).

% image_insert
tff(fact_1948_image__insert,axiom,
    ! [F: fun_nat_nat,A: nat,B5: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(insert_nat(A),B5)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(F,A)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ).

% image_insert
tff(fact_1949_image__insert,axiom,
    ! [F: fun_nat_Product_unit,A: nat,B5: set_nat] : image_8730104196221521654t_unit(F,aa_set_nat_set_nat(insert_nat(A),B5)) = aa_set8324764978855220220t_unit(insert_Product_unit(aa_nat_Product_unit(F,A)),image_8730104196221521654t_unit(F,B5)) ).

% image_insert
tff(fact_1950_image__insert,axiom,
    ! [F: fun_nat_o,A: nat,B5: set_nat] : image_nat_o(F,aa_set_nat_set_nat(insert_nat(A),B5)) = aa_set_o_set_o(insert_o(aa_nat_o(F,A)),image_nat_o(F,B5)) ).

% image_insert
tff(fact_1951_image__insert,axiom,
    ! [F: fun_Product_unit_nat,A: product_unit,B5: set_Product_unit] : image_875570014554754200it_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),B5)) = aa_set_nat_set_nat(insert_nat(aa_Product_unit_nat(F,A)),image_875570014554754200it_nat(F,B5)) ).

% image_insert
tff(fact_1952_image__insert,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A: product_unit,B5: set_Product_unit] : image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),B5)) = aa_set8324764978855220220t_unit(insert_Product_unit(aa_Pro8294289767544744956t_unit(F,A)),image_405062704495631173t_unit(F,B5)) ).

% image_insert
tff(fact_1953_image__insert,axiom,
    ! [F: fun_Product_unit_o,A: product_unit,B5: set_Product_unit] : image_Product_unit_o(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),B5)) = aa_set_o_set_o(insert_o(aa_Product_unit_o(F,A)),image_Product_unit_o(F,B5)) ).

% image_insert
tff(fact_1954_insert__image,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_nat] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_nat_set_nat(insert_nat(aa_o_nat(F,(Xa))),image_o_nat(F,A4)) = image_o_nat(F,A4) ) ) ).

% insert_image
tff(fact_1955_insert__image,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set8324764978855220220t_unit(insert_Product_unit(aa_o_Product_unit(F,(Xa))),image_o_Product_unit(F,A4)) = image_o_Product_unit(F,A4) ) ) ).

% insert_image
tff(fact_1956_insert__image,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_o_set_o(insert_o(aa_o_o(F,(Xa))),aa_set_o_set_o(image_o_o(F),A4)) = aa_set_o_set_o(image_o_o(F),A4) ) ) ).

% insert_image
tff(fact_1957_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( insert_rat(aa_nat_rat(F,Xa),aa_set_nat_set_rat(image_nat_rat(F),A4)) = aa_set_nat_set_rat(image_nat_rat(F),A4) ) ) ).

% insert_image
tff(fact_1958_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_int_set_int(insert_int(aa_nat_int(F,Xa)),aa_set_nat_set_int(image_nat_int(F),A4)) = aa_set_nat_set_int(image_nat_int(F),A4) ) ) ).

% insert_image
tff(fact_1959_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_set_nat(insert_nat(aa_nat_nat(F,Xa)),aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_set_nat_set_nat(image_nat_nat(F),A4) ) ) ).

% insert_image
tff(fact_1960_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set8324764978855220220t_unit(insert_Product_unit(aa_nat_Product_unit(F,Xa)),image_8730104196221521654t_unit(F,A4)) = image_8730104196221521654t_unit(F,A4) ) ) ).

% insert_image
tff(fact_1961_insert__image,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_o_set_o(insert_o(aa_nat_o(F,Xa)),image_nat_o(F,A4)) = image_nat_o(F,A4) ) ) ).

% insert_image
tff(fact_1962_insert__image,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_set_int(insert_int(aa_int_int(F,Xa)),aa_set_int_set_int(image_int_int(F),A4)) = aa_set_int_set_int(image_int_int(F),A4) ) ) ).

% insert_image
tff(fact_1963_insert__image,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_nat_set_nat(insert_nat(aa_int_nat(F,Xa)),aa_set_int_set_nat(image_int_nat(F),A4)) = aa_set_int_set_nat(image_int_nat(F),A4) ) ) ).

% insert_image
tff(fact_1964_SUP__identity__eq,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_ap),A4)) = aa_set_nat_nat(complete_Sup_Sup_nat,A4) ).

% SUP_identity_eq
tff(fact_1965_SUP__identity__eq,axiom,
    ! [A4: set_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_ld),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4) ).

% SUP_identity_eq
tff(fact_1966_SUP__identity__eq,axiom,
    ! [A4: set_int] : aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aTP_Lamm_aq),A4)) = aa_set_int_int(complete_Sup_Sup_int,A4) ).

% SUP_identity_eq
tff(fact_1967_SUP__identity__eq,axiom,
    ! [A4: set_Product_unit] : comple4687483117567863418t_unit(image_405062704495631173t_unit(aTP_Lamm_le,A4)) = comple4687483117567863418t_unit(A4) ).

% SUP_identity_eq
tff(fact_1968_INF__identity__eq,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(aTP_Lamm_ap),A4)) = aa_set_nat_nat(complete_Inf_Inf_nat,A4) ).

% INF_identity_eq
tff(fact_1969_INF__identity__eq,axiom,
    ! [A4: set_set_nat] : comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_ld),A4)) = comple7806235888213564991et_nat(A4) ).

% INF_identity_eq
tff(fact_1970_INF__identity__eq,axiom,
    ! [A4: set_int] : aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aTP_Lamm_aq),A4)) = aa_set_int_int(complete_Inf_Inf_int,A4) ).

% INF_identity_eq
tff(fact_1971_INF__identity__eq,axiom,
    ! [A4: set_Product_unit] : comple2584293577114468500t_unit(image_405062704495631173t_unit(aTP_Lamm_le,A4)) = comple2584293577114468500t_unit(A4) ).

% INF_identity_eq
tff(fact_1972_UN__I,axiom,
    ! [A: $o,A4: set_o,B: $o,B5: fun_o_set_o] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_o_o(member_o((B)),aa_o_set_o(B5,(A)))
       => aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_o_set_o(B5,A4))) ) ) ).

% UN_I
tff(fact_1973_UN__I,axiom,
    ! [A: $o,A4: set_o,B: int,B5: fun_o_set_int] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_int_o(member_int(B),aa_o_set_int(B5,(A)))
       => aa_set_int_o(member_int(B),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(B5,A4))) ) ) ).

% UN_I
tff(fact_1974_UN__I,axiom,
    ! [A: nat,A4: set_nat,B: $o,B5: fun_nat_set_o] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_o_o(member_o((B)),aa_nat_set_o(B5,A))
       => aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_nat_set_o(B5,A4))) ) ) ).

% UN_I
tff(fact_1975_UN__I,axiom,
    ! [A: nat,A4: set_nat,B: int,B5: fun_nat_set_int] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_int_o(member_int(B),aa_nat_set_int(B5,A))
       => aa_set_int_o(member_int(B),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(B5,A4))) ) ) ).

% UN_I
tff(fact_1976_UN__I,axiom,
    ! [A: int,A4: set_int,B: $o,B5: fun_int_set_o] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_o_o(member_o((B)),aa_int_set_o(B5,A))
       => aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_int_set_o(B5,A4))) ) ) ).

% UN_I
tff(fact_1977_UN__I,axiom,
    ! [A: int,A4: set_int,B: int,B5: fun_int_set_int] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_int_o(member_int(B),aa_int_set_int(B5,A))
       => aa_set_int_o(member_int(B),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(B5,A4))) ) ) ).

% UN_I
tff(fact_1978_UN__I,axiom,
    ! [A: $o,A4: set_o,B: nat,B5: fun_o_set_nat] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(A)))
       => aa_set_nat_o(member_nat(B),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4))) ) ) ).

% UN_I
tff(fact_1979_UN__I,axiom,
    ! [A: nat,A4: set_nat,B: nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A))
       => aa_set_nat_o(member_nat(B),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4))) ) ) ).

% UN_I
tff(fact_1980_UN__I,axiom,
    ! [A: int,A4: set_int,B: nat,B5: fun_int_set_nat] :
      ( aa_set_int_o(member_int(A),A4)
     => ( aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,A))
       => aa_set_nat_o(member_nat(B),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,A4))) ) ) ).

% UN_I
tff(fact_1981_UN__I,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: $o,B5: fun_Pr8467005832278515962_set_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( aa_set_o_o(member_o((B)),aa_Pro5970036531051701619_set_o(B5,A))
       => aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_501298686397505212_set_o(B5,A4))) ) ) ).

% UN_I
tff(fact_1982_INT__I,axiom,
    ! [A4: set_o,B: $o,B5: fun_o_set_o] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_o_o(member_o((B)),aa_o_set_o(B5,(X))) )
     => aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_o_set_o(B5,A4))) ) ).

% INT_I
tff(fact_1983_INT__I,axiom,
    ! [A4: set_o,B: int,B5: fun_o_set_int] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_int_o(member_int(B),aa_o_set_int(B5,(X))) )
     => aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_o_set_int(B5,A4))) ) ).

% INT_I
tff(fact_1984_INT__I,axiom,
    ! [A4: set_nat,B: $o,B5: fun_nat_set_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_o_o(member_o((B)),aa_nat_set_o(B5,X)) )
     => aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_nat_set_o(B5,A4))) ) ).

% INT_I
tff(fact_1985_INT__I,axiom,
    ! [A4: set_nat,B: int,B5: fun_nat_set_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_int_o(member_int(B),aa_nat_set_int(B5,X)) )
     => aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_nat_set_int(B5,A4))) ) ).

% INT_I
tff(fact_1986_INT__I,axiom,
    ! [A4: set_int,B: $o,B5: fun_int_set_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_o_o(member_o((B)),aa_int_set_o(B5,X)) )
     => aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_int_set_o(B5,A4))) ) ).

% INT_I
tff(fact_1987_INT__I,axiom,
    ! [A4: set_int,B: int,B5: fun_int_set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_int_o(member_int(B),aa_int_set_int(B5,X)) )
     => aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_int_set_int(B5,A4))) ) ).

% INT_I
tff(fact_1988_INT__I,axiom,
    ! [A4: set_o,B: nat,B5: fun_o_set_nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(X))) )
     => aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_o_set_nat(B5,A4))) ) ).

% INT_I
tff(fact_1989_INT__I,axiom,
    ! [A4: set_nat,B: nat,B5: fun_nat_set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,X)) )
     => aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))) ) ).

% INT_I
tff(fact_1990_INT__I,axiom,
    ! [A4: set_int,B: nat,B5: fun_int_set_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,X)) )
     => aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_int_set_nat(B5,A4))) ) ).

% INT_I
tff(fact_1991_INT__I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: $o,B5: fun_Pr8467005832278515962_set_o] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
         => aa_set_o_o(member_o((B)),aa_Pro5970036531051701619_set_o(B5,X)) )
     => aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_501298686397505212_set_o(B5,A4))) ) ).

% INT_I
tff(fact_1992_SUP__id__eq,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(image_o_o(id_o),A4))
    <=> aa_set_o_o(complete_Sup_Sup_o,A4) ) ).

% SUP_id_eq
tff(fact_1993_SUP__id__eq,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(id_nat),A4)) = aa_set_nat_nat(complete_Sup_Sup_nat,A4) ).

% SUP_id_eq
tff(fact_1994_SUP__id__eq,axiom,
    ! [A4: set_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(id_set_nat),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4) ).

% SUP_id_eq
tff(fact_1995_SUP__id__eq,axiom,
    ! [A4: set_int] : aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(id_int),A4)) = aa_set_int_int(complete_Sup_Sup_int,A4) ).

% SUP_id_eq
tff(fact_1996_SUP__id__eq,axiom,
    ! [A4: set_Product_unit] : comple4687483117567863418t_unit(image_405062704495631173t_unit(id_Product_unit,A4)) = comple4687483117567863418t_unit(A4) ).

% SUP_id_eq
tff(fact_1997_INF__id__eq,axiom,
    ! [A4: set_o] :
      ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(image_o_o(id_o),A4))
    <=> aa_set_o_o(complete_Inf_Inf_o,A4) ) ).

% INF_id_eq
tff(fact_1998_INF__id__eq,axiom,
    ! [A4: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(id_nat),A4)) = aa_set_nat_nat(complete_Inf_Inf_nat,A4) ).

% INF_id_eq
tff(fact_1999_INF__id__eq,axiom,
    ! [A4: set_set_nat] : comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(id_set_nat),A4)) = comple7806235888213564991et_nat(A4) ).

% INF_id_eq
tff(fact_2000_INF__id__eq,axiom,
    ! [A4: set_int] : aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(id_int),A4)) = aa_set_int_int(complete_Inf_Inf_int,A4) ).

% INF_id_eq
tff(fact_2001_INF__id__eq,axiom,
    ! [A4: set_Product_unit] : comple2584293577114468500t_unit(image_405062704495631173t_unit(id_Product_unit,A4)) = comple2584293577114468500t_unit(A4) ).

% INF_id_eq
tff(fact_2002_SUP__const,axiom,
    ! [A4: set_Product_unit,F: set_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aTP_Lamm_jr(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2003_SUP__const,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: set_nat] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(aTP_Lamm_lf(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2004_SUP__const,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: set_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(aTP_Lamm_lg(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2005_SUP__const,axiom,
    ! [A4: set_o,F: set_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aTP_Lamm_jd(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2006_SUP__const,axiom,
    ! [A4: set_nat,F: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aTP_Lamm_gm(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2007_SUP__const,axiom,
    ! [A4: set_Product_unit,F: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(aTP_Lamm_lh(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2008_SUP__const,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: product_unit] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => ( comple4687483117567863418t_unit(image_7082456443403096939t_unit(aTP_Lamm_li(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2009_SUP__const,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_unit] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( comple4687483117567863418t_unit(image_6169909464789311569t_unit(aTP_Lamm_lj(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2010_SUP__const,axiom,
    ! [A4: set_o,F: product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( comple4687483117567863418t_unit(image_o_Product_unit(aTP_Lamm_lk(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2011_SUP__const,axiom,
    ! [A4: set_nat,F: product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(aTP_Lamm_kt(F),A4)) = F ) ) ).

% SUP_const
tff(fact_2012_INF__const,axiom,
    ! [A4: set_Product_unit,F: set_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(aTP_Lamm_jr(F),A4)) = F ) ) ).

% INF_const
tff(fact_2013_INF__const,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: set_nat] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => ( comple7806235888213564991et_nat(image_3160451284896210088et_nat(aTP_Lamm_lf(F),A4)) = F ) ) ).

% INF_const
tff(fact_2014_INF__const,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: set_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( comple7806235888213564991et_nat(image_15824709712370754et_nat(aTP_Lamm_lg(F),A4)) = F ) ) ).

% INF_const
tff(fact_2015_INF__const,axiom,
    ! [A4: set_o,F: set_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( comple7806235888213564991et_nat(image_o_set_nat(aTP_Lamm_jd(F),A4)) = F ) ) ).

% INF_const
tff(fact_2016_INF__const,axiom,
    ! [A4: set_nat,F: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_gm(F),A4)) = F ) ) ).

% INF_const
tff(fact_2017_INF__const,axiom,
    ! [A4: set_Product_unit,F: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(aTP_Lamm_lh(F),A4)) = F ) ) ).

% INF_const
tff(fact_2018_INF__const,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: product_unit] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => ( comple2584293577114468500t_unit(image_7082456443403096939t_unit(aTP_Lamm_li(F),A4)) = F ) ) ).

% INF_const
tff(fact_2019_INF__const,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_unit] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( comple2584293577114468500t_unit(image_6169909464789311569t_unit(aTP_Lamm_lj(F),A4)) = F ) ) ).

% INF_const
tff(fact_2020_INF__const,axiom,
    ! [A4: set_o,F: product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( comple2584293577114468500t_unit(image_o_Product_unit(aTP_Lamm_lk(F),A4)) = F ) ) ).

% INF_const
tff(fact_2021_INF__const,axiom,
    ! [A4: set_nat,F: product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(aTP_Lamm_kt(F),A4)) = F ) ) ).

% INF_const
tff(fact_2022_UN__constant,axiom,
    ! [C: set_Product_unit,A4: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(aTP_Lamm_jp(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,C) ).

% UN_constant
tff(fact_2023_UN__constant,axiom,
    ! [C: set_o,A4: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(aTP_Lamm_jq(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_o,C) ).

% UN_constant
tff(fact_2024_UN__constant,axiom,
    ! [C: set_Product_unit,A4: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(aTP_Lamm_js(C),A4)) = $ite(A4 = bot_bot_set_o,bot_bo3957492148770167129t_unit,C) ).

% UN_constant
tff(fact_2025_UN__constant,axiom,
    ! [C: set_o,A4: set_o] :
      comple90263536869209701_set_o(image_o_set_o(aTP_Lamm_jc(C),A4)) = $ite(A4 = bot_bot_set_o,bot_bot_set_o,C) ).

% UN_constant
tff(fact_2026_UN__constant,axiom,
    ! [C: set_Product_unit,A4: set_nat] :
      comple3349607820504768346t_unit(image_4824033694279209942t_unit(aTP_Lamm_jt(C),A4)) = $ite(A4 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,C) ).

% UN_constant
tff(fact_2027_UN__constant,axiom,
    ! [C: set_o,A4: set_nat] :
      comple90263536869209701_set_o(image_nat_set_o(aTP_Lamm_jf(C),A4)) = $ite(A4 = bot_bot_set_nat,bot_bot_set_o,C) ).

% UN_constant
tff(fact_2028_UN__constant,axiom,
    ! [C: set_nat,A4: set_Product_unit] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aTP_Lamm_jr(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,C) ).

% UN_constant
tff(fact_2029_UN__constant,axiom,
    ! [C: set_nat,A4: set_o] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aTP_Lamm_jd(C),A4)) = $ite(A4 = bot_bot_set_o,bot_bot_set_nat,C) ).

% UN_constant
tff(fact_2030_UN__constant,axiom,
    ! [C: set_nat,A4: set_nat] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aTP_Lamm_gm(C),A4)) = $ite(A4 = bot_bot_set_nat,bot_bot_set_nat,C) ).

% UN_constant
tff(fact_2031_UN__constant,axiom,
    ! [C: set_Pr1261947904930325089at_nat,A4: set_Product_unit] :
      comple5685304695842803022at_nat(image_3617315916088684441at_nat(aTP_Lamm_jx(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bo2099793752762293965at_nat,C) ).

% UN_constant
tff(fact_2032_UN__simps_I1_J,axiom,
    ! [A: product_unit,B5: fun_Pr3013102738158121955t_unit,C3: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(aa_fun8117516366632191236t_unit(aTP_Lamm_ll(A),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_7778121751606633509t_unit(B5,C3)))) ).

% UN_simps(1)
tff(fact_2033_UN__simps_I1_J,axiom,
    ! [A: $o,B5: fun_Pr6739565440344758894_set_o,C3: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(aa_fun5942798580971356698_set_o(aTP_Lamm_lm((A)),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bot_set_o,aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_6930536745178679216_set_o(B5,C3)))) ).

% UN_simps(1)
tff(fact_2034_UN__simps_I1_J,axiom,
    ! [A: product_unit,B5: fun_o_1833192751255691672t_unit,C3: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ln(A),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bo3957492148770167129t_unit,aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,C3)))) ).

% UN_simps(1)
tff(fact_2035_UN__simps_I1_J,axiom,
    ! [A: $o,B5: fun_o_set_o,C3: set_o] :
      comple90263536869209701_set_o(image_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_lo((A)),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bot_set_o,aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_o_set_o(B5,C3)))) ).

% UN_simps(1)
tff(fact_2036_UN__simps_I1_J,axiom,
    ! [A: product_unit,B5: fun_na896991816813928310t_unit,C3: set_nat] :
      comple3349607820504768346t_unit(image_4824033694279209942t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_lp(A),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_4824033694279209942t_unit(B5,C3)))) ).

% UN_simps(1)
tff(fact_2037_UN__simps_I1_J,axiom,
    ! [A: $o,B5: fun_nat_set_o,C3: set_nat] :
      comple90263536869209701_set_o(image_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_lq((A)),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bot_set_o,aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_nat_set_o(B5,C3)))) ).

% UN_simps(1)
tff(fact_2038_UN__simps_I1_J,axiom,
    ! [A: nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_lr(A),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,C3)))) ).

% UN_simps(1)
tff(fact_2039_UN__simps_I1_J,axiom,
    ! [A: nat,B5: fun_o_set_nat,C3: set_o] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_ls(A),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,C3)))) ).

% UN_simps(1)
tff(fact_2040_UN__simps_I1_J,axiom,
    ! [A: nat,B5: fun_nat_set_nat,C3: set_nat] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_lt(A),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,C3)))) ).

% UN_simps(1)
tff(fact_2041_UN__simps_I1_J,axiom,
    ! [A: product_prod_nat_nat,B5: fun_Pr3332065415989118679at_nat,C3: set_Product_unit] :
      comple5685304695842803022at_nat(image_3617315916088684441at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_lu(A),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bo2099793752762293965at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),comple5685304695842803022at_nat(image_3617315916088684441at_nat(B5,C3)))) ).

% UN_simps(1)
tff(fact_2042_UN__singleton,axiom,
    ! [A4: set_Product_unit] : comple3349607820504768346t_unit(image_7778121751606633509t_unit(aTP_Lamm_lv,A4)) = A4 ).

% UN_singleton
tff(fact_2043_UN__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(aTP_Lamm_lw),A4)) = A4 ).

% UN_singleton
tff(fact_2044_UN__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : comple5685304695842803022at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(aTP_Lamm_lx),A4)) = A4 ).

% UN_singleton
tff(fact_2045_UN__singleton,axiom,
    ! [A4: set_o] : comple90263536869209701_set_o(image_o_set_o(aTP_Lamm_ly,A4)) = A4 ).

% UN_singleton
tff(fact_2046_UN__singleton,axiom,
    ! [A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aTP_Lamm_lz,A4)) = A4 ).

% UN_singleton
tff(fact_2047_refl__on__singleton,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat] : refl_o6272981002401706822at_nat(insert2290795007606087959at_nat(Xa,bot_bo1534825418459847827at_nat),insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Xa),bot_bo4948859079157340979at_nat)) ).

% refl_on_singleton
tff(fact_2048_refl__on__singleton,axiom,
    ! [Xa: int] : refl_on_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xa),bot_bo1796632182523588997nt_int)) ).

% refl_on_singleton
tff(fact_2049_refl__on__singleton,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Xa)),bot_bo228742789529271731at_nat)) ).

% refl_on_singleton
tff(fact_2050_refl__on__singleton,axiom,
    ! [Xa: product_unit] : refl_on_Product_unit(aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit),insert4616891024742550839t_unit(produc5692694829384537815t_unit(Xa,Xa),bot_bo1654022762305692595t_unit)) ).

% refl_on_singleton
tff(fact_2051_refl__on__singleton,axiom,
    ! [Xa: produc3843707927480180839at_nat] : refl_o7446464703481863270at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Xa),bot_bo228742789529271731at_nat),insert398606708087465655at_nat(produc2335810615602371415at_nat(Xa,Xa),bot_bo7193250159860175667at_nat)) ).

% refl_on_singleton
tff(fact_2052_refl__on__singleton,axiom,
    ! [Xa: product_prod_nat_nat] : refl_o7122697186039111552at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),bot_bo2099793752762293965at_nat),insert5050368324300391991at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Xa),bot_bo5327735625951526323at_nat)) ).

% refl_on_singleton
tff(fact_2053_refl__on__singleton,axiom,
    ! [Xa: $o] : refl_on_o(aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o),insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa)),(Xa)),bot_bo7073875226086086771od_o_o)) ).

% refl_on_singleton
tff(fact_2054_refl__on__singleton,axiom,
    ! [Xa: nat] : aa_set3702741264012733747_nat_o(refl_on_nat(aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Xa)),bot_bo2099793752762293965at_nat)) ).

% refl_on_singleton
tff(fact_2055_INT__insert__distrib,axiom,
    ! [U: $o,A4: set_o,A: product_unit,B5: fun_o_1833192751255691672t_unit] :
      ( aa_set_o_o(member_o((U)),A4)
     => ( comple7216939928495492468t_unit(image_663195431845683034t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ln(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple7216939928495492468t_unit(image_663195431845683034t_unit(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2056_INT__insert__distrib,axiom,
    ! [U: $o,A4: set_o,A: $o,B5: fun_o_set_o] :
      ( aa_set_o_o(member_o((U)),A4)
     => ( comple3063163877087187839_set_o(image_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_lo((A)),B5),A4)) = aa_set_o_set_o(insert_o((A)),comple3063163877087187839_set_o(image_o_set_o(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2057_INT__insert__distrib,axiom,
    ! [U: nat,A4: set_nat,A: product_unit,B5: fun_na896991816813928310t_unit] :
      ( aa_set_nat_o(member_nat(U),A4)
     => ( comple7216939928495492468t_unit(image_4824033694279209942t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_lp(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple7216939928495492468t_unit(image_4824033694279209942t_unit(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2058_INT__insert__distrib,axiom,
    ! [U: nat,A4: set_nat,A: $o,B5: fun_nat_set_o] :
      ( aa_set_nat_o(member_nat(U),A4)
     => ( comple3063163877087187839_set_o(image_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_lq((A)),B5),A4)) = aa_set_o_set_o(insert_o((A)),comple3063163877087187839_set_o(image_nat_set_o(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2059_INT__insert__distrib,axiom,
    ! [U: int,A4: set_int,A: product_unit,B5: fun_in4731230917697954842t_unit] :
      ( aa_set_int_o(member_int(U),A4)
     => ( comple7216939928495492468t_unit(image_8658272795163236474t_unit(aa_fun7057315224923419708t_unit(aTP_Lamm_ma(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple7216939928495492468t_unit(image_8658272795163236474t_unit(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2060_INT__insert__distrib,axiom,
    ! [U: int,A4: set_int,A: $o,B5: fun_int_set_o] :
      ( aa_set_int_o(member_int(U),A4)
     => ( comple3063163877087187839_set_o(image_int_set_o(aa_fun9190404755809275900_set_o(aTP_Lamm_mb((A)),B5),A4)) = aa_set_o_set_o(insert_o((A)),comple3063163877087187839_set_o(image_int_set_o(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2061_INT__insert__distrib,axiom,
    ! [U: $o,A4: set_o,A: nat,B5: fun_o_set_nat] :
      ( aa_set_o_o(member_o((U)),A4)
     => ( comple7806235888213564991et_nat(image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_ls(A),B5),A4)) = aa_set_nat_set_nat(insert_nat(A),comple7806235888213564991et_nat(image_o_set_nat(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2062_INT__insert__distrib,axiom,
    ! [U: nat,A4: set_nat,A: nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(U),A4)
     => ( comple7806235888213564991et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_lt(A),B5),A4)) = aa_set_nat_set_nat(insert_nat(A),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2063_INT__insert__distrib,axiom,
    ! [U: int,A4: set_int,A: nat,B5: fun_int_set_nat] :
      ( aa_set_int_o(member_int(U),A4)
     => ( comple7806235888213564991et_nat(image_int_set_nat(aa_fun5565777959088800668et_nat(aTP_Lamm_mc(A),B5),A4)) = aa_set_nat_set_nat(insert_nat(A),comple7806235888213564991et_nat(image_int_set_nat(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2064_INT__insert__distrib,axiom,
    ! [U: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,A: product_unit,B5: fun_Pr2387170064179968111t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(U),A4)
     => ( comple7216939928495492468t_unit(image_121980125215734065t_unit(aa_fun7169552063946493212t_unit(aTP_Lamm_md(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple7216939928495492468t_unit(image_121980125215734065t_unit(B5,A4))) ) ) ).

% INT_insert_distrib
tff(fact_2065_UN__insert__distrib,axiom,
    ! [U: $o,A4: set_o,A: product_unit,B5: fun_o_1833192751255691672t_unit] :
      ( aa_set_o_o(member_o((U)),A4)
     => ( comple3349607820504768346t_unit(image_663195431845683034t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ln(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2066_UN__insert__distrib,axiom,
    ! [U: $o,A4: set_o,A: $o,B5: fun_o_set_o] :
      ( aa_set_o_o(member_o((U)),A4)
     => ( comple90263536869209701_set_o(image_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_lo((A)),B5),A4)) = aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_o_set_o(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2067_UN__insert__distrib,axiom,
    ! [U: nat,A4: set_nat,A: product_unit,B5: fun_na896991816813928310t_unit] :
      ( aa_set_nat_o(member_nat(U),A4)
     => ( comple3349607820504768346t_unit(image_4824033694279209942t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_lp(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_4824033694279209942t_unit(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2068_UN__insert__distrib,axiom,
    ! [U: nat,A4: set_nat,A: $o,B5: fun_nat_set_o] :
      ( aa_set_nat_o(member_nat(U),A4)
     => ( comple90263536869209701_set_o(image_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_lq((A)),B5),A4)) = aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_nat_set_o(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2069_UN__insert__distrib,axiom,
    ! [U: int,A4: set_int,A: product_unit,B5: fun_in4731230917697954842t_unit] :
      ( aa_set_int_o(member_int(U),A4)
     => ( comple3349607820504768346t_unit(image_8658272795163236474t_unit(aa_fun7057315224923419708t_unit(aTP_Lamm_ma(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_8658272795163236474t_unit(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2070_UN__insert__distrib,axiom,
    ! [U: int,A4: set_int,A: $o,B5: fun_int_set_o] :
      ( aa_set_int_o(member_int(U),A4)
     => ( comple90263536869209701_set_o(image_int_set_o(aa_fun9190404755809275900_set_o(aTP_Lamm_mb((A)),B5),A4)) = aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_int_set_o(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2071_UN__insert__distrib,axiom,
    ! [U: $o,A4: set_o,A: nat,B5: fun_o_set_nat] :
      ( aa_set_o_o(member_o((U)),A4)
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_ls(A),B5),A4)) = aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2072_UN__insert__distrib,axiom,
    ! [U: nat,A4: set_nat,A: nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(U),A4)
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_lt(A),B5),A4)) = aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2073_UN__insert__distrib,axiom,
    ! [U: int,A4: set_int,A: nat,B5: fun_int_set_nat] :
      ( aa_set_int_o(member_int(U),A4)
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(aa_fun5565777959088800668et_nat(aTP_Lamm_mc(A),B5),A4)) = aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2074_UN__insert__distrib,axiom,
    ! [U: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,A: product_unit,B5: fun_Pr2387170064179968111t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(U),A4)
     => ( comple3349607820504768346t_unit(image_121980125215734065t_unit(aa_fun7169552063946493212t_unit(aTP_Lamm_md(A),B5),A4)) = aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_121980125215734065t_unit(B5,A4))) ) ) ).

% UN_insert_distrib
tff(fact_2075_Sigma__Union,axiom,
    ! [X3: set_set_nat,B5: fun_nat_set_nat] : produc457027306803732586at_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,X3),B5) = comple5685304695842803022at_nat(image_6521575952113991828at_nat(aTP_Lamm_me(B5),X3)) ).

% Sigma_Union
tff(fact_2076_INF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_nat,D: fun_nat_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_nat(C3,X) = aa_nat_nat(D,X) ) )
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(C3),A4)) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(D),B5)) ) ) ) ).

% INF_cong
tff(fact_2077_INF__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_nat,D: fun_int_nat] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_nat(C3,X) = aa_int_nat(D,X) ) )
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(C3),A4)) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(D),B5)) ) ) ) ).

% INF_cong
tff(fact_2078_INF__cong,axiom,
    ! [A4: set_o,B5: set_o,C3: fun_o_int,D: fun_o_int] :
      ( ( A4 = B5 )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_int(C3,(X)) = aa_o_int(D,(X)) ) )
       => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(C3,A4)) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(D,B5)) ) ) ) ).

% INF_cong
tff(fact_2079_INF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_int,D: fun_nat_int] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_int(C3,X) = aa_nat_int(D,X) ) )
       => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(C3),A4)) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(D),B5)) ) ) ) ).

% INF_cong
tff(fact_2080_INF__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_int,D: fun_int_int] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_int(C3,X) = aa_int_int(D,X) ) )
       => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(C3),A4)) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(D),B5)) ) ) ) ).

% INF_cong
tff(fact_2081_INF__cong,axiom,
    ! [A4: set_o,B5: set_o,C3: fun_o_Product_unit,D: fun_o_Product_unit] :
      ( ( A4 = B5 )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_Product_unit(C3,(X)) = aa_o_Product_unit(D,(X)) ) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit(C3,A4)) = comple2584293577114468500t_unit(image_o_Product_unit(D,B5)) ) ) ) ).

% INF_cong
tff(fact_2082_INF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_Product_unit,D: fun_nat_Product_unit] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_Product_unit(C3,X) = aa_nat_Product_unit(D,X) ) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(C3,A4)) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(D,B5)) ) ) ) ).

% INF_cong
tff(fact_2083_INF__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_Product_unit,D: fun_int_Product_unit] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_Product_unit(C3,X) = aa_int_Product_unit(D,X) ) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(C3,A4)) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(D,B5)) ) ) ) ).

% INF_cong
tff(fact_2084_INF__cong,axiom,
    ! [A4: set_o,B5: set_o,C3: fun_o_set_nat,D: fun_o_set_nat] :
      ( ( A4 = B5 )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_set_nat(C3,(X)) = aa_o_set_nat(D,(X)) ) )
       => ( comple7806235888213564991et_nat(image_o_set_nat(C3,A4)) = comple7806235888213564991et_nat(image_o_set_nat(D,B5)) ) ) ) ).

% INF_cong
tff(fact_2085_INF__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_set_nat,D: fun_nat_set_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_set_nat(C3,X) = aa_nat_set_nat(D,X) ) )
       => ( comple7806235888213564991et_nat(image_nat_set_nat(C3,A4)) = comple7806235888213564991et_nat(image_nat_set_nat(D,B5)) ) ) ) ).

% INF_cong
tff(fact_2086_SUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_nat,D: fun_nat_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_nat(C3,X) = aa_nat_nat(D,X) ) )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(C3),A4)) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(D),B5)) ) ) ) ).

% SUP_cong
tff(fact_2087_SUP__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_nat,D: fun_int_nat] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_nat(C3,X) = aa_int_nat(D,X) ) )
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(C3),A4)) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(D),B5)) ) ) ) ).

% SUP_cong
tff(fact_2088_SUP__cong,axiom,
    ! [A4: set_o,B5: set_o,C3: fun_o_int,D: fun_o_int] :
      ( ( A4 = B5 )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_int(C3,(X)) = aa_o_int(D,(X)) ) )
       => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(C3,A4)) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(D,B5)) ) ) ) ).

% SUP_cong
tff(fact_2089_SUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_int,D: fun_nat_int] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_int(C3,X) = aa_nat_int(D,X) ) )
       => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(C3),A4)) = aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(D),B5)) ) ) ) ).

% SUP_cong
tff(fact_2090_SUP__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_int,D: fun_int_int] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_int(C3,X) = aa_int_int(D,X) ) )
       => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(C3),A4)) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(D),B5)) ) ) ) ).

% SUP_cong
tff(fact_2091_SUP__cong,axiom,
    ! [A4: set_o,B5: set_o,C3: fun_o_Product_unit,D: fun_o_Product_unit] :
      ( ( A4 = B5 )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_Product_unit(C3,(X)) = aa_o_Product_unit(D,(X)) ) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit(C3,A4)) = comple4687483117567863418t_unit(image_o_Product_unit(D,B5)) ) ) ) ).

% SUP_cong
tff(fact_2092_SUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_Product_unit,D: fun_nat_Product_unit] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_Product_unit(C3,X) = aa_nat_Product_unit(D,X) ) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(C3,A4)) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(D,B5)) ) ) ) ).

% SUP_cong
tff(fact_2093_SUP__cong,axiom,
    ! [A4: set_int,B5: set_int,C3: fun_int_Product_unit,D: fun_int_Product_unit] :
      ( ( A4 = B5 )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => ( aa_int_Product_unit(C3,X) = aa_int_Product_unit(D,X) ) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(C3,A4)) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(D,B5)) ) ) ) ).

% SUP_cong
tff(fact_2094_SUP__cong,axiom,
    ! [A4: set_o,B5: set_o,C3: fun_o_set_nat,D: fun_o_set_nat] :
      ( ( A4 = B5 )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => ( aa_o_set_nat(C3,(X)) = aa_o_set_nat(D,(X)) ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(C3,A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(D,B5)) ) ) ) ).

% SUP_cong
tff(fact_2095_SUP__cong,axiom,
    ! [A4: set_nat,B5: set_nat,C3: fun_nat_set_nat,D: fun_nat_set_nat] :
      ( ( A4 = B5 )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => ( aa_nat_set_nat(C3,X) = aa_nat_set_nat(D,X) ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(C3,A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(D,B5)) ) ) ) ).

% SUP_cong
tff(fact_2096_insert__subsetI,axiom,
    ! [Xa: product_unit,A4: set_Product_unit,X3: set_Product_unit] :
      ( member_Product_unit(Xa,A4)
     => ( ord_le3507040750410214029t_unit(X3,A4)
       => ord_le3507040750410214029t_unit(aa_set8324764978855220220t_unit(insert_Product_unit(Xa),X3),A4) ) ) ).

% insert_subsetI
tff(fact_2097_insert__subsetI,axiom,
    ! [Xa: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),A4)
     => ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,X3),A4)
       => aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Xa),X3)),A4) ) ) ).

% insert_subsetI
tff(fact_2098_insert__subsetI,axiom,
    ! [Xa: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X3),A4)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),X3)),A4) ) ) ).

% insert_subsetI
tff(fact_2099_insert__subsetI,axiom,
    ! [Xa: $o,A4: set_o,X3: set_o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_o_o(ord_less_eq_set_o(X3),A4)
       => aa_set_o_o(ord_less_eq_set_o(aa_set_o_set_o(insert_o((Xa)),X3)),A4) ) ) ).

% insert_subsetI
tff(fact_2100_insert__subsetI,axiom,
    ! [Xa: nat,A4: set_nat,X3: set_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(X3),A4)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(insert_nat(Xa),X3)),A4) ) ) ).

% insert_subsetI
tff(fact_2101_insert__subsetI,axiom,
    ! [Xa: int,A4: set_int,X3: set_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(X3),A4)
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(insert_int(Xa),X3)),A4) ) ) ).

% insert_subsetI
tff(fact_2102_UN__extend__simps_I1_J,axiom,
    ! [A: product_unit,B5: fun_Pr3013102738158121955t_unit,C3: set_Product_unit] :
      aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_7778121751606633509t_unit(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit),comple3349607820504768346t_unit(image_7778121751606633509t_unit(aa_fun8117516366632191236t_unit(aTP_Lamm_ll(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2103_UN__extend__simps_I1_J,axiom,
    ! [A: $o,B5: fun_Pr6739565440344758894_set_o,C3: set_Product_unit] :
      aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_6930536745178679216_set_o(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,aa_set_o_set_o(insert_o((A)),bot_bot_set_o),comple90263536869209701_set_o(image_6930536745178679216_set_o(aa_fun5942798580971356698_set_o(aTP_Lamm_lm((A)),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2104_UN__extend__simps_I1_J,axiom,
    ! [A: product_unit,B5: fun_o_1833192751255691672t_unit,C3: set_o] :
      aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,C3))) = $ite(C3 = bot_bot_set_o,aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit),comple3349607820504768346t_unit(image_663195431845683034t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ln(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2105_UN__extend__simps_I1_J,axiom,
    ! [A: $o,B5: fun_o_set_o,C3: set_o] :
      aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_o_set_o(B5,C3))) = $ite(C3 = bot_bot_set_o,aa_set_o_set_o(insert_o((A)),bot_bot_set_o),comple90263536869209701_set_o(image_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_lo((A)),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2106_UN__extend__simps_I1_J,axiom,
    ! [A: product_unit,B5: fun_na896991816813928310t_unit,C3: set_nat] :
      aa_set8324764978855220220t_unit(insert_Product_unit(A),comple3349607820504768346t_unit(image_4824033694279209942t_unit(B5,C3))) = $ite(C3 = bot_bot_set_nat,aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit),comple3349607820504768346t_unit(image_4824033694279209942t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_lp(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2107_UN__extend__simps_I1_J,axiom,
    ! [A: $o,B5: fun_nat_set_o,C3: set_nat] :
      aa_set_o_set_o(insert_o((A)),comple90263536869209701_set_o(image_nat_set_o(B5,C3))) = $ite(C3 = bot_bot_set_nat,aa_set_o_set_o(insert_o((A)),bot_bot_set_o),comple90263536869209701_set_o(image_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_lq((A)),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2108_UN__extend__simps_I1_J,axiom,
    ! [A: nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_lr(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2109_UN__extend__simps_I1_J,axiom,
    ! [A: nat,B5: fun_o_set_nat,C3: set_o] :
      aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_ls(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2110_UN__extend__simps_I1_J,axiom,
    ! [A: nat,B5: fun_nat_set_nat,C3: set_nat] :
      aa_set_nat_set_nat(insert_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_lt(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2111_UN__extend__simps_I1_J,axiom,
    ! [A: product_prod_nat_nat,B5: fun_Pr3332065415989118679at_nat,C3: set_Product_unit] :
      aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),comple5685304695842803022at_nat(image_3617315916088684441at_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),bot_bo2099793752762293965at_nat),comple5685304695842803022at_nat(image_3617315916088684441at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_lu(A),B5),C3))) ).

% UN_extend_simps(1)
tff(fact_2112_Sigma__def,axiom,
    ! [A4: set_se5479201105039950375at_nat,B5: fun_se3430100612660269923at_nat] : produc6258361011560675592at_nat(A4,B5) = comple7342073961973170484at_nat(image_1124969593608450885at_nat(aTP_Lamm_mg(B5),A4)) ).

% Sigma_def
tff(fact_2113_Sigma__def,axiom,
    ! [A4: set_nat,B5: fun_nat_set_num] : produc6237731129814287028at_num(A4,B5) = comple1400524285282149784at_num(image_2893549341467670548at_num(aTP_Lamm_mi(B5),A4)) ).

% Sigma_def
tff(fact_2114_Sigma__def,axiom,
    ! [A4: set_int,B5: fun_int_set_int] : produc1453890548195736866nt_int(A4,B5) = comple5382143125604098054nt_int(image_8635204845542730022nt_int(aTP_Lamm_mk(B5),A4)) ).

% Sigma_def
tff(fact_2115_Sigma__def,axiom,
    ! [A4: set_se7855581050983116737at_nat,B5: fun_se953073389798102371at_nat] : produc7726050920695093128at_nat(A4,B5) = comple4364388969068068532at_nat(image_6898982319499645995at_nat(aTP_Lamm_mm(B5),A4)) ).

% Sigma_def
tff(fact_2116_Sigma__def,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat] : produc457027306803732586at_nat(A4,B5) = comple5685304695842803022at_nat(image_7178329752028323786at_nat(aTP_Lamm_mo(B5),A4)) ).

% Sigma_def
tff(fact_2117_UN__Times__distrib,axiom,
    ! [E2: fun_nat_set_nat,F3: fun_nat_set_nat,A4: set_nat,B5: set_nat] : comple5685304695842803022at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(produc8197505143624133779at_nat(aa_fun2373699841287868962at_nat(aTP_Lamm_mq(E2),F3))),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5)))) = produc457027306803732586at_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(E2,A4)),aa_set562305657478724375et_nat(aTP_Lamm_mr(F3),B5)) ).

% UN_Times_distrib
tff(fact_2118_right__total__OO,axiom,
    ! [A4: fun_Pr2028404973800837934_nat_o,B5: fun_Pr4738963024209204295_int_o] :
      ( right_4094035864313884895at_nat(A4)
     => ( right_1157779049177982926at_int(B5)
       => right_1157779049177982926at_int(relcom8911704206367525917at_int(A4,B5)) ) ) ).

% right_total_OO
tff(fact_2119_right__total__OO,axiom,
    ! [A4: fun_int_fun_int_o,B5: fun_in4891666191172872833eger_o] :
      ( right_total_int_int(A4)
     => ( right_7281159819619925802nteger(B5)
       => right_7281159819619925802nteger(relcom967497705283160040nteger(A4,B5)) ) ) ).

% right_total_OO
tff(fact_2120_refl__onD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Ra: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( refl_o7446464703481863270at_nat(A4,Ra)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
       => member6009668390874684176at_nat(produc2335810615602371415at_nat(A,A),Ra) ) ) ).

% refl_onD
tff(fact_2121_refl__onD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( refl_o7122697186039111552at_nat(A4,Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),A),Ra) ) ) ).

% refl_onD
tff(fact_2122_refl__onD,axiom,
    ! [A4: set_o,Ra: set_Product_prod_o_o,A: $o] :
      ( refl_on_o(A4,Ra)
     => ( aa_set_o_o(member_o((A)),A4)
       => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(A)),Ra) ) ) ).

% refl_onD
tff(fact_2123_refl__onD,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( refl_o6272981002401706822at_nat(A4,Ra)
     => ( member6273874620674182256at_nat(A,A4)
       => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),A),Ra) ) ) ).

% refl_onD
tff(fact_2124_refl__onD,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(A4),Ra)
     => ( member2643936169264416010at_nat(A,A4)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),A)),Ra) ) ) ).

% refl_onD
tff(fact_2125_refl__onD,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat,A: nat] :
      ( aa_set3702741264012733747_nat_o(refl_on_nat(A4),Ra)
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),A)),Ra) ) ) ).

% refl_onD
tff(fact_2126_refl__onD,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int,A: int] :
      ( refl_on_int(A4,Ra)
     => ( aa_set_int_o(member_int(A),A4)
       => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),A)),Ra) ) ) ).

% refl_onD
tff(fact_2127_refl__onD1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Ra: set_Pr5564308138774400199at_nat,Xa: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
      ( refl_o7446464703481863270at_nat(A4,Ra)
     => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(Xa,Y2),Ra)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),A4) ) ) ).

% refl_onD1
tff(fact_2128_refl__onD1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( refl_o7122697186039111552at_nat(A4,Ra)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),Ra)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),A4) ) ) ).

% refl_onD1
tff(fact_2129_refl__onD1,axiom,
    ! [A4: set_o,Ra: set_Product_prod_o_o,Xa: $o,Y2: $o] :
      ( refl_on_o(A4,Ra)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa)),(Y2)),Ra)
       => aa_set_o_o(member_o((Xa)),A4) ) ) ).

% refl_onD1
tff(fact_2130_refl__onD1,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat,Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( refl_o6272981002401706822at_nat(A4,Ra)
     => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2),Ra)
       => member6273874620674182256at_nat(Xa,A4) ) ) ).

% refl_onD1
tff(fact_2131_refl__onD1,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat,Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(A4),Ra)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),Ra)
       => member2643936169264416010at_nat(Xa,A4) ) ) ).

% refl_onD1
tff(fact_2132_refl__onD1,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat,Xa: nat,Y2: nat] :
      ( aa_set3702741264012733747_nat_o(refl_on_nat(A4),Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),Ra)
       => aa_set_nat_o(member_nat(Xa),A4) ) ) ).

% refl_onD1
tff(fact_2133_refl__onD1,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int,Xa: int,Y2: int] :
      ( refl_on_int(A4,Ra)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),Ra)
       => aa_set_int_o(member_int(Xa),A4) ) ) ).

% refl_onD1
tff(fact_2134_refl__onD2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Ra: set_Pr5564308138774400199at_nat,Xa: produc3843707927480180839at_nat,Y2: produc3843707927480180839at_nat] :
      ( refl_o7446464703481863270at_nat(A4,Ra)
     => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(Xa,Y2),Ra)
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y2),A4) ) ) ).

% refl_onD2
tff(fact_2135_refl__onD2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat,Xa: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( refl_o7122697186039111552at_nat(A4,Ra)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),Ra)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),A4) ) ) ).

% refl_onD2
tff(fact_2136_refl__onD2,axiom,
    ! [A4: set_o,Ra: set_Product_prod_o_o,Xa: $o,Y2: $o] :
      ( refl_on_o(A4,Ra)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa)),(Y2)),Ra)
       => aa_set_o_o(member_o((Y2)),A4) ) ) ).

% refl_onD2
tff(fact_2137_refl__onD2,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat,Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( refl_o6272981002401706822at_nat(A4,Ra)
     => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2),Ra)
       => member6273874620674182256at_nat(Y2,A4) ) ) ).

% refl_onD2
tff(fact_2138_refl__onD2,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat,Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(A4),Ra)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)),Ra)
       => member2643936169264416010at_nat(Y2,A4) ) ) ).

% refl_onD2
tff(fact_2139_refl__onD2,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat,Xa: nat,Y2: nat] :
      ( aa_set3702741264012733747_nat_o(refl_on_nat(A4),Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),Ra)
       => aa_set_nat_o(member_nat(Y2),A4) ) ) ).

% refl_onD2
tff(fact_2140_refl__onD2,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int,Xa: int,Y2: int] :
      ( refl_on_int(A4,Ra)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)),Ra)
       => aa_set_int_o(member_int(Y2),A4) ) ) ).

% refl_onD2
tff(fact_2141_INT__extend__simps_I8_J,axiom,
    ! [B5: fun_nat_set_nat,A4: set_set_nat] : comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_ms(B5)),A4)) = comple7806235888213564991et_nat(image_nat_set_nat(B5,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))) ).

% INT_extend_simps(8)
tff(fact_2142_Id__on__def,axiom,
    ! [A4: set_se5479201105039950375at_nat] : id_on_3275838214495744996at_nat(A4) = comple7342073961973170484at_nat(image_1124969593608450885at_nat(aTP_Lamm_mt,A4)) ).

% Id_on_def
tff(fact_2143_Id__on__def,axiom,
    ! [A4: set_int] : id_on_int(A4) = comple5382143125604098054nt_int(image_8635204845542730022nt_int(aTP_Lamm_mu,A4)) ).

% Id_on_def
tff(fact_2144_Id__on__def,axiom,
    ! [A4: set_se7855581050983116737at_nat] : id_on_6713504163309144702at_nat(A4) = comple4364388969068068532at_nat(image_6898982319499645995at_nat(aTP_Lamm_mv,A4)) ).

% Id_on_def
tff(fact_2145_Id__on__def,axiom,
    ! [A4: set_nat] : id_on_nat(A4) = comple5685304695842803022at_nat(image_7178329752028323786at_nat(aTP_Lamm_mw,A4)) ).

% Id_on_def
tff(fact_2146_INF__le__SUP,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,A4))) ) ).

% INF_le_SUP
tff(fact_2147_INF__le__SUP,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4))) ) ).

% INF_le_SUP
tff(fact_2148_INF__le__SUP,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))) ) ).

% INF_le_SUP
tff(fact_2149_INF__le__SUP,axiom,
    ! [A4: set_o,F: fun_o_set_nat] :
      ( ( A4 != bot_bot_set_o )
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))) ) ).

% INF_le_SUP
tff(fact_2150_INF__le__SUP,axiom,
    ! [A4: set_nat,F: fun_nat_set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))) ) ).

% INF_le_SUP
tff(fact_2151_INF__le__SUP,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))) ) ).

% INF_le_SUP
tff(fact_2152_INF__le__SUP,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit] :
      ( ( A4 != bot_bo228742789529271731at_nat )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_7082456443403096939t_unit(F,A4))),comple4687483117567863418t_unit(image_7082456443403096939t_unit(F,A4))) ) ).

% INF_le_SUP
tff(fact_2153_INF__le__SUP,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4))),comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4))) ) ).

% INF_le_SUP
tff(fact_2154_INF__le__SUP,axiom,
    ! [A4: set_o,F: fun_o_Product_unit] :
      ( ( A4 != bot_bot_set_o )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ) ).

% INF_le_SUP
tff(fact_2155_INF__le__SUP,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ) ).

% INF_le_SUP
tff(fact_2156_right__unique__OO,axiom,
    ! [A4: fun_Pr2028404973800837934_nat_o,B5: fun_Pr4738963024209204295_int_o] :
      ( right_1383565826979811350at_nat(A4)
     => ( right_8621877368619751767at_int(B5)
       => right_8621877368619751767at_int(relcom8911704206367525917at_int(A4,B5)) ) ) ).

% right_unique_OO
tff(fact_2157_right__unique__OO,axiom,
    ! [A4: fun_int_fun_int_o,B5: fun_in4891666191172872833eger_o] :
      ( right_unique_int_int(A4)
     => ( right_8475562212078014689nteger(B5)
       => right_8475562212078014689nteger(relcom967497705283160040nteger(A4,B5)) ) ) ).

% right_unique_OO
tff(fact_2158_INT__E,axiom,
    ! [B: $o,B5: fun_o_set_o,A4: set_o,A: $o] :
      ( aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_o_set_o(B5,A4)))
     => ( ~ aa_set_o_o(member_o((B)),aa_o_set_o(B5,(A)))
       => ~ aa_set_o_o(member_o((A)),A4) ) ) ).

% INT_E
tff(fact_2159_INT__E,axiom,
    ! [B: $o,B5: fun_nat_set_o,A4: set_nat,A: nat] :
      ( aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_nat_set_o(B5,A4)))
     => ( ~ aa_set_o_o(member_o((B)),aa_nat_set_o(B5,A))
       => ~ aa_set_nat_o(member_nat(A),A4) ) ) ).

% INT_E
tff(fact_2160_INT__E,axiom,
    ! [B: $o,B5: fun_int_set_o,A4: set_int,A: int] :
      ( aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_int_set_o(B5,A4)))
     => ( ~ aa_set_o_o(member_o((B)),aa_int_set_o(B5,A))
       => ~ aa_set_int_o(member_int(A),A4) ) ) ).

% INT_E
tff(fact_2161_INT__E,axiom,
    ! [B: int,B5: fun_o_set_int,A4: set_o,A: $o] :
      ( aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_o_set_int(B5,A4)))
     => ( ~ aa_set_int_o(member_int(B),aa_o_set_int(B5,(A)))
       => ~ aa_set_o_o(member_o((A)),A4) ) ) ).

% INT_E
tff(fact_2162_INT__E,axiom,
    ! [B: int,B5: fun_nat_set_int,A4: set_nat,A: nat] :
      ( aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_nat_set_int(B5,A4)))
     => ( ~ aa_set_int_o(member_int(B),aa_nat_set_int(B5,A))
       => ~ aa_set_nat_o(member_nat(A),A4) ) ) ).

% INT_E
tff(fact_2163_INT__E,axiom,
    ! [B: int,B5: fun_int_set_int,A4: set_int,A: int] :
      ( aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_int_set_int(B5,A4)))
     => ( ~ aa_set_int_o(member_int(B),aa_int_set_int(B5,A))
       => ~ aa_set_int_o(member_int(A),A4) ) ) ).

% INT_E
tff(fact_2164_INT__E,axiom,
    ! [B: nat,B5: fun_o_set_nat,A4: set_o,A: $o] :
      ( aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_o_set_nat(B5,A4)))
     => ( ~ aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(A)))
       => ~ aa_set_o_o(member_o((A)),A4) ) ) ).

% INT_E
tff(fact_2165_INT__E,axiom,
    ! [B: nat,B5: fun_nat_set_nat,A4: set_nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4)))
     => ( ~ aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A))
       => ~ aa_set_nat_o(member_nat(A),A4) ) ) ).

% INT_E
tff(fact_2166_INT__E,axiom,
    ! [B: nat,B5: fun_int_set_nat,A4: set_int,A: int] :
      ( aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_int_set_nat(B5,A4)))
     => ( ~ aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,A))
       => ~ aa_set_int_o(member_int(A),A4) ) ) ).

% INT_E
tff(fact_2167_INT__E,axiom,
    ! [B: product_prod_nat_nat,B5: fun_o_4630291819367765772at_nat,A4: set_o,A: $o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),comple1930127330929219688at_nat(image_7077428525161401678at_nat(B5,A4)))
     => ( ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),aa_o_s3322794332960822277at_nat(B5,(A)))
       => ~ aa_set_o_o(member_o((A)),A4) ) ) ).

% INT_E
tff(fact_2168_INT__D,axiom,
    ! [B: $o,B5: fun_o_set_o,A4: set_o,A: $o] :
      ( aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_o_set_o(B5,A4)))
     => ( aa_set_o_o(member_o((A)),A4)
       => aa_set_o_o(member_o((B)),aa_o_set_o(B5,(A))) ) ) ).

% INT_D
tff(fact_2169_INT__D,axiom,
    ! [B: $o,B5: fun_nat_set_o,A4: set_nat,A: nat] :
      ( aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_nat_set_o(B5,A4)))
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_set_o_o(member_o((B)),aa_nat_set_o(B5,A)) ) ) ).

% INT_D
tff(fact_2170_INT__D,axiom,
    ! [B: $o,B5: fun_int_set_o,A4: set_int,A: int] :
      ( aa_set_o_o(member_o((B)),comple3063163877087187839_set_o(image_int_set_o(B5,A4)))
     => ( aa_set_int_o(member_int(A),A4)
       => aa_set_o_o(member_o((B)),aa_int_set_o(B5,A)) ) ) ).

% INT_D
tff(fact_2171_INT__D,axiom,
    ! [B: int,B5: fun_o_set_int,A4: set_o,A: $o] :
      ( aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_o_set_int(B5,A4)))
     => ( aa_set_o_o(member_o((A)),A4)
       => aa_set_int_o(member_int(B),aa_o_set_int(B5,(A))) ) ) ).

% INT_D
tff(fact_2172_INT__D,axiom,
    ! [B: int,B5: fun_nat_set_int,A4: set_nat,A: nat] :
      ( aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_nat_set_int(B5,A4)))
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_set_int_o(member_int(B),aa_nat_set_int(B5,A)) ) ) ).

% INT_D
tff(fact_2173_INT__D,axiom,
    ! [B: int,B5: fun_int_set_int,A4: set_int,A: int] :
      ( aa_set_int_o(member_int(B),comple3628384868704368283et_int(image_int_set_int(B5,A4)))
     => ( aa_set_int_o(member_int(A),A4)
       => aa_set_int_o(member_int(B),aa_int_set_int(B5,A)) ) ) ).

% INT_D
tff(fact_2174_INT__D,axiom,
    ! [B: nat,B5: fun_o_set_nat,A4: set_o,A: $o] :
      ( aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_o_set_nat(B5,A4)))
     => ( aa_set_o_o(member_o((A)),A4)
       => aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(A))) ) ) ).

% INT_D
tff(fact_2175_INT__D,axiom,
    ! [B: nat,B5: fun_nat_set_nat,A4: set_nat,A: nat] :
      ( aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4)))
     => ( aa_set_nat_o(member_nat(A),A4)
       => aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,A)) ) ) ).

% INT_D
tff(fact_2176_INT__D,axiom,
    ! [B: nat,B5: fun_int_set_nat,A4: set_int,A: int] :
      ( aa_set_nat_o(member_nat(B),comple7806235888213564991et_nat(image_int_set_nat(B5,A4)))
     => ( aa_set_int_o(member_int(A),A4)
       => aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,A)) ) ) ).

% INT_D
tff(fact_2177_INT__D,axiom,
    ! [B: product_prod_nat_nat,B5: fun_o_4630291819367765772at_nat,A4: set_o,A: $o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),comple1930127330929219688at_nat(image_7077428525161401678at_nat(B5,A4)))
     => ( aa_set_o_o(member_o((A)),A4)
       => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),aa_o_s3322794332960822277at_nat(B5,(A))) ) ) ).

% INT_D
tff(fact_2178_UN__E,axiom,
    ! [B: $o,B5: fun_o_set_o,A4: set_o] :
      ( aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_o_set_o(B5,A4)))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ~ aa_set_o_o(member_o((B)),aa_o_set_o(B5,(X))) ) ) ).

% UN_E
tff(fact_2179_UN__E,axiom,
    ! [B: $o,B5: fun_nat_set_o,A4: set_nat] :
      ( aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_nat_set_o(B5,A4)))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ~ aa_set_o_o(member_o((B)),aa_nat_set_o(B5,X)) ) ) ).

% UN_E
tff(fact_2180_UN__E,axiom,
    ! [B: $o,B5: fun_int_set_o,A4: set_int] :
      ( aa_set_o_o(member_o((B)),comple90263536869209701_set_o(image_int_set_o(B5,A4)))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ~ aa_set_o_o(member_o((B)),aa_int_set_o(B5,X)) ) ) ).

% UN_E
tff(fact_2181_UN__E,axiom,
    ! [B: int,B5: fun_o_set_int,A4: set_o] :
      ( aa_set_int_o(member_int(B),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(B5,A4)))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ~ aa_set_int_o(member_int(B),aa_o_set_int(B5,(X))) ) ) ).

% UN_E
tff(fact_2182_UN__E,axiom,
    ! [B: int,B5: fun_nat_set_int,A4: set_nat] :
      ( aa_set_int_o(member_int(B),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(B5,A4)))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ~ aa_set_int_o(member_int(B),aa_nat_set_int(B5,X)) ) ) ).

% UN_E
tff(fact_2183_UN__E,axiom,
    ! [B: int,B5: fun_int_set_int,A4: set_int] :
      ( aa_set_int_o(member_int(B),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(B5,A4)))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ~ aa_set_int_o(member_int(B),aa_int_set_int(B5,X)) ) ) ).

% UN_E
tff(fact_2184_UN__E,axiom,
    ! [B: nat,B5: fun_o_set_nat,A4: set_o] :
      ( aa_set_nat_o(member_nat(B),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4)))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ~ aa_set_nat_o(member_nat(B),aa_o_set_nat(B5,(X))) ) ) ).

% UN_E
tff(fact_2185_UN__E,axiom,
    ! [B: nat,B5: fun_nat_set_nat,A4: set_nat] :
      ( aa_set_nat_o(member_nat(B),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4)))
     => ~ ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ~ aa_set_nat_o(member_nat(B),aa_nat_set_nat(B5,X)) ) ) ).

% UN_E
tff(fact_2186_UN__E,axiom,
    ! [B: nat,B5: fun_int_set_nat,A4: set_int] :
      ( aa_set_nat_o(member_nat(B),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,A4)))
     => ~ ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ~ aa_set_nat_o(member_nat(B),aa_int_set_nat(B5,X)) ) ) ).

% UN_E
tff(fact_2187_UN__E,axiom,
    ! [B: product_prod_nat_nat,B5: fun_o_4630291819367765772at_nat,A4: set_o] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),comple5685304695842803022at_nat(image_7077428525161401678at_nat(B5,A4)))
     => ~ ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),aa_o_s3322794332960822277at_nat(B5,(X))) ) ) ).

% UN_E
tff(fact_2188_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_rat,A4: set_nat] : aa_set6107595668591848464et_rat(comple3890839924845867745et_rat,image_nat_set_rat(aTP_Lamm_mx(F),A4)) = aa_set_nat_set_rat(image_nat_rat(F),A4) ).

% UNION_singleton_eq_range
tff(fact_2189_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_int,A4: set_nat] : aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(aTP_Lamm_my(F),A4)) = aa_set_nat_set_int(image_nat_int(F),A4) ).

% UNION_singleton_eq_range
tff(fact_2190_UNION__singleton__eq__range,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aTP_Lamm_mz(F),A4)) = aa_set_int_set_int(image_int_int(F),A4) ).

% UNION_singleton_eq_range
tff(fact_2191_UNION__singleton__eq__range,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aTP_Lamm_na(F),A4)) = aa_set_nat_set_nat(image_nat_nat(F),A4) ).

% UNION_singleton_eq_range
tff(fact_2192_UNION__singleton__eq__range,axiom,
    ! [F: fun_int_nat,A4: set_int] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(aTP_Lamm_nb(F),A4)) = aa_set_int_set_nat(image_int_nat(F),A4) ).

% UNION_singleton_eq_range
tff(fact_2193_INF__eq,axiom,
    ! [A4: set_o,B5: set_o,G: fun_o_Product_unit,F: fun_o_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(G,(X5))),aa_o_Product_unit(F,(I4))) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(X5))),aa_o_Product_unit(G,(J2))) ) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit(F,A4)) = comple2584293577114468500t_unit(image_o_Product_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2194_INF__eq,axiom,
    ! [A4: set_o,B5: set_nat,G: fun_nat_Product_unit,F: fun_o_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(G,X5)),aa_o_Product_unit(F,(I4))) ) )
     => ( ! [J2: nat] :
            ( aa_set_nat_o(member_nat(J2),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(X5))),aa_nat_Product_unit(G,J2)) ) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit(F,A4)) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2195_INF__eq,axiom,
    ! [A4: set_o,B5: set_int,G: fun_int_Product_unit,F: fun_o_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: int] :
              ( aa_set_int_o(member_int(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,X5)),aa_o_Product_unit(F,(I4))) ) )
     => ( ! [J2: int] :
            ( aa_set_int_o(member_int(J2),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(X5))),aa_int_Product_unit(G,J2)) ) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit(F,A4)) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2196_INF__eq,axiom,
    ! [A4: set_nat,B5: set_o,G: fun_o_Product_unit,F: fun_nat_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(G,(X5))),aa_nat_Product_unit(F,I4)) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,X5)),aa_o_Product_unit(G,(J2))) ) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4)) = comple2584293577114468500t_unit(image_o_Product_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2197_INF__eq,axiom,
    ! [A4: set_nat,B5: set_nat,G: fun_nat_Product_unit,F: fun_nat_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(G,X5)),aa_nat_Product_unit(F,I4)) ) )
     => ( ! [J2: nat] :
            ( aa_set_nat_o(member_nat(J2),B5)
           => ? [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,X5)),aa_nat_Product_unit(G,J2)) ) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4)) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2198_INF__eq,axiom,
    ! [A4: set_nat,B5: set_int,G: fun_int_Product_unit,F: fun_nat_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => ? [X5: int] :
              ( aa_set_int_o(member_int(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,X5)),aa_nat_Product_unit(F,I4)) ) )
     => ( ! [J2: int] :
            ( aa_set_int_o(member_int(J2),B5)
           => ? [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,X5)),aa_int_Product_unit(G,J2)) ) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4)) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2199_INF__eq,axiom,
    ! [A4: set_int,B5: set_o,G: fun_o_Product_unit,F: fun_int_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(G,(X5))),aa_int_Product_unit(F,I4)) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X5)),aa_o_Product_unit(G,(J2))) ) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4)) = comple2584293577114468500t_unit(image_o_Product_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2200_INF__eq,axiom,
    ! [A4: set_int,B5: set_nat,G: fun_nat_Product_unit,F: fun_int_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(G,X5)),aa_int_Product_unit(F,I4)) ) )
     => ( ! [J2: nat] :
            ( aa_set_nat_o(member_nat(J2),B5)
           => ? [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X5)),aa_nat_Product_unit(G,J2)) ) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4)) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2201_INF__eq,axiom,
    ! [A4: set_int,B5: set_int,G: fun_int_Product_unit,F: fun_int_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => ? [X5: int] :
              ( aa_set_int_o(member_int(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,X5)),aa_int_Product_unit(F,I4)) ) )
     => ( ! [J2: int] :
            ( aa_set_int_o(member_int(J2),B5)
           => ? [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X5)),aa_int_Product_unit(G,J2)) ) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4)) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2202_INF__eq,axiom,
    ! [A4: set_o,B5: set_o,G: fun_o_set_nat,F: fun_o_set_nat] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(G,(X5))),aa_o_set_nat(F,(I4))) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(X5))),aa_o_set_nat(G,(J2))) ) )
       => ( comple7806235888213564991et_nat(image_o_set_nat(F,A4)) = comple7806235888213564991et_nat(image_o_set_nat(G,B5)) ) ) ) ).

% INF_eq
tff(fact_2203_SUP__eq,axiom,
    ! [A4: set_o,B5: set_o,F: fun_o_Product_unit,G: fun_o_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I4))),aa_o_Product_unit(G,(X5))) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(G,(J2))),aa_o_Product_unit(F,(X5))) ) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit(F,A4)) = comple4687483117567863418t_unit(image_o_Product_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2204_SUP__eq,axiom,
    ! [A4: set_o,B5: set_nat,F: fun_o_Product_unit,G: fun_nat_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I4))),aa_nat_Product_unit(G,X5)) ) )
     => ( ! [J2: nat] :
            ( aa_set_nat_o(member_nat(J2),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(G,J2)),aa_o_Product_unit(F,(X5))) ) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit(F,A4)) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2205_SUP__eq,axiom,
    ! [A4: set_o,B5: set_int,F: fun_o_Product_unit,G: fun_int_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: int] :
              ( aa_set_int_o(member_int(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I4))),aa_int_Product_unit(G,X5)) ) )
     => ( ! [J2: int] :
            ( aa_set_int_o(member_int(J2),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,J2)),aa_o_Product_unit(F,(X5))) ) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit(F,A4)) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2206_SUP__eq,axiom,
    ! [A4: set_nat,B5: set_o,F: fun_nat_Product_unit,G: fun_o_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I4)),aa_o_Product_unit(G,(X5))) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(G,(J2))),aa_nat_Product_unit(F,X5)) ) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4)) = comple4687483117567863418t_unit(image_o_Product_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2207_SUP__eq,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_Product_unit,G: fun_nat_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I4)),aa_nat_Product_unit(G,X5)) ) )
     => ( ! [J2: nat] :
            ( aa_set_nat_o(member_nat(J2),B5)
           => ? [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(G,J2)),aa_nat_Product_unit(F,X5)) ) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4)) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2208_SUP__eq,axiom,
    ! [A4: set_nat,B5: set_int,F: fun_nat_Product_unit,G: fun_int_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => ? [X5: int] :
              ( aa_set_int_o(member_int(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I4)),aa_int_Product_unit(G,X5)) ) )
     => ( ! [J2: int] :
            ( aa_set_int_o(member_int(J2),B5)
           => ? [X5: nat] :
                ( aa_set_nat_o(member_nat(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,J2)),aa_nat_Product_unit(F,X5)) ) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4)) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2209_SUP__eq,axiom,
    ! [A4: set_int,B5: set_o,F: fun_int_Product_unit,G: fun_o_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I4)),aa_o_Product_unit(G,(X5))) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(G,(J2))),aa_int_Product_unit(F,X5)) ) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4)) = comple4687483117567863418t_unit(image_o_Product_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2210_SUP__eq,axiom,
    ! [A4: set_int,B5: set_nat,F: fun_int_Product_unit,G: fun_nat_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => ? [X5: nat] :
              ( aa_set_nat_o(member_nat(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I4)),aa_nat_Product_unit(G,X5)) ) )
     => ( ! [J2: nat] :
            ( aa_set_nat_o(member_nat(J2),B5)
           => ? [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(G,J2)),aa_int_Product_unit(F,X5)) ) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4)) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2211_SUP__eq,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_Product_unit,G: fun_int_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => ? [X5: int] :
              ( aa_set_int_o(member_int(X5),B5)
              & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I4)),aa_int_Product_unit(G,X5)) ) )
     => ( ! [J2: int] :
            ( aa_set_int_o(member_int(J2),B5)
           => ? [X5: int] :
                ( aa_set_int_o(member_int(X5),A4)
                & aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,J2)),aa_int_Product_unit(F,X5)) ) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4)) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2212_SUP__eq,axiom,
    ! [A4: set_o,B5: set_o,F: fun_o_set_nat,G: fun_o_set_nat] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => ? [X5: $o] :
              ( aa_set_o_o(member_o((X5)),B5)
              & aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I4))),aa_o_set_nat(G,(X5))) ) )
     => ( ! [J2: $o] :
            ( aa_set_o_o(member_o((J2)),B5)
           => ? [X5: $o] :
                ( aa_set_o_o(member_o((X5)),A4)
                & aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(G,(J2))),aa_o_set_nat(F,(X5))) ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(G,B5)) ) ) ) ).

% SUP_eq
tff(fact_2213_INF__eq__const,axiom,
    ! [I3: set_int,F: fun_int_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => ( aa_int_Product_unit(F,I4) = Xa ) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2214_INF__eq__const,axiom,
    ! [I3: set_Product_unit,F: fun_Pr2191293291710260995t_unit,Xa: product_unit] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => ( aa_Pro8294289767544744956t_unit(F,I4) = Xa ) )
       => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2215_INF__eq__const,axiom,
    ! [I3: set_o,F: fun_o_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => ( aa_o_Product_unit(F,(I4)) = Xa ) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2216_INF__eq__const,axiom,
    ! [I3: set_nat,F: fun_nat_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => ( aa_nat_Product_unit(F,I4) = Xa ) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2217_INF__eq__const,axiom,
    ! [I3: set_int,F: fun_int_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => ( aa_int_set_nat(F,I4) = Xa ) )
       => ( comple7806235888213564991et_nat(image_int_set_nat(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2218_INF__eq__const,axiom,
    ! [I3: set_Product_unit,F: fun_Pr1628945989756059814et_nat,Xa: set_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => ( aa_Pro6064029160526250199et_nat(F,I4) = Xa ) )
       => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2219_INF__eq__const,axiom,
    ! [I3: set_o,F: fun_o_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => ( aa_o_set_nat(F,(I4)) = Xa ) )
       => ( comple7806235888213564991et_nat(image_o_set_nat(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2220_INF__eq__const,axiom,
    ! [I3: set_nat,F: fun_nat_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => ( aa_nat_set_nat(F,I4) = Xa ) )
       => ( comple7806235888213564991et_nat(image_nat_set_nat(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2221_INF__eq__const,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,Xa: product_unit] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => ( aa_Pro676242055593665672t_unit(F,I4) = Xa ) )
       => ( comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2222_INF__eq__const,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,Xa: set_nat] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => ( aa_Pro9123573313595863371et_nat(F,I4) = Xa ) )
       => ( comple7806235888213564991et_nat(image_15824709712370754et_nat(F,I3)) = Xa ) ) ) ).

% INF_eq_const
tff(fact_2223_SUP__eq__const,axiom,
    ! [I3: set_int,F: fun_int_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => ( aa_int_Product_unit(F,I4) = Xa ) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2224_SUP__eq__const,axiom,
    ! [I3: set_Product_unit,F: fun_Pr2191293291710260995t_unit,Xa: product_unit] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => ( aa_Pro8294289767544744956t_unit(F,I4) = Xa ) )
       => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2225_SUP__eq__const,axiom,
    ! [I3: set_o,F: fun_o_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => ( aa_o_Product_unit(F,(I4)) = Xa ) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2226_SUP__eq__const,axiom,
    ! [I3: set_nat,F: fun_nat_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => ( aa_nat_Product_unit(F,I4) = Xa ) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2227_SUP__eq__const,axiom,
    ! [I3: set_int,F: fun_int_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => ( aa_int_set_nat(F,I4) = Xa ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2228_SUP__eq__const,axiom,
    ! [I3: set_Product_unit,F: fun_Pr1628945989756059814et_nat,Xa: set_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => ( aa_Pro6064029160526250199et_nat(F,I4) = Xa ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2229_SUP__eq__const,axiom,
    ! [I3: set_o,F: fun_o_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => ( aa_o_set_nat(F,(I4)) = Xa ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2230_SUP__eq__const,axiom,
    ! [I3: set_nat,F: fun_nat_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => ( aa_nat_set_nat(F,I4) = Xa ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2231_SUP__eq__const,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,Xa: product_unit] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => ( aa_Pro676242055593665672t_unit(F,I4) = Xa ) )
       => ( comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2232_SUP__eq__const,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,Xa: set_nat] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => ( aa_Pro9123573313595863371et_nat(F,I4) = Xa ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,I3)) = Xa ) ) ) ).

% SUP_eq_const
tff(fact_2233_INF__image,axiom,
    ! [G: fun_o_o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(complete_Inf_Inf_o,aa_set_o_set_o(image_o_o(G),image_nat_o(F,A4)))
    <=> aa_set_o_o(complete_Inf_Inf_o,image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F),A4)) ) ).

% INF_image
tff(fact_2234_INF__image,axiom,
    ! [G: fun_rat_nat,F: fun_nat_rat,A4: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,image_rat_nat(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(G,F)),A4)) ).

% INF_image
tff(fact_2235_INF__image,axiom,
    ! [G: fun_nat_nat,F: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),A4)) ).

% INF_image
tff(fact_2236_INF__image,axiom,
    ! [G: fun_nat_nat,F: fun_int_nat,A4: set_int] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),F)),A4)) ).

% INF_image
tff(fact_2237_INF__image,axiom,
    ! [G: fun_int_nat,F: fun_nat_int,A4: set_nat] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(G),aa_set_nat_set_int(image_nat_int(F),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(comp_int_nat_nat(G,F)),A4)) ).

% INF_image
tff(fact_2238_INF__image,axiom,
    ! [G: fun_int_nat,F: fun_int_int,A4: set_int] : aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(G),aa_set_int_set_int(image_int_int(F),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),A4)) ).

% INF_image
tff(fact_2239_INF__image,axiom,
    ! [G: fun_rat_int,F: fun_nat_rat,A4: set_nat] : aa_set_int_int(complete_Inf_Inf_int,image_rat_int(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(G,F)),A4)) ).

% INF_image
tff(fact_2240_INF__image,axiom,
    ! [G: fun_nat_int,F: fun_nat_nat,A4: set_nat] : aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(comp_nat_int_nat(G,F)),A4)) ).

% INF_image
tff(fact_2241_INF__image,axiom,
    ! [G: fun_nat_int,F: fun_int_nat,A4: set_int] : aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(comp_nat_int_int(G,F)),A4)) ).

% INF_image
tff(fact_2242_INF__image,axiom,
    ! [G: fun_int_int,F: fun_nat_int,A4: set_nat] : aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(G),aa_set_nat_set_int(image_nat_int(F),A4))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(comp_int_int_nat(G,F)),A4)) ).

% INF_image
tff(fact_2243_SUP__image,axiom,
    ! [G: fun_o_o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(complete_Sup_Sup_o,aa_set_o_set_o(image_o_o(G),image_nat_o(F,A4)))
    <=> aa_set_o_o(complete_Sup_Sup_o,image_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F),A4)) ) ).

% SUP_image
tff(fact_2244_SUP__image,axiom,
    ! [G: fun_rat_nat,F: fun_nat_rat,A4: set_nat] : aa_set_nat_nat(complete_Sup_Sup_nat,image_rat_nat(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(comp_rat_nat_nat(G,F)),A4)) ).

% SUP_image
tff(fact_2245_SUP__image,axiom,
    ! [G: fun_nat_nat,F: fun_nat_nat,A4: set_nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),A4)) ).

% SUP_image
tff(fact_2246_SUP__image,axiom,
    ! [G: fun_nat_nat,F: fun_int_nat,A4: set_int] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun8757566081149294410nt_nat(comp_nat_nat_int,G),F)),A4)) ).

% SUP_image
tff(fact_2247_SUP__image,axiom,
    ! [G: fun_int_nat,F: fun_nat_int,A4: set_nat] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),aa_set_nat_set_int(image_nat_int(F),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(comp_int_nat_nat(G,F)),A4)) ).

% SUP_image
tff(fact_2248_SUP__image,axiom,
    ! [G: fun_int_nat,F: fun_int_int,A4: set_int] : aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),aa_set_int_set_int(image_int_int(F),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),A4)) ).

% SUP_image
tff(fact_2249_SUP__image,axiom,
    ! [G: fun_rat_int,F: fun_nat_rat,A4: set_nat] : aa_set_int_int(complete_Sup_Sup_int,image_rat_int(G,aa_set_nat_set_rat(image_nat_rat(F),A4))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(comp_rat_int_nat(G,F)),A4)) ).

% SUP_image
tff(fact_2250_SUP__image,axiom,
    ! [G: fun_nat_int,F: fun_nat_nat,A4: set_nat] : aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(G),aa_set_nat_set_nat(image_nat_nat(F),A4))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(comp_nat_int_nat(G,F)),A4)) ).

% SUP_image
tff(fact_2251_SUP__image,axiom,
    ! [G: fun_nat_int,F: fun_int_nat,A4: set_int] : aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(G),aa_set_int_set_nat(image_int_nat(F),A4))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(comp_nat_int_int(G,F)),A4)) ).

% SUP_image
tff(fact_2252_SUP__image,axiom,
    ! [G: fun_int_int,F: fun_nat_int,A4: set_nat] : aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(G),aa_set_nat_set_int(image_nat_int(F),A4))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(comp_int_int_nat(G,F)),A4)) ).

% SUP_image
tff(fact_2253_INF__greatest,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,U: set_nat,F: fun_Pr6764980218223570944et_nat] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(U),aa_Pro1692590229789857841et_nat(F,I4)) )
     => aa_set_nat_o(ord_less_eq_set_nat(U),comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,A4))) ) ).

% INF_greatest
tff(fact_2254_INF__greatest,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,U: set_nat,F: fun_Pr9038118721296872730et_nat] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(U),aa_Pro9123573313595863371et_nat(F,I4)) )
     => aa_set_nat_o(ord_less_eq_set_nat(U),comple7806235888213564991et_nat(image_15824709712370754et_nat(F,A4))) ) ).

% INF_greatest
tff(fact_2255_INF__greatest,axiom,
    ! [A4: set_o,U: set_nat,F: fun_o_set_nat] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(U),aa_o_set_nat(F,(I4))) )
     => aa_set_nat_o(ord_less_eq_set_nat(U),comple7806235888213564991et_nat(image_o_set_nat(F,A4))) ) ).

% INF_greatest
tff(fact_2256_INF__greatest,axiom,
    ! [A4: set_nat,U: set_nat,F: fun_nat_set_nat] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(U),aa_nat_set_nat(F,I4)) )
     => aa_set_nat_o(ord_less_eq_set_nat(U),comple7806235888213564991et_nat(image_nat_set_nat(F,A4))) ) ).

% INF_greatest
tff(fact_2257_INF__greatest,axiom,
    ! [A4: set_int,U: set_nat,F: fun_int_set_nat] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(U),aa_int_set_nat(F,I4)) )
     => aa_set_nat_o(ord_less_eq_set_nat(U),comple7806235888213564991et_nat(image_int_set_nat(F,A4))) ) ).

% INF_greatest
tff(fact_2258_INF__greatest,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,U: product_unit,F: fun_Pr8319229360018371625t_unit] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_Pro1279885795167260962t_unit(F,I4)) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_7082456443403096939t_unit(F,A4))) ) ).

% INF_greatest
tff(fact_2259_INF__greatest,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,U: product_unit,F: fun_Pr2481279687482350479t_unit] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_Pro676242055593665672t_unit(F,I4)) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4))) ) ).

% INF_greatest
tff(fact_2260_INF__greatest,axiom,
    ! [A4: set_o,U: product_unit,F: fun_o_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_o_Product_unit(F,(I4))) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_o_Product_unit(F,A4))) ) ).

% INF_greatest
tff(fact_2261_INF__greatest,axiom,
    ! [A4: set_nat,U: product_unit,F: fun_nat_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_nat_Product_unit(F,I4)) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))) ) ).

% INF_greatest
tff(fact_2262_INF__greatest,axiom,
    ! [A4: set_int,U: product_unit,F: fun_int_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_int_Product_unit(F,I4)) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))) ) ).

% INF_greatest
tff(fact_2263_INF__lower2,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,U: set_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,I)),U)
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2264_INF__lower2,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,U: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,I)),U)
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2265_INF__lower2,axiom,
    ! [I: $o,A4: set_o,F: fun_o_set_nat,U: set_nat] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I))),U)
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2266_INF__lower2,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_set_nat,U: set_nat] :
      ( aa_set_nat_o(member_nat(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,I)),U)
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2267_INF__lower2,axiom,
    ! [I: int,A4: set_int,F: fun_int_set_nat,U: set_nat] :
      ( aa_set_int_o(member_int(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,I)),U)
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_int_set_nat(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2268_INF__lower2,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit,U: product_unit] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,I)),U)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_7082456443403096939t_unit(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2269_INF__lower2,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,U: product_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,I)),U)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2270_INF__lower2,axiom,
    ! [I: $o,A4: set_o,F: fun_o_Product_unit,U: product_unit] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I))),U)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2271_INF__lower2,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_Product_unit,U: product_unit] :
      ( aa_set_nat_o(member_nat(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I)),U)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2272_INF__lower2,axiom,
    ! [I: int,A4: set_int,F: fun_int_Product_unit,U: product_unit] :
      ( aa_set_int_o(member_int(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I)),U)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))),U) ) ) ).

% INF_lower2
tff(fact_2273_INF__lower,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,A4))),aa_Pro1692590229789857841et_nat(F,I)) ) ).

% INF_lower
tff(fact_2274_INF__lower,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(F,A4))),aa_Pro9123573313595863371et_nat(F,I)) ) ).

% INF_lower
tff(fact_2275_INF__lower,axiom,
    ! [I: $o,A4: set_o,F: fun_o_set_nat] :
      ( aa_set_o_o(member_o((I)),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,A4))),aa_o_set_nat(F,(I))) ) ).

% INF_lower
tff(fact_2276_INF__lower,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(F,A4))),aa_nat_set_nat(F,I)) ) ).

% INF_lower
tff(fact_2277_INF__lower,axiom,
    ! [I: int,A4: set_int,F: fun_int_set_nat] :
      ( aa_set_int_o(member_int(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_int_set_nat(F,A4))),aa_int_set_nat(F,I)) ) ).

% INF_lower
tff(fact_2278_INF__lower,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_7082456443403096939t_unit(F,A4))),aa_Pro1279885795167260962t_unit(F,I)) ) ).

% INF_lower
tff(fact_2279_INF__lower,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4))),aa_Pro676242055593665672t_unit(F,I)) ) ).

% INF_lower
tff(fact_2280_INF__lower,axiom,
    ! [I: $o,A4: set_o,F: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((I)),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),aa_o_Product_unit(F,(I))) ) ).

% INF_lower
tff(fact_2281_INF__lower,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),aa_nat_Product_unit(F,I)) ) ).

% INF_lower
tff(fact_2282_INF__lower,axiom,
    ! [I: int,A4: set_int,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))),aa_int_Product_unit(F,I)) ) ).

% INF_lower
tff(fact_2283_INF__eqI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Xa: set_nat,F: fun_Pr6764980218223570944et_nat] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_Pro1692590229789857841et_nat(F,I4)) )
     => ( ! [Y: set_nat] :
            ( ! [I5: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(Y),aa_Pro1692590229789857841et_nat(F,I5)) )
           => aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) )
       => ( comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2284_INF__eqI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Xa: set_nat,F: fun_Pr9038118721296872730et_nat] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_Pro9123573313595863371et_nat(F,I4)) )
     => ( ! [Y: set_nat] :
            ( ! [I5: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(Y),aa_Pro9123573313595863371et_nat(F,I5)) )
           => aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) )
       => ( comple7806235888213564991et_nat(image_15824709712370754et_nat(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2285_INF__eqI,axiom,
    ! [A4: set_o,Xa: set_nat,F: fun_o_set_nat] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_o_set_nat(F,(I4))) )
     => ( ! [Y: set_nat] :
            ( ! [I5: $o] :
                ( aa_set_o_o(member_o((I5)),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(Y),aa_o_set_nat(F,(I5))) )
           => aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) )
       => ( comple7806235888213564991et_nat(image_o_set_nat(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2286_INF__eqI,axiom,
    ! [A4: set_nat,Xa: set_nat,F: fun_nat_set_nat] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_nat_set_nat(F,I4)) )
     => ( ! [Y: set_nat] :
            ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(Y),aa_nat_set_nat(F,I5)) )
           => aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) )
       => ( comple7806235888213564991et_nat(image_nat_set_nat(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2287_INF__eqI,axiom,
    ! [A4: set_int,Xa: set_nat,F: fun_int_set_nat] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(Xa),aa_int_set_nat(F,I4)) )
     => ( ! [Y: set_nat] :
            ( ! [I5: int] :
                ( aa_set_int_o(member_int(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(Y),aa_int_set_nat(F,I5)) )
           => aa_set_nat_o(ord_less_eq_set_nat(Y),Xa) )
       => ( comple7806235888213564991et_nat(image_int_set_nat(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2288_INF__eqI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Xa: product_unit,F: fun_Pr8319229360018371625t_unit] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),aa_Pro1279885795167260962t_unit(F,I4)) )
     => ( ! [Y: product_unit] :
            ( ! [I5: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_Pro1279885795167260962t_unit(F,I5)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),Xa) )
       => ( comple2584293577114468500t_unit(image_7082456443403096939t_unit(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2289_INF__eqI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Xa: product_unit,F: fun_Pr2481279687482350479t_unit] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),aa_Pro676242055593665672t_unit(F,I4)) )
     => ( ! [Y: product_unit] :
            ( ! [I5: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_Pro676242055593665672t_unit(F,I5)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),Xa) )
       => ( comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2290_INF__eqI,axiom,
    ! [A4: set_o,Xa: product_unit,F: fun_o_Product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),aa_o_Product_unit(F,(I4))) )
     => ( ! [Y: product_unit] :
            ( ! [I5: $o] :
                ( aa_set_o_o(member_o((I5)),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_o_Product_unit(F,(I5))) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),Xa) )
       => ( comple2584293577114468500t_unit(image_o_Product_unit(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2291_INF__eqI,axiom,
    ! [A4: set_nat,Xa: product_unit,F: fun_nat_Product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),aa_nat_Product_unit(F,I4)) )
     => ( ! [Y: product_unit] :
            ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_nat_Product_unit(F,I5)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),Xa) )
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2292_INF__eqI,axiom,
    ! [A4: set_int,Xa: product_unit,F: fun_int_Product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),aa_int_Product_unit(F,I4)) )
     => ( ! [Y: product_unit] :
            ( ! [I5: int] :
                ( aa_set_int_o(member_int(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),aa_int_Product_unit(F,I5)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Y),Xa) )
       => ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4)) = Xa ) ) ) ).

% INF_eqI
tff(fact_2293_SUP__upper2,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,U: set_nat,F: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(U),aa_Pro1692590229789857841et_nat(F,I))
       => aa_set_nat_o(ord_less_eq_set_nat(U),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2294_SUP__upper2,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,U: set_nat,F: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(U),aa_Pro9123573313595863371et_nat(F,I))
       => aa_set_nat_o(ord_less_eq_set_nat(U),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2295_SUP__upper2,axiom,
    ! [I: $o,A4: set_o,U: set_nat,F: fun_o_set_nat] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(U),aa_o_set_nat(F,(I)))
       => aa_set_nat_o(ord_less_eq_set_nat(U),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2296_SUP__upper2,axiom,
    ! [I: nat,A4: set_nat,U: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(U),aa_nat_set_nat(F,I))
       => aa_set_nat_o(ord_less_eq_set_nat(U),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2297_SUP__upper2,axiom,
    ! [I: int,A4: set_int,U: set_nat,F: fun_int_set_nat] :
      ( aa_set_int_o(member_int(I),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(U),aa_int_set_nat(F,I))
       => aa_set_nat_o(ord_less_eq_set_nat(U),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2298_SUP__upper2,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,U: product_unit,F: fun_Pr8319229360018371625t_unit] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_Pro1279885795167260962t_unit(F,I))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_7082456443403096939t_unit(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2299_SUP__upper2,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,U: product_unit,F: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_Pro676242055593665672t_unit(F,I))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2300_SUP__upper2,axiom,
    ! [I: $o,A4: set_o,U: product_unit,F: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((I)),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_o_Product_unit(F,(I)))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2301_SUP__upper2,axiom,
    ! [I: nat,A4: set_nat,U: product_unit,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_nat_Product_unit(F,I))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2302_SUP__upper2,axiom,
    ! [I: int,A4: set_int,U: product_unit,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(I),A4)
     => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_int_Product_unit(F,I))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))) ) ) ).

% SUP_upper2
tff(fact_2303_SUP__upper,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,I)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4))) ) ).

% SUP_upper
tff(fact_2304_SUP__upper,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,I)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))) ) ).

% SUP_upper
tff(fact_2305_SUP__upper,axiom,
    ! [I: $o,A4: set_o,F: fun_o_set_nat] :
      ( aa_set_o_o(member_o((I)),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))) ) ).

% SUP_upper
tff(fact_2306_SUP__upper,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,I)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))) ) ).

% SUP_upper
tff(fact_2307_SUP__upper,axiom,
    ! [I: int,A4: set_int,F: fun_int_set_nat] :
      ( aa_set_int_o(member_int(I),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,I)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4))) ) ).

% SUP_upper
tff(fact_2308_SUP__upper,axiom,
    ! [I: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,I)),comple4687483117567863418t_unit(image_7082456443403096939t_unit(F,A4))) ) ).

% SUP_upper
tff(fact_2309_SUP__upper,axiom,
    ! [I: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,I)),comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4))) ) ).

% SUP_upper
tff(fact_2310_SUP__upper,axiom,
    ! [I: $o,A4: set_o,F: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((I)),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I))),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ) ).

% SUP_upper
tff(fact_2311_SUP__upper,axiom,
    ! [I: nat,A4: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I)),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ) ).

% SUP_upper
tff(fact_2312_SUP__upper,axiom,
    ! [I: int,A4: set_int,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(I),A4)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I)),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))) ) ).

% SUP_upper
tff(fact_2313_SUP__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,U: set_nat] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,I4)),U) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4))),U) ) ).

% SUP_least
tff(fact_2314_SUP__least,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,U: set_nat] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,I4)),U) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))),U) ) ).

% SUP_least
tff(fact_2315_SUP__least,axiom,
    ! [A4: set_o,F: fun_o_set_nat,U: set_nat] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I4))),U) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))),U) ) ).

% SUP_least
tff(fact_2316_SUP__least,axiom,
    ! [A4: set_nat,F: fun_nat_set_nat,U: set_nat] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,I4)),U) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))),U) ) ).

% SUP_least
tff(fact_2317_SUP__least,axiom,
    ! [A4: set_int,F: fun_int_set_nat,U: set_nat] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,I4)),U) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4))),U) ) ).

% SUP_least
tff(fact_2318_SUP__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit,U: product_unit] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,I4)),U) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_7082456443403096939t_unit(F,A4))),U) ) ).

% SUP_least
tff(fact_2319_SUP__least,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,U: product_unit] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,I4)),U) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4))),U) ) ).

% SUP_least
tff(fact_2320_SUP__least,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,U: product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I4))),U) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_o_Product_unit(F,A4))),U) ) ).

% SUP_least
tff(fact_2321_SUP__least,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,U: product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I4)),U) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))),U) ) ).

% SUP_least
tff(fact_2322_SUP__least,axiom,
    ! [A4: set_int,F: fun_int_Product_unit,U: product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I4)),U) )
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))),U) ) ).

% SUP_least
tff(fact_2323_SUP__eqI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,Xa: set_nat] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,I4)),Xa) )
     => ( ! [Y: set_nat] :
            ( ! [I5: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,I5)),Y) )
           => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2324_SUP__eqI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,Xa: set_nat] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,I4)),Xa) )
     => ( ! [Y: set_nat] :
            ( ! [I5: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,I5)),Y) )
           => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2325_SUP__eqI,axiom,
    ! [A4: set_o,F: fun_o_set_nat,Xa: set_nat] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I4))),Xa) )
     => ( ! [Y: set_nat] :
            ( ! [I5: $o] :
                ( aa_set_o_o(member_o((I5)),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I5))),Y) )
           => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2326_SUP__eqI,axiom,
    ! [A4: set_nat,F: fun_nat_set_nat,Xa: set_nat] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,I4)),Xa) )
     => ( ! [Y: set_nat] :
            ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,I5)),Y) )
           => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2327_SUP__eqI,axiom,
    ! [A4: set_int,F: fun_int_set_nat,Xa: set_nat] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,I4)),Xa) )
     => ( ! [Y: set_nat] :
            ( ! [I5: int] :
                ( aa_set_int_o(member_int(I5),A4)
               => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,I5)),Y) )
           => aa_set_nat_o(ord_less_eq_set_nat(Xa),Y) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2328_SUP__eqI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit,Xa: product_unit] :
      ( ! [I4: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,I4)),Xa) )
     => ( ! [Y: product_unit] :
            ( ! [I5: produc3843707927480180839at_nat] :
                ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,I5)),Y) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),Y) )
       => ( comple4687483117567863418t_unit(image_7082456443403096939t_unit(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2329_SUP__eqI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,Xa: product_unit] :
      ( ! [I4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,I4)),Xa) )
     => ( ! [Y: product_unit] :
            ( ! [I5: product_prod_nat_nat] :
                ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,I5)),Y) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),Y) )
       => ( comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2330_SUP__eqI,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,Xa: product_unit] :
      ( ! [I4: $o] :
          ( aa_set_o_o(member_o((I4)),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I4))),Xa) )
     => ( ! [Y: product_unit] :
            ( ! [I5: $o] :
                ( aa_set_o_o(member_o((I5)),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I5))),Y) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),Y) )
       => ( comple4687483117567863418t_unit(image_o_Product_unit(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2331_SUP__eqI,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,Xa: product_unit] :
      ( ! [I4: nat] :
          ( aa_set_nat_o(member_nat(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I4)),Xa) )
     => ( ! [Y: product_unit] :
            ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I5)),Y) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),Y) )
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2332_SUP__eqI,axiom,
    ! [A4: set_int,F: fun_int_Product_unit,Xa: product_unit] :
      ( ! [I4: int] :
          ( aa_set_int_o(member_int(I4),A4)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I4)),Xa) )
     => ( ! [Y: product_unit] :
            ( ! [I5: int] :
                ( aa_set_int_o(member_int(I5),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I5)),Y) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(Xa),Y) )
       => ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4)) = Xa ) ) ) ).

% SUP_eqI
tff(fact_2333_INT__extend__simps_I10_J,axiom,
    ! [B5: fun_rat_set_nat,F: fun_nat_rat,A4: set_nat] : comple7806235888213564991et_nat(image_nat_set_nat(aa_fun7287459611079639094et_nat(aTP_Lamm_nc(B5),F),A4)) = comple7806235888213564991et_nat(image_rat_set_nat(B5,aa_set_nat_set_rat(image_nat_rat(F),A4))) ).

% INT_extend_simps(10)
tff(fact_2334_INT__extend__simps_I10_J,axiom,
    ! [B5: fun_nat_set_nat,F: fun_nat_nat,A4: set_nat] : comple7806235888213564991et_nat(image_nat_set_nat(aa_fun7909995953855020590et_nat(aTP_Lamm_nd(B5),F),A4)) = comple7806235888213564991et_nat(image_nat_set_nat(B5,aa_set_nat_set_nat(image_nat_nat(F),A4))) ).

% INT_extend_simps(10)
tff(fact_2335_INT__extend__simps_I10_J,axiom,
    ! [B5: fun_int_set_nat,F: fun_nat_int,A4: set_nat] : comple7806235888213564991et_nat(image_nat_set_nat(aa_fun3079086916031139466et_nat(aTP_Lamm_ne(B5),F),A4)) = comple7806235888213564991et_nat(image_int_set_nat(B5,aa_set_nat_set_int(image_nat_int(F),A4))) ).

% INT_extend_simps(10)
tff(fact_2336_INT__extend__simps_I10_J,axiom,
    ! [B5: fun_nat_set_nat,F: fun_int_nat,A4: set_int] : comple7806235888213564991et_nat(image_int_set_nat(aa_fun7439376938756338406et_nat(aTP_Lamm_nf(B5),F),A4)) = comple7806235888213564991et_nat(image_nat_set_nat(B5,aa_set_int_set_nat(image_int_nat(F),A4))) ).

% INT_extend_simps(10)
tff(fact_2337_INT__extend__simps_I10_J,axiom,
    ! [B5: fun_int_set_nat,F: fun_int_int,A4: set_int] : comple7806235888213564991et_nat(image_int_set_nat(aa_fun2608467900932457282et_nat(aTP_Lamm_ng(B5),F),A4)) = comple7806235888213564991et_nat(image_int_set_nat(B5,aa_set_int_set_int(image_int_int(F),A4))) ).

% INT_extend_simps(10)
tff(fact_2338_INT__lower,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(B5,A4))),aa_Pro1692590229789857841et_nat(B5,A)) ) ).

% INT_lower
tff(fact_2339_INT__lower,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(B5,A4))),aa_Pro9123573313595863371et_nat(B5,A)) ) ).

% INT_lower
tff(fact_2340_INT__lower,axiom,
    ! [A: $o,A4: set_o,B5: fun_o_set_nat] :
      ( aa_set_o_o(member_o((A)),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(B5,A4))),aa_o_set_nat(B5,(A))) ) ).

% INT_lower
tff(fact_2341_INT__lower,axiom,
    ! [A: nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))),aa_nat_set_nat(B5,A)) ) ).

% INT_lower
tff(fact_2342_INT__lower,axiom,
    ! [A: int,A4: set_int,B5: fun_int_set_nat] :
      ( aa_set_int_o(member_int(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_int_set_nat(B5,A4))),aa_int_set_nat(B5,A)) ) ).

% INT_lower
tff(fact_2343_INT__greatest,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C3: set_nat,B5: fun_Pr6764980218223570944et_nat] :
      ( ! [X: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(C3),aa_Pro1692590229789857841et_nat(B5,X)) )
     => aa_set_nat_o(ord_less_eq_set_nat(C3),comple7806235888213564991et_nat(image_3160451284896210088et_nat(B5,A4))) ) ).

% INT_greatest
tff(fact_2344_INT__greatest,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C3: set_nat,B5: fun_Pr9038118721296872730et_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(C3),aa_Pro9123573313595863371et_nat(B5,X)) )
     => aa_set_nat_o(ord_less_eq_set_nat(C3),comple7806235888213564991et_nat(image_15824709712370754et_nat(B5,A4))) ) ).

% INT_greatest
tff(fact_2345_INT__greatest,axiom,
    ! [A4: set_o,C3: set_nat,B5: fun_o_set_nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(C3),aa_o_set_nat(B5,(X))) )
     => aa_set_nat_o(ord_less_eq_set_nat(C3),comple7806235888213564991et_nat(image_o_set_nat(B5,A4))) ) ).

% INT_greatest
tff(fact_2346_INT__greatest,axiom,
    ! [A4: set_nat,C3: set_nat,B5: fun_nat_set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(C3),aa_nat_set_nat(B5,X)) )
     => aa_set_nat_o(ord_less_eq_set_nat(C3),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))) ) ).

% INT_greatest
tff(fact_2347_INT__greatest,axiom,
    ! [A4: set_int,C3: set_nat,B5: fun_int_set_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(C3),aa_int_set_nat(B5,X)) )
     => aa_set_nat_o(ord_less_eq_set_nat(C3),comple7806235888213564991et_nat(image_int_set_nat(B5,A4))) ) ).

% INT_greatest
tff(fact_2348_INT__anti__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,G: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),B5)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,X)),aa_Pro1692590229789857841et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,B5))),comple7806235888213564991et_nat(image_3160451284896210088et_nat(G,A4))) ) ) ).

% INT_anti_mono
tff(fact_2349_INT__anti__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,G: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B5)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,X)),aa_Pro9123573313595863371et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(F,B5))),comple7806235888213564991et_nat(image_15824709712370754et_nat(G,A4))) ) ) ).

% INT_anti_mono
tff(fact_2350_INT__anti__mono,axiom,
    ! [A4: set_o,B5: set_o,F: fun_o_set_nat,G: fun_o_set_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(X))),aa_o_set_nat(G,(X))) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,B5))),comple7806235888213564991et_nat(image_o_set_nat(G,A4))) ) ) ).

% INT_anti_mono
tff(fact_2351_INT__anti__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(F,B5))),comple7806235888213564991et_nat(image_nat_set_nat(G,A4))) ) ) ).

% INT_anti_mono
tff(fact_2352_INT__anti__mono,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_set_nat,G: fun_int_set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,X)),aa_int_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_int_set_nat(F,B5))),comple7806235888213564991et_nat(image_int_set_nat(G,A4))) ) ) ).

% INT_anti_mono
tff(fact_2353_image__constant__conv,axiom,
    ! [C: int,A4: set_int] :
      aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_dt,C)),A4) = $ite(A4 = bot_bot_set_int,bot_bot_set_int,aa_set_int_set_int(insert_int(C),bot_bot_set_int)) ).

% image_constant_conv
tff(fact_2354_image__constant__conv,axiom,
    ! [C: nat,A4: set_int] :
      aa_set_int_set_nat(image_int_nat(aTP_Lamm_fo(C)),A4) = $ite(A4 = bot_bot_set_int,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat)) ).

% image_constant_conv
tff(fact_2355_image__constant__conv,axiom,
    ! [C: product_unit,A4: set_Product_unit] :
      image_405062704495631173t_unit(aTP_Lamm_lh(C),A4) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,aa_set8324764978855220220t_unit(insert_Product_unit(C),bot_bo3957492148770167129t_unit)) ).

% image_constant_conv
tff(fact_2356_image__constant__conv,axiom,
    ! [C: $o,A4: set_Product_unit] :
      image_Product_unit_o(aTP_Lamm_nh((C)),A4) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_o,aa_set_o_set_o(insert_o((C)),bot_bot_set_o)) ).

% image_constant_conv
tff(fact_2357_image__constant__conv,axiom,
    ! [C: nat,A4: set_Product_unit] :
      image_875570014554754200it_nat(aTP_Lamm_ni(C),A4) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat)) ).

% image_constant_conv
tff(fact_2358_image__constant__conv,axiom,
    ! [C: product_unit,A4: set_o] :
      image_o_Product_unit(aTP_Lamm_lk(C),A4) = $ite(A4 = bot_bot_set_o,bot_bo3957492148770167129t_unit,aa_set8324764978855220220t_unit(insert_Product_unit(C),bot_bo3957492148770167129t_unit)) ).

% image_constant_conv
tff(fact_2359_image__constant__conv,axiom,
    ! [C: $o,A4: set_o] :
      aa_set_o_set_o(image_o_o(aa_o_fun_o_o(aTP_Lamm_fl,(C))),A4) = $ite(A4 = bot_bot_set_o,bot_bot_set_o,aa_set_o_set_o(insert_o((C)),bot_bot_set_o)) ).

% image_constant_conv
tff(fact_2360_image__constant__conv,axiom,
    ! [C: nat,A4: set_o] :
      image_o_nat(aTP_Lamm_nj(C),A4) = $ite(A4 = bot_bot_set_o,bot_bot_set_nat,aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat)) ).

% image_constant_conv
tff(fact_2361_image__constant__conv,axiom,
    ! [C: rat,A4: set_nat] :
      aa_set_nat_set_rat(image_nat_rat(aTP_Lamm_kr(C)),A4) = $ite(A4 = bot_bot_set_nat,bot_bot_set_rat,insert_rat(C,bot_bot_set_rat)) ).

% image_constant_conv
tff(fact_2362_image__constant__conv,axiom,
    ! [C: int,A4: set_nat] :
      aa_set_nat_set_int(image_nat_int(aTP_Lamm_ks(C)),A4) = $ite(A4 = bot_bot_set_nat,bot_bot_set_int,aa_set_int_set_int(insert_int(C),bot_bot_set_int)) ).

% image_constant_conv
tff(fact_2363_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C: rat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_set_rat(image_nat_rat(aTP_Lamm_kr(C)),A4) = insert_rat(C,bot_bot_set_rat) ) ) ).

% image_constant
tff(fact_2364_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C: int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_set_int(image_nat_int(aTP_Lamm_ks(C)),A4) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ).

% image_constant
tff(fact_2365_image__constant,axiom,
    ! [Xa: int,A4: set_int,C: int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_dt,C)),A4) = aa_set_int_set_int(insert_int(C),bot_bot_set_int) ) ) ).

% image_constant
tff(fact_2366_image__constant,axiom,
    ! [Xa: $o,A4: set_o,C: product_unit] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( image_o_Product_unit(aTP_Lamm_lk(C),A4) = aa_set8324764978855220220t_unit(insert_Product_unit(C),bot_bo3957492148770167129t_unit) ) ) ).

% image_constant
tff(fact_2367_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C: product_unit] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( image_8730104196221521654t_unit(aTP_Lamm_kt(C),A4) = aa_set8324764978855220220t_unit(insert_Product_unit(C),bot_bo3957492148770167129t_unit) ) ) ).

% image_constant
tff(fact_2368_image__constant,axiom,
    ! [Xa: int,A4: set_int,C: product_unit] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( image_5846526143423701914t_unit(aTP_Lamm_nk(C),A4) = aa_set8324764978855220220t_unit(insert_Product_unit(C),bot_bo3957492148770167129t_unit) ) ) ).

% image_constant
tff(fact_2369_image__constant,axiom,
    ! [Xa: $o,A4: set_o,C: $o] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_o_set_o(image_o_o(aa_o_fun_o_o(aTP_Lamm_fl,(C))),A4) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) ) ) ).

% image_constant
tff(fact_2370_image__constant,axiom,
    ! [Xa: nat,A4: set_nat,C: $o] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( image_nat_o(aa_o_fun_nat_o(aTP_Lamm_ku,(C)),A4) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) ) ) ).

% image_constant
tff(fact_2371_image__constant,axiom,
    ! [Xa: int,A4: set_int,C: $o] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( image_int_o(aa_o_fun_int_o(aTP_Lamm_nl,(C)),A4) = aa_set_o_set_o(insert_o((C)),bot_bot_set_o) ) ) ).

% image_constant
tff(fact_2372_image__constant,axiom,
    ! [Xa: $o,A4: set_o,C: nat] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( image_o_nat(aTP_Lamm_nj(C),A4) = aa_set_nat_set_nat(insert_nat(C),bot_bot_set_nat) ) ) ).

% image_constant
tff(fact_2373_UN__extend__simps_I10_J,axiom,
    ! [B5: fun_rat_set_nat,F: fun_nat_rat,A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun7287459611079639094et_nat(aTP_Lamm_nc(B5),F),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_rat_set_nat(B5,aa_set_nat_set_rat(image_nat_rat(F),A4))) ).

% UN_extend_simps(10)
tff(fact_2374_UN__extend__simps_I10_J,axiom,
    ! [B5: fun_nat_set_nat,F: fun_nat_nat,A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun7909995953855020590et_nat(aTP_Lamm_nd(B5),F),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,aa_set_nat_set_nat(image_nat_nat(F),A4))) ).

% UN_extend_simps(10)
tff(fact_2375_UN__extend__simps_I10_J,axiom,
    ! [B5: fun_int_set_nat,F: fun_nat_int,A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun3079086916031139466et_nat(aTP_Lamm_ne(B5),F),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,aa_set_nat_set_int(image_nat_int(F),A4))) ).

% UN_extend_simps(10)
tff(fact_2376_UN__extend__simps_I10_J,axiom,
    ! [B5: fun_nat_set_nat,F: fun_int_nat,A4: set_int] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(aa_fun7439376938756338406et_nat(aTP_Lamm_nf(B5),F),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,aa_set_int_set_nat(image_int_nat(F),A4))) ).

% UN_extend_simps(10)
tff(fact_2377_UN__extend__simps_I10_J,axiom,
    ! [B5: fun_int_set_nat,F: fun_int_int,A4: set_int] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(aa_fun2608467900932457282et_nat(aTP_Lamm_ng(B5),F),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,aa_set_int_set_int(image_int_int(F),A4))) ).

% UN_extend_simps(10)
tff(fact_2378_UN__empty,axiom,
    ! [B5: fun_Pr3013102738158121955t_unit] : comple3349607820504768346t_unit(image_7778121751606633509t_unit(B5,bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit ).

% UN_empty
tff(fact_2379_UN__empty,axiom,
    ! [B5: fun_Pr6739565440344758894_set_o] : comple90263536869209701_set_o(image_6930536745178679216_set_o(B5,bot_bo3957492148770167129t_unit)) = bot_bot_set_o ).

% UN_empty
tff(fact_2380_UN__empty,axiom,
    ! [B5: fun_o_1833192751255691672t_unit] : comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,bot_bot_set_o)) = bot_bo3957492148770167129t_unit ).

% UN_empty
tff(fact_2381_UN__empty,axiom,
    ! [B5: fun_o_set_o] : comple90263536869209701_set_o(image_o_set_o(B5,bot_bot_set_o)) = bot_bot_set_o ).

% UN_empty
tff(fact_2382_UN__empty,axiom,
    ! [B5: fun_na896991816813928310t_unit] : comple3349607820504768346t_unit(image_4824033694279209942t_unit(B5,bot_bot_set_nat)) = bot_bo3957492148770167129t_unit ).

% UN_empty
tff(fact_2383_UN__empty,axiom,
    ! [B5: fun_nat_set_o] : comple90263536869209701_set_o(image_nat_set_o(B5,bot_bot_set_nat)) = bot_bot_set_o ).

% UN_empty
tff(fact_2384_UN__empty,axiom,
    ! [B5: fun_Pr1628945989756059814et_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,bot_bo3957492148770167129t_unit)) = bot_bot_set_nat ).

% UN_empty
tff(fact_2385_UN__empty,axiom,
    ! [B5: fun_o_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,bot_bot_set_o)) = bot_bot_set_nat ).

% UN_empty
tff(fact_2386_UN__empty,axiom,
    ! [B5: fun_nat_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,bot_bot_set_nat)) = bot_bot_set_nat ).

% UN_empty
tff(fact_2387_UN__empty,axiom,
    ! [B5: fun_Pr3332065415989118679at_nat] : comple5685304695842803022at_nat(image_3617315916088684441at_nat(B5,bot_bo3957492148770167129t_unit)) = bot_bo2099793752762293965at_nat ).

% UN_empty
tff(fact_2388_UN__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,G: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),B5)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,X)),aa_Pro1692590229789857841et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(G,B5))) ) ) ).

% UN_mono
tff(fact_2389_UN__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,G: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B5)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,X)),aa_Pro9123573313595863371et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(G,B5))) ) ) ).

% UN_mono
tff(fact_2390_UN__mono,axiom,
    ! [A4: set_o,B5: set_o,F: fun_o_set_nat,G: fun_o_set_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(X))),aa_o_set_nat(G,(X))) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(G,B5))) ) ) ).

% UN_mono
tff(fact_2391_UN__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(G,B5))) ) ) ).

% UN_mono
tff(fact_2392_UN__mono,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_set_nat,G: fun_int_set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,X)),aa_int_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(G,B5))) ) ) ).

% UN_mono
tff(fact_2393_UN__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: fun_Pr6764980218223570944et_nat,C3: set_nat] :
      ( ! [X: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(B5,X)),C3) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(B5,A4))),C3) ) ).

% UN_least
tff(fact_2394_UN__least,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr9038118721296872730et_nat,C3: set_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(B5,X)),C3) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,A4))),C3) ) ).

% UN_least
tff(fact_2395_UN__least,axiom,
    ! [A4: set_o,B5: fun_o_set_nat,C3: set_nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(B5,(X))),C3) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4))),C3) ) ).

% UN_least
tff(fact_2396_UN__least,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat,C3: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(B5,X)),C3) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4))),C3) ) ).

% UN_least
tff(fact_2397_UN__least,axiom,
    ! [A4: set_int,B5: fun_int_set_nat,C3: set_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(B5,X)),C3) )
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,A4))),C3) ) ).

% UN_least
tff(fact_2398_UN__upper,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B5: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(B5,A4))) ) ).

% UN_upper
tff(fact_2399_UN__upper,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B5: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,A4))) ) ).

% UN_upper
tff(fact_2400_UN__upper,axiom,
    ! [A: $o,A4: set_o,B5: fun_o_set_nat] :
      ( aa_set_o_o(member_o((A)),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(B5,(A))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4))) ) ).

% UN_upper
tff(fact_2401_UN__upper,axiom,
    ! [A: nat,A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4))) ) ).

% UN_upper
tff(fact_2402_UN__upper,axiom,
    ! [A: int,A4: set_int,B5: fun_int_set_nat] :
      ( aa_set_int_o(member_int(A),A4)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,A4))) ) ).

% UN_upper
tff(fact_2403_Union__natural,axiom,
    ! [F: fun_int_int] : comp_s8709038167382007543et_int(comple3221217463730067765et_int,image_524474410958335435et_int(image_int_int(F))) = comp_s2487324123861929025et_int(image_int_int(F),comple3221217463730067765et_int) ).

% Union_natural
tff(fact_2404_Union__natural,axiom,
    ! [F: fun_nat_rat] : comp_s703482200030188019et_nat(comple3890839924845867745et_rat,image_4408659257933336347et_rat(image_nat_rat(F))) = comp_s6435139744279249717et_nat(image_nat_rat(F),comple7399068483239264473et_nat) ).

% Union_natural
tff(fact_2405_Union__natural,axiom,
    ! [F: fun_nat_int] : comp_s1975749275458958491et_nat(comple3221217463730067765et_int,image_3739036796817536367et_int(image_nat_int(F))) = comp_s3768223441818958217et_nat(image_nat_int(F),comple7399068483239264473et_nat) ).

% Union_natural
tff(fact_2406_Union__natural,axiom,
    ! [F: fun_int_nat] : comp_s6907669228194913343et_int(comple7399068483239264473et_nat,image_4702325430467532143et_nat(image_int_nat(F))) = comp_s5712175470073906149et_int(image_int_nat(F),comple3221217463730067765et_int) ).

% Union_natural
tff(fact_2407_Union__natural,axiom,
    ! [F: fun_nat_nat] : comp_s174380336271864291et_nat(comple7399068483239264473et_nat,image_7916887816326733075et_nat(image_nat_nat(F))) = comp_s6993074788030935341et_nat(image_nat_nat(F),comple7399068483239264473et_nat) ).

% Union_natural
tff(fact_2408_image__Union,axiom,
    ! [F: fun_int_int,S3: set_set_int] : aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,S3)) = aa_set4034802472124477112et_int(comple3221217463730067765et_int,aa_set5070719292279795054et_int(image_524474410958335435et_int(image_int_int(F)),S3)) ).

% image_Union
tff(fact_2409_image__Union,axiom,
    ! [F: fun_int_nat,S3: set_set_int] : aa_set_int_set_nat(image_int_nat(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,S3)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set7560802437211521810et_nat(image_4702325430467532143et_nat(image_int_nat(F)),S3)) ).

% image_Union
tff(fact_2410_image__Union,axiom,
    ! [F: fun_nat_rat,S3: set_set_nat] : aa_set_nat_set_rat(image_nat_rat(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,S3)) = aa_set6107595668591848464et_rat(comple3890839924845867745et_rat,aa_set3856643248234747582et_rat(image_4408659257933336347et_rat(image_nat_rat(F)),S3)) ).

% image_Union
tff(fact_2411_image__Union,axiom,
    ! [F: fun_nat_int,S3: set_set_nat] : aa_set_nat_set_int(image_nat_int(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,S3)) = aa_set4034802472124477112et_int(comple3221217463730067765et_int,aa_set7746546440780698386et_int(image_3739036796817536367et_int(image_nat_int(F)),S3)) ).

% image_Union
tff(fact_2412_image__Union,axiom,
    ! [F: fun_nat_nat,S3: set_set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,S3)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(image_nat_nat(F)),S3)) ).

% image_Union
tff(fact_2413_UN__extend__simps_I8_J,axiom,
    ! [B5: fun_nat_set_nat,A4: set_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_nm(B5)),A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))) ).

% UN_extend_simps(8)
tff(fact_2414_INF__eq__iff,axiom,
    ! [I3: set_int,F: fun_int_Product_unit,C: product_unit] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,I4)),C) )
       => ( ( comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,I3)) = C )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),I3)
             => ( aa_int_Product_unit(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2415_INF__eq__iff,axiom,
    ! [I3: set_Product_unit,F: fun_Pr2191293291710260995t_unit,C: product_unit] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(F,I4)),C) )
       => ( ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,I3)) = C )
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,I3)
             => ( aa_Pro8294289767544744956t_unit(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2416_INF__eq__iff,axiom,
    ! [I3: set_o,F: fun_o_Product_unit,C: product_unit] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(I4))),C) )
       => ( ( comple2584293577114468500t_unit(image_o_Product_unit(F,I3)) = C )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),I3)
             => ( aa_o_Product_unit(F,(X4)) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2417_INF__eq__iff,axiom,
    ! [I3: set_nat,F: fun_nat_Product_unit,C: product_unit] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,I4)),C) )
       => ( ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,I3)) = C )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),I3)
             => ( aa_nat_Product_unit(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2418_INF__eq__iff,axiom,
    ! [I3: set_int,F: fun_int_set_nat,C: set_nat] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,I4)),C) )
       => ( ( comple7806235888213564991et_nat(image_int_set_nat(F,I3)) = C )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),I3)
             => ( aa_int_set_nat(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2419_INF__eq__iff,axiom,
    ! [I3: set_Product_unit,F: fun_Pr1628945989756059814et_nat,C: set_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro6064029160526250199et_nat(F,I4)),C) )
       => ( ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,I3)) = C )
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,I3)
             => ( aa_Pro6064029160526250199et_nat(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2420_INF__eq__iff,axiom,
    ! [I3: set_o,F: fun_o_set_nat,C: set_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(I4))),C) )
       => ( ( comple7806235888213564991et_nat(image_o_set_nat(F,I3)) = C )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),I3)
             => ( aa_o_set_nat(F,(X4)) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2421_INF__eq__iff,axiom,
    ! [I3: set_nat,F: fun_nat_set_nat,C: set_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,I4)),C) )
       => ( ( comple7806235888213564991et_nat(image_nat_set_nat(F,I3)) = C )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),I3)
             => ( aa_nat_set_nat(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2422_INF__eq__iff,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,C: product_unit] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,I4)),C) )
       => ( ( comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,I3)) = C )
        <=> ! [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),I3)
             => ( aa_Pro676242055593665672t_unit(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2423_INF__eq__iff,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,C: set_nat] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,I4)),C) )
       => ( ( comple7806235888213564991et_nat(image_15824709712370754et_nat(F,I3)) = C )
        <=> ! [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),I3)
             => ( aa_Pro9123573313595863371et_nat(F,X4) = C ) ) ) ) ) ).

% INF_eq_iff
tff(fact_2424_SUP__eq__iff,axiom,
    ! [I3: set_int,C: product_unit,F: fun_int_Product_unit] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(C),aa_int_Product_unit(F,I4)) )
       => ( ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,I3)) = C )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),I3)
             => ( aa_int_Product_unit(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2425_SUP__eq__iff,axiom,
    ! [I3: set_Product_unit,C: product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(C),aa_Pro8294289767544744956t_unit(F,I4)) )
       => ( ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,I3)) = C )
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,I3)
             => ( aa_Pro8294289767544744956t_unit(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2426_SUP__eq__iff,axiom,
    ! [I3: set_o,C: product_unit,F: fun_o_Product_unit] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(C),aa_o_Product_unit(F,(I4))) )
       => ( ( comple4687483117567863418t_unit(image_o_Product_unit(F,I3)) = C )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),I3)
             => ( aa_o_Product_unit(F,(X4)) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2427_SUP__eq__iff,axiom,
    ! [I3: set_nat,C: product_unit,F: fun_nat_Product_unit] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(C),aa_nat_Product_unit(F,I4)) )
       => ( ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,I3)) = C )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),I3)
             => ( aa_nat_Product_unit(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2428_SUP__eq__iff,axiom,
    ! [I3: set_int,C: set_nat,F: fun_int_set_nat] :
      ( ( I3 != bot_bot_set_int )
     => ( ! [I4: int] :
            ( aa_set_int_o(member_int(I4),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(C),aa_int_set_nat(F,I4)) )
       => ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,I3)) = C )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),I3)
             => ( aa_int_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2429_SUP__eq__iff,axiom,
    ! [I3: set_Product_unit,C: set_nat,F: fun_Pr1628945989756059814et_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( ! [I4: product_unit] :
            ( member_Product_unit(I4,I3)
           => aa_set_nat_o(ord_less_eq_set_nat(C),aa_Pro6064029160526250199et_nat(F,I4)) )
       => ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,I3)) = C )
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,I3)
             => ( aa_Pro6064029160526250199et_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2430_SUP__eq__iff,axiom,
    ! [I3: set_o,C: set_nat,F: fun_o_set_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( ! [I4: $o] :
            ( aa_set_o_o(member_o((I4)),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(C),aa_o_set_nat(F,(I4))) )
       => ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,I3)) = C )
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),I3)
             => ( aa_o_set_nat(F,(X4)) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2431_SUP__eq__iff,axiom,
    ! [I3: set_nat,C: set_nat,F: fun_nat_set_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( ! [I4: nat] :
            ( aa_set_nat_o(member_nat(I4),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(C),aa_nat_set_nat(F,I4)) )
       => ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,I3)) = C )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),I3)
             => ( aa_nat_set_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2432_SUP__eq__iff,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,C: product_unit,F: fun_Pr2481279687482350479t_unit] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(C),aa_Pro676242055593665672t_unit(F,I4)) )
       => ( ( comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,I3)) = C )
        <=> ! [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),I3)
             => ( aa_Pro676242055593665672t_unit(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2433_SUP__eq__iff,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,C: set_nat,F: fun_Pr9038118721296872730et_nat] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( ! [I4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I4),I3)
           => aa_set_nat_o(ord_less_eq_set_nat(C),aa_Pro9123573313595863371et_nat(F,I4)) )
       => ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,I3)) = C )
        <=> ! [X4: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),I3)
             => ( aa_Pro9123573313595863371et_nat(F,X4) = C ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_2434_range__eq__singletonD,axiom,
    ! [F: fun_ch5401695571711387349t_unit,A: product_unit,Xa: char] :
      ( ( image_4337914935673063477t_unit(F,top_top_set_char) = aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit) )
     => ( aa_char_Product_unit(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2435_range__eq__singletonD,axiom,
    ! [F: fun_char_o,A: $o,Xa: char] :
      ( ( image_char_o(F,top_top_set_char) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) )
     => ( aa_char_o(F,Xa)
      <=> (A) ) ) ).

% range_eq_singletonD
tff(fact_2436_range__eq__singletonD,axiom,
    ! [F: fun_char_nat,A: nat,Xa: char] :
      ( ( image_char_nat(F,top_top_set_char) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) )
     => ( aa_char_nat(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2437_range__eq__singletonD,axiom,
    ! [F: fun_rat_Product_unit,A: product_unit,Xa: rat] :
      ( ( image_2327579789247107822t_unit(F,top_top_set_rat) = aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit) )
     => ( aa_rat_Product_unit(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2438_range__eq__singletonD,axiom,
    ! [F: fun_rat_o,A: $o,Xa: rat] :
      ( ( image_rat_o(F,top_top_set_rat) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) )
     => ( aa_rat_o(F,Xa)
      <=> (A) ) ) ).

% range_eq_singletonD
tff(fact_2439_range__eq__singletonD,axiom,
    ! [F: fun_rat_nat,A: nat,Xa: rat] :
      ( ( image_rat_nat(F,top_top_set_rat) = aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat) )
     => ( aa_rat_nat(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2440_range__eq__singletonD,axiom,
    ! [F: fun_nat_rat,A: rat,Xa: nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = insert_rat(A,bot_bot_set_rat) )
     => ( aa_nat_rat(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2441_range__eq__singletonD,axiom,
    ! [F: fun_nat_int,A: int,Xa: nat] :
      ( ( aa_set_nat_set_int(image_nat_int(F),top_top_set_nat) = aa_set_int_set_int(insert_int(A),bot_bot_set_int) )
     => ( aa_nat_int(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2442_range__eq__singletonD,axiom,
    ! [F: fun_nat_Product_unit,A: product_unit,Xa: nat] :
      ( ( image_8730104196221521654t_unit(F,top_top_set_nat) = aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit) )
     => ( aa_nat_Product_unit(F,Xa) = A ) ) ).

% range_eq_singletonD
tff(fact_2443_range__eq__singletonD,axiom,
    ! [F: fun_nat_o,A: $o,Xa: nat] :
      ( ( image_nat_o(F,top_top_set_nat) = aa_set_o_set_o(insert_o((A)),bot_bot_set_o) )
     => ( aa_nat_o(F,Xa)
      <=> (A) ) ) ).

% range_eq_singletonD
tff(fact_2444_INF__superset__mono,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,G: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,B5),A4)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B5)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,X)),aa_Pro1692590229789857841et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(F,A4))),comple7806235888213564991et_nat(image_3160451284896210088et_nat(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2445_INF__superset__mono,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,G: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),A4)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B5)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,X)),aa_Pro9123573313595863371et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(F,A4))),comple7806235888213564991et_nat(image_15824709712370754et_nat(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2446_INF__superset__mono,axiom,
    ! [B5: set_o,A4: set_o,F: fun_o_set_nat,G: fun_o_set_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(B5),A4)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(X))),aa_o_set_nat(G,(X))) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,A4))),comple7806235888213564991et_nat(image_o_set_nat(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2447_INF__superset__mono,axiom,
    ! [B5: set_nat,A4: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(F,A4))),comple7806235888213564991et_nat(image_nat_set_nat(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2448_INF__superset__mono,axiom,
    ! [B5: set_int,A4: set_int,F: fun_int_set_nat,G: fun_int_set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,X)),aa_int_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_int_set_nat(F,A4))),comple7806235888213564991et_nat(image_int_set_nat(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2449_INF__superset__mono,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit,G: fun_Pr8319229360018371625t_unit] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,B5),A4)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),B5)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,X)),aa_Pro1279885795167260962t_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_7082456443403096939t_unit(F,A4))),comple2584293577114468500t_unit(image_7082456443403096939t_unit(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2450_INF__superset__mono,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,G: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),A4)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),B5)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,X)),aa_Pro676242055593665672t_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4))),comple2584293577114468500t_unit(image_6169909464789311569t_unit(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2451_INF__superset__mono,axiom,
    ! [B5: set_o,A4: set_o,F: fun_o_Product_unit,G: fun_o_Product_unit] :
      ( aa_set_o_o(ord_less_eq_set_o(B5),A4)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),B5)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(X))),aa_o_Product_unit(G,(X))) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),comple2584293577114468500t_unit(image_o_Product_unit(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2452_INF__superset__mono,axiom,
    ! [B5: set_nat,A4: set_nat,F: fun_nat_Product_unit,G: fun_nat_Product_unit] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B5),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B5)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,X)),aa_nat_Product_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),comple2584293577114468500t_unit(image_8730104196221521654t_unit(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2453_INF__superset__mono,axiom,
    ! [B5: set_int,A4: set_int,F: fun_int_Product_unit,G: fun_int_Product_unit] :
      ( aa_set_int_o(ord_less_eq_set_int(B5),A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B5)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X)),aa_int_Product_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))),comple2584293577114468500t_unit(image_5846526143423701914t_unit(G,B5))) ) ) ).

% INF_superset_mono
tff(fact_2454_SUP__subset__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,G: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),B5)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro1692590229789857841et_nat(F,X)),aa_Pro1692590229789857841et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2455_SUP__subset__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,G: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B5)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro9123573313595863371et_nat(F,X)),aa_Pro9123573313595863371et_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2456_SUP__subset__mono,axiom,
    ! [A4: set_o,B5: set_o,F: fun_o_set_nat,G: fun_o_set_nat] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(X))),aa_o_set_nat(G,(X))) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2457_SUP__subset__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_set_nat,G: fun_nat_set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,X)),aa_nat_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2458_SUP__subset__mono,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_set_nat,G: fun_int_set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_int_set_nat(F,X)),aa_int_set_nat(G,X)) )
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2459_SUP__subset__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,F: fun_Pr8319229360018371625t_unit,G: fun_Pr8319229360018371625t_unit] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A4),B5)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro1279885795167260962t_unit(F,X)),aa_Pro1279885795167260962t_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_7082456443403096939t_unit(F,A4))),comple4687483117567863418t_unit(image_7082456443403096939t_unit(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2460_SUP__subset__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,G: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),B5)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro676242055593665672t_unit(F,X)),aa_Pro676242055593665672t_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4))),comple4687483117567863418t_unit(image_6169909464789311569t_unit(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2461_SUP__subset__mono,axiom,
    ! [A4: set_o,B5: set_o,F: fun_o_Product_unit,G: fun_o_Product_unit] :
      ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(X))),aa_o_Product_unit(G,(X))) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_o_Product_unit(F,A4))),comple4687483117567863418t_unit(image_o_Product_unit(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2462_SUP__subset__mono,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_Product_unit,G: fun_nat_Product_unit] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,X)),aa_nat_Product_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))),comple4687483117567863418t_unit(image_8730104196221521654t_unit(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2463_SUP__subset__mono,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_Product_unit,G: fun_int_Product_unit] :
      ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X)),aa_int_Product_unit(G,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))),comple4687483117567863418t_unit(image_5846526143423701914t_unit(G,B5))) ) ) ).

% SUP_subset_mono
tff(fact_2464_INF__constant,axiom,
    ! [C: product_unit,A4: set_Product_unit] :
      comple2584293577114468500t_unit(image_405062704495631173t_unit(aTP_Lamm_lh(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_Product_unit,C) ).

% INF_constant
tff(fact_2465_INF__constant,axiom,
    ! [C: product_unit,A4: set_o] :
      comple2584293577114468500t_unit(image_o_Product_unit(aTP_Lamm_lk(C),A4)) = $ite(A4 = bot_bot_set_o,top_top_Product_unit,C) ).

% INF_constant
tff(fact_2466_INF__constant,axiom,
    ! [C: product_unit,A4: set_nat] :
      comple2584293577114468500t_unit(image_8730104196221521654t_unit(aTP_Lamm_kt(C),A4)) = $ite(A4 = bot_bot_set_nat,top_top_Product_unit,C) ).

% INF_constant
tff(fact_2467_INF__constant,axiom,
    ! [C: set_char,A4: set_Product_unit] :
      comple7965528767983648192t_char(image_5325685772115642575t_char(aTP_Lamm_kv(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_char,C) ).

% INF_constant
tff(fact_2468_INF__constant,axiom,
    ! [C: set_char,A4: set_o] :
      comple7965528767983648192t_char(image_o_set_char(aTP_Lamm_kw(C),A4)) = $ite(A4 = bot_bot_set_o,top_top_set_char,C) ).

% INF_constant
tff(fact_2469_INF__constant,axiom,
    ! [C: set_char,A4: set_nat] :
      comple7965528767983648192t_char(image_nat_set_char(aTP_Lamm_kx(C),A4)) = $ite(A4 = bot_bot_set_nat,top_top_set_char,C) ).

% INF_constant
tff(fact_2470_INF__constant,axiom,
    ! [C: set_rat,A4: set_Product_unit] :
      comple4298007329820168263et_rat(image_4665621433434589782et_rat(aTP_Lamm_ky(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_rat,C) ).

% INF_constant
tff(fact_2471_INF__constant,axiom,
    ! [C: set_rat,A4: set_o] :
      comple4298007329820168263et_rat(image_o_set_rat(aTP_Lamm_kz(C),A4)) = $ite(A4 = bot_bot_set_o,top_top_set_rat,C) ).

% INF_constant
tff(fact_2472_INF__constant,axiom,
    ! [C: set_rat,A4: set_nat] :
      comple4298007329820168263et_rat(image_nat_set_rat(aTP_Lamm_gl(C),A4)) = $ite(A4 = bot_bot_set_nat,top_top_set_rat,C) ).

% INF_constant
tff(fact_2473_INF__constant,axiom,
    ! [C: set_int,A4: set_Product_unit] :
      comple3628384868704368283et_int(image_3995998972318789802et_int(aTP_Lamm_la(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,top_top_set_int,C) ).

% INF_constant
tff(fact_2474_INF__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit] : comple2584293577114468500t_unit(image_405062704495631173t_unit(F,bot_bo3957492148770167129t_unit)) = top_top_Product_unit ).

% INF_empty
tff(fact_2475_INF__empty,axiom,
    ! [F: fun_o_Product_unit] : comple2584293577114468500t_unit(image_o_Product_unit(F,bot_bot_set_o)) = top_top_Product_unit ).

% INF_empty
tff(fact_2476_INF__empty,axiom,
    ! [F: fun_nat_Product_unit] : comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,bot_bot_set_nat)) = top_top_Product_unit ).

% INF_empty
tff(fact_2477_INF__empty,axiom,
    ! [F: fun_Pr533032343101338535t_char] : comple7965528767983648192t_char(image_5325685772115642575t_char(F,bot_bo3957492148770167129t_unit)) = top_top_set_char ).

% INF_empty
tff(fact_2478_INF__empty,axiom,
    ! [F: fun_o_set_char] : comple7965528767983648192t_char(image_o_set_char(F,bot_bot_set_o)) = top_top_set_char ).

% INF_empty
tff(fact_2479_INF__empty,axiom,
    ! [F: fun_nat_set_char] : comple7965528767983648192t_char(image_nat_set_char(F,bot_bot_set_nat)) = top_top_set_char ).

% INF_empty
tff(fact_2480_INF__empty,axiom,
    ! [F: fun_Pr4472331689133158062et_rat] : comple4298007329820168263et_rat(image_4665621433434589782et_rat(F,bot_bo3957492148770167129t_unit)) = top_top_set_rat ).

% INF_empty
tff(fact_2481_INF__empty,axiom,
    ! [F: fun_o_set_rat] : comple4298007329820168263et_rat(image_o_set_rat(F,bot_bot_set_o)) = top_top_set_rat ).

% INF_empty
tff(fact_2482_INF__empty,axiom,
    ! [F: fun_nat_set_rat] : comple4298007329820168263et_rat(image_nat_set_rat(F,bot_bot_set_nat)) = top_top_set_rat ).

% INF_empty
tff(fact_2483_INF__empty,axiom,
    ! [F: fun_Pr8362234881679108866et_int] : comple3628384868704368283et_int(image_3995998972318789802et_int(F,bot_bo3957492148770167129t_unit)) = top_top_set_int ).

% INF_empty
tff(fact_2484_SUP__constant,axiom,
    ! [C: product_unit,A4: set_Product_unit] :
      comple4687483117567863418t_unit(image_405062704495631173t_unit(aTP_Lamm_lh(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_Product_unit,C) ).

% SUP_constant
tff(fact_2485_SUP__constant,axiom,
    ! [C: product_unit,A4: set_o] :
      comple4687483117567863418t_unit(image_o_Product_unit(aTP_Lamm_lk(C),A4)) = $ite(A4 = bot_bot_set_o,bot_bot_Product_unit,C) ).

% SUP_constant
tff(fact_2486_SUP__constant,axiom,
    ! [C: product_unit,A4: set_nat] :
      comple4687483117567863418t_unit(image_8730104196221521654t_unit(aTP_Lamm_kt(C),A4)) = $ite(A4 = bot_bot_set_nat,bot_bot_Product_unit,C) ).

% SUP_constant
tff(fact_2487_SUP__constant,axiom,
    ! [C: set_Product_unit,A4: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(aTP_Lamm_jp(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,C) ).

% SUP_constant
tff(fact_2488_SUP__constant,axiom,
    ! [C: set_o,A4: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(aTP_Lamm_jq(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_o,C) ).

% SUP_constant
tff(fact_2489_SUP__constant,axiom,
    ! [C: set_Product_unit,A4: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(aTP_Lamm_js(C),A4)) = $ite(A4 = bot_bot_set_o,bot_bo3957492148770167129t_unit,C) ).

% SUP_constant
tff(fact_2490_SUP__constant,axiom,
    ! [C: set_o,A4: set_o] :
      comple90263536869209701_set_o(image_o_set_o(aTP_Lamm_jc(C),A4)) = $ite(A4 = bot_bot_set_o,bot_bot_set_o,C) ).

% SUP_constant
tff(fact_2491_SUP__constant,axiom,
    ! [C: set_Product_unit,A4: set_nat] :
      comple3349607820504768346t_unit(image_4824033694279209942t_unit(aTP_Lamm_jt(C),A4)) = $ite(A4 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,C) ).

% SUP_constant
tff(fact_2492_SUP__constant,axiom,
    ! [C: set_o,A4: set_nat] :
      comple90263536869209701_set_o(image_nat_set_o(aTP_Lamm_jf(C),A4)) = $ite(A4 = bot_bot_set_nat,bot_bot_set_o,C) ).

% SUP_constant
tff(fact_2493_SUP__constant,axiom,
    ! [C: set_nat,A4: set_Product_unit] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aTP_Lamm_jr(C),A4)) = $ite(A4 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,C) ).

% SUP_constant
tff(fact_2494_SUP__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit] : comple4687483117567863418t_unit(image_405062704495631173t_unit(F,bot_bo3957492148770167129t_unit)) = bot_bot_Product_unit ).

% SUP_empty
tff(fact_2495_SUP__empty,axiom,
    ! [F: fun_o_Product_unit] : comple4687483117567863418t_unit(image_o_Product_unit(F,bot_bot_set_o)) = bot_bot_Product_unit ).

% SUP_empty
tff(fact_2496_SUP__empty,axiom,
    ! [F: fun_nat_Product_unit] : comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,bot_bot_set_nat)) = bot_bot_Product_unit ).

% SUP_empty
tff(fact_2497_SUP__empty,axiom,
    ! [F: fun_Pr3013102738158121955t_unit] : comple3349607820504768346t_unit(image_7778121751606633509t_unit(F,bot_bo3957492148770167129t_unit)) = bot_bo3957492148770167129t_unit ).

% SUP_empty
tff(fact_2498_SUP__empty,axiom,
    ! [F: fun_Pr6739565440344758894_set_o] : comple90263536869209701_set_o(image_6930536745178679216_set_o(F,bot_bo3957492148770167129t_unit)) = bot_bot_set_o ).

% SUP_empty
tff(fact_2499_SUP__empty,axiom,
    ! [F: fun_o_1833192751255691672t_unit] : comple3349607820504768346t_unit(image_663195431845683034t_unit(F,bot_bot_set_o)) = bot_bo3957492148770167129t_unit ).

% SUP_empty
tff(fact_2500_SUP__empty,axiom,
    ! [F: fun_o_set_o] : comple90263536869209701_set_o(image_o_set_o(F,bot_bot_set_o)) = bot_bot_set_o ).

% SUP_empty
tff(fact_2501_SUP__empty,axiom,
    ! [F: fun_na896991816813928310t_unit] : comple3349607820504768346t_unit(image_4824033694279209942t_unit(F,bot_bot_set_nat)) = bot_bo3957492148770167129t_unit ).

% SUP_empty
tff(fact_2502_SUP__empty,axiom,
    ! [F: fun_nat_set_o] : comple90263536869209701_set_o(image_nat_set_o(F,bot_bot_set_nat)) = bot_bot_set_o ).

% SUP_empty
tff(fact_2503_SUP__empty,axiom,
    ! [F: fun_Pr1628945989756059814et_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,bot_bo3957492148770167129t_unit)) = bot_bot_set_nat ).

% SUP_empty
tff(fact_2504_INT__empty,axiom,
    ! [B5: fun_Pr533032343101338535t_char] : comple7965528767983648192t_char(image_5325685772115642575t_char(B5,bot_bo3957492148770167129t_unit)) = top_top_set_char ).

% INT_empty
tff(fact_2505_INT__empty,axiom,
    ! [B5: fun_o_set_char] : comple7965528767983648192t_char(image_o_set_char(B5,bot_bot_set_o)) = top_top_set_char ).

% INT_empty
tff(fact_2506_INT__empty,axiom,
    ! [B5: fun_nat_set_char] : comple7965528767983648192t_char(image_nat_set_char(B5,bot_bot_set_nat)) = top_top_set_char ).

% INT_empty
tff(fact_2507_INT__empty,axiom,
    ! [B5: fun_Pr4472331689133158062et_rat] : comple4298007329820168263et_rat(image_4665621433434589782et_rat(B5,bot_bo3957492148770167129t_unit)) = top_top_set_rat ).

% INT_empty
tff(fact_2508_INT__empty,axiom,
    ! [B5: fun_o_set_rat] : comple4298007329820168263et_rat(image_o_set_rat(B5,bot_bot_set_o)) = top_top_set_rat ).

% INT_empty
tff(fact_2509_INT__empty,axiom,
    ! [B5: fun_nat_set_rat] : comple4298007329820168263et_rat(image_nat_set_rat(B5,bot_bot_set_nat)) = top_top_set_rat ).

% INT_empty
tff(fact_2510_INT__empty,axiom,
    ! [B5: fun_Pr8362234881679108866et_int] : comple3628384868704368283et_int(image_3995998972318789802et_int(B5,bot_bo3957492148770167129t_unit)) = top_top_set_int ).

% INT_empty
tff(fact_2511_INT__empty,axiom,
    ! [B5: fun_o_set_int] : comple3628384868704368283et_int(image_o_set_int(B5,bot_bot_set_o)) = top_top_set_int ).

% INT_empty
tff(fact_2512_INT__empty,axiom,
    ! [B5: fun_nat_set_int] : comple3628384868704368283et_int(image_nat_set_int(B5,bot_bot_set_nat)) = top_top_set_int ).

% INT_empty
tff(fact_2513_INT__empty,axiom,
    ! [B5: fun_Pr1628945989756059814et_nat] : comple7806235888213564991et_nat(image_8173849991827986510et_nat(B5,bot_bo3957492148770167129t_unit)) = top_top_set_nat ).

% INT_empty
tff(fact_2514_cINF__const,axiom,
    ! [A4: set_int,C: nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(aTP_Lamm_fo(C)),A4)) = C ) ) ).

% cINF_const
tff(fact_2515_cINF__const,axiom,
    ! [A4: set_nat,C: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fn,C)),A4)) = C ) ) ).

% cINF_const
tff(fact_2516_cINF__const,axiom,
    ! [A4: set_int,C: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_dt,C)),A4)) = C ) ) ).

% cINF_const
tff(fact_2517_cINF__const,axiom,
    ! [A4: set_Product_unit,C: int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(aTP_Lamm_nn(C),A4)) = C ) ) ).

% cINF_const
tff(fact_2518_cINF__const,axiom,
    ! [A4: set_o,C: int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(aTP_Lamm_no(C),A4)) = C ) ) ).

% cINF_const
tff(fact_2519_cINF__const,axiom,
    ! [A4: set_nat,C: int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(aTP_Lamm_ks(C)),A4)) = C ) ) ).

% cINF_const
tff(fact_2520_cINF__const,axiom,
    ! [A4: set_Product_unit,C: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(aTP_Lamm_lh(C),A4)) = C ) ) ).

% cINF_const
tff(fact_2521_cINF__const,axiom,
    ! [A4: set_o,C: product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( comple2584293577114468500t_unit(image_o_Product_unit(aTP_Lamm_lk(C),A4)) = C ) ) ).

% cINF_const
tff(fact_2522_cINF__const,axiom,
    ! [A4: set_nat,C: product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(aTP_Lamm_kt(C),A4)) = C ) ) ).

% cINF_const
tff(fact_2523_cINF__const,axiom,
    ! [A4: set_Product_unit,C: set_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(aTP_Lamm_jr(C),A4)) = C ) ) ).

% cINF_const
tff(fact_2524_cSUP__const,axiom,
    ! [A4: set_int,C: nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aTP_Lamm_fo(C)),A4)) = C ) ) ).

% cSUP_const
tff(fact_2525_cSUP__const,axiom,
    ! [A4: set_nat,C: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fn,C)),A4)) = C ) ) ).

% cSUP_const
tff(fact_2526_cSUP__const,axiom,
    ! [A4: set_int,C: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_dt,C)),A4)) = C ) ) ).

% cSUP_const
tff(fact_2527_cSUP__const,axiom,
    ! [A4: set_Product_unit,C: int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(aTP_Lamm_nn(C),A4)) = C ) ) ).

% cSUP_const
tff(fact_2528_cSUP__const,axiom,
    ! [A4: set_o,C: int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(aTP_Lamm_no(C),A4)) = C ) ) ).

% cSUP_const
tff(fact_2529_cSUP__const,axiom,
    ! [A4: set_nat,C: int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(aTP_Lamm_ks(C)),A4)) = C ) ) ).

% cSUP_const
tff(fact_2530_cSUP__const,axiom,
    ! [A4: set_Product_unit,C: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(aTP_Lamm_lh(C),A4)) = C ) ) ).

% cSUP_const
tff(fact_2531_cSUP__const,axiom,
    ! [A4: set_o,C: product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( comple4687483117567863418t_unit(image_o_Product_unit(aTP_Lamm_lk(C),A4)) = C ) ) ).

% cSUP_const
tff(fact_2532_cSUP__const,axiom,
    ! [A4: set_nat,C: product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(aTP_Lamm_kt(C),A4)) = C ) ) ).

% cSUP_const
tff(fact_2533_cSUP__const,axiom,
    ! [A4: set_Product_unit,C: set_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aTP_Lamm_jr(C),A4)) = C ) ) ).

% cSUP_const
tff(fact_2534_SUP__INF,axiom,
    ! [P2: fun_ch5514019212406764701t_unit] : comple4687483117567863418t_unit(image_4337914935673063477t_unit(aTP_Lamm_nq(P2),top_top_set_char)) = comple2584293577114468500t_unit(image_398633649777079655t_unit(aTP_Lamm_ns(P2),top_to1024782381248034053r_char)) ).

% SUP_INF
tff(fact_2535_SUP__INF,axiom,
    ! [P2: fun_ch7403422310664315094t_unit] : comple4687483117567863418t_unit(image_2327579789247107822t_unit(aTP_Lamm_nu(P2),top_top_set_rat)) = comple2584293577114468500t_unit(image_6378637578383855136t_unit(aTP_Lamm_nw(P2),top_to8358049600343314572t_char)) ).

% SUP_INF
tff(fact_2536_SUP__INF,axiom,
    ! [P2: fun_ch9120586758495803614t_unit] : comple4687483117567863418t_unit(image_8730104196221521654t_unit(aTP_Lamm_ny(P2),top_top_set_nat)) = comple2584293577114468500t_unit(image_4038039149101736488t_unit(aTP_Lamm_oa(P2),top_to3021313259269883524t_char)) ).

% SUP_INF
tff(fact_2537_SUP__INF,axiom,
    ! [P2: fun_ch1452000917118098306t_unit] : comple4687483117567863418t_unit(image_5846526143423701914t_unit(aTP_Lamm_oc(P2),top_top_set_int)) = comple2584293577114468500t_unit(image_3847333233869472460t_unit(aTP_Lamm_oe(P2),top_to7900224466070196960t_char)) ).

% SUP_INF
tff(fact_2538_SUP__INF,axiom,
    ! [P2: fun_ra2707149721221939542t_unit] : comple4687483117567863418t_unit(image_4337914935673063477t_unit(aTP_Lamm_og(P2),top_top_set_char)) = comple2584293577114468500t_unit(image_7053167900234035360t_unit(aTP_Lamm_oi(P2),top_to4351144125366144524ar_rat)) ).

% SUP_INF
tff(fact_2539_SUP__INF,axiom,
    ! [P2: fun_ra3623852323372631695t_unit] : comple4687483117567863418t_unit(image_2327579789247107822t_unit(aTP_Lamm_ok(P2),top_top_set_rat)) = comple2584293577114468500t_unit(image_1018659401590874713t_unit(aTP_Lamm_om(P2),top_to5307049543647809683at_rat)) ).

% SUP_INF
tff(fact_2540_SUP__INF,axiom,
    ! [P2: fun_ra5341016771204120215t_unit] : comple4687483117567863418t_unit(image_8730104196221521654t_unit(aTP_Lamm_oo(P2),top_top_set_nat)) = comple2584293577114468500t_unit(image_165855287175413345t_unit(aTP_Lamm_oq(P2),top_to484361697302036107at_rat)) ).

% SUP_INF
tff(fact_2541_SUP__INF,axiom,
    ! [P2: fun_ra6895802966681190715t_unit] : comple4687483117567863418t_unit(image_5846526143423701914t_unit(aTP_Lamm_os(P2),top_top_set_int)) = comple2584293577114468500t_unit(image_4000094388059439877t_unit(aTP_Lamm_ou(P2),top_to2671283271995057895nt_rat)) ).

% SUP_INF
tff(fact_2542_SUP__INF,axiom,
    ! [P2: fun_na6561490796278613342t_unit] : comple4687483117567863418t_unit(image_4337914935673063477t_unit(aTP_Lamm_ow(P2),top_top_set_char)) = comple2584293577114468500t_unit(image_8770332348065523880t_unit(aTP_Lamm_oy(P2),top_to1507758425989046276ar_nat)) ).

% SUP_INF
tff(fact_2543_SUP__INF,axiom,
    ! [P2: fun_na1234882596965831319t_unit] : comple4687483117567863418t_unit(image_2327579789247107822t_unit(aTP_Lamm_pa(P2),top_top_set_rat)) = comple2584293577114468500t_unit(image_2735823849422363233t_unit(aTP_Lamm_pc(P2),top_to2463663844270711435at_nat)) ).

% SUP_INF
tff(fact_2544_INF__SUP,axiom,
    ! [P2: fun_ch5514019212406764701t_unit] : comple2584293577114468500t_unit(image_4337914935673063477t_unit(aTP_Lamm_pd(P2),top_top_set_char)) = comple4687483117567863418t_unit(image_398633649777079655t_unit(aTP_Lamm_pe(P2),top_to1024782381248034053r_char)) ).

% INF_SUP
tff(fact_2545_INF__SUP,axiom,
    ! [P2: fun_ch7403422310664315094t_unit] : comple2584293577114468500t_unit(image_2327579789247107822t_unit(aTP_Lamm_pf(P2),top_top_set_rat)) = comple4687483117567863418t_unit(image_6378637578383855136t_unit(aTP_Lamm_pg(P2),top_to8358049600343314572t_char)) ).

% INF_SUP
tff(fact_2546_INF__SUP,axiom,
    ! [P2: fun_ch9120586758495803614t_unit] : comple2584293577114468500t_unit(image_8730104196221521654t_unit(aTP_Lamm_ph(P2),top_top_set_nat)) = comple4687483117567863418t_unit(image_4038039149101736488t_unit(aTP_Lamm_pi(P2),top_to3021313259269883524t_char)) ).

% INF_SUP
tff(fact_2547_INF__SUP,axiom,
    ! [P2: fun_ch1452000917118098306t_unit] : comple2584293577114468500t_unit(image_5846526143423701914t_unit(aTP_Lamm_pj(P2),top_top_set_int)) = comple4687483117567863418t_unit(image_3847333233869472460t_unit(aTP_Lamm_pk(P2),top_to7900224466070196960t_char)) ).

% INF_SUP
tff(fact_2548_INF__SUP,axiom,
    ! [P2: fun_ra2707149721221939542t_unit] : comple2584293577114468500t_unit(image_4337914935673063477t_unit(aTP_Lamm_pl(P2),top_top_set_char)) = comple4687483117567863418t_unit(image_7053167900234035360t_unit(aTP_Lamm_pm(P2),top_to4351144125366144524ar_rat)) ).

% INF_SUP
tff(fact_2549_INF__SUP,axiom,
    ! [P2: fun_ra3623852323372631695t_unit] : comple2584293577114468500t_unit(image_2327579789247107822t_unit(aTP_Lamm_pn(P2),top_top_set_rat)) = comple4687483117567863418t_unit(image_1018659401590874713t_unit(aTP_Lamm_po(P2),top_to5307049543647809683at_rat)) ).

% INF_SUP
tff(fact_2550_INF__SUP,axiom,
    ! [P2: fun_ra5341016771204120215t_unit] : comple2584293577114468500t_unit(image_8730104196221521654t_unit(aTP_Lamm_pp(P2),top_top_set_nat)) = comple4687483117567863418t_unit(image_165855287175413345t_unit(aTP_Lamm_pq(P2),top_to484361697302036107at_rat)) ).

% INF_SUP
tff(fact_2551_INF__SUP,axiom,
    ! [P2: fun_ra6895802966681190715t_unit] : comple2584293577114468500t_unit(image_5846526143423701914t_unit(aTP_Lamm_pr(P2),top_top_set_int)) = comple4687483117567863418t_unit(image_4000094388059439877t_unit(aTP_Lamm_ps(P2),top_to2671283271995057895nt_rat)) ).

% INF_SUP
tff(fact_2552_INF__SUP,axiom,
    ! [P2: fun_na6561490796278613342t_unit] : comple2584293577114468500t_unit(image_4337914935673063477t_unit(aTP_Lamm_pt(P2),top_top_set_char)) = comple4687483117567863418t_unit(image_8770332348065523880t_unit(aTP_Lamm_pu(P2),top_to1507758425989046276ar_nat)) ).

% INF_SUP
tff(fact_2553_INF__SUP,axiom,
    ! [P2: fun_na1234882596965831319t_unit] : comple2584293577114468500t_unit(image_2327579789247107822t_unit(aTP_Lamm_pv(P2),top_top_set_rat)) = comple4687483117567863418t_unit(image_2735823849422363233t_unit(aTP_Lamm_pw(P2),top_to2463663844270711435at_nat)) ).

% INF_SUP
tff(fact_2554_cINF__greatest,axiom,
    ! [A4: set_int,M3: nat,F: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_int_nat(F,X)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ).

% cINF_greatest
tff(fact_2555_cINF__greatest,axiom,
    ! [A4: set_Product_unit,M3: nat,F: fun_Product_unit_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_Product_unit_nat(F,X)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A4))) ) ) ).

% cINF_greatest
tff(fact_2556_cINF__greatest,axiom,
    ! [A4: set_o,M3: nat,F: fun_o_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_o_nat(F,(X))) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(F,A4))) ) ) ).

% cINF_greatest
tff(fact_2557_cINF__greatest,axiom,
    ! [A4: set_nat,M3: nat,F: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_nat_nat(F,X)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ).

% cINF_greatest
tff(fact_2558_cINF__greatest,axiom,
    ! [A4: set_int,M3: int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_int_int(F,X)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ).

% cINF_greatest
tff(fact_2559_cINF__greatest,axiom,
    ! [A4: set_Product_unit,M3: int,F: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_Product_unit_int(F,X)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,A4))) ) ) ).

% cINF_greatest
tff(fact_2560_cINF__greatest,axiom,
    ! [A4: set_o,M3: int,F: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_o_int(F,(X))) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))) ) ) ).

% cINF_greatest
tff(fact_2561_cINF__greatest,axiom,
    ! [A4: set_nat,M3: int,F: fun_nat_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_nat_int(F,X)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ).

% cINF_greatest
tff(fact_2562_cINF__greatest,axiom,
    ! [A4: set_int,M3: product_unit,F: fun_int_Product_unit] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(M3),aa_int_Product_unit(F,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(M3),comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))) ) ) ).

% cINF_greatest
tff(fact_2563_cINF__greatest,axiom,
    ! [A4: set_Product_unit,M3: product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(M3),aa_Pro8294289767544744956t_unit(F,X)) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(M3),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))) ) ) ).

% cINF_greatest
tff(fact_2564_cSUP__least,axiom,
    ! [A4: set_int,F: fun_int_nat,M: nat] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),M) ) ) ).

% cSUP_least
tff(fact_2565_cSUP__least,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_nat,M: nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,X)),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A4))),M) ) ) ).

% cSUP_least
tff(fact_2566_cSUP__least,axiom,
    ! [A4: set_o,F: fun_o_nat,M: nat] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))),M) ) ) ).

% cSUP_least
tff(fact_2567_cSUP__least,axiom,
    ! [A4: set_nat,F: fun_nat_nat,M: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),M) ) ) ).

% cSUP_least
tff(fact_2568_cSUP__least,axiom,
    ! [A4: set_int,F: fun_int_int,M: int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X)),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),M) ) ) ).

% cSUP_least
tff(fact_2569_cSUP__least,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,M: int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(F,X)),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))),M) ) ) ).

% cSUP_least
tff(fact_2570_cSUP__least,axiom,
    ! [A4: set_o,F: fun_o_int,M: int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X))),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))),M) ) ) ).

% cSUP_least
tff(fact_2571_cSUP__least,axiom,
    ! [A4: set_nat,F: fun_nat_int,M: int] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X)),M) )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))),M) ) ) ).

% cSUP_least
tff(fact_2572_cSUP__least,axiom,
    ! [A4: set_int,F: fun_int_Product_unit,M: product_unit] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X)),M) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))),M) ) ) ).

% cSUP_least
tff(fact_2573_cSUP__least,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,M: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(F,X)),M) )
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))),M) ) ) ).

% cSUP_least
tff(fact_2574_UN__constant__eq,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: fun_Pr6764980218223570944et_nat,C: set_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
     => ( ! [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
           => ( aa_Pro1692590229789857841et_nat(F,X) = C ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(F,A4)) = C ) ) ) ).

% UN_constant_eq
tff(fact_2575_UN__constant__eq,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: fun_Pr9038118721296872730et_nat,C: set_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
     => ( ! [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
           => ( aa_Pro9123573313595863371et_nat(F,X) = C ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4)) = C ) ) ) ).

% UN_constant_eq
tff(fact_2576_UN__constant__eq,axiom,
    ! [A: $o,A4: set_o,F: fun_o_set_nat,C: set_nat] :
      ( aa_set_o_o(member_o((A)),A4)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_set_nat(F,(X)) = C ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4)) = C ) ) ) ).

% UN_constant_eq
tff(fact_2577_UN__constant__eq,axiom,
    ! [A: nat,A4: set_nat,F: fun_nat_set_nat,C: set_nat] :
      ( aa_set_nat_o(member_nat(A),A4)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ( aa_nat_set_nat(F,X) = C ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4)) = C ) ) ) ).

% UN_constant_eq
tff(fact_2578_UN__constant__eq,axiom,
    ! [A: int,A4: set_int,F: fun_int_set_nat,C: set_nat] :
      ( aa_set_int_o(member_int(A),A4)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int_set_nat(F,X) = C ) )
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(F,A4)) = C ) ) ) ).

% UN_constant_eq
tff(fact_2579_refl__on__domain,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Ra: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( refl_o7446464703481863270at_nat(A4,Ra)
     => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(A,B),Ra)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
          & aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A4) ) ) ) ).

% refl_on_domain
tff(fact_2580_refl__on__domain,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( refl_o7122697186039111552at_nat(A4,Ra)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),Ra)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
          & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A4) ) ) ) ).

% refl_on_domain
tff(fact_2581_refl__on__domain,axiom,
    ! [A4: set_o,Ra: set_Product_prod_o_o,A: $o,B: $o] :
      ( refl_on_o(A4,Ra)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),Ra)
       => ( aa_set_o_o(member_o((A)),A4)
          & aa_set_o_o(member_o((B)),A4) ) ) ) ).

% refl_on_domain
tff(fact_2582_refl__on__domain,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( refl_o6272981002401706822at_nat(A4,Ra)
     => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),Ra)
       => ( member6273874620674182256at_nat(A,A4)
          & member6273874620674182256at_nat(B,A4) ) ) ) ).

% refl_on_domain
tff(fact_2583_refl__on__domain,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(refl_o6955724340487615712at_nat(A4),Ra)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),Ra)
       => ( member2643936169264416010at_nat(A,A4)
          & member2643936169264416010at_nat(B,A4) ) ) ) ).

% refl_on_domain
tff(fact_2584_refl__on__domain,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(refl_on_nat(A4),Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),Ra)
       => ( aa_set_nat_o(member_nat(A),A4)
          & aa_set_nat_o(member_nat(B),A4) ) ) ) ).

% refl_on_domain
tff(fact_2585_refl__on__domain,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int,A: int,B: int] :
      ( refl_on_int(A4,Ra)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),Ra)
       => ( aa_set_int_o(member_int(A),A4)
          & aa_set_int_o(member_int(B),A4) ) ) ) ).

% refl_on_domain
tff(fact_2586_INT__simps_I4_J,axiom,
    ! [A4: set_char,B5: fun_Pr533032343101338535t_char,C3: set_Product_unit] :
      comple7965528767983648192t_char(image_5325685772115642575t_char(aa_fun3708311092942545532t_char(aTP_Lamm_px(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_char,minus_minus_set_char(A4,comple4338141846124676698t_char(image_5325685772115642575t_char(B5,C3)))) ).

% INT_simps(4)
tff(fact_2587_INT__simps_I4_J,axiom,
    ! [A4: set_char,B5: fun_o_set_char,C3: set_o] :
      comple7965528767983648192t_char(image_o_set_char(aa_fun8606171068324639164t_char(aTP_Lamm_py(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_char,minus_minus_set_char(A4,comple4338141846124676698t_char(image_o_set_char(B5,C3)))) ).

% INT_simps(4)
tff(fact_2588_INT__simps_I4_J,axiom,
    ! [A4: set_char,B5: fun_nat_set_char,C3: set_nat] :
      comple7965528767983648192t_char(image_nat_set_char(aa_fun6435767835650977254t_char(aTP_Lamm_pz(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_char,minus_minus_set_char(A4,comple4338141846124676698t_char(image_nat_set_char(B5,C3)))) ).

% INT_simps(4)
tff(fact_2589_INT__simps_I4_J,axiom,
    ! [A4: set_rat,B5: fun_Pr4472331689133158062et_rat,C3: set_Product_unit] :
      comple4298007329820168263et_rat(image_4665621433434589782et_rat(aa_fun1353959048015916348et_rat(aTP_Lamm_qa(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_rat,minus_minus_set_rat(A4,aa_set6107595668591848464et_rat(comple3890839924845867745et_rat,image_4665621433434589782et_rat(B5,C3)))) ).

% INT_simps(4)
tff(fact_2590_INT__simps_I4_J,axiom,
    ! [A4: set_rat,B5: fun_o_set_rat,C3: set_o] :
      comple4298007329820168263et_rat(image_o_set_rat(aa_fun5436817229860788988et_rat(aTP_Lamm_qb(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_rat,minus_minus_set_rat(A4,aa_set6107595668591848464et_rat(comple3890839924845867745et_rat,image_o_set_rat(B5,C3)))) ).

% INT_simps(4)
tff(fact_2591_INT__simps_I4_J,axiom,
    ! [A4: set_rat,B5: fun_nat_set_rat,C3: set_nat] :
      comple4298007329820168263et_rat(image_nat_set_rat(aa_fun57241613143730932et_rat(aTP_Lamm_qc(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_rat,minus_minus_set_rat(A4,aa_set6107595668591848464et_rat(comple3890839924845867745et_rat,image_nat_set_rat(B5,C3)))) ).

% INT_simps(4)
tff(fact_2592_INT__simps_I4_J,axiom,
    ! [A4: set_int,B5: fun_Pr8362234881679108866et_int,C3: set_Product_unit] :
      comple3628384868704368283et_int(image_3995998972318789802et_int(aa_fun3247719957694404156et_int(aTP_Lamm_qd(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_int,minus_minus_set_int(A4,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(B5,C3)))) ).

% INT_simps(4)
tff(fact_2593_INT__simps_I4_J,axiom,
    ! [A4: set_int,B5: fun_o_set_int,C3: set_o] :
      comple3628384868704368283et_int(image_o_set_int(aa_fun1981618354277381628et_int(aTP_Lamm_qe(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_int,minus_minus_set_int(A4,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(B5,C3)))) ).

% INT_simps(4)
tff(fact_2594_INT__simps_I4_J,axiom,
    ! [A4: set_int,B5: fun_nat_set_int,C3: set_nat] :
      comple3628384868704368283et_int(image_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_qf(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_int,minus_minus_set_int(A4,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(B5,C3)))) ).

% INT_simps(4)
tff(fact_2595_INT__simps_I4_J,axiom,
    ! [A4: set_nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_qg(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,C3)))) ).

% INT_simps(4)
tff(fact_2596_relcompp__bot2,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o] : relcom8911704206367525917at_int(R2,bot_bo638547379360335067_int_o) = bot_bo638547379360335067_int_o ).

% relcompp_bot2
tff(fact_2597_relcompp__bot2,axiom,
    ! [R2: fun_int_fun_int_o] : relcom967497705283160040nteger(R2,bot_bo1552053798131012333eger_o) = bot_bo1552053798131012333eger_o ).

% relcompp_bot2
tff(fact_2598_relcompp__bot1,axiom,
    ! [R2: fun_Pr4738963024209204295_int_o] : relcom8911704206367525917at_int(bot_bo5970965355748268442_nat_o,R2) = bot_bo638547379360335067_int_o ).

% relcompp_bot1
tff(fact_2599_relcompp__bot1,axiom,
    ! [R2: fun_in4891666191172872833eger_o] : relcom967497705283160040nteger(bot_bo428638681919004382_int_o,R2) = bot_bo1552053798131012333eger_o ).

% relcompp_bot1
tff(fact_2600_surj__diff__right,axiom,
    ! [A: int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),top_top_set_int) = top_top_set_int ).

% surj_diff_right
tff(fact_2601_surj__diff__right,axiom,
    ! [A: rat] : image_rat_rat(aTP_Lamm_qi(A),top_top_set_rat) = top_top_set_rat ).

% surj_diff_right
tff(fact_2602_surj__diff__right,axiom,
    ! [A: code_integer] : image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),top_to4645266643341252675nteger) = top_to4645266643341252675nteger ).

% surj_diff_right
tff(fact_2603_INT__simps_I3_J,axiom,
    ! [A4: fun_Pr533032343101338535t_char,B5: set_char,C3: set_Product_unit] :
      comple7965528767983648192t_char(image_5325685772115642575t_char(aa_set6291933337142777762t_char(aTP_Lamm_qk(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_char,minus_minus_set_char(comple7965528767983648192t_char(image_5325685772115642575t_char(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2604_INT__simps_I3_J,axiom,
    ! [A4: fun_o_set_char,B5: set_char,C3: set_o] :
      comple7965528767983648192t_char(image_o_set_char(aa_set4325695674101962541t_char(aTP_Lamm_ql(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_char,minus_minus_set_char(comple7965528767983648192t_char(image_o_set_char(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2605_INT__simps_I3_J,axiom,
    ! [A4: fun_nat_set_char,B5: set_char,C3: set_nat] :
      comple7965528767983648192t_char(image_nat_set_char(aa_set8633863032000686489t_char(aTP_Lamm_qm(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_char,minus_minus_set_char(comple7965528767983648192t_char(image_nat_set_char(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2606_INT__simps_I3_J,axiom,
    ! [A4: fun_Pr4472331689133158062et_rat,B5: set_rat,C3: set_Product_unit] :
      comple4298007329820168263et_rat(image_4665621433434589782et_rat(aa_set3117609736885039074et_rat(aTP_Lamm_qn(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_rat,minus_minus_set_rat(comple4298007329820168263et_rat(image_4665621433434589782et_rat(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2607_INT__simps_I3_J,axiom,
    ! [A4: fun_o_set_rat,B5: set_rat,C3: set_o] :
      comple4298007329820168263et_rat(image_o_set_rat(aa_set6880101219114951661et_rat(aTP_Lamm_qo(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_rat,minus_minus_set_rat(comple4298007329820168263et_rat(image_o_set_rat(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2608_INT__simps_I3_J,axiom,
    ! [A4: fun_nat_set_rat,B5: set_rat,C3: set_nat] :
      comple4298007329820168263et_rat(image_nat_set_rat(aa_set2783155014080441127et_rat(aTP_Lamm_qp(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_rat,minus_minus_set_rat(comple4298007329820168263et_rat(image_nat_set_rat(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2609_INT__simps_I3_J,axiom,
    ! [A4: fun_Pr8362234881679108866et_int,B5: set_int,C3: set_Product_unit] :
      comple3628384868704368283et_int(image_3995998972318789802et_int(aa_set8624687020482735842et_int(aTP_Lamm_qq(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_int,minus_minus_set_int(comple3628384868704368283et_int(image_3995998972318789802et_int(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2610_INT__simps_I3_J,axiom,
    ! [A4: fun_o_set_int,B5: set_int,C3: set_o] :
      comple3628384868704368283et_int(image_o_set_int(aa_set8238700067146519789et_int(aTP_Lamm_qr(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_int,minus_minus_set_int(comple3628384868704368283et_int(image_o_set_int(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2611_INT__simps_I3_J,axiom,
    ! [A4: fun_nat_set_int,B5: set_int,C3: set_nat] :
      comple3628384868704368283et_int(image_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_qs(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_int,minus_minus_set_int(comple3628384868704368283et_int(image_nat_set_int(A4,C3)),B5)) ).

% INT_simps(3)
tff(fact_2612_INT__simps_I3_J,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,B5: set_nat,C3: set_Product_unit] :
      comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_qt(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_nat,aa_set_nat_set_nat(minus_minus_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(A4,C3))),B5)) ).

% INT_simps(3)
tff(fact_2613_Sigma__Diff__distrib1,axiom,
    ! [I3: set_nat,J3: set_nat,C3: fun_nat_set_nat] : produc457027306803732586at_nat(aa_set_nat_set_nat(minus_minus_set_nat(I3),J3),C3) = aa_set8868280134176360700at_nat(minus_1356011639430497352at_nat(produc457027306803732586at_nat(I3,C3)),produc457027306803732586at_nat(J3,C3)) ).

% Sigma_Diff_distrib1
tff(fact_2614_translation__subtract__diff,axiom,
    ! [A: int,S: set_int,T: set_int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),minus_minus_set_int(S,T)) = minus_minus_set_int(aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),S),aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),T)) ).

% translation_subtract_diff
tff(fact_2615_translation__subtract__diff,axiom,
    ! [A: rat,S: set_rat,T: set_rat] : image_rat_rat(aTP_Lamm_qi(A),minus_minus_set_rat(S,T)) = minus_minus_set_rat(image_rat_rat(aTP_Lamm_qi(A),S),image_rat_rat(aTP_Lamm_qi(A),T)) ).

% translation_subtract_diff
tff(fact_2616_translation__subtract__diff,axiom,
    ! [A: code_integer,S: set_Code_integer,T: set_Code_integer] : image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),minus_2355218937544613996nteger(S,T)) = minus_2355218937544613996nteger(image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),S),image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),T)) ).

% translation_subtract_diff
tff(fact_2617_Times__Diff__distrib1,axiom,
    ! [A4: set_nat,B5: set_nat,C3: set_nat] : produc457027306803732586at_nat(aa_set_nat_set_nat(minus_minus_set_nat(A4),B5),aTP_Lamm_gm(C3)) = aa_set8868280134176360700at_nat(minus_1356011639430497352at_nat(produc457027306803732586at_nat(A4,aTP_Lamm_gm(C3))),produc457027306803732586at_nat(B5,aTP_Lamm_gm(C3))) ).

% Times_Diff_distrib1
tff(fact_2618_Sigma__Diff__distrib2,axiom,
    ! [I3: set_nat,A4: fun_nat_set_nat,B5: fun_nat_set_nat] : produc457027306803732586at_nat(I3,aa_fun850356524293783268et_nat(aTP_Lamm_qu(A4),B5)) = aa_set8868280134176360700at_nat(minus_1356011639430497352at_nat(produc457027306803732586at_nat(I3,A4)),produc457027306803732586at_nat(I3,B5)) ).

% Sigma_Diff_distrib2
tff(fact_2619_Sup__SUP__eq,axiom,
    ! [S3: set_fu1867049230074397546_nat_o,X5: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(comple8985044907624445703_nat_o(S3),X5)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),comple4364388969068068532at_nat(image_2888349181732859468at_nat(collec6321179662152712658at_nat,S3))) ) ).

% Sup_SUP_eq
tff(fact_2620_Sup__SUP__eq,axiom,
    ! [S3: set_fun_o_o,X5: $o] :
      ( aa_o_o(comple7262668920879777174un_o_o(S3),(X5))
    <=> aa_set_o_o(member_o((X5)),comple90263536869209701_set_o(image_fun_o_o_set_o(collect_o,S3))) ) ).

% Sup_SUP_eq
tff(fact_2621_Sup__SUP__eq,axiom,
    ! [S3: set_fun_int_o,X5: int] :
      ( aa_int_o(comple3527310399393502258_int_o(S3),X5)
    <=> aa_set_int_o(member_int(X5),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3849661908695225314et_int(collect_int,S3))) ) ).

% Sup_SUP_eq
tff(fact_2622_Sup__SUP__eq,axiom,
    ! [S3: set_fu8838435424971791696_nat_o,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(comple7581358851531288557_nat_o(S3),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),comple5685304695842803022at_nat(image_19007421832380416at_nat(collec3392354462482085612at_nat,S3))) ) ).

% Sup_SUP_eq
tff(fact_2623_Sup__SUP__eq,axiom,
    ! [S3: set_fu2082804241934847128_int_o,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(comple6022789916901393205_int_o(S3),X5)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X5),comple5382143125604098054nt_int(image_6564931188981925232nt_int(collec213857154873943460nt_int,S3))) ) ).

% Sup_SUP_eq
tff(fact_2624_Sup__SUP__eq,axiom,
    ! [S3: set_fun_nat_o,X5: nat] :
      ( aa_nat_o(comple5348352744826190222_nat_o(S3),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_4516309135669182690et_nat(collect_nat,S3))) ) ).

% Sup_SUP_eq
tff(fact_2625_Inf__INT__eq,axiom,
    ! [S3: set_fu1867049230074397546_nat_o,X5: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(comple3522540304083821677_nat_o(S3),X5)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),comple6751905152567503566at_nat(image_2888349181732859468at_nat(collec6321179662152712658at_nat,S3))) ) ).

% Inf_INT_eq
tff(fact_2626_Inf__INT__eq,axiom,
    ! [S3: set_fun_o_o,X5: $o] :
      ( aa_o_o(comple7101583973641323004un_o_o(S3),(X5))
    <=> aa_set_o_o(member_o((X5)),comple3063163877087187839_set_o(image_fun_o_o_set_o(collect_o,S3))) ) ).

% Inf_INT_eq
tff(fact_2627_Inf__INT__eq,axiom,
    ! [S3: set_fun_int_o,X5: int] :
      ( aa_int_o(comple7892175240008826188_int_o(S3),X5)
    <=> aa_set_int_o(member_int(X5),comple3628384868704368283et_int(image_3849661908695225314et_int(collect_int,S3))) ) ).

% Inf_INT_eq
tff(fact_2628_Inf__INT__eq,axiom,
    ! [S3: set_fu8838435424971791696_nat_o,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(comple5316338724409556563_nat_o(S3),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),comple1930127330929219688at_nat(image_19007421832380416at_nat(collec3392354462482085612at_nat,S3))) ) ).

% Inf_INT_eq
tff(fact_2629_Inf__INT__eq,axiom,
    ! [S3: set_fu2082804241934847128_int_o,X5: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(comple3757769789779661211_int_o(S3),X5)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X5),comple1626965760690514720nt_int(image_6564931188981925232nt_int(collec213857154873943460nt_int,S3))) ) ).

% Inf_INT_eq
tff(fact_2630_Inf__INT__eq,axiom,
    ! [S3: set_fun_nat_o,X5: nat] :
      ( aa_nat_o(comple489845548586738344_nat_o(S3),X5)
    <=> aa_set_nat_o(member_nat(X5),comple7806235888213564991et_nat(image_4516309135669182690et_nat(collect_nat,S3))) ) ).

% Inf_INT_eq
tff(fact_2631_INF__Int__eq2,axiom,
    ! [S3: set_se7592288975875140775at_nat,X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(comple8846512533470300341_nat_o(image_6840955169035328556_nat_o(aTP_Lamm_ep,S3)),X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),comple4403251139954110286at_nat(S3)) ) ).

% INF_Int_eq2
tff(fact_2632_INF__Int__eq2,axiom,
    ! [S3: set_se5479201105039950375at_nat,X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(comple531699222623097269_nat_o(image_8577875265703384492_nat_o(aTP_Lamm_eq,S3)),X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),comple6751905152567503566at_nat(S3)) ) ).

% INF_Int_eq2
tff(fact_2633_INF__Int__eq2,axiom,
    ! [S3: set_se4826145725398303499at_num,X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(comple7634033968075569255_num_o(image_6356701302038092666_num_o(aTP_Lamm_er,S3)),X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),comple6868718957223342258at_num(S3)) ) ).

% INF_Int_eq2
tff(fact_2634_INF__Int__eq2,axiom,
    ! [S3: set_se7855581050983116737at_nat,X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(comple6875853405106058289_nat_o(image_8883634599237391886_nat_o(aTP_Lamm_es,S3)),X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),comple1930127330929219688at_nat(S3)) ) ).

% INF_Int_eq2
tff(fact_2635_INF__Int__eq2,axiom,
    ! [S3: set_se6260736226359567993nt_int,X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(comple6185494632912682873_int_o(image_6990949236391375006_int_o(aTP_Lamm_et,S3)),X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),comple1626965760690514720nt_int(S3)) ) ).

% INF_Int_eq2
tff(fact_2636_SUP__Sup__eq2,axiom,
    ! [S3: set_se7592288975875140775at_nat,X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(comple1612470374839605403_nat_o(image_6840955169035328556_nat_o(aTP_Lamm_ep,S3)),X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),comple7342073961973170484at_nat(S3)) ) ).

% SUP_Sup_eq2
tff(fact_2637_SUP__Sup__eq2,axiom,
    ! [S3: set_se5479201105039950375at_nat,X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(comple5471869746640610203_nat_o(image_8577875265703384492_nat_o(aTP_Lamm_eq,S3)),X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),comple4364388969068068532at_nat(S3)) ) ).

% SUP_Sup_eq2
tff(fact_2638_SUP__Sup__eq2,axiom,
    ! [S3: set_se4826145725398303499at_num,X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(comple6119627921928170317_num_o(image_6356701302038092666_num_o(aTP_Lamm_er,S3)),X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),comple1400524285282149784at_num(S3)) ) ).

% SUP_Sup_eq2
tff(fact_2639_SUP__Sup__eq2,axiom,
    ! [S3: set_se7855581050983116737at_nat,X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(comple5361447358958659351_nat_o(image_8883634599237391886_nat_o(aTP_Lamm_es,S3)),X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),comple5685304695842803022at_nat(S3)) ) ).

% SUP_Sup_eq2
tff(fact_2640_SUP__Sup__eq2,axiom,
    ! [S3: set_se6260736226359567993nt_int,X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(comple4671088586765283935_int_o(image_6990949236391375006_int_o(aTP_Lamm_et,S3)),X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),comple5382143125604098054nt_int(S3)) ) ).

% SUP_Sup_eq2
tff(fact_2641_SUP__Sup__eq,axiom,
    ! [S3: set_se5479201105039950375at_nat,X5: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(comple8985044907624445703_nat_o(image_7275882091196097974_nat_o(aTP_Lamm_a,S3)),X5)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),comple4364388969068068532at_nat(S3)) ) ).

% SUP_Sup_eq
tff(fact_2642_SUP__Sup__eq,axiom,
    ! [S3: set_se7855581050983116737at_nat,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(comple7581358851531288557_nat_o(image_4053334768481158710_nat_o(aTP_Lamm_ac,S3)),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),comple5685304695842803022at_nat(S3)) ) ).

% SUP_Sup_eq
tff(fact_2643_SUP__Sup__eq,axiom,
    ! [S3: set_set_o,X5: $o] :
      ( aa_o_o(comple7262668920879777174un_o_o(image_set_o_fun_o_o(aTP_Lamm_aa,S3)),(X5))
    <=> aa_set_o_o(member_o((X5)),comple90263536869209701_set_o(S3)) ) ).

% SUP_Sup_eq
tff(fact_2644_SUP__Sup__eq,axiom,
    ! [S3: set_set_int,X5: int] :
      ( aa_int_o(comple3527310399393502258_int_o(image_7455838903466834460_int_o(aTP_Lamm_ab,S3)),X5)
    <=> aa_set_int_o(member_int(X5),aa_set4034802472124477112et_int(comple3221217463730067765et_int,S3)) ) ).

% SUP_Sup_eq
tff(fact_2645_SUP__Sup__eq,axiom,
    ! [S3: set_set_nat,X5: nat] :
      ( aa_nat_o(comple5348352744826190222_nat_o(image_6547276679300706004_nat_o(aTP_Lamm_ad,S3)),X5)
    <=> aa_set_nat_o(member_nat(X5),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,S3)) ) ).

% SUP_Sup_eq
tff(fact_2646_INF__Int__eq,axiom,
    ! [S3: set_se5479201105039950375at_nat,X5: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(comple3522540304083821677_nat_o(image_7275882091196097974_nat_o(aTP_Lamm_a,S3)),X5)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X5),comple6751905152567503566at_nat(S3)) ) ).

% INF_Int_eq
tff(fact_2647_INF__Int__eq,axiom,
    ! [S3: set_se7855581050983116737at_nat,X5: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(comple5316338724409556563_nat_o(image_4053334768481158710_nat_o(aTP_Lamm_ac,S3)),X5)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X5),comple1930127330929219688at_nat(S3)) ) ).

% INF_Int_eq
tff(fact_2648_INF__Int__eq,axiom,
    ! [S3: set_set_o,X5: $o] :
      ( aa_o_o(comple7101583973641323004un_o_o(image_set_o_fun_o_o(aTP_Lamm_aa,S3)),(X5))
    <=> aa_set_o_o(member_o((X5)),comple3063163877087187839_set_o(S3)) ) ).

% INF_Int_eq
tff(fact_2649_INF__Int__eq,axiom,
    ! [S3: set_set_int,X5: int] :
      ( aa_int_o(comple7892175240008826188_int_o(image_7455838903466834460_int_o(aTP_Lamm_ab,S3)),X5)
    <=> aa_set_int_o(member_int(X5),comple3628384868704368283et_int(S3)) ) ).

% INF_Int_eq
tff(fact_2650_INF__Int__eq,axiom,
    ! [S3: set_set_nat,X5: nat] :
      ( aa_nat_o(comple489845548586738344_nat_o(image_6547276679300706004_nat_o(aTP_Lamm_ad,S3)),X5)
    <=> aa_set_nat_o(member_nat(X5),comple7806235888213564991et_nat(S3)) ) ).

% INF_Int_eq
tff(fact_2651_Sup__set__def,axiom,
    ! [A4: set_se5479201105039950375at_nat] : comple4364388969068068532at_nat(A4) = aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aTP_Lamm_qv(A4)) ).

% Sup_set_def
tff(fact_2652_Sup__set__def,axiom,
    ! [A4: set_set_o] : comple90263536869209701_set_o(A4) = aa_fun_o_o_set_o(collect_o,aTP_Lamm_qw(A4)) ).

% Sup_set_def
tff(fact_2653_Sup__set__def,axiom,
    ! [A4: set_set_int] : aa_set4034802472124477112et_int(comple3221217463730067765et_int,A4) = aa_fun_int_o_set_int(collect_int,aTP_Lamm_qx(A4)) ).

% Sup_set_def
tff(fact_2654_Sup__set__def,axiom,
    ! [A4: set_se7855581050983116737at_nat] : comple5685304695842803022at_nat(A4) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aTP_Lamm_qy(A4)) ).

% Sup_set_def
tff(fact_2655_Sup__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int] : comple5382143125604098054nt_int(A4) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aTP_Lamm_qz(A4)) ).

% Sup_set_def
tff(fact_2656_Sup__set__def,axiom,
    ! [A4: set_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4) = aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_ra(A4)) ).

% Sup_set_def
tff(fact_2657_Inf__set__def,axiom,
    ! [A4: set_se5479201105039950375at_nat] : comple6751905152567503566at_nat(A4) = aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aTP_Lamm_rb(A4)) ).

% Inf_set_def
tff(fact_2658_Inf__set__def,axiom,
    ! [A4: set_set_o] : comple3063163877087187839_set_o(A4) = aa_fun_o_o_set_o(collect_o,aTP_Lamm_rc(A4)) ).

% Inf_set_def
tff(fact_2659_Inf__set__def,axiom,
    ! [A4: set_set_int] : comple3628384868704368283et_int(A4) = aa_fun_int_o_set_int(collect_int,aTP_Lamm_rd(A4)) ).

% Inf_set_def
tff(fact_2660_Inf__set__def,axiom,
    ! [A4: set_se7855581050983116737at_nat] : comple1930127330929219688at_nat(A4) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aTP_Lamm_re(A4)) ).

% Inf_set_def
tff(fact_2661_Inf__set__def,axiom,
    ! [A4: set_se6260736226359567993nt_int] : comple1626965760690514720nt_int(A4) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aTP_Lamm_rf(A4)) ).

% Inf_set_def
tff(fact_2662_Inf__set__def,axiom,
    ! [A4: set_set_nat] : comple7806235888213564991et_nat(A4) = aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_rg(A4)) ).

% Inf_set_def
tff(fact_2663_image__diff__subset,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] : aa_set_rat_o(ord_less_eq_set_rat(minus_minus_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5))),aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5))) ).

% image_diff_subset
tff(fact_2664_image__diff__subset,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5))),aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5))) ).

% image_diff_subset
tff(fact_2665_image__diff__subset,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5))),aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5))) ).

% image_diff_subset
tff(fact_2666_image__diff__subset,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5))),aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,B5))) ).

% image_diff_subset
tff(fact_2667_image__diff__subset,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5))),aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,B5))) ).

% image_diff_subset
tff(fact_2668_Inf__INT__eq2,axiom,
    ! [S3: set_fu3296753518148528270_nat_o,X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(comple8846512533470300341_nat_o(S3),X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),comple4403251139954110286at_nat(image_878628511295945292at_nat(collec7872792499061297490at_nat,image_5187068165612269341_nat_o(produc9020218426428501292_nat_o,S3)))) ) ).

% Inf_INT_eq2
tff(fact_2669_Inf__INT__eq2,axiom,
    ! [S3: set_fu5472951507345886094_nat_o,X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(comple531699222623097269_nat_o(S3),X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),comple6751905152567503566at_nat(image_2888349181732859468at_nat(collec6321179662152712658at_nat,image_3264509002305780637_nat_o(produc410239310623530412_nat_o,S3)))) ) ).

% Inf_INT_eq2
tff(fact_2670_Inf__INT__eq2,axiom,
    ! [S3: set_fu8054534348295464512_num_o,X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(comple7634033968075569255_num_o(S3),X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),comple6868718957223342258at_num(image_8698227314257897620at_num(collec4100598399430923318at_num,image_5750617934288055275_num_o(produc4927758841916487424_num_o,S3)))) ) ).

% Inf_INT_eq2
tff(fact_2671_Inf__INT__eq2,axiom,
    ! [S3: set_fu7727944524835118346_nat_o,X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(comple6875853405106058289_nat_o(S3),X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),comple1930127330929219688at_nat(image_19007421832380416at_nat(collec3392354462482085612at_nat,image_8786112330847644287_nat_o(produc6081775807080527818_nat_o,S3)))) ) ).

% Inf_INT_eq2
tff(fact_2672_Inf__INT__eq2,axiom,
    ! [S3: set_fu1306127549802120274_int_o,X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(comple6185494632912682873_int_o(S3),X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),comple1626965760690514720nt_int(image_6564931188981925232nt_int(collec213857154873943460nt_int,image_8550835289491772687_int_o(produc4947309494688390418_int_o,S3)))) ) ).

% Inf_INT_eq2
tff(fact_2673_Sup__SUP__eq2,axiom,
    ! [S3: set_fu3296753518148528270_nat_o,X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(comple1612470374839605403_nat_o(S3),X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),comple7342073961973170484at_nat(image_878628511295945292at_nat(collec7872792499061297490at_nat,image_5187068165612269341_nat_o(produc9020218426428501292_nat_o,S3)))) ) ).

% Sup_SUP_eq2
tff(fact_2674_Sup__SUP__eq2,axiom,
    ! [S3: set_fu5472951507345886094_nat_o,X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(comple5471869746640610203_nat_o(S3),X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),comple4364388969068068532at_nat(image_2888349181732859468at_nat(collec6321179662152712658at_nat,image_3264509002305780637_nat_o(produc410239310623530412_nat_o,S3)))) ) ).

% Sup_SUP_eq2
tff(fact_2675_Sup__SUP__eq2,axiom,
    ! [S3: set_fu8054534348295464512_num_o,X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(comple6119627921928170317_num_o(S3),X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),comple1400524285282149784at_num(image_8698227314257897620at_num(collec4100598399430923318at_num,image_5750617934288055275_num_o(produc4927758841916487424_num_o,S3)))) ) ).

% Sup_SUP_eq2
tff(fact_2676_Sup__SUP__eq2,axiom,
    ! [S3: set_fu7727944524835118346_nat_o,X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(comple5361447358958659351_nat_o(S3),X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),comple5685304695842803022at_nat(image_19007421832380416at_nat(collec3392354462482085612at_nat,image_8786112330847644287_nat_o(produc6081775807080527818_nat_o,S3)))) ) ).

% Sup_SUP_eq2
tff(fact_2677_Sup__SUP__eq2,axiom,
    ! [S3: set_fu1306127549802120274_int_o,X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(comple4671088586765283935_int_o(S3),X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),comple5382143125604098054nt_int(image_6564931188981925232nt_int(collec213857154873943460nt_int,image_8550835289491772687_int_o(produc4947309494688390418_int_o,S3)))) ) ).

% Sup_SUP_eq2
tff(fact_2678_bot__empty__eq2,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(bot_bo8614507336562527642_nat_o,X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),bot_bo4948859079157340979at_nat) ) ).

% bot_empty_eq2
tff(fact_2679_bot__empty__eq2,axiom,
    ! [X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_2680_bot__empty__eq2,axiom,
    ! [X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_2681_bot__empty__eq2,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(bot_bo1070391751820557722_nat_o,X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),bot_bo228742789529271731at_nat) ) ).

% bot_empty_eq2
tff(fact_2682_bot__empty__eq2,axiom,
    ! [X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_2683_top__empty__eq2,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(top_to393751346391201918_nat_o,X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),top_to5281548843312042519at_nat) ) ).

% top_empty_eq2
tff(fact_2684_top__empty__eq2,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(top_to7888465585648448638_nat_o,X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),top_to6833984726390702231at_nat) ) ).

% top_empty_eq2
tff(fact_2685_top__empty__eq2,axiom,
    ! [X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(top_to6046883302655497392_num_o,X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),top_to385025497714130939at_num) ) ).

% top_empty_eq2
tff(fact_2686_top__empty__eq2,axiom,
    ! [X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(top_to5288702739685986426_nat_o,X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),top_to4669805908274784177at_nat) ) ).

% top_empty_eq2
tff(fact_2687_top__empty__eq2,axiom,
    ! [X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(top_to4598343967492611010_int_o,X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),top_to4366644338036079209nt_int) ) ).

% top_empty_eq2
tff(fact_2688_in__image__insert__iff,axiom,
    ! [B5: set_set_int,Xa: int,A4: set_int] :
      ( ! [C2: set_int] :
          ( member_set_int(C2,B5)
         => ~ aa_set_int_o(member_int(Xa),C2) )
     => ( member_set_int(A4,aa_set5070719292279795054et_int(image_524474410958335435et_int(insert_int(Xa)),B5))
      <=> ( aa_set_int_o(member_int(Xa),A4)
          & member_set_int(minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_2689_in__image__insert__iff,axiom,
    ! [B5: set_set_Product_unit,Xa: product_unit,A4: set_Product_unit] :
      ( ! [C2: set_Product_unit] :
          ( member5877623283571906838t_unit(C2,B5)
         => ~ member_Product_unit(Xa,C2) )
     => ( member5877623283571906838t_unit(A4,image_4019945301284682565t_unit(insert_Product_unit(Xa),B5))
      <=> ( member_Product_unit(Xa,A4)
          & member5877623283571906838t_unit(minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_2690_in__image__insert__iff,axiom,
    ! [B5: set_se5479201105039950375at_nat,Xa: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ! [C2: set_Pr4329608150637261639at_nat] :
          ( member6273874620674182256at_nat(C2,B5)
         => ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),C2) )
     => ( member6273874620674182256at_nat(A4,image_4303274092540115909at_nat(insert9069300056098147895at_nat(Xa),B5))
      <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa),A4)
          & member6273874620674182256at_nat(aa_set4883462369968598652at_nat(minus_3314409938677909166at_nat(A4),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Xa),bot_bo228742789529271731at_nat)),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_2691_in__image__insert__iff,axiom,
    ! [B5: set_se7855581050983116737at_nat,Xa: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ! [C2: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(C2,B5)
         => ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),C2) )
     => ( member2643936169264416010at_nat(A4,image_3684629450409544005at_nat(insert8211810215607154385at_nat(Xa),B5))
      <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),A4)
          & member2643936169264416010at_nat(aa_set8868280134176360700at_nat(minus_1356011639430497352at_nat(A4),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),bot_bo2099793752762293965at_nat)),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_2692_in__image__insert__iff,axiom,
    ! [B5: set_set_o,Xa: $o,A4: set_o] :
      ( ! [C2: set_o] :
          ( member_set_o(C2,B5)
         => ~ aa_set_o_o(member_o((Xa)),C2) )
     => ( member_set_o(A4,image_set_o_set_o(insert_o((Xa)),B5))
      <=> ( aa_set_o_o(member_o((Xa)),A4)
          & member_set_o(minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_2693_in__image__insert__iff,axiom,
    ! [B5: set_set_nat,Xa: nat,A4: set_nat] :
      ( ! [C2: set_nat] :
          ( member_set_nat(C2,B5)
         => ~ aa_set_nat_o(member_nat(Xa),C2) )
     => ( member_set_nat(A4,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(insert_nat(Xa)),B5))
      <=> ( aa_set_nat_o(member_nat(Xa),A4)
          & member_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_2694_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_Pr533032343101338535t_char,C3: set_Product_unit,B5: set_char] :
      minus_minus_set_char(comple7965528767983648192t_char(image_5325685772115642575t_char(A4,C3)),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,minus_minus_set_char(top_top_set_char,B5),comple7965528767983648192t_char(image_5325685772115642575t_char(aa_set6291933337142777762t_char(aTP_Lamm_qk(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2695_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_o_set_char,C3: set_o,B5: set_char] :
      minus_minus_set_char(comple7965528767983648192t_char(image_o_set_char(A4,C3)),B5) = $ite(C3 = bot_bot_set_o,minus_minus_set_char(top_top_set_char,B5),comple7965528767983648192t_char(image_o_set_char(aa_set4325695674101962541t_char(aTP_Lamm_ql(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2696_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_nat_set_char,C3: set_nat,B5: set_char] :
      minus_minus_set_char(comple7965528767983648192t_char(image_nat_set_char(A4,C3)),B5) = $ite(C3 = bot_bot_set_nat,minus_minus_set_char(top_top_set_char,B5),comple7965528767983648192t_char(image_nat_set_char(aa_set8633863032000686489t_char(aTP_Lamm_qm(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2697_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_Pr4472331689133158062et_rat,C3: set_Product_unit,B5: set_rat] :
      minus_minus_set_rat(comple4298007329820168263et_rat(image_4665621433434589782et_rat(A4,C3)),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,minus_minus_set_rat(top_top_set_rat,B5),comple4298007329820168263et_rat(image_4665621433434589782et_rat(aa_set3117609736885039074et_rat(aTP_Lamm_qn(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2698_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_o_set_rat,C3: set_o,B5: set_rat] :
      minus_minus_set_rat(comple4298007329820168263et_rat(image_o_set_rat(A4,C3)),B5) = $ite(C3 = bot_bot_set_o,minus_minus_set_rat(top_top_set_rat,B5),comple4298007329820168263et_rat(image_o_set_rat(aa_set6880101219114951661et_rat(aTP_Lamm_qo(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2699_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_nat_set_rat,C3: set_nat,B5: set_rat] :
      minus_minus_set_rat(comple4298007329820168263et_rat(image_nat_set_rat(A4,C3)),B5) = $ite(C3 = bot_bot_set_nat,minus_minus_set_rat(top_top_set_rat,B5),comple4298007329820168263et_rat(image_nat_set_rat(aa_set2783155014080441127et_rat(aTP_Lamm_qp(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2700_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_Pr8362234881679108866et_int,C3: set_Product_unit,B5: set_int] :
      minus_minus_set_int(comple3628384868704368283et_int(image_3995998972318789802et_int(A4,C3)),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,minus_minus_set_int(top_top_set_int,B5),comple3628384868704368283et_int(image_3995998972318789802et_int(aa_set8624687020482735842et_int(aTP_Lamm_qq(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2701_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_o_set_int,C3: set_o,B5: set_int] :
      minus_minus_set_int(comple3628384868704368283et_int(image_o_set_int(A4,C3)),B5) = $ite(C3 = bot_bot_set_o,minus_minus_set_int(top_top_set_int,B5),comple3628384868704368283et_int(image_o_set_int(aa_set8238700067146519789et_int(aTP_Lamm_qr(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2702_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_nat_set_int,C3: set_nat,B5: set_int] :
      minus_minus_set_int(comple3628384868704368283et_int(image_nat_set_int(A4,C3)),B5) = $ite(C3 = bot_bot_set_nat,minus_minus_set_int(top_top_set_int,B5),comple3628384868704368283et_int(image_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_qs(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2703_INT__extend__simps_I3_J,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,C3: set_Product_unit,B5: set_nat] :
      aa_set_nat_set_nat(minus_minus_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(A4,C3))),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,aa_set_nat_set_nat(minus_minus_set_nat(top_top_set_nat),B5),comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_qt(A4),B5),C3))) ).

% INT_extend_simps(3)
tff(fact_2704_INT__extend__simps_I4_J,axiom,
    ! [A4: set_nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,A4,comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_qg(A4),B5),C3))) ).

% INT_extend_simps(4)
tff(fact_2705_INT__extend__simps_I4_J,axiom,
    ! [A4: set_nat,B5: fun_Pr6764980218223570944et_nat,C3: set_Pr4329608150637261639at_nat] :
      aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(B5,C3))) = $ite(C3 = bot_bo228742789529271731at_nat,A4,comple7806235888213564991et_nat(image_3160451284896210088et_nat(aa_fun2592980488468313788et_nat(aTP_Lamm_rh(A4),B5),C3))) ).

% INT_extend_simps(4)
tff(fact_2706_INT__extend__simps_I4_J,axiom,
    ! [A4: set_nat,B5: fun_Pr9038118721296872730et_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,C3))) = $ite(C3 = bot_bo2099793752762293965at_nat,A4,comple7806235888213564991et_nat(image_15824709712370754et_nat(aa_fun994922058233850812et_nat(aTP_Lamm_ri(A4),B5),C3))) ).

% INT_extend_simps(4)
tff(fact_2707_INT__extend__simps_I4_J,axiom,
    ! [A4: set_nat,B5: fun_o_set_nat,C3: set_o] :
      aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,A4,comple7806235888213564991et_nat(image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_rj(A4),B5),C3))) ).

% INT_extend_simps(4)
tff(fact_2708_INT__extend__simps_I4_J,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat,C3: set_nat] :
      aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_nat,A4,comple7806235888213564991et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_rk(A4),B5),C3))) ).

% INT_extend_simps(4)
tff(fact_2709_linear__order__on__singleton,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat] : order_1878278401035043452at_nat(insert2290795007606087959at_nat(Xa,bot_bo1534825418459847827at_nat),insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Xa),bot_bo4948859079157340979at_nat)) ).

% linear_order_on_singleton
tff(fact_2710_linear__order__on__singleton,axiom,
    ! [Xa: int] : order_4471489696718655927on_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xa),bot_bo1796632182523588997nt_int)) ).

% linear_order_on_singleton
tff(fact_2711_linear__order__on__singleton,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(order_5942792088423767318at_nat(insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Xa)),bot_bo228742789529271731at_nat)) ).

% linear_order_on_singleton
tff(fact_2712_linear__order__on__singleton,axiom,
    ! [Xa: product_unit] : order_888282469287196226t_unit(aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit),insert4616891024742550839t_unit(produc5692694829384537815t_unit(Xa,Xa),bot_bo1654022762305692595t_unit)) ).

% linear_order_on_singleton
tff(fact_2713_linear__order__on__singleton,axiom,
    ! [Xa: produc3843707927480180839at_nat] : order_2711500226769404828at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Xa),bot_bo228742789529271731at_nat),insert398606708087465655at_nat(produc2335810615602371415at_nat(Xa,Xa),bot_bo7193250159860175667at_nat)) ).

% linear_order_on_singleton
tff(fact_2714_linear__order__on__singleton,axiom,
    ! [Xa: product_prod_nat_nat] : order_7480980593162297270at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),bot_bo2099793752762293965at_nat),insert5050368324300391991at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Xa),bot_bo5327735625951526323at_nat)) ).

% linear_order_on_singleton
tff(fact_2715_linear__order__on__singleton,axiom,
    ! [Xa: $o] : order_6238756238976269133r_on_o(aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o),insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa)),(Xa)),bot_bo7073875226086086771od_o_o)) ).

% linear_order_on_singleton
tff(fact_2716_linear__order__on__singleton,axiom,
    ! [Xa: nat] : aa_set3702741264012733747_nat_o(order_4473980167227706203on_nat(aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Xa)),bot_bo2099793752762293965at_nat)) ).

% linear_order_on_singleton
tff(fact_2717_total__on__singleton,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat] : total_5117575981256876747at_nat(insert2290795007606087959at_nat(Xa,bot_bo1534825418459847827at_nat),insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Xa),bot_bo4948859079157340979at_nat)) ).

% total_on_singleton
tff(fact_2718_total__on__singleton,axiom,
    ! [Xa: int] : total_on_int(aa_set_int_set_int(insert_int(Xa),bot_bot_set_int),insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Xa),bot_bo1796632182523588997nt_int)) ).

% total_on_singleton
tff(fact_2719_total__on__singleton,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(total_7212417256332265445at_nat(insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Xa)),bot_bo228742789529271731at_nat)) ).

% total_on_singleton
tff(fact_2720_total__on__singleton,axiom,
    ! [Xa: product_unit] : total_5665700600833547665t_unit(aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit),insert4616891024742550839t_unit(produc5692694829384537815t_unit(Xa,Xa),bot_bo1654022762305692595t_unit)) ).

% total_on_singleton
tff(fact_2721_total__on__singleton,axiom,
    ! [Xa: produc3843707927480180839at_nat] : total_1513685709984341995at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Xa),bot_bo228742789529271731at_nat),insert398606708087465655at_nat(produc2335810615602371415at_nat(Xa,Xa),bot_bo7193250159860175667at_nat)) ).

% total_on_singleton
tff(fact_2722_total__on__singleton,axiom,
    ! [Xa: product_prod_nat_nat] : total_3592101749530773125at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),bot_bo2099793752762293965at_nat),insert5050368324300391991at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Xa),bot_bo5327735625951526323at_nat)) ).

% total_on_singleton
tff(fact_2723_total__on__singleton,axiom,
    ! [Xa: $o] : total_on_o(aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o),insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Xa)),(Xa)),bot_bo7073875226086086771od_o_o)) ).

% total_on_singleton
tff(fact_2724_total__on__singleton,axiom,
    ! [Xa: nat] : aa_set3702741264012733747_nat_o(total_on_nat(aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Xa)),bot_bo2099793752762293965at_nat)) ).

% total_on_singleton
tff(fact_2725_relation__of__def,axiom,
    ! [P2: fun_se5672428230568430382_nat_o,A4: set_se5479201105039950375at_nat] : order_8308734813930759573at_nat(P2,A4) = aa_fun4050081229871760341at_nat(collec7872792499061297490at_nat,aa_fun1900873372130133414_nat_o(produc9020218426428501292_nat_o,aa_set5310703806256024963_nat_o(aTP_Lamm_rl(P2),A4))) ).

% relation_of_def
tff(fact_2726_relation__of__def,axiom,
    ! [P2: fun_se5639961202347202350_nat_o,A4: set_se7855581050983116737at_nat] : order_8809246210994205103at_nat(P2,A4) = aa_fun1965866159956205013at_nat(collec6321179662152712658at_nat,aa_fun762164075915671590_nat_o(produc410239310623530412_nat_o,aa_set5424408346876907241_nat_o(aTP_Lamm_rm(P2),A4))) ).

% relation_of_def
tff(fact_2727_relation__of__def,axiom,
    ! [P2: fun_nat_fun_nat_o,A4: set_nat] : order_1962693441708344834of_nat(P2,A4) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_set8149993881734264422_nat_o(aTP_Lamm_rn(P2),A4))) ).

% relation_of_def
tff(fact_2728_relation__of__def,axiom,
    ! [P2: fun_int_fun_int_o,A4: set_int] : order_1960202971199294558of_int(P2,A4) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_set5204721951351315026_int_o(aTP_Lamm_ro(P2),A4))) ).

% relation_of_def
tff(fact_2729_insert__Times__insert,axiom,
    ! [A: nat,A4: set_nat,B: product_unit,B5: set_Product_unit] : produc5506539147108004339t_unit(aa_set_nat_set_nat(insert_nat(A),A4),aa_set4829896985964406405t_unit(aTP_Lamm_rp(B),B5)) = insert6804906068341198690t_unit(aa_Pro3283014634955171971t_unit(produc3434450474625521892t_unit(A),B),sup_su4209953369189666580t_unit(produc5506539147108004339t_unit(A4,aa_set4829896985964406405t_unit(aTP_Lamm_rp(B),B5)),produc5506539147108004339t_unit(aa_set_nat_set_nat(insert_nat(A),A4),aTP_Lamm_jt(B5)))) ).

% insert_Times_insert
tff(fact_2730_insert__Times__insert,axiom,
    ! [A: nat,A4: set_nat,B: $o,B5: set_o] : product_Sigma_nat_o(aa_set_nat_set_nat(insert_nat(A),A4),aa_set557987488673405253_set_o(aTP_Lamm_rq((B)),B5)) = insert2233377226759017069_nat_o(aa_o_P6899409326285396377_nat_o(product_Pair_nat_o(A),(B)),sup_su503371503184139039_nat_o(product_Sigma_nat_o(A4,aa_set557987488673405253_set_o(aTP_Lamm_rq((B)),B5)),product_Sigma_nat_o(aa_set_nat_set_nat(insert_nat(A),A4),aTP_Lamm_jf(B5)))) ).

% insert_Times_insert
tff(fact_2731_insert__Times__insert,axiom,
    ! [A: product_unit,A4: set_Product_unit,B: nat,B5: set_nat] : produc6875377002296012693it_nat(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4),aa_set7097633925648574946et_nat(aTP_Lamm_rr(B),B5)) = insert6879346373837673660it_nat(produc4803288329813530246it_nat(A,B),sup_su1639320892072265582it_nat(produc6875377002296012693it_nat(A4,aa_set7097633925648574946et_nat(aTP_Lamm_rr(B),B5)),produc6875377002296012693it_nat(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4),aTP_Lamm_jr(B5)))) ).

% insert_Times_insert
tff(fact_2732_insert__Times__insert,axiom,
    ! [A: product_unit,A4: set_Product_unit,B: product_unit,B5: set_Product_unit] : produc6121607364029222536t_unit(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4),aa_set7533529402238418324t_unit(aTP_Lamm_rs(B),B5)) = insert4616891024742550839t_unit(produc5692694829384537815t_unit(A,B),sup_su6481247232241170203t_unit(produc6121607364029222536t_unit(A4,aa_set7533529402238418324t_unit(aTP_Lamm_rs(B),B5)),produc6121607364029222536t_unit(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4),aTP_Lamm_jp(B5)))) ).

% insert_Times_insert
tff(fact_2733_insert__Times__insert,axiom,
    ! [A: product_unit,A4: set_Product_unit,B: $o,B5: set_o] : produc4996980317674386579unit_o(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4),aa_set5869638957405543594_set_o(aTP_Lamm_rt((B)),B5)) = insert3586912282507785154unit_o(produc8487917018914037602unit_o(A,(B)),sup_su2851927382625014182unit_o(produc4996980317674386579unit_o(A4,aa_set5869638957405543594_set_o(aTP_Lamm_rt((B)),B5)),produc4996980317674386579unit_o(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4),aTP_Lamm_jq(B5)))) ).

% insert_Times_insert
tff(fact_2734_insert__Times__insert,axiom,
    ! [A: $o,A4: set_o,B: nat,B5: set_nat] : product_Sigma_o_nat(aa_set_o_set_o(insert_o((A)),A4),aa_set1696117090455734765et_nat(aTP_Lamm_ru(B),B5)) = insert7948214739558299847_o_nat(aa_nat139401504800077733_o_nat(product_Pair_o_nat((A)),B),sup_su8679140417860611193_o_nat(product_Sigma_o_nat(A4,aa_set1696117090455734765et_nat(aTP_Lamm_ru(B),B5)),product_Sigma_o_nat(aa_set_o_set_o(insert_o((A)),A4),aTP_Lamm_jd(B5)))) ).

% insert_Times_insert
tff(fact_2735_insert__Times__insert,axiom,
    ! [A: $o,A4: set_o,B: product_unit,B5: set_Product_unit] : produc411009820408249917t_unit(aa_set_o_set_o(insert_o((A)),A4),aa_set1706710359383640265t_unit(aTP_Lamm_rv(B),B5)) = insert9112848208840649964t_unit(aa_Pro7405480915597867403t_unit(produc3901946521647900940t_unit((A)),B),sup_su4058753791541417424t_unit(produc411009820408249917t_unit(A4,aa_set1706710359383640265t_unit(aTP_Lamm_rv(B),B5)),produc411009820408249917t_unit(aa_set_o_set_o(insert_o((A)),A4),aTP_Lamm_js(B5)))) ).

% insert_Times_insert
tff(fact_2736_insert__Times__insert,axiom,
    ! [A: $o,A4: set_o,B: $o,B5: set_o] : product_Sigma_o_o(aa_set_o_set_o(insert_o((A)),A4),aa_set_o_fun_o_set_o(aTP_Lamm_rw((B)),B5)) = insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),sup_su5769328420594410459od_o_o(product_Sigma_o_o(A4,aa_set_o_fun_o_set_o(aTP_Lamm_rw((B)),B5)),product_Sigma_o_o(aa_set_o_set_o(insert_o((A)),A4),aTP_Lamm_jc(B5)))) ).

% insert_Times_insert
tff(fact_2737_insert__Times__insert,axiom,
    ! [A: nat,A4: set_nat,B: num,B5: set_num] : produc6237731129814287028at_num(aa_set_nat_set_nat(insert_nat(A),A4),aa_set6533290090869974187et_num(aTP_Lamm_rx(B),B5)) = insert8920054152555992091at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B),sup_su2042722026077122175at_num(produc6237731129814287028at_num(A4,aa_set6533290090869974187et_num(aTP_Lamm_rx(B),B5)),produc6237731129814287028at_num(aa_set_nat_set_nat(insert_nat(A),A4),aTP_Lamm_jl(B5)))) ).

% insert_Times_insert
tff(fact_2738_insert__Times__insert,axiom,
    ! [A: int,A4: set_int,B: int,B5: set_int] : produc1453890548195736866nt_int(aa_set_int_set_int(insert_int(A),A4),aa_set2836339071814736939et_int(aTP_Lamm_ry(B),B5)) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),sup_su6024340866399070445nt_int(produc1453890548195736866nt_int(A4,aa_set2836339071814736939et_int(aTP_Lamm_ry(B),B5)),produc1453890548195736866nt_int(aa_set_int_set_int(insert_int(A),A4),aTP_Lamm_gf(B5)))) ).

% insert_Times_insert
tff(fact_2739_INT__simps_I2_J,axiom,
    ! [A4: set_char,B5: fun_Pr533032343101338535t_char,C3: set_Product_unit] :
      comple7965528767983648192t_char(image_5325685772115642575t_char(aa_fun3708311092942545532t_char(aTP_Lamm_rz(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_char,inf_inf_set_char(A4,comple7965528767983648192t_char(image_5325685772115642575t_char(B5,C3)))) ).

% INT_simps(2)
tff(fact_2740_INT__simps_I2_J,axiom,
    ! [A4: set_char,B5: fun_o_set_char,C3: set_o] :
      comple7965528767983648192t_char(image_o_set_char(aa_fun8606171068324639164t_char(aTP_Lamm_sa(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_char,inf_inf_set_char(A4,comple7965528767983648192t_char(image_o_set_char(B5,C3)))) ).

% INT_simps(2)
tff(fact_2741_INT__simps_I2_J,axiom,
    ! [A4: set_char,B5: fun_nat_set_char,C3: set_nat] :
      comple7965528767983648192t_char(image_nat_set_char(aa_fun6435767835650977254t_char(aTP_Lamm_sb(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_char,inf_inf_set_char(A4,comple7965528767983648192t_char(image_nat_set_char(B5,C3)))) ).

% INT_simps(2)
tff(fact_2742_INT__simps_I2_J,axiom,
    ! [A4: set_rat,B5: fun_Pr4472331689133158062et_rat,C3: set_Product_unit] :
      comple4298007329820168263et_rat(image_4665621433434589782et_rat(aa_fun1353959048015916348et_rat(aTP_Lamm_sc(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_rat,inf_inf_set_rat(A4,comple4298007329820168263et_rat(image_4665621433434589782et_rat(B5,C3)))) ).

% INT_simps(2)
tff(fact_2743_INT__simps_I2_J,axiom,
    ! [A4: set_rat,B5: fun_o_set_rat,C3: set_o] :
      comple4298007329820168263et_rat(image_o_set_rat(aa_fun5436817229860788988et_rat(aTP_Lamm_sd(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_rat,inf_inf_set_rat(A4,comple4298007329820168263et_rat(image_o_set_rat(B5,C3)))) ).

% INT_simps(2)
tff(fact_2744_INT__simps_I2_J,axiom,
    ! [A4: set_rat,B5: fun_nat_set_rat,C3: set_nat] :
      comple4298007329820168263et_rat(image_nat_set_rat(aa_fun57241613143730932et_rat(aTP_Lamm_se(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_rat,inf_inf_set_rat(A4,comple4298007329820168263et_rat(image_nat_set_rat(B5,C3)))) ).

% INT_simps(2)
tff(fact_2745_INT__simps_I2_J,axiom,
    ! [A4: set_int,B5: fun_Pr8362234881679108866et_int,C3: set_Product_unit] :
      comple3628384868704368283et_int(image_3995998972318789802et_int(aa_fun3247719957694404156et_int(aTP_Lamm_sf(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_int,inf_inf_set_int(A4,comple3628384868704368283et_int(image_3995998972318789802et_int(B5,C3)))) ).

% INT_simps(2)
tff(fact_2746_INT__simps_I2_J,axiom,
    ! [A4: set_int,B5: fun_o_set_int,C3: set_o] :
      comple3628384868704368283et_int(image_o_set_int(aa_fun1981618354277381628et_int(aTP_Lamm_sg(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_int,inf_inf_set_int(A4,comple3628384868704368283et_int(image_o_set_int(B5,C3)))) ).

% INT_simps(2)
tff(fact_2747_INT__simps_I2_J,axiom,
    ! [A4: set_int,B5: fun_nat_set_int,C3: set_nat] :
      comple3628384868704368283et_int(image_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_sh(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_int,inf_inf_set_int(A4,comple3628384868704368283et_int(image_nat_set_int(B5,C3)))) ).

% INT_simps(2)
tff(fact_2748_INT__simps_I2_J,axiom,
    ! [A4: set_nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_si(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_nat,aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(image_8173849991827986510et_nat(B5,C3)))) ).

% INT_simps(2)
tff(fact_2749_INT__simps_I1_J,axiom,
    ! [A4: fun_Pr533032343101338535t_char,B5: set_char,C3: set_Product_unit] :
      comple7965528767983648192t_char(image_5325685772115642575t_char(aa_set6291933337142777762t_char(aTP_Lamm_sj(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_char,inf_inf_set_char(comple7965528767983648192t_char(image_5325685772115642575t_char(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2750_INT__simps_I1_J,axiom,
    ! [A4: fun_o_set_char,B5: set_char,C3: set_o] :
      comple7965528767983648192t_char(image_o_set_char(aa_set4325695674101962541t_char(aTP_Lamm_sk(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_char,inf_inf_set_char(comple7965528767983648192t_char(image_o_set_char(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2751_INT__simps_I1_J,axiom,
    ! [A4: fun_nat_set_char,B5: set_char,C3: set_nat] :
      comple7965528767983648192t_char(image_nat_set_char(aa_set8633863032000686489t_char(aTP_Lamm_sl(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_char,inf_inf_set_char(comple7965528767983648192t_char(image_nat_set_char(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2752_INT__simps_I1_J,axiom,
    ! [A4: fun_Pr4472331689133158062et_rat,B5: set_rat,C3: set_Product_unit] :
      comple4298007329820168263et_rat(image_4665621433434589782et_rat(aa_set3117609736885039074et_rat(aTP_Lamm_sm(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_rat,inf_inf_set_rat(comple4298007329820168263et_rat(image_4665621433434589782et_rat(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2753_INT__simps_I1_J,axiom,
    ! [A4: fun_o_set_rat,B5: set_rat,C3: set_o] :
      comple4298007329820168263et_rat(image_o_set_rat(aa_set6880101219114951661et_rat(aTP_Lamm_sn(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_rat,inf_inf_set_rat(comple4298007329820168263et_rat(image_o_set_rat(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2754_INT__simps_I1_J,axiom,
    ! [A4: fun_nat_set_rat,B5: set_rat,C3: set_nat] :
      comple4298007329820168263et_rat(image_nat_set_rat(aa_set2783155014080441127et_rat(aTP_Lamm_so(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_rat,inf_inf_set_rat(comple4298007329820168263et_rat(image_nat_set_rat(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2755_INT__simps_I1_J,axiom,
    ! [A4: fun_Pr8362234881679108866et_int,B5: set_int,C3: set_Product_unit] :
      comple3628384868704368283et_int(image_3995998972318789802et_int(aa_set8624687020482735842et_int(aTP_Lamm_sp(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_int,inf_inf_set_int(comple3628384868704368283et_int(image_3995998972318789802et_int(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2756_INT__simps_I1_J,axiom,
    ! [A4: fun_o_set_int,B5: set_int,C3: set_o] :
      comple3628384868704368283et_int(image_o_set_int(aa_set8238700067146519789et_int(aTP_Lamm_sq(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,top_top_set_int,inf_inf_set_int(comple3628384868704368283et_int(image_o_set_int(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2757_INT__simps_I1_J,axiom,
    ! [A4: fun_nat_set_int,B5: set_int,C3: set_nat] :
      comple3628384868704368283et_int(image_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_sr(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,top_top_set_int,inf_inf_set_int(comple3628384868704368283et_int(image_nat_set_int(A4,C3)),B5)) ).

% INT_simps(1)
tff(fact_2758_INT__simps_I1_J,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,B5: set_nat,C3: set_Product_unit] :
      comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_ss(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,top_top_set_nat,aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(A4,C3))),B5)) ).

% INT_simps(1)
tff(fact_2759_cSUP__UNION,axiom,
    ! [A4: set_int,B5: fun_int_set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int_set_int(B5,X) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aa_fun118384756000730526et_int(aTP_Lamm_st(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_su(B5),F)),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2760_cSUP__UNION,axiom,
    ! [A4: set_int,B5: fun_in4731230917697954842t_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int6548451963861500163t_unit(B5,X) != bot_bo3957492148770167129t_unit ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aa_fun7610309373818225051et_int(aTP_Lamm_sv(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,comple3349607820504768346t_unit(image_8658272795163236474t_unit(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aa_fun5430718389669344997nt_int(aTP_Lamm_sw(B5),F)),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2761_cSUP__UNION,axiom,
    ! [A4: set_int,B5: fun_int_set_o,F: fun_o_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int_set_o(B5,X) != bot_bot_set_o ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aa_fun4704607026696346150et_int(aTP_Lamm_sx(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,comple90263536869209701_set_o(image_int_set_o(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aa_fun1998554717657234288nt_int(aTP_Lamm_sy(B5),F)),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2762_cSUP__UNION,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr8362234881679108866et_int,F: fun_int_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => ( aa_Pro1886178141017053491et_int(B5,X) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(aa_fun339368410111030063et_int(aTP_Lamm_sz(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(aa_fun2487178249965271161it_int(aTP_Lamm_ta(B5),F),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2763_cSUP__UNION,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr3013102738158121955t_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => ( aa_Pro234974305641502940t_unit(B5,X) != bot_bo3957492148770167129t_unit ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(aa_fun2049105527398774130et_int(aTP_Lamm_tb(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,comple3349607820504768346t_unit(image_7778121751606633509t_unit(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(aa_fun3641207808737740476it_int(aTP_Lamm_tc(B5),F),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2764_cSUP__UNION,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr6739565440344758894_set_o,F: fun_o_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => ( aa_Pro536174410435844839_set_o(B5,X) != bot_bot_set_o ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(aa_fun7391356117566996647et_int(aTP_Lamm_td(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,comple90263536869209701_set_o(image_6930536745178679216_set_o(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(aa_fun3625413194728170993it_int(aTP_Lamm_te(B5),F),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2765_cSUP__UNION,axiom,
    ! [A4: set_o,B5: fun_o_set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_set_int(B5,(X)) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(aa_fun1632018885420323002et_int(aTP_Lamm_tf(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(aa_fun2290039529095201284_o_int(aTP_Lamm_tg(B5),F),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2766_cSUP__UNION,axiom,
    ! [A4: set_o,B5: fun_o_1833192751255691672t_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_s3492205133957624465t_unit(B5,(X)) != bot_bo3957492148770167129t_unit ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(aa_fun5570681940130100093et_int(aTP_Lamm_th(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(aa_fun4152134955913539783_o_int(aTP_Lamm_ti(B5),F),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2767_cSUP__UNION,axiom,
    ! [A4: set_o,B5: fun_o_set_o,F: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_set_o(B5,(X)) != bot_bot_set_o ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(aa_fun4317264022763090482et_int(aTP_Lamm_tj(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,comple90263536869209701_set_o(image_o_set_o(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_tk(B5),F),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2768_cSUP__UNION,axiom,
    ! [A4: set_nat,B5: fun_nat_set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ( aa_nat_set_int(B5,X) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(aa_fun8970103232618661698et_int(aTP_Lamm_tl(B5),F),A4)))
         => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(B5,A4)))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(aa_fun5280992047749994636at_int(aTP_Lamm_tm(B5),F)),A4)) ) ) ) ) ).

% cSUP_UNION
tff(fact_2769_cINF__UNION,axiom,
    ! [A4: set_int,B5: fun_int_set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int_set_int(B5,X) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aa_fun118384756000730526et_int(aTP_Lamm_st(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tn(B5),F)),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2770_cINF__UNION,axiom,
    ! [A4: set_int,B5: fun_in4731230917697954842t_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int6548451963861500163t_unit(B5,X) != bot_bo3957492148770167129t_unit ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aa_fun7610309373818225051et_int(aTP_Lamm_sv(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,comple3349607820504768346t_unit(image_8658272795163236474t_unit(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aa_fun5430718389669344997nt_int(aTP_Lamm_to(B5),F)),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2771_cINF__UNION,axiom,
    ! [A4: set_int,B5: fun_int_set_o,F: fun_o_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => ( aa_int_set_o(B5,X) != bot_bot_set_o ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_int_set_int(aa_fun4704607026696346150et_int(aTP_Lamm_sx(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,comple90263536869209701_set_o(image_int_set_o(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aa_fun1998554717657234288nt_int(aTP_Lamm_tp(B5),F)),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2772_cINF__UNION,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr8362234881679108866et_int,F: fun_int_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => ( aa_Pro1886178141017053491et_int(B5,X) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(aa_fun339368410111030063et_int(aTP_Lamm_sz(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(aa_fun2487178249965271161it_int(aTP_Lamm_tq(B5),F),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2773_cINF__UNION,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr3013102738158121955t_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => ( aa_Pro234974305641502940t_unit(B5,X) != bot_bo3957492148770167129t_unit ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(aa_fun2049105527398774130et_int(aTP_Lamm_tb(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,comple3349607820504768346t_unit(image_7778121751606633509t_unit(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(aa_fun3641207808737740476it_int(aTP_Lamm_tr(B5),F),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2774_cINF__UNION,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr6739565440344758894_set_o,F: fun_o_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( ! [X: product_unit] :
            ( member_Product_unit(X,A4)
           => ( aa_Pro536174410435844839_set_o(B5,X) != bot_bot_set_o ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_3995998972318789802et_int(aa_fun7391356117566996647et_int(aTP_Lamm_td(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,comple90263536869209701_set_o(image_6930536745178679216_set_o(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(aa_fun3625413194728170993it_int(aTP_Lamm_ts(B5),F),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2775_cINF__UNION,axiom,
    ! [A4: set_o,B5: fun_o_set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_set_int(B5,(X)) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(aa_fun1632018885420323002et_int(aTP_Lamm_tf(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(aa_fun2290039529095201284_o_int(aTP_Lamm_tt(B5),F),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2776_cINF__UNION,axiom,
    ! [A4: set_o,B5: fun_o_1833192751255691672t_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_s3492205133957624465t_unit(B5,(X)) != bot_bo3957492148770167129t_unit ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(aa_fun5570681940130100093et_int(aTP_Lamm_th(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(aa_fun4152134955913539783_o_int(aTP_Lamm_tu(B5),F),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2777_cINF__UNION,axiom,
    ! [A4: set_o,B5: fun_o_set_o,F: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => ( aa_o_set_o(B5,(X)) != bot_bot_set_o ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_o_set_int(aa_fun4317264022763090482et_int(aTP_Lamm_tj(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,comple90263536869209701_set_o(image_o_set_o(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_tv(B5),F),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2778_cINF__UNION,axiom,
    ! [A4: set_nat,B5: fun_nat_set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => ( aa_nat_set_int(B5,X) != bot_bot_set_int ) )
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(aa_fun8970103232618661698et_int(aTP_Lamm_tl(B5),F),A4)))
         => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,image_nat_set_int(B5,A4)))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(aa_fun5280992047749994636at_int(aTP_Lamm_tw(B5),F)),A4)) ) ) ) ) ).

% cINF_UNION
tff(fact_2779_Ex__inj__on__UNION__Sigma,axiom,
    ! [A4: fun_nat_set_nat,I3: set_nat] :
    ? [F4: fun_na1476630412504255114at_nat] :
      ( aa_set_nat_o(inj_on5538052773655684606at_nat(F4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3)))
      & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F4,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3)))),produc457027306803732586at_nat(I3,A4)) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_2780_prod__set__simps_I1_J,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : basic_1274512288150691911at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = insert2290795007606087959at_nat(Xa,bot_bo1534825418459847827at_nat) ).

% prod_set_simps(1)
tff(fact_2781_prod__set__simps_I1_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : basic_3661821009137050567at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = insert9200635055090092081at_nat(Xa,bot_bo3083307316010499117at_nat) ).

% prod_set_simps(1)
tff(fact_2782_prod__set__simps_I1_J,axiom,
    ! [Xa: int,Y2: int] : basic_fsts_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = aa_set_int_set_int(insert_int(Xa),bot_bot_set_int) ).

% prod_set_simps(1)
tff(fact_2783_prod__set__simps_I1_J,axiom,
    ! [Xa: nat,Y2: num] : basic_fsts_nat_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat) ).

% prod_set_simps(1)
tff(fact_2784_prod__set__simps_I1_J,axiom,
    ! [Xa: nat,Y2: nat] : aa_Pro9123573313595863371et_nat(basic_fsts_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat) ).

% prod_set_simps(1)
tff(fact_2785_if__image__distrib,axiom,
    ! [P2: fun_int_o,F: fun_int_nat,G: fun_int_nat,S3: set_int] : aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_tx(P2),F),G)),S3) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set_int_set_nat(image_int_nat(F),inf_inf_set_int(S3,aa_fun_int_o_set_int(collect_int,P2)))),aa_set_int_set_nat(image_int_nat(G),inf_inf_set_int(S3,aa_fun_int_o_set_int(collect_int,aTP_Lamm_ty(P2))))) ).

% if_image_distrib
tff(fact_2786_if__image__distrib,axiom,
    ! [P2: fun_int_o,F: fun_int_int,G: fun_int_int,S3: set_int] : aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_tz(P2),F),G)),S3) = sup_sup_set_int(aa_set_int_set_int(image_int_int(F),inf_inf_set_int(S3,aa_fun_int_o_set_int(collect_int,P2))),aa_set_int_set_int(image_int_int(G),inf_inf_set_int(S3,aa_fun_int_o_set_int(collect_int,aTP_Lamm_ty(P2))))) ).

% if_image_distrib
tff(fact_2787_if__image__distrib,axiom,
    ! [P2: fun_nat_o,F: fun_nat_rat,G: fun_nat_rat,S3: set_nat] : aa_set_nat_set_rat(image_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ua(P2),F),G)),S3) = sup_sup_set_rat(aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,P2))),aa_set_nat_set_rat(image_nat_rat(G),aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_ub(P2))))) ).

% if_image_distrib
tff(fact_2788_if__image__distrib,axiom,
    ! [P2: fun_nat_o,F: fun_nat_nat,G: fun_nat_nat,S3: set_nat] : aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uc(P2),F),G)),S3) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,P2)))),aa_set_nat_set_nat(image_nat_nat(G),aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_ub(P2))))) ).

% if_image_distrib
tff(fact_2789_if__image__distrib,axiom,
    ! [P2: fun_nat_o,F: fun_nat_int,G: fun_nat_int,S3: set_nat] : aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ud(P2),F),G)),S3) = sup_sup_set_int(aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,P2))),aa_set_nat_set_int(image_nat_int(G),aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_ub(P2))))) ).

% if_image_distrib
tff(fact_2790_if__image__distrib,axiom,
    ! [P2: fun_nat_o,F: fun_na3202795955365919216at_nat,G: fun_na3202795955365919216at_nat,S3: set_nat] : image_3276603626957510736at_nat(aa_fun2085067056331341500at_nat(aa_fun2877972393441065591at_nat(aTP_Lamm_ue(P2),F),G),S3) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(image_3276603626957510736at_nat(F,aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,P2)))),image_3276603626957510736at_nat(G,aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_ub(P2))))) ).

% if_image_distrib
tff(fact_2791_if__image__distrib,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,F: fun_Pr3818778794767778097at_nat,G: fun_Pr3818778794767778097at_nat,S3: set_Pr958786334691620121nt_int] : image_8858061497167895027at_nat(aa_fun2261957746281396896at_nat(aa_fun1759909663267528014at_nat(aTP_Lamm_uf(P2),F),G),S3) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(image_8858061497167895027at_nat(F,inf_in2269163501485487111nt_int(S3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2)))),image_8858061497167895027at_nat(G,inf_in2269163501485487111nt_int(S3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aTP_Lamm_ug(P2))))) ).

% if_image_distrib
tff(fact_2792_if__image__distrib,axiom,
    ! [P2: fun_Pr4796943543331071514_nat_o,F: fun_Pr5303981572562559465at_nat,G: fun_Pr5303981572562559465at_nat,S3: set_Pr1261947904930325089at_nat] : image_1119892238107900587at_nat(aa_fun5472858705586414608at_nat(aa_fun8506152981838162806at_nat(aTP_Lamm_uh(P2),F),G),S3) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(image_1119892238107900587at_nat(F,aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(S3),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,P2)))),image_1119892238107900587at_nat(G,aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(S3),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun359341089595923442_nat_o(aTP_Lamm_ui,P2))))) ).

% if_image_distrib
tff(fact_2793_if__image__distrib,axiom,
    ! [P2: fun_nat_o,F: fun_na367238361360896496at_nat,G: fun_na367238361360896496at_nat,S3: set_nat] : image_2499013659892586576at_nat(aa_fun2206166324314207932at_nat(aa_fun4824349113117479543at_nat(aTP_Lamm_uj(P2),F),G),S3) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(image_2499013659892586576at_nat(F,aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,P2)))),image_2499013659892586576at_nat(G,aa_set_nat_set_nat(inf_inf_set_nat(S3),aa_fun_nat_o_set_nat(collect_nat,aTP_Lamm_ub(P2))))) ).

% if_image_distrib
tff(fact_2794_if__image__distrib,axiom,
    ! [P2: fun_Pr3238374608701176162_int_o,F: fun_Pr6044903459216281905at_nat,G: fun_Pr6044903459216281905at_nat,S3: set_Pr958786334691620121nt_int] : image_6029125644611338739at_nat(aa_fun5130711716664441504at_nat(aa_fun4358810537669686606at_nat(aTP_Lamm_uk(P2),F),G),S3) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(image_6029125644611338739at_nat(F,inf_in2269163501485487111nt_int(S3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,P2)))),image_6029125644611338739at_nat(G,inf_in2269163501485487111nt_int(S3,aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aTP_Lamm_ug(P2))))) ).

% if_image_distrib
tff(fact_2795_UN__Un,axiom,
    ! [M: fun_Pr1737042432193527651at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple2748534634209756852at_nat(image_6654316230217496229at_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_6654316230217496229at_nat(M,A4))),comple2748534634209756852at_nat(image_6654316230217496229at_nat(M,B5))) ).

% UN_Un
tff(fact_2796_UN__Un,axiom,
    ! [M: fun_Pr2954218192831387491at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple4364388969068068532at_nat(image_3876558564303283877at_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple4364388969068068532at_nat(image_3876558564303283877at_nat(M,A4))),comple4364388969068068532at_nat(image_3876558564303283877at_nat(M,B5))) ).

% UN_Un
tff(fact_2797_UN__Un,axiom,
    ! [M: fun_Pr7460830217888665443at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple2748534634209756852at_nat(image_7260680879217540773at_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_7260680879217540773at_nat(M,A4))),comple2748534634209756852at_nat(image_7260680879217540773at_nat(M,B5))) ).

% UN_Un
tff(fact_2798_UN__Un,axiom,
    ! [M: fun_Pr5191738088909055843at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(M),aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(M),A4))),comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(M),B5))) ).

% UN_Un
tff(fact_2799_UN__Un,axiom,
    ! [M: fun_Pr5666111689324150784et_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_2819269535336293544et_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_2819269535336293544et_nat(M,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_2819269535336293544et_nat(M,B5))) ).

% UN_Un
tff(fact_2800_UN__Un,axiom,
    ! [M: fun_Pr6764980218223570944et_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(M,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(M,B5))) ).

% UN_Un
tff(fact_2801_bdd__below__image__inf,axiom,
    ! [F: fun_nat_rat,G: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ul(F),G)),A4))
    <=> ( aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
        & aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(G),A4)) ) ) ).

% bdd_below_image_inf
tff(fact_2802_bdd__below__image__inf,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_um(F),G)),A4))
    <=> ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
        & aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(G),A4)) ) ) ).

% bdd_below_image_inf
tff(fact_2803_bdd__below__image__inf,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_un(F),G)),A4))
    <=> ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
        & aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(G),A4)) ) ) ).

% bdd_below_image_inf
tff(fact_2804_bdd__below__image__inf,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_uo(F),G)),A4))
    <=> ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
        & aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(G),A4)) ) ) ).

% bdd_below_image_inf
tff(fact_2805_bdd__below__image__inf,axiom,
    ! [F: fun_int_int,G: fun_int_int,A4: set_int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_up(F),G)),A4))
    <=> ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
        & aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(G),A4)) ) ) ).

% bdd_below_image_inf
tff(fact_2806_bdd__above__image__sup,axiom,
    ! [F: fun_nat_rat,G: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(condit1579696412822616692ve_rat,aa_set_nat_set_rat(image_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_uq(F),G)),A4))
    <=> ( aa_set_rat_o(condit1579696412822616692ve_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
        & aa_set_rat_o(condit1579696412822616692ve_rat,aa_set_nat_set_rat(image_nat_rat(G),A4)) ) ) ).

% bdd_above_image_sup
tff(fact_2807_bdd__above__image__sup,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ur(F),G)),A4))
    <=> ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
        & aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),A4)) ) ) ).

% bdd_above_image_sup
tff(fact_2808_bdd__above__image__sup,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_us(F),G)),A4))
    <=> ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
        & aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(G),A4)) ) ) ).

% bdd_above_image_sup
tff(fact_2809_bdd__above__image__sup,axiom,
    ! [F: fun_int_nat,G: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ut(F),G)),A4))
    <=> ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
        & aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),A4)) ) ) ).

% bdd_above_image_sup
tff(fact_2810_bdd__above__image__sup,axiom,
    ! [F: fun_int_int,G: fun_int_int,A4: set_int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_uu(F),G)),A4))
    <=> ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
        & aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(G),A4)) ) ) ).

% bdd_above_image_sup
tff(fact_2811_UN__simps_I3_J,axiom,
    ! [A4: set_Product_unit,B5: fun_Pr3013102738158121955t_unit,C3: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(aa_fun8117516366632191236t_unit(aTP_Lamm_uv(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,sup_su793286257634532545t_unit(A4,comple3349607820504768346t_unit(image_7778121751606633509t_unit(B5,C3)))) ).

% UN_simps(3)
tff(fact_2812_UN__simps_I3_J,axiom,
    ! [A4: set_o,B5: fun_Pr6739565440344758894_set_o,C3: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(aa_fun5942798580971356698_set_o(aTP_Lamm_uw(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bot_set_o,sup_sup_set_o(A4,comple90263536869209701_set_o(image_6930536745178679216_set_o(B5,C3)))) ).

% UN_simps(3)
tff(fact_2813_UN__simps_I3_J,axiom,
    ! [A4: set_Product_unit,B5: fun_o_1833192751255691672t_unit,C3: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ux(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bo3957492148770167129t_unit,sup_su793286257634532545t_unit(A4,comple3349607820504768346t_unit(image_663195431845683034t_unit(B5,C3)))) ).

% UN_simps(3)
tff(fact_2814_UN__simps_I3_J,axiom,
    ! [A4: set_o,B5: fun_o_set_o,C3: set_o] :
      comple90263536869209701_set_o(image_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_uy(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bot_set_o,sup_sup_set_o(A4,comple90263536869209701_set_o(image_o_set_o(B5,C3)))) ).

% UN_simps(3)
tff(fact_2815_UN__simps_I3_J,axiom,
    ! [A4: set_Product_unit,B5: fun_na896991816813928310t_unit,C3: set_nat] :
      comple3349607820504768346t_unit(image_4824033694279209942t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_uz(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,sup_su793286257634532545t_unit(A4,comple3349607820504768346t_unit(image_4824033694279209942t_unit(B5,C3)))) ).

% UN_simps(3)
tff(fact_2816_UN__simps_I3_J,axiom,
    ! [A4: set_o,B5: fun_nat_set_o,C3: set_nat] :
      comple90263536869209701_set_o(image_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_va(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bot_set_o,sup_sup_set_o(A4,comple90263536869209701_set_o(image_nat_set_o(B5,C3)))) ).

% UN_simps(3)
tff(fact_2817_UN__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_vb(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,C3)))) ).

% UN_simps(3)
tff(fact_2818_UN__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_o_set_nat,C3: set_o] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_vc(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bot_set_nat,aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,C3)))) ).

% UN_simps(3)
tff(fact_2819_UN__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat,C3: set_nat] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_vd(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bot_set_nat,aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,C3)))) ).

% UN_simps(3)
tff(fact_2820_UN__simps_I3_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr3332065415989118679at_nat,C3: set_Product_unit] :
      comple5685304695842803022at_nat(image_3617315916088684441at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_ve(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bo2099793752762293965at_nat,aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(A4),comple5685304695842803022at_nat(image_3617315916088684441at_nat(B5,C3)))) ).

% UN_simps(3)
tff(fact_2821_UN__simps_I2_J,axiom,
    ! [A4: fun_Pr3013102738158121955t_unit,B5: set_Product_unit,C3: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(aa_set7533529402238418324t_unit(aTP_Lamm_vf(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bo3957492148770167129t_unit,sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_7778121751606633509t_unit(A4,C3)),B5)) ).

% UN_simps(2)
tff(fact_2822_UN__simps_I2_J,axiom,
    ! [A4: fun_Pr6739565440344758894_set_o,B5: set_o,C3: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(aa_set5869638957405543594_set_o(aTP_Lamm_vg(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bot_set_o,sup_sup_set_o(comple90263536869209701_set_o(image_6930536745178679216_set_o(A4,C3)),B5)) ).

% UN_simps(2)
tff(fact_2823_UN__simps_I2_J,axiom,
    ! [A4: fun_o_1833192751255691672t_unit,B5: set_Product_unit,C3: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(aa_set1706710359383640265t_unit(aTP_Lamm_vh(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bo3957492148770167129t_unit,sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_663195431845683034t_unit(A4,C3)),B5)) ).

% UN_simps(2)
tff(fact_2824_UN__simps_I2_J,axiom,
    ! [A4: fun_o_set_o,B5: set_o,C3: set_o] :
      comple90263536869209701_set_o(image_o_set_o(aa_set_o_fun_o_set_o(aTP_Lamm_vi(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bot_set_o,sup_sup_set_o(comple90263536869209701_set_o(image_o_set_o(A4,C3)),B5)) ).

% UN_simps(2)
tff(fact_2825_UN__simps_I2_J,axiom,
    ! [A4: fun_na896991816813928310t_unit,B5: set_Product_unit,C3: set_nat] :
      comple3349607820504768346t_unit(image_4824033694279209942t_unit(aa_set4829896985964406405t_unit(aTP_Lamm_vj(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bo3957492148770167129t_unit,sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_4824033694279209942t_unit(A4,C3)),B5)) ).

% UN_simps(2)
tff(fact_2826_UN__simps_I2_J,axiom,
    ! [A4: fun_nat_set_o,B5: set_o,C3: set_nat] :
      comple90263536869209701_set_o(image_nat_set_o(aa_set557987488673405253_set_o(aTP_Lamm_vk(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bot_set_o,sup_sup_set_o(comple90263536869209701_set_o(image_nat_set_o(A4,C3)),B5)) ).

% UN_simps(2)
tff(fact_2827_UN__simps_I2_J,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,B5: set_nat,C3: set_Product_unit] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_vl(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bot_set_nat,aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(A4,C3))),B5)) ).

% UN_simps(2)
tff(fact_2828_UN__simps_I2_J,axiom,
    ! [A4: fun_o_set_nat,B5: set_nat,C3: set_o] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_vm(A4),B5),C3)) = $ite(C3 = bot_bot_set_o,bot_bot_set_nat,aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,C3))),B5)) ).

% UN_simps(2)
tff(fact_2829_UN__simps_I2_J,axiom,
    ! [A4: fun_nat_set_nat,B5: set_nat,C3: set_nat] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_vn(A4),B5),C3)) = $ite(C3 = bot_bot_set_nat,bot_bot_set_nat,aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,C3))),B5)) ).

% UN_simps(2)
tff(fact_2830_UN__simps_I2_J,axiom,
    ! [A4: fun_Pr3332065415989118679at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Product_unit] :
      comple5685304695842803022at_nat(image_3617315916088684441at_nat(aa_set9196459572129487356at_nat(aTP_Lamm_vo(A4),B5),C3)) = $ite(C3 = bot_bo3957492148770167129t_unit,bot_bo2099793752762293965at_nat,aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(comple5685304695842803022at_nat(image_3617315916088684441at_nat(A4,C3))),B5)) ).

% UN_simps(2)
tff(fact_2831_UN__insert,axiom,
    ! [B5: fun_nat_set_nat,A: nat,A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4))) ).

% UN_insert
tff(fact_2832_UN__insert,axiom,
    ! [B5: fun_Pr1628945989756059814et_nat,A: product_unit,A4: set_Product_unit] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro6064029160526250199et_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,A4))) ).

% UN_insert
tff(fact_2833_UN__insert,axiom,
    ! [B5: fun_o_set_nat,A: $o,A4: set_o] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,aa_set_o_set_o(insert_o((A)),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(B5,(A))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4))) ).

% UN_insert
tff(fact_2834_UN__insert,axiom,
    ! [B5: fun_Pr9038118721296872730et_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro9123573313595863371et_nat(B5,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,A4))) ).

% UN_insert
tff(fact_2835_UN__insert,axiom,
    ! [B5: fun_na317059118361677008at_nat,A: nat,A4: set_nat] : comple2748534634209756852at_nat(image_4592875511192039216at_nat(B5,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_nat1866645186517922489at_nat(B5,A)),comple2748534634209756852at_nat(image_4592875511192039216at_nat(B5,A4))) ).

% UN_insert
tff(fact_2836_UN__insert,axiom,
    ! [B5: fun_Pr53831763054929981at_nat,A: product_unit,A4: set_Product_unit] : comple2748534634209756852at_nat(image_7288820656315755647at_nat(B5,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro2353044888477550134at_nat(B5,A)),comple2748534634209756852at_nat(image_7288820656315755647at_nat(B5,A4))) ).

% UN_insert
tff(fact_2837_UN__insert,axiom,
    ! [B5: fun_o_7500966680246433010at_nat,A: $o,A4: set_o] : comple2748534634209756852at_nat(image_8493167801588156084at_nat(B5,aa_set_o_set_o(insert_o((A)),A4))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(B5,(A))),comple2748534634209756852at_nat(image_8493167801588156084at_nat(B5,A4))) ).

% UN_insert
tff(fact_2838_UN__insert,axiom,
    ! [B5: fun_Pr6108405906935189705at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : comple2748534634209756852at_nat(image_5803077947706746251at_nat(B5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro1853492592809407170at_nat(B5,A)),comple2748534634209756852at_nat(image_5803077947706746251at_nat(B5,A4))) ).

% UN_insert
tff(fact_2839_UN__insert,axiom,
    ! [B5: fun_na5670658592214072528at_nat,A: nat,A4: set_nat] : comple4364388969068068532at_nat(image_1087905878993176368at_nat(B5,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_nat8843416860741599929at_nat(B5,A)),comple4364388969068068532at_nat(image_1087905878993176368at_nat(B5,A4))) ).

% UN_insert
tff(fact_2840_UN__insert,axiom,
    ! [B5: fun_Pr1686037257031381053at_nat,A: product_unit,A4: set_Product_unit] : comple4364388969068068532at_nat(image_5172358348707908735at_nat(B5,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_Pro2875264725485350454at_nat(B5,A)),comple4364388969068068532at_nat(image_5172358348707908735at_nat(B5,A4))) ).

% UN_insert
tff(fact_2841_inj__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger] :
      ( inj_on20428373856169009nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),top_to2363841527343591755nteger)
    <=> inj_on1158064722259180589nteger(F,top_to4645266643341252675nteger) ) ).

% inj_apsnd
tff(fact_2842_INT__insert,axiom,
    ! [B5: fun_na7044673412300895210at_nat,A: nat,A4: set_nat] : comple1930127330929219688at_nat(image_7178329752028323786at_nat(B5,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_nat7062706319057040595at_nat(B5,A)),comple1930127330929219688at_nat(image_7178329752028323786at_nat(B5,A4))) ).

% INT_insert
tff(fact_2843_INT__insert,axiom,
    ! [B5: fun_Pr1469247711798767971at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(B5),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro2745879313366468188at_nat(B5,A)),comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(B5),A4))) ).

% INT_insert
tff(fact_2844_INT__insert,axiom,
    ! [B5: fun_Pr6212429452034653949at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : comple1930127330929219688at_nat(image_7979982168039674559at_nat(B5,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro441628658106787830at_nat(B5,A)),comple1930127330929219688at_nat(image_7979982168039674559at_nat(B5,A4))) ).

% INT_insert
tff(fact_2845_INT__insert,axiom,
    ! [B5: fun_Pr3332065415989118679at_nat,A: product_unit,A4: set_Product_unit] : comple1930127330929219688at_nat(image_3617315916088684441at_nat(B5,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro5062749757247690704at_nat(B5,A)),comple1930127330929219688at_nat(image_3617315916088684441at_nat(B5,A4))) ).

% INT_insert
tff(fact_2846_INT__insert,axiom,
    ! [B5: fun_o_4630291819367765772at_nat,A: $o,A4: set_o] : comple1930127330929219688at_nat(image_7077428525161401678at_nat(B5,aa_set_o_set_o(insert_o((A)),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(B5,(A))),comple1930127330929219688at_nat(image_7077428525161401678at_nat(B5,A4))) ).

% INT_insert
tff(fact_2847_INT__insert,axiom,
    ! [B5: fun_nat_set_nat,A: nat,A4: set_nat] : comple7806235888213564991et_nat(image_nat_set_nat(B5,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_nat_set_nat(B5,A)),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))) ).

% INT_insert
tff(fact_2848_INT__insert,axiom,
    ! [B5: fun_Pr9038118721296872730et_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : comple7806235888213564991et_nat(image_15824709712370754et_nat(B5,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro9123573313595863371et_nat(B5,A)),comple7806235888213564991et_nat(image_15824709712370754et_nat(B5,A4))) ).

% INT_insert
tff(fact_2849_INT__insert,axiom,
    ! [B5: fun_Pr6764980218223570944et_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : comple7806235888213564991et_nat(image_3160451284896210088et_nat(B5,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro1692590229789857841et_nat(B5,A)),comple7806235888213564991et_nat(image_3160451284896210088et_nat(B5,A4))) ).

% INT_insert
tff(fact_2850_INT__insert,axiom,
    ! [B5: fun_Pr1628945989756059814et_nat,A: product_unit,A4: set_Product_unit] : comple7806235888213564991et_nat(image_8173849991827986510et_nat(B5,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro6064029160526250199et_nat(B5,A)),comple7806235888213564991et_nat(image_8173849991827986510et_nat(B5,A4))) ).

% INT_insert
tff(fact_2851_INT__insert,axiom,
    ! [B5: fun_o_set_nat,A: $o,A4: set_o] : comple7806235888213564991et_nat(image_o_set_nat(B5,aa_set_o_set_o(insert_o((A)),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(B5,(A))),comple7806235888213564991et_nat(image_o_set_nat(B5,A4))) ).

% INT_insert
tff(fact_2852_inj__on__apsnd,axiom,
    ! [F: fun_Co8499238673658658103nteger,A4: set_Code_integer] :
      ( inj_on20428373856169009nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,F),produc5882431272649516420nteger(top_to4645266643341252675nteger,aTP_Lamm_vp(A4)))
    <=> inj_on1158064722259180589nteger(F,A4) ) ).

% inj_on_apsnd
tff(fact_2853_inj__on__insert,axiom,
    ! [F: fun_int_nat,A: int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),aa_set_int_set_int(insert_int(A),A4))
    <=> ( aa_set_int_o(inj_on_int_nat(F),A4)
        & ~ aa_set_nat_o(member_nat(aa_int_nat(F,A)),aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int)))) ) ) ).

% inj_on_insert
tff(fact_2854_inj__on__insert,axiom,
    ! [F: fun_int_int,A: int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),aa_set_int_set_int(insert_int(A),A4))
    <=> ( aa_set_int_o(inj_on_int_int(F),A4)
        & ~ aa_set_int_o(member_int(aa_int_int(F,A)),aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(A),bot_bot_set_int)))) ) ) ).

% inj_on_insert
tff(fact_2855_inj__on__insert,axiom,
    ! [F: fun_Product_unit_o,A: product_unit,A4: set_Product_unit] :
      ( inj_on8854722734121536060unit_o(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))
    <=> ( inj_on8854722734121536060unit_o(F,A4)
        & ~ aa_set_o_o(member_o(aa_Product_unit_o(F,A)),image_Product_unit_o(F,minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_2856_inj__on__insert,axiom,
    ! [F: fun_Product_unit_nat,A: product_unit,A4: set_Product_unit] :
      ( inj_on8430439091780834860it_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))
    <=> ( inj_on8430439091780834860it_nat(F,A4)
        & ~ aa_set_nat_o(member_nat(aa_Product_unit_nat(F,A)),image_875570014554754200it_nat(F,minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_2857_inj__on__insert,axiom,
    ! [F: fun_Product_unit_int,A: product_unit,A4: set_Product_unit] :
      ( inj_on8427948621271784584it_int(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))
    <=> ( inj_on8427948621271784584it_int(F,A4)
        & ~ aa_set_int_o(member_int(aa_Product_unit_int(F,A)),image_873079544045703924it_int(F,minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(A),bot_bo3957492148770167129t_unit)))) ) ) ).

% inj_on_insert
tff(fact_2858_inj__on__insert,axiom,
    ! [F: fun_o_o,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),aa_set_o_set_o(insert_o((A)),A4))
    <=> ( aa_set_o_o(inj_on_o_o(F),A4)
        & ~ aa_set_o_o(member_o(aa_o_o(F,(A))),aa_set_o_set_o(image_o_o(F),minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_2859_inj__on__insert,axiom,
    ! [F: fun_o_nat,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_nat(F),aa_set_o_set_o(insert_o((A)),A4))
    <=> ( aa_set_o_o(inj_on_o_nat(F),A4)
        & ~ aa_set_nat_o(member_nat(aa_o_nat(F,(A))),image_o_nat(F,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_2860_inj__on__insert,axiom,
    ! [F: fun_o_int,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_int(F),aa_set_o_set_o(insert_o((A)),A4))
    <=> ( aa_set_o_o(inj_on_o_int(F),A4)
        & ~ aa_set_int_o(member_int(aa_o_int(F,(A))),image_o_int(F,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((A)),bot_bot_set_o)))) ) ) ).

% inj_on_insert
tff(fact_2861_inj__on__insert,axiom,
    ! [F: fun_nat_rat,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),aa_set_nat_set_nat(insert_nat(A),A4))
    <=> ( aa_set_nat_o(inj_on_nat_rat(F),A4)
        & ~ aa_set_rat_o(member_rat(aa_nat_rat(F,A)),aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% inj_on_insert
tff(fact_2862_inj__on__insert,axiom,
    ! [F: fun_nat_o,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(F),aa_set_nat_set_nat(insert_nat(A),A4))
    <=> ( aa_set_nat_o(inj_on_nat_o(F),A4)
        & ~ aa_set_o_o(member_o(aa_nat_o(F,A)),image_nat_o(F,aa_set_nat_set_nat(minus_minus_set_nat(A4),aa_set_nat_set_nat(insert_nat(A),bot_bot_set_nat)))) ) ) ).

% inj_on_insert
tff(fact_2863_inj__on__Un__image__eq__iff,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))
     => ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = aa_set_nat_set_rat(image_nat_rat(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_2864_inj__on__Un__image__eq__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))
     => ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = aa_set_nat_set_nat(image_nat_nat(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_2865_inj__on__Un__image__eq__iff,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))
     => ( ( aa_set_nat_set_int(image_nat_int(F),A4) = aa_set_nat_set_int(image_nat_int(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_2866_inj__on__Un__image__eq__iff,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),sup_sup_set_int(A4,B5))
     => ( ( aa_set_int_set_nat(image_int_nat(F),A4) = aa_set_int_set_nat(image_int_nat(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_2867_inj__on__Un__image__eq__iff,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),sup_sup_set_int(A4,B5))
     => ( ( aa_set_int_set_int(image_int_int(F),A4) = aa_set_int_set_int(image_int_int(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_2868_Sigma__Un__distrib1,axiom,
    ! [I3: set_nat,J3: set_nat,C3: fun_nat_set_nat] : produc457027306803732586at_nat(aa_set_nat_set_nat(sup_sup_set_nat(I3),J3),C3) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(produc457027306803732586at_nat(I3,C3)),produc457027306803732586at_nat(J3,C3)) ).

% Sigma_Un_distrib1
tff(fact_2869_Sigma__Un__distrib1,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,C3: fun_Pr1469247711798767971at_nat] : produc2761391749766926216at_nat(aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(I3),J3),C3) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(produc2761391749766926216at_nat(I3,C3)),produc2761391749766926216at_nat(J3,C3)) ).

% Sigma_Un_distrib1
tff(fact_2870_Sigma__Un__distrib1,axiom,
    ! [I3: set_se7855581050983116737at_nat,J3: set_se7855581050983116737at_nat,C3: fun_se953073389798102371at_nat] : produc7726050920695093128at_nat(sup_su3642409539654194069at_nat(I3,J3),C3) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(produc7726050920695093128at_nat(I3,C3)),produc7726050920695093128at_nat(J3,C3)) ).

% Sigma_Un_distrib1
tff(fact_2871_Sigma__Int__distrib1,axiom,
    ! [I3: set_nat,J3: set_nat,C3: fun_nat_set_nat] : produc457027306803732586at_nat(aa_set_nat_set_nat(inf_inf_set_nat(I3),J3),C3) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(produc457027306803732586at_nat(I3,C3)),produc457027306803732586at_nat(J3,C3)) ).

% Sigma_Int_distrib1
tff(fact_2872_inj__on__disjoint__Un,axiom,
    ! [F: fun_nat_rat,A4: set_nat,G: fun_nat_rat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_nat_o(inj_on_nat_rat(G),B5)
       => ( ( inf_inf_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(G),B5)) = bot_bot_set_rat )
         => aa_set_nat_o(inj_on_nat_rat(aa_fun765758790203053448at_rat(aa_set8926561665770780043at_rat(aTP_Lamm_vq(F),A4),G)),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2873_inj__on__disjoint__Un,axiom,
    ! [F: fun_nat_int,A4: set_nat,G: fun_nat_int,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_nat_o(inj_on_nat_int(G),B5)
       => ( ( inf_inf_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(G),B5)) = bot_bot_set_int )
         => aa_set_nat_o(inj_on_nat_int(aa_fun5565043122120104496at_int(aa_set9061999614494806067at_int(aTP_Lamm_vr(F),A4),G)),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2874_inj__on__disjoint__Un,axiom,
    ! [F: fun_int_int,A4: set_int,G: fun_int_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(inj_on_int_int(G),B5)
       => ( ( inf_inf_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(G),B5)) = bot_bot_set_int )
         => aa_set_int_o(inj_on_int_int(aa_fun6280345759651049192nt_int(aa_set7789521219411994439nt_int(aTP_Lamm_vs(F),A4),G)),sup_sup_set_int(A4,B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2875_inj__on__disjoint__Un,axiom,
    ! [F: fun_o_Product_unit,A4: set_o,G: fun_o_Product_unit,B5: set_o] :
      ( aa_set_o_o(inj_on4268752236855399398t_unit(F),A4)
     => ( aa_set_o_o(inj_on4268752236855399398t_unit(G),B5)
       => ( ( inf_in4660618365625256667t_unit(image_o_Product_unit(F,A4),image_o_Product_unit(G,B5)) = bot_bo3957492148770167129t_unit )
         => aa_set_o_o(inj_on4268752236855399398t_unit(aa_fun4820891174006887598t_unit(aa_set5856121796315333033t_unit(aTP_Lamm_vt(F),A4),G)),sup_sup_set_o(A4,B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2876_inj__on__disjoint__Un,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat,G: fun_nat_Product_unit,B5: set_nat] :
      ( aa_set_nat_o(inj_on7061601236592826506t_unit(F),A4)
     => ( aa_set_nat_o(inj_on7061601236592826506t_unit(G),B5)
       => ( ( inf_in4660618365625256667t_unit(image_8730104196221521654t_unit(F,A4),image_8730104196221521654t_unit(G,B5)) = bot_bo3957492148770167129t_unit )
         => aa_set_nat_o(inj_on7061601236592826506t_unit(aa_fun4858661924651670332t_unit(aa_set7546214604051456903t_unit(aTP_Lamm_vu(F),A4),G)),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2877_inj__on__disjoint__Un,axiom,
    ! [F: fun_int_Product_unit,A4: set_int,G: fun_int_Product_unit,B5: set_int] :
      ( aa_set_int_o(inj_on4178023183795006766t_unit(F),A4)
     => ( aa_set_int_o(inj_on4178023183795006766t_unit(G),B5)
       => ( ( inf_in4660618365625256667t_unit(image_5846526143423701914t_unit(F,A4),image_5846526143423701914t_unit(G,B5)) = bot_bo3957492148770167129t_unit )
         => aa_set_int_o(inj_on4178023183795006766t_unit(aa_fun3184267698828548156t_unit(aa_set2257548544956115171t_unit(aTP_Lamm_vv(F),A4),G)),sup_sup_set_int(A4,B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2878_inj__on__disjoint__Un,axiom,
    ! [F: fun_o_o,A4: set_o,G: fun_o_o,B5: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),A4)
     => ( aa_set_o_o(inj_on_o_o(G),B5)
       => ( ( inf_inf_set_o(aa_set_o_set_o(image_o_o(F),A4),aa_set_o_set_o(image_o_o(G),B5)) = bot_bot_set_o )
         => aa_set_o_o(inj_on_o_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_vw(F),A4),G)),sup_sup_set_o(A4,B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2879_inj__on__disjoint__Un,axiom,
    ! [F: fun_nat_o,A4: set_nat,G: fun_nat_o,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(F),A4)
     => ( aa_set_nat_o(inj_on_nat_o(G),B5)
       => ( ( inf_inf_set_o(image_nat_o(F,A4),image_nat_o(G,B5)) = bot_bot_set_o )
         => aa_set_nat_o(inj_on_nat_o(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_vx(F),A4),G)),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2880_inj__on__disjoint__Un,axiom,
    ! [F: fun_int_o,A4: set_int,G: fun_int_o,B5: set_int] :
      ( aa_set_int_o(inj_on_int_o(F),A4)
     => ( aa_set_int_o(inj_on_int_o(G),B5)
       => ( ( inf_inf_set_o(image_int_o(F,A4),image_int_o(G,B5)) = bot_bot_set_o )
         => aa_set_int_o(inj_on_int_o(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_vy(F),A4),G)),sup_sup_set_int(A4,B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2881_inj__on__disjoint__Un,axiom,
    ! [F: fun_o_nat,A4: set_o,G: fun_o_nat,B5: set_o] :
      ( aa_set_o_o(inj_on_o_nat(F),A4)
     => ( aa_set_o_o(inj_on_o_nat(G),B5)
       => ( ( aa_set_nat_set_nat(inf_inf_set_nat(image_o_nat(F,A4)),image_o_nat(G,B5)) = bot_bot_set_nat )
         => aa_set_o_o(inj_on_o_nat(aa_fun6176012476734143100_o_nat(aa_set4039056531748102335_o_nat(aTP_Lamm_vz(F),A4),G)),sup_sup_set_o(A4,B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_2882_Sigma__Un__distrib2,axiom,
    ! [I3: set_nat,A4: fun_nat_set_nat,B5: fun_nat_set_nat] : produc457027306803732586at_nat(I3,aa_fun850356524293783268et_nat(aTP_Lamm_wa(A4),B5)) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(produc457027306803732586at_nat(I3,A4)),produc457027306803732586at_nat(I3,B5)) ).

% Sigma_Un_distrib2
tff(fact_2883_Sigma__Un__distrib2,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,A4: fun_Pr1469247711798767971at_nat,B5: fun_Pr1469247711798767971at_nat] : produc2761391749766926216at_nat(I3,aa_fun5344049784256257540at_nat(aTP_Lamm_wb(A4),B5)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(produc2761391749766926216at_nat(I3,A4)),produc2761391749766926216at_nat(I3,B5)) ).

% Sigma_Un_distrib2
tff(fact_2884_Sigma__Un__distrib2,axiom,
    ! [I3: set_se7855581050983116737at_nat,A4: fun_se953073389798102371at_nat,B5: fun_se953073389798102371at_nat] : produc7726050920695093128at_nat(I3,aa_fun3786497492886271492at_nat(aTP_Lamm_wc(A4),B5)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(produc7726050920695093128at_nat(I3,A4)),produc7726050920695093128at_nat(I3,B5)) ).

% Sigma_Un_distrib2
tff(fact_2885_Times__Un__distrib1,axiom,
    ! [A4: set_nat,B5: set_nat,C3: set_nat] : produc457027306803732586at_nat(aa_set_nat_set_nat(sup_sup_set_nat(A4),B5),aTP_Lamm_gm(C3)) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(produc457027306803732586at_nat(A4,aTP_Lamm_gm(C3))),produc457027306803732586at_nat(B5,aTP_Lamm_gm(C3))) ).

% Times_Un_distrib1
tff(fact_2886_Times__Un__distrib1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] : produc2761391749766926216at_nat(aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(A4),B5),aTP_Lamm_lc(C3)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(produc2761391749766926216at_nat(A4,aTP_Lamm_lc(C3))),produc2761391749766926216at_nat(B5,aTP_Lamm_lc(C3))) ).

% Times_Un_distrib1
tff(fact_2887_Times__Un__distrib1,axiom,
    ! [A4: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat,C3: set_se7855581050983116737at_nat] : produc7726050920695093128at_nat(sup_su3642409539654194069at_nat(A4,B5),aTP_Lamm_jj(C3)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(produc7726050920695093128at_nat(A4,aTP_Lamm_jj(C3))),produc7726050920695093128at_nat(B5,aTP_Lamm_jj(C3))) ).

% Times_Un_distrib1
tff(fact_2888_Times__Int__Times,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] : inf_in4302113700860409141at_nat(produc2761391749766926216at_nat(A4,aTP_Lamm_lc(B5)),produc2761391749766926216at_nat(C3,aTP_Lamm_lc(D))) = produc2761391749766926216at_nat(aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),C3),aa_set6004322782020247304at_nat(aTP_Lamm_wd(B5),D)) ).

% Times_Int_Times
tff(fact_2889_Times__Int__Times,axiom,
    ! [A4: set_nat,B5: set_nat,C3: set_nat,D: set_nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))),produc457027306803732586at_nat(C3,aTP_Lamm_gm(D))) = produc457027306803732586at_nat(aa_set_nat_set_nat(inf_inf_set_nat(A4),C3),aa_set562305657478724375et_nat(aTP_Lamm_we(B5),D)) ).

% Times_Int_Times
tff(fact_2890_Sigma__Int__distrib2,axiom,
    ! [I3: set_nat,A4: fun_nat_set_nat,B5: fun_nat_set_nat] : produc457027306803732586at_nat(I3,aa_fun850356524293783268et_nat(aTP_Lamm_wf(A4),B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(produc457027306803732586at_nat(I3,A4)),produc457027306803732586at_nat(I3,B5)) ).

% Sigma_Int_distrib2
tff(fact_2891_Times__Int__distrib1,axiom,
    ! [A4: set_nat,B5: set_nat,C3: set_nat] : produc457027306803732586at_nat(aa_set_nat_set_nat(inf_inf_set_nat(A4),B5),aTP_Lamm_gm(C3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(produc457027306803732586at_nat(A4,aTP_Lamm_gm(C3))),produc457027306803732586at_nat(B5,aTP_Lamm_gm(C3))) ).

% Times_Int_distrib1
tff(fact_2892_inj__on__Un,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))
    <=> ( aa_set_nat_o(inj_on_nat_rat(F),A4)
        & aa_set_nat_o(inj_on_nat_rat(F),B5)
        & ( inf_inf_set_rat(aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)),aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(minus_minus_set_nat(B5),A4))) = bot_bot_set_rat ) ) ) ).

% inj_on_Un
tff(fact_2893_inj__on__Un,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))
    <=> ( aa_set_nat_o(inj_on_nat_int(F),A4)
        & aa_set_nat_o(inj_on_nat_int(F),B5)
        & ( inf_inf_set_int(aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)),aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(minus_minus_set_nat(B5),A4))) = bot_bot_set_int ) ) ) ).

% inj_on_Un
tff(fact_2894_inj__on__Un,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),sup_sup_set_int(A4,B5))
    <=> ( aa_set_int_o(inj_on_int_int(F),A4)
        & aa_set_int_o(inj_on_int_int(F),B5)
        & ( inf_inf_set_int(aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,B5)),aa_set_int_set_int(image_int_int(F),minus_minus_set_int(B5,A4))) = bot_bot_set_int ) ) ) ).

% inj_on_Un
tff(fact_2895_inj__on__Un,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))
    <=> ( aa_set_nat_o(inj_on_nat_nat(F),A4)
        & aa_set_nat_o(inj_on_nat_nat(F),B5)
        & ( aa_set_nat_set_nat(inf_inf_set_nat(aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5))),aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(B5),A4))) = bot_bot_set_nat ) ) ) ).

% inj_on_Un
tff(fact_2896_inj__on__Un,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),sup_sup_set_int(A4,B5))
    <=> ( aa_set_int_o(inj_on_int_nat(F),A4)
        & aa_set_int_o(inj_on_int_nat(F),B5)
        & ( aa_set_nat_set_nat(inf_inf_set_nat(aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,B5))),aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(B5,A4))) = bot_bot_set_nat ) ) ) ).

% inj_on_Un
tff(fact_2897_inj__on__Un,axiom,
    ! [F: fun_Pr5236084471349978153t_unit,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( inj_on1041401102088758999t_unit(F,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))
    <=> ( inj_on1041401102088758999t_unit(F,A4)
        & inj_on1041401102088758999t_unit(F,B5)
        & ( inf_in4660618365625256667t_unit(image_8487516615895632747t_unit(F,minus_8321449233255521966at_nat(A4,B5)),image_8487516615895632747t_unit(F,minus_8321449233255521966at_nat(B5,A4))) = bot_bo3957492148770167129t_unit ) ) ) ).

% inj_on_Un
tff(fact_2898_inj__on__Un,axiom,
    ! [F: fun_Pr363642936170290612_nat_o,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( inj_on991952103320416098_nat_o(F,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))
    <=> ( inj_on991952103320416098_nat_o(F,A4)
        & inj_on991952103320416098_nat_o(F,B5)
        & ( inf_inf_set_o(image_3275415580317762038_nat_o(F,minus_8321449233255521966at_nat(A4,B5)),image_3275415580317762038_nat_o(F,minus_8321449233255521966at_nat(B5,A4))) = bot_bot_set_o ) ) ) ).

% inj_on_Un
tff(fact_2899_inj__on__Un,axiom,
    ! [F: fun_Pr2871540613828922186at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( inj_on490725545064285830at_nat(F,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))
    <=> ( inj_on490725545064285830at_nat(F,A4)
        & inj_on490725545064285830at_nat(F,B5)
        & ( aa_set_nat_set_nat(inf_inf_set_nat(image_6160980552072806642at_nat(F,minus_8321449233255521966at_nat(A4,B5))),image_6160980552072806642at_nat(F,minus_8321449233255521966at_nat(B5,A4))) = bot_bot_set_nat ) ) ) ).

% inj_on_Un
tff(fact_2900_inj__on__Un,axiom,
    ! [F: fun_Pr8319229360018371625t_unit,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(inj_on3108384905993948887t_unit(F),aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))
    <=> ( aa_set5844364881195198285_nat_o(inj_on3108384905993948887t_unit(F),A4)
        & aa_set5844364881195198285_nat_o(inj_on3108384905993948887t_unit(F),B5)
        & ( inf_in4660618365625256667t_unit(image_7082456443403096939t_unit(F,aa_set4883462369968598652at_nat(minus_3314409938677909166at_nat(A4),B5)),image_7082456443403096939t_unit(F,aa_set4883462369968598652at_nat(minus_3314409938677909166at_nat(B5),A4))) = bot_bo3957492148770167129t_unit ) ) ) ).

% inj_on_Un
tff(fact_2901_inj__on__Un,axiom,
    ! [F: fun_Pr3189796393087291805at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( inj_on229527429751349963at_nat(F,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))
    <=> ( inj_on229527429751349963at_nat(F,A4)
        & inj_on229527429751349963at_nat(F,B5)
        & ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(image_6706873702294942047at_nat(F,minus_8321449233255521966at_nat(A4,B5))),image_6706873702294942047at_nat(F,minus_8321449233255521966at_nat(B5,A4))) = bot_bo2099793752762293965at_nat ) ) ) ).

% inj_on_Un
tff(fact_2902_inj__on__image,axiom,
    ! [F: fun_int_nat,A4: set_set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,A4))
     => inj_on1389844818000382683et_nat(image_int_nat(F),A4) ) ).

% inj_on_image
tff(fact_2903_inj__on__image,axiom,
    ! [F: fun_int_int,A4: set_set_int] :
      ( aa_set_int_o(inj_on_int_int(F),aa_set4034802472124477112et_int(comple3221217463730067765et_int,A4))
     => inj_on6435365835345961783et_int(image_int_int(F),A4) ) ).

% inj_on_image
tff(fact_2904_inj__on__image,axiom,
    ! [F: fun_nat_rat,A4: set_set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))
     => inj_on1096178645466186887et_rat(image_nat_rat(F),A4) ) ).

% inj_on_image
tff(fact_2905_inj__on__image,axiom,
    ! [F: fun_nat_nat,A4: set_set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))
     => inj_on4604407203859583615et_nat(image_nat_nat(F),A4) ) ).

% inj_on_image
tff(fact_2906_inj__on__image,axiom,
    ! [F: fun_nat_int,A4: set_set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))
     => inj_on426556184350386907et_int(image_nat_int(F),A4) ) ).

% inj_on_image
tff(fact_2907_cSUP__union,axiom,
    ! [A4: set_int,F: fun_int_nat,B5: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( ( B5 != bot_bot_set_int )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),B5))
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),sup_sup_set_int(A4,B5))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2908_cSUP__union,axiom,
    ! [A4: set_nat,F: fun_nat_nat,B5: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( ( B5 != bot_bot_set_nat )
         => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),B5))
           => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2909_cSUP__union,axiom,
    ! [A4: set_int,F: fun_int_int,B5: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( ( B5 != bot_bot_set_int )
         => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),B5))
           => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),sup_sup_set_int(A4,B5))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2910_cSUP__union,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,B5: set_Product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(F,A4))
       => ( ( B5 != bot_bo3957492148770167129t_unit )
         => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(F,B5))
           => ( aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,sup_su793286257634532545t_unit(A4,B5))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))),aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2911_cSUP__union,axiom,
    ! [A4: set_o,F: fun_o_int,B5: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
       => ( ( B5 != bot_bot_set_o )
         => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,B5))
           => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,sup_sup_set_o(A4,B5))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))),aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2912_cSUP__union,axiom,
    ! [A4: set_nat,F: fun_nat_int,B5: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( ( B5 != bot_bot_set_nat )
         => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),B5))
           => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2913_cSUP__union,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,B5: set_Product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(F,A4))
       => ( ( B5 != bot_bo3957492148770167129t_unit )
         => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(F,B5))
           => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,sup_su793286257634532545t_unit(A4,B5))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2914_cSUP__union,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,B5: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
       => ( ( B5 != bot_bot_set_o )
         => ( condit2080475194861589617t_unit(image_o_Product_unit(F,B5))
           => ( comple4687483117567863418t_unit(image_o_Product_unit(F,sup_sup_set_o(A4,B5))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_o_Product_unit(F,A4))),comple4687483117567863418t_unit(image_o_Product_unit(F,B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2915_cSUP__union,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,B5: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,A4))
       => ( ( B5 != bot_bot_set_nat )
         => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,B5))
           => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2916_cSUP__union,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat,B5: set_Product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,A4))
       => ( ( B5 != bot_bo3957492148770167129t_unit )
         => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,B5))
           => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,sup_su793286257634532545t_unit(A4,B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,B5))) ) ) ) ) ) ).

% cSUP_union
tff(fact_2917_cINF__union,axiom,
    ! [A4: set_int,F: fun_int_nat,B5: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( ( B5 != bot_bot_set_int )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),B5))
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),sup_sup_set_int(A4,B5))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2918_cINF__union,axiom,
    ! [A4: set_nat,F: fun_nat_nat,B5: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( ( B5 != bot_bot_set_nat )
         => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),B5))
           => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2919_cINF__union,axiom,
    ! [A4: set_int,F: fun_int_int,B5: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( ( B5 != bot_bot_set_int )
         => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),B5))
           => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),sup_sup_set_int(A4,B5))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2920_cINF__union,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,B5: set_Product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(F,A4))
       => ( ( B5 != bot_bo3957492148770167129t_unit )
         => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(F,B5))
           => ( aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,sup_su793286257634532545t_unit(A4,B5))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,A4))),aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2921_cINF__union,axiom,
    ! [A4: set_o,F: fun_o_int,B5: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
       => ( ( B5 != bot_bot_set_o )
         => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,B5))
           => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,sup_sup_set_o(A4,B5))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))),aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2922_cINF__union,axiom,
    ! [A4: set_nat,F: fun_nat_int,B5: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( ( B5 != bot_bot_set_nat )
         => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),B5))
           => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2923_cINF__union,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,B5: set_Product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(F,A4))
       => ( ( B5 != bot_bo3957492148770167129t_unit )
         => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(F,B5))
           => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,sup_su793286257634532545t_unit(A4,B5))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2924_cINF__union,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,B5: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
       => ( ( B5 != bot_bot_set_o )
         => ( condit9161856309968323677t_unit(image_o_Product_unit(F,B5))
           => ( comple2584293577114468500t_unit(image_o_Product_unit(F,sup_sup_set_o(A4,B5))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),comple2584293577114468500t_unit(image_o_Product_unit(F,B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2925_cINF__union,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,B5: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,A4))
       => ( ( B5 != bot_bot_set_nat )
         => ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,B5))
           => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(sup_sup_set_nat(A4),B5))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2926_cINF__union,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat,B5: set_Product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,A4))
       => ( ( B5 != bot_bo3957492148770167129t_unit )
         => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,B5))
           => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,sup_su793286257634532545t_unit(A4,B5))) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A4))),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,B5))) ) ) ) ) ) ).

% cINF_union
tff(fact_2927_inj__on__image__Int,axiom,
    ! [F: fun_nat_rat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = inf_inf_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_2928_inj__on__image__Int,axiom,
    ! [F: fun_nat_nat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_2929_inj__on__image__Int,axiom,
    ! [F: fun_nat_int,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = inf_inf_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_2930_inj__on__image__Int,axiom,
    ! [F: fun_int_nat,C3: set_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C3)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C3)
         => ( aa_set_int_set_nat(image_int_nat(F),inf_inf_set_int(A4,B5)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_2931_inj__on__image__Int,axiom,
    ! [F: fun_int_int,C3: set_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C3)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C3)
         => ( aa_set_int_set_int(image_int_int(F),inf_inf_set_int(A4,B5)) = inf_inf_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_2932_inj__on__image__Int,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,C3: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),C3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),C3)
       => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),C3)
         => ( aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4)),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_2933_image__Int,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = inf_inf_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5)) ) ) ).

% image_Int
tff(fact_2934_image__Int,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ) ) ).

% image_Int
tff(fact_2935_image__Int,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = inf_inf_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5)) ) ) ).

% image_Int
tff(fact_2936_image__Int,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_int_set_nat(image_int_nat(F),inf_inf_set_int(A4,B5)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5)) ) ) ).

% image_Int
tff(fact_2937_image__Int,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( aa_set_int_set_int(image_int_int(F),inf_inf_set_int(A4,B5)) = inf_inf_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5)) ) ) ).

% image_Int
tff(fact_2938_image__Int,axiom,
    ! [F: fun_ch7922468903353698889at_nat,A4: set_char,B5: set_char] :
      ( inj_on1021528964648911549at_nat(F,top_top_set_char)
     => ( image_8382823969816519465at_nat(F,inf_inf_set_char(A4,B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(image_8382823969816519465at_nat(F,A4)),image_8382823969816519465at_nat(F,B5)) ) ) ).

% image_Int
tff(fact_2939_image__Int,axiom,
    ! [F: fun_ra4806459883200776834at_nat,A4: set_rat,B5: set_rat] :
      ( aa_set_rat_o(inj_on8867882244352206326at_nat(F),top_top_set_rat)
     => ( image_9175953278516507234at_nat(F,inf_inf_set_rat(A4,B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(image_9175953278516507234at_nat(F,A4)),image_9175953278516507234at_nat(F,B5)) ) ) ).

% image_Int
tff(fact_2940_image__Int,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on5538052773655684606at_nat(F),top_top_set_nat)
     => ( image_5846123807819985514at_nat(F,aa_set_nat_set_nat(inf_inf_set_nat(A4),B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(image_5846123807819985514at_nat(F,A4)),image_5846123807819985514at_nat(F,B5)) ) ) ).

% image_Int
tff(fact_2941_image__Int,axiom,
    ! [F: fun_in4514472496968388910at_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on8575894858119818402at_nat(F),top_top_set_int)
     => ( image_8883965892284119310at_nat(F,inf_inf_set_int(A4,B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(image_8883965892284119310at_nat(F,A4)),image_8883965892284119310at_nat(F,B5)) ) ) ).

% image_Int
tff(fact_2942_image__Int,axiom,
    ! [F: fun_li4460313466401262074at_nat,A4: set_list_nat,B5: set_list_nat] :
      ( inj_on7574789179980466158at_nat(F,top_top_set_list_nat)
     => ( image_2795870212791037530at_nat(F,inf_inf_set_list_nat(A4,B5)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(image_2795870212791037530at_nat(F,A4)),image_2795870212791037530at_nat(F,B5)) ) ) ).

% image_Int
tff(fact_2943_inj__on__image__Pow,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => inj_on1096178645466186887et_rat(image_nat_rat(F),pow_nat(A4)) ) ).

% inj_on_image_Pow
tff(fact_2944_inj__on__image__Pow,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => inj_on4604407203859583615et_nat(image_nat_nat(F),pow_nat(A4)) ) ).

% inj_on_image_Pow
tff(fact_2945_inj__on__image__Pow,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => inj_on426556184350386907et_int(image_nat_int(F),pow_nat(A4)) ) ).

% inj_on_image_Pow
tff(fact_2946_inj__on__image__Pow,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => inj_on1389844818000382683et_nat(image_int_nat(F),pow_int(A4)) ) ).

% inj_on_image_Pow
tff(fact_2947_inj__on__image__Pow,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => inj_on6435365835345961783et_int(image_int_int(F),pow_int(A4)) ) ).

% inj_on_image_Pow
tff(fact_2948_inj__on__image__Fpow,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => inj_on1096178645466186887et_rat(image_nat_rat(F),finite_Fpow_nat(A4)) ) ).

% inj_on_image_Fpow
tff(fact_2949_inj__on__image__Fpow,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => inj_on4604407203859583615et_nat(image_nat_nat(F),finite_Fpow_nat(A4)) ) ).

% inj_on_image_Fpow
tff(fact_2950_inj__on__image__Fpow,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => inj_on426556184350386907et_int(image_nat_int(F),finite_Fpow_nat(A4)) ) ).

% inj_on_image_Fpow
tff(fact_2951_inj__on__image__Fpow,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => inj_on1389844818000382683et_nat(image_int_nat(F),finite_Fpow_int(A4)) ) ).

% inj_on_image_Fpow
tff(fact_2952_inj__on__image__Fpow,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => inj_on6435365835345961783et_int(image_int_int(F),finite_Fpow_int(A4)) ) ).

% inj_on_image_Fpow
tff(fact_2953_inj__on__imageI2,axiom,
    ! [F2: fun_o_o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F2),F)),A4)
     => aa_set_nat_o(inj_on_nat_o(F),A4) ) ).

% inj_on_imageI2
tff(fact_2954_inj__on__imageI2,axiom,
    ! [F2: fun_nat_num,F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F2),F)),A4)
     => aa_set_int_o(inj_on_int_nat(F),A4) ) ).

% inj_on_imageI2
tff(fact_2955_inj__on__imageI2,axiom,
    ! [F2: fun_nat_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F2),F)),A4)
     => aa_set_nat_o(inj_on_nat_nat(F),A4) ) ).

% inj_on_imageI2
tff(fact_2956_inj__on__imageI2,axiom,
    ! [F2: fun_int_nat,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F2),F)),A4)
     => aa_set_int_o(inj_on_int_int(F),A4) ) ).

% inj_on_imageI2
tff(fact_2957_inj__on__imageI2,axiom,
    ! [F2: fun_fu4800041728187187127nteger,F: fun_Co1981900195452921617nteger,A4: set_Code_integer] :
      ( inj_on736898771723690579nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F2),F),A4)
     => inj_on428699138970331399nteger(F,A4) ) ).

% inj_on_imageI2
tff(fact_2958_SUP__union,axiom,
    ! [M: fun_Pr1737042432193527651at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple2748534634209756852at_nat(image_6654316230217496229at_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_6654316230217496229at_nat(M,A4))),comple2748534634209756852at_nat(image_6654316230217496229at_nat(M,B5))) ).

% SUP_union
tff(fact_2959_SUP__union,axiom,
    ! [M: fun_Pr2954218192831387491at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple4364388969068068532at_nat(image_3876558564303283877at_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple4364388969068068532at_nat(image_3876558564303283877at_nat(M,A4))),comple4364388969068068532at_nat(image_3876558564303283877at_nat(M,B5))) ).

% SUP_union
tff(fact_2960_SUP__union,axiom,
    ! [M: fun_Pr7460830217888665443at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple2748534634209756852at_nat(image_7260680879217540773at_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_7260680879217540773at_nat(M,A4))),comple2748534634209756852at_nat(image_7260680879217540773at_nat(M,B5))) ).

% SUP_union
tff(fact_2961_SUP__union,axiom,
    ! [M: fun_Pr5191738088909055843at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(M),aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(M),A4))),comple4364388969068068532at_nat(aa_set7518756442812098908at_nat(image_7364710425340685989at_nat(M),B5))) ).

% SUP_union
tff(fact_2962_SUP__union,axiom,
    ! [M: fun_Pr5666111689324150784et_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_2819269535336293544et_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_2819269535336293544et_nat(M,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_2819269535336293544et_nat(M,B5))) ).

% SUP_union
tff(fact_2963_SUP__union,axiom,
    ! [M: fun_Pr6764980218223570944et_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(M,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(M,B5))) ).

% SUP_union
tff(fact_2964_SUP__union,axiom,
    ! [M: fun_Pr5236084471349978153t_unit,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple4687483117567863418t_unit(image_8487516615895632747t_unit(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_8487516615895632747t_unit(M,A4))),comple4687483117567863418t_unit(image_8487516615895632747t_unit(M,B5))) ).

% SUP_union
tff(fact_2965_SUP__union,axiom,
    ! [M: fun_Pr8319229360018371625t_unit,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple4687483117567863418t_unit(image_7082456443403096939t_unit(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_7082456443403096939t_unit(M,A4))),comple4687483117567863418t_unit(image_7082456443403096939t_unit(M,B5))) ).

% SUP_union
tff(fact_2966_INF__union,axiom,
    ! [M: fun_Pr8880351011014127357at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple1930127330929219688at_nat(image_1384942819424619199at_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_1384942819424619199at_nat(M,A4))),comple1930127330929219688at_nat(image_1384942819424619199at_nat(M,B5))) ).

% INF_union
tff(fact_2967_INF__union,axiom,
    ! [M: fun_Pr6212429452034653949at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple1930127330929219688at_nat(image_7979982168039674559at_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7979982168039674559at_nat(M,A4))),comple1930127330929219688at_nat(image_7979982168039674559at_nat(M,B5))) ).

% INF_union
tff(fact_2968_INF__union,axiom,
    ! [M: fun_Pr5666111689324150784et_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple7806235888213564991et_nat(image_2819269535336293544et_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_2819269535336293544et_nat(M,A4))),comple7806235888213564991et_nat(image_2819269535336293544et_nat(M,B5))) ).

% INF_union
tff(fact_2969_INF__union,axiom,
    ! [M: fun_Pr6764980218223570944et_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple7806235888213564991et_nat(image_3160451284896210088et_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(M,A4))),comple7806235888213564991et_nat(image_3160451284896210088et_nat(M,B5))) ).

% INF_union
tff(fact_2970_INF__union,axiom,
    ! [M: fun_Pr5236084471349978153t_unit,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple2584293577114468500t_unit(image_8487516615895632747t_unit(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_8487516615895632747t_unit(M,A4))),comple2584293577114468500t_unit(image_8487516615895632747t_unit(M,B5))) ).

% INF_union
tff(fact_2971_INF__union,axiom,
    ! [M: fun_Pr8319229360018371625t_unit,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple2584293577114468500t_unit(image_7082456443403096939t_unit(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_7082456443403096939t_unit(M,A4))),comple2584293577114468500t_unit(image_7082456443403096939t_unit(M,B5))) ).

% INF_union
tff(fact_2972_image__Un,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] : aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) = sup_sup_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5)) ).

% image_Un
tff(fact_2973_image__Un,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ).

% image_Un
tff(fact_2974_image__Un,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] : aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(sup_sup_set_nat(A4),B5)) = sup_sup_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5)) ).

% image_Un
tff(fact_2975_image__Un,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] : aa_set_int_set_nat(image_int_nat(F),sup_sup_set_int(A4,B5)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5)) ).

% image_Un
tff(fact_2976_image__Un,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] : aa_set_int_set_int(image_int_int(F),sup_sup_set_int(A4,B5)) = sup_sup_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5)) ).

% image_Un
tff(fact_2977_image__Un,axiom,
    ! [F: fun_Pr6278524904407988355at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : image_6321802852832981957at_nat(F,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(image_6321802852832981957at_nat(F,A4)),image_6321802852832981957at_nat(F,B5)) ).

% image_Un
tff(fact_2978_image__Un,axiom,
    ! [F: fun_Pr678129486068332675at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : image_7637461350386011077at_nat(F,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(image_7637461350386011077at_nat(F,A4)),image_7637461350386011077at_nat(F,B5)) ).

% image_Un
tff(fact_2979_image__Un,axiom,
    ! [F: fun_Pr2096352117387331715at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : image_5653576389946358725at_nat(F,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(image_5653576389946358725at_nat(F,A4)),image_5653576389946358725at_nat(F,B5)) ).

% image_Un
tff(fact_2980_image__Un,axiom,
    ! [F: fun_Pr9034069120582568067at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(image_960689587819713477at_nat(F),aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_set4883462369968598652at_nat(image_960689587819713477at_nat(F),A4)),aa_set4883462369968598652at_nat(image_960689587819713477at_nat(F),B5)) ).

% image_Un
tff(fact_2981_INT__Un,axiom,
    ! [M: fun_Pr8880351011014127357at_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple1930127330929219688at_nat(image_1384942819424619199at_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_1384942819424619199at_nat(M,A4))),comple1930127330929219688at_nat(image_1384942819424619199at_nat(M,B5))) ).

% INT_Un
tff(fact_2982_INT__Un,axiom,
    ! [M: fun_Pr6212429452034653949at_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple1930127330929219688at_nat(image_7979982168039674559at_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7979982168039674559at_nat(M,A4))),comple1930127330929219688at_nat(image_7979982168039674559at_nat(M,B5))) ).

% INT_Un
tff(fact_2983_INT__Un,axiom,
    ! [M: fun_Pr5666111689324150784et_nat,A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : comple7806235888213564991et_nat(image_2819269535336293544et_nat(M,aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5))) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_2819269535336293544et_nat(M,A4))),comple7806235888213564991et_nat(image_2819269535336293544et_nat(M,B5))) ).

% INT_Un
tff(fact_2984_INT__Un,axiom,
    ! [M: fun_Pr6764980218223570944et_nat,A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : comple7806235888213564991et_nat(image_3160451284896210088et_nat(M,aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5))) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(M,A4))),comple7806235888213564991et_nat(image_3160451284896210088et_nat(M,B5))) ).

% INT_Un
tff(fact_2985_inj__on__convol__ident,axiom,
    ! [F: fun_se6846035212209387651at_nat,X3: set_se5479201105039950375at_nat] : inj_on1468925692805905105at_nat(aTP_Lamm_wg(F),X3) ).

% inj_on_convol_ident
tff(fact_2986_inj__on__convol__ident,axiom,
    ! [F: fun_se320340636488258563at_nat,X3: set_se7855581050983116737at_nat] : inj_on699426629415580343at_nat(aTP_Lamm_wh(F),X3) ).

% inj_on_convol_ident
tff(fact_2987_inj__on__convol__ident,axiom,
    ! [F: fun_nat_num,X3: set_nat] : aa_set_nat_o(inj_on6246296710604522312at_num(aTP_Lamm_wi(F)),X3) ).

% inj_on_convol_ident
tff(fact_2988_inj__on__convol__ident,axiom,
    ! [F: fun_nat_nat,X3: set_nat] : aa_set_nat_o(inj_on5538052773655684606at_nat(aTP_Lamm_wj(F)),X3) ).

% inj_on_convol_ident
tff(fact_2989_inj__on__convol__ident,axiom,
    ! [F: fun_int_int,X3: set_int] : aa_set_int_o(inj_on5397397550511676250nt_int(aTP_Lamm_wk(F)),X3) ).

% inj_on_convol_ident
tff(fact_2990_total__onI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Ra: set_Pr5564308138774400199at_nat] :
      ( ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
          ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),A4)
         => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Y),A4)
           => ( ( X != Y )
             => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(X,Y),Ra)
                | member6009668390874684176at_nat(produc2335810615602371415at_nat(Y,X),Ra) ) ) ) )
     => total_1513685709984341995at_nat(A4,Ra) ) ).

% total_onI
tff(fact_2991_total__onI,axiom,
    ! [A4: set_o,Ra: set_Product_prod_o_o] :
      ( ! [X: $o,Y: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => ( aa_set_o_o(member_o((Y)),A4)
           => ( ~ ( (X)
                <=> (Y) )
             => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X)),(Y)),Ra)
                | member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Y)),(X)),Ra) ) ) ) )
     => total_on_o(A4,Ra) ) ).

% total_onI
tff(fact_2992_total__onI,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat] :
      ( ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
          ( member6273874620674182256at_nat(X,A4)
         => ( member6273874620674182256at_nat(Y,A4)
           => ( ( X != Y )
             => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),Y),Ra)
                | member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Y),X),Ra) ) ) ) )
     => total_5117575981256876747at_nat(A4,Ra) ) ).

% total_onI
tff(fact_2993_total__onI,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat] :
      ( ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X,A4)
         => ( member2643936169264416010at_nat(Y,A4)
           => ( ( X != Y )
             => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),Y)),Ra)
                | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Y),X)),Ra) ) ) ) )
     => aa_set5844364881195198285_nat_o(total_7212417256332265445at_nat(A4),Ra) ) ).

% total_onI
tff(fact_2994_total__onI,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => ( aa_set_nat_o(member_nat(Y),A4)
           => ( ( X != Y )
             => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y)),Ra)
                | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y),X)),Ra) ) ) ) )
     => aa_set3702741264012733747_nat_o(total_on_nat(A4),Ra) ) ).

% total_onI
tff(fact_2995_total__onI,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Y: int] :
          ( aa_set_int_o(member_int(X),A4)
         => ( aa_set_int_o(member_int(Y),A4)
           => ( ( X != Y )
             => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y)),Ra)
                | aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y),X)),Ra) ) ) ) )
     => total_on_int(A4,Ra) ) ).

% total_onI
tff(fact_2996_total__onI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat] :
      ( ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
         => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),A4)
           => ( ( X != Y )
             => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X),Y),Ra)
                | member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Y),X),Ra) ) ) ) )
     => total_3592101749530773125at_nat(A4,Ra) ) ).

% total_onI
tff(fact_2997_total__on__def,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat] :
      ( total_5117575981256876747at_nat(A4,Ra)
    <=> ! [X4: set_Pr4329608150637261639at_nat] :
          ( member6273874620674182256at_nat(X4,A4)
         => ! [Xa3: set_Pr4329608150637261639at_nat] :
              ( member6273874620674182256at_nat(Xa3,A4)
             => ( ( X4 != Xa3 )
               => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X4),Xa3),Ra)
                  | member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa3),X4),Ra) ) ) ) ) ) ).

% total_on_def
tff(fact_2998_total__on__def,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(total_7212417256332265445at_nat(A4),Ra)
    <=> ! [X4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X4,A4)
         => ! [Xa3: set_Pr1261947904930325089at_nat] :
              ( member2643936169264416010at_nat(Xa3,A4)
             => ( ( X4 != Xa3 )
               => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X4),Xa3)),Ra)
                  | aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa3),X4)),Ra) ) ) ) ) ) ).

% total_on_def
tff(fact_2999_total__on__def,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(total_on_nat(A4),Ra)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A4)
         => ! [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),A4)
             => ( ( X4 != Xa3 )
               => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa3)),Ra)
                  | aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa3),X4)),Ra) ) ) ) ) ) ).

% total_on_def
tff(fact_3000_total__on__def,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int] :
      ( total_on_int(A4,Ra)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A4)
         => ! [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),A4)
             => ( ( X4 != Xa3 )
               => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),Ra)
                  | aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa3),X4)),Ra) ) ) ) ) ) ).

% total_on_def
tff(fact_3001_total__on__def,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat] :
      ( total_3592101749530773125at_nat(A4,Ra)
    <=> ! [X4: product_prod_nat_nat] :
          ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
         => ! [Xa3: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),A4)
             => ( ( X4 != Xa3 )
               => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X4),Xa3),Ra)
                  | member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa3),X4),Ra) ) ) ) ) ) ).

% total_on_def
tff(fact_3002_SUP__UNIV__bool__expand,axiom,
    ! [A4: fun_o_7500966680246433010at_nat] : comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,top_top_set_o)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(A4,$true)),aa_o_s7415647395592896235at_nat(A4,$false)) ).

% SUP_UNIV_bool_expand
tff(fact_3003_SUP__UNIV__bool__expand,axiom,
    ! [A4: fun_o_7855627885593154802at_nat] : comple4364388969068068532at_nat(image_2496312346330513076at_nat(A4,top_top_set_o)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_o_s4772616771316612843at_nat(A4,$true)),aa_o_s4772616771316612843at_nat(A4,$false)) ).

% SUP_UNIV_bool_expand
tff(fact_3004_SUP__UNIV__bool__expand,axiom,
    ! [A4: fun_o_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,top_top_set_o)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(A4,$true)),aa_o_set_nat(A4,$false)) ).

% SUP_UNIV_bool_expand
tff(fact_3005_SUP__UNIV__bool__expand,axiom,
    ! [A4: fun_o_Product_unit] : comple4687483117567863418t_unit(image_o_Product_unit(A4,top_top_set_o)) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_o_Product_unit(A4,$true)),aa_o_Product_unit(A4,$false)) ).

% SUP_UNIV_bool_expand
tff(fact_3006_INF__UNIV__bool__expand,axiom,
    ! [A4: fun_o_4630291819367765772at_nat] : comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,top_top_set_o)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(A4,$true)),aa_o_s3322794332960822277at_nat(A4,$false)) ).

% INF_UNIV_bool_expand
tff(fact_3007_INF__UNIV__bool__expand,axiom,
    ! [A4: fun_o_set_nat] : comple7806235888213564991et_nat(image_o_set_nat(A4,top_top_set_o)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(A4,$true)),aa_o_set_nat(A4,$false)) ).

% INF_UNIV_bool_expand
tff(fact_3008_INF__UNIV__bool__expand,axiom,
    ! [A4: fun_o_Product_unit] : comple2584293577114468500t_unit(image_o_Product_unit(A4,top_top_set_o)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_o_Product_unit(A4,$true)),aa_o_Product_unit(A4,$false)) ).

% INF_UNIV_bool_expand
tff(fact_3009_fun_Oinj__map,axiom,
    ! [F: fun_o_o] :
      ( aa_set_o_o(inj_on_o_o(F),top_top_set_o)
     => inj_on3273395367213011633_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),top_to8708385046839970129_nat_o) ) ).

% fun.inj_map
tff(fact_3010_fun_Oinj__map,axiom,
    ! [F: fun_fu4800041728187187127nteger] :
      ( inj_on5093811489028909101nteger(F,top_to805621317456759325nteger)
     => inj_on4668255421698215009nteger(comp_f6739083541385432444nteger(F),top_to4960787393209669879nteger) ) ).

% fun.inj_map
tff(fact_3011_fun_Oinj__map,axiom,
    ! [F: fun_nat_num] :
      ( aa_set_nat_o(inj_on_nat_num(F),top_top_set_nat)
     => inj_on2745149795294922415nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),top_to9051269609472735455nt_nat) ) ).

% fun.inj_map
tff(fact_3012_fun_Oinj__map,axiom,
    ! [F: fun_nat_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => inj_on1321603220815140013at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),top_to6864348034779713667at_nat) ) ).

% fun.inj_map
tff(fact_3013_fun_Oinj__map,axiom,
    ! [F: fun_int_nat] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => inj_on5988350486761793985nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),top_to6561186464541008699nt_int) ) ).

% fun.inj_map
tff(fact_3014_Un__eq__UN,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),B5) = comple2748534634209756852at_nat(image_8493167801588156084at_nat(aa_set4424263627741053053at_nat(aTP_Lamm_wl(A4),B5),top_top_set_o)) ).

% Un_eq_UN
tff(fact_3015_Un__eq__UN,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),B5) = comple4364388969068068532at_nat(image_2496312346330513076at_nat(aa_set7164036131237892221at_nat(aTP_Lamm_wm(A4),B5),top_top_set_o)) ).

% Un_eq_UN
tff(fact_3016_Un__eq__UN,axiom,
    ! [A4: set_nat,B5: set_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A4),B5) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_wn(A4),B5),top_top_set_o)) ).

% Un_eq_UN
tff(fact_3017_UN__bool__eq,axiom,
    ! [A4: fun_o_7500966680246433010at_nat] : comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,top_top_set_o)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(A4,$true)),aa_o_s7415647395592896235at_nat(A4,$false)) ).

% UN_bool_eq
tff(fact_3018_UN__bool__eq,axiom,
    ! [A4: fun_o_7855627885593154802at_nat] : comple4364388969068068532at_nat(image_2496312346330513076at_nat(A4,top_top_set_o)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_o_s4772616771316612843at_nat(A4,$true)),aa_o_s4772616771316612843at_nat(A4,$false)) ).

% UN_bool_eq
tff(fact_3019_UN__bool__eq,axiom,
    ! [A4: fun_o_set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,top_top_set_o)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(A4,$true)),aa_o_set_nat(A4,$false)) ).

% UN_bool_eq
tff(fact_3020_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),A4))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ut(F),G)),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3021_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),A4))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),A4))) = aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ur(F),G)),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3022_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(G),A4))
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(G),A4))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_uu(F),G)),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3023_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,G: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(F,A4))
       => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(G,A4))
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))),aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(G,A4))) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(aa_fun3641207808737740476it_int(aTP_Lamm_wo(F),G),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3024_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_o,F: fun_o_int,G: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
       => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(G,A4))
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))),aa_set_int_int(complete_Sup_Sup_int,image_o_int(G,A4))) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_wp(F),G),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3025_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(G),A4))
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(G),A4))) = aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_us(F),G)),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3026_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,G: fun_Pr2191293291710260995t_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(F,A4))
       => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(G,A4))
         => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))),comple4687483117567863418t_unit(image_405062704495631173t_unit(G,A4))) = comple4687483117567863418t_unit(image_405062704495631173t_unit(aa_fun6153996563283759940t_unit(aTP_Lamm_wq(F),G),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3027_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,G: fun_o_Product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
       => ( condit2080475194861589617t_unit(image_o_Product_unit(G,A4))
         => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_o_Product_unit(F,A4))),comple4687483117567863418t_unit(image_o_Product_unit(G,A4))) = comple4687483117567863418t_unit(image_o_Product_unit(aa_fun4820891174006887598t_unit(aTP_Lamm_wr(F),G),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3028_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,G: fun_nat_Product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,A4))
       => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(G,A4))
         => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))),comple4687483117567863418t_unit(image_8730104196221521654t_unit(G,A4))) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(aa_fun4858661924651670332t_unit(aTP_Lamm_ws(F),G),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3029_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat,G: fun_Pr1628945989756059814et_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,A4))
       => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(G,A4))
         => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,A4))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(G,A4))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_wt(F),G),A4)) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_3030_cINF__inf__distrib,axiom,
    ! [A4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(G),A4))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(G),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_uo(F),G)),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3031_cINF__inf__distrib,axiom,
    ! [A4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(G),A4))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(G),A4))) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_um(F),G)),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3032_cINF__inf__distrib,axiom,
    ! [A4: set_int,F: fun_int_int,G: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(G),A4))
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(G),A4))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_up(F),G)),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3033_cINF__inf__distrib,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,G: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(F,A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(G,A4))
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,A4))),aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(G,A4))) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(aa_fun3641207808737740476it_int(aTP_Lamm_wu(F),G),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3034_cINF__inf__distrib,axiom,
    ! [A4: set_o,F: fun_o_int,G: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(G,A4))
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))),aa_set_int_int(complete_Inf_Inf_int,image_o_int(G,A4))) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_wv(F),G),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3035_cINF__inf__distrib,axiom,
    ! [A4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(G),A4))
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(G),A4))) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_un(F),G)),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3036_cINF__inf__distrib,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,G: fun_Pr2191293291710260995t_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(F,A4))
       => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(G,A4))
         => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))),comple2584293577114468500t_unit(image_405062704495631173t_unit(G,A4))) = comple2584293577114468500t_unit(image_405062704495631173t_unit(aa_fun6153996563283759940t_unit(aTP_Lamm_ww(F),G),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3037_cINF__inf__distrib,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,G: fun_o_Product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
       => ( condit9161856309968323677t_unit(image_o_Product_unit(G,A4))
         => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),comple2584293577114468500t_unit(image_o_Product_unit(G,A4))) = comple2584293577114468500t_unit(image_o_Product_unit(aa_fun4820891174006887598t_unit(aTP_Lamm_wx(F),G),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3038_cINF__inf__distrib,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,G: fun_nat_Product_unit] :
      ( ( A4 != bot_bot_set_nat )
     => ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,A4))
       => ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(G,A4))
         => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),comple2584293577114468500t_unit(image_8730104196221521654t_unit(G,A4))) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_fun4858661924651670332t_unit(aTP_Lamm_wy(F),G),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3039_cINF__inf__distrib,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat,G: fun_Pr1628945989756059814et_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,A4))
       => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(G,A4))
         => ( aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A4))),comple7806235888213564991et_nat(image_8173849991827986510et_nat(G,A4))) = comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_wz(F),G),A4)) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_3040_fsts_Ointros,axiom,
    ! [P: produc7812382424243267045t_unit] : member_ref_a(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P),basic_685971451221398055t_unit(P)) ).

% fsts.intros
tff(fact_3041_fsts_Osimps,axiom,
    ! [A: ref_a,P: produc7812382424243267045t_unit] :
      ( member_ref_a(A,basic_685971451221398055t_unit(P))
    <=> ( A = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P) ) ) ).

% fsts.simps
tff(fact_3042_fsts_Ocases,axiom,
    ! [A: ref_a,P: produc7812382424243267045t_unit] :
      ( member_ref_a(A,basic_685971451221398055t_unit(P))
     => ( A = aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,P) ) ) ).

% fsts.cases
tff(fact_3043_INT__bool__eq,axiom,
    ! [A4: fun_o_4630291819367765772at_nat] : comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,top_top_set_o)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(A4,$true)),aa_o_s3322794332960822277at_nat(A4,$false)) ).

% INT_bool_eq
tff(fact_3044_INT__bool__eq,axiom,
    ! [A4: fun_o_set_nat] : comple7806235888213564991et_nat(image_o_set_nat(A4,top_top_set_o)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(A4,$true)),aa_o_set_nat(A4,$false)) ).

% INT_bool_eq
tff(fact_3045_cSUP__eq__cINF__D,axiom,
    ! [F: fun_nat_nat,A4: set_nat,A: nat] :
      ( ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
         => ( aa_set_nat_o(member_nat(A),A4)
           => ( aa_nat_nat(F,A) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3046_cSUP__eq__cINF__D,axiom,
    ! [F: fun_int_nat,A4: set_int,A: int] :
      ( ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4)) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4)) )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
         => ( aa_set_int_o(member_int(A),A4)
           => ( aa_int_nat(F,A) = aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3047_cSUP__eq__cINF__D,axiom,
    ! [F: fun_o_int,A4: set_o,A: $o] :
      ( ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4)) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4)) )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
         => ( aa_set_o_o(member_o((A)),A4)
           => ( aa_o_int(F,(A)) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3048_cSUP__eq__cINF__D,axiom,
    ! [F: fun_nat_int,A4: set_nat,A: nat] :
      ( ( aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4)) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4)) )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
         => ( aa_set_nat_o(member_nat(A),A4)
           => ( aa_nat_int(F,A) = aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3049_cSUP__eq__cINF__D,axiom,
    ! [F: fun_int_int,A4: set_int,A: int] :
      ( ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4)) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4)) )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
         => ( aa_set_int_o(member_int(A),A4)
           => ( aa_int_int(F,A) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3050_cSUP__eq__cINF__D,axiom,
    ! [F: fun_o_Product_unit,A4: set_o,A: $o] :
      ( ( comple4687483117567863418t_unit(image_o_Product_unit(F,A4)) = comple2584293577114468500t_unit(image_o_Product_unit(F,A4)) )
     => ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
       => ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
         => ( aa_set_o_o(member_o((A)),A4)
           => ( aa_o_Product_unit(F,(A)) = comple2584293577114468500t_unit(image_o_Product_unit(F,A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3051_cSUP__eq__cINF__D,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat,A: nat] :
      ( ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4)) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4)) )
     => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,A4))
       => ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,A4))
         => ( aa_set_nat_o(member_nat(A),A4)
           => ( aa_nat_Product_unit(F,A) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3052_cSUP__eq__cINF__D,axiom,
    ! [F: fun_int_Product_unit,A4: set_int,A: int] :
      ( ( comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4)) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4)) )
     => ( condit2080475194861589617t_unit(image_5846526143423701914t_unit(F,A4))
       => ( condit9161856309968323677t_unit(image_5846526143423701914t_unit(F,A4))
         => ( aa_set_int_o(member_int(A),A4)
           => ( aa_int_Product_unit(F,A) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3053_cSUP__eq__cINF__D,axiom,
    ! [F: fun_o_set_nat,A4: set_o,A: $o] :
      ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4)) = comple7806235888213564991et_nat(image_o_set_nat(F,A4)) )
     => ( condit5477540289124974626et_nat(image_o_set_nat(F,A4))
       => ( condit68592940725977398et_nat(image_o_set_nat(F,A4))
         => ( aa_set_o_o(member_o((A)),A4)
           => ( aa_o_set_nat(F,(A)) = comple7806235888213564991et_nat(image_o_set_nat(F,A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3054_cSUP__eq__cINF__D,axiom,
    ! [F: fun_nat_set_nat,A4: set_nat,A: nat] :
      ( ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4)) = comple7806235888213564991et_nat(image_nat_set_nat(F,A4)) )
     => ( condit5477540289124974626et_nat(image_nat_set_nat(F,A4))
       => ( condit68592940725977398et_nat(image_nat_set_nat(F,A4))
         => ( aa_set_nat_o(member_nat(A),A4)
           => ( aa_nat_set_nat(F,A) = comple7806235888213564991et_nat(image_nat_set_nat(F,A4)) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_3055_cSUP__insert,axiom,
    ! [A4: set_int,F: fun_int_nat,A: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),aa_set_int_set_int(insert_int(A),A4))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_int_nat(F,A)),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ) ).

% cSUP_insert
tff(fact_3056_cSUP__insert,axiom,
    ! [A4: set_nat,F: fun_nat_nat,A: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(insert_nat(A),A4))) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(F,A)),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ) ).

% cSUP_insert
tff(fact_3057_cSUP__insert,axiom,
    ! [A4: set_int,F: fun_int_int,A: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(insert_int(A),A4))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(F,A)),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ) ).

% cSUP_insert
tff(fact_3058_cSUP__insert,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,A: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(F,A4))
       => ( aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_Product_unit_int(F,A)),aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))) ) ) ) ).

% cSUP_insert
tff(fact_3059_cSUP__insert,axiom,
    ! [A4: set_o,F: fun_o_int,A: $o] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
       => ( aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_o_int(F,(A))),aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))) ) ) ) ).

% cSUP_insert
tff(fact_3060_cSUP__insert,axiom,
    ! [A4: set_nat,F: fun_nat_int,A: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(insert_nat(A),A4))) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_nat_int(F,A)),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ) ).

% cSUP_insert
tff(fact_3061_cSUP__insert,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,A: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(F,A4))
       => ( comple4687483117567863418t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro8294289767544744956t_unit(F,A)),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))) ) ) ) ).

% cSUP_insert
tff(fact_3062_cSUP__insert,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,A: $o] :
      ( ( A4 != bot_bot_set_o )
     => ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
       => ( comple4687483117567863418t_unit(image_o_Product_unit(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_o_Product_unit(F,(A))),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ) ) ) ).

% cSUP_insert
tff(fact_3063_cSUP__insert,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,A: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,A4))
       => ( comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_nat_Product_unit(F,A)),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ) ) ) ).

% cSUP_insert
tff(fact_3064_cSUP__insert,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat,A: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit5477540289124974626et_nat(image_8173849991827986510et_nat(F,A4))
       => ( aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro6064029160526250199et_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,A4))) ) ) ) ).

% cSUP_insert
tff(fact_3065_cINF__insert,axiom,
    ! [A4: set_int,F: fun_int_nat,A: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),aa_set_int_set_int(insert_int(A),A4))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_int_nat(F,A)),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ) ).

% cINF_insert
tff(fact_3066_cINF__insert,axiom,
    ! [A4: set_nat,F: fun_nat_nat,A: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(insert_nat(A),A4))) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(F,A)),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ) ).

% cINF_insert
tff(fact_3067_cINF__insert,axiom,
    ! [A4: set_int,F: fun_int_int,A: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(insert_int(A),A4))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(F,A)),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ) ).

% cINF_insert
tff(fact_3068_cINF__insert,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,A: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(F,A4))
       => ( aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_Product_unit_int(F,A)),aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,A4))) ) ) ) ).

% cINF_insert
tff(fact_3069_cINF__insert,axiom,
    ! [A4: set_o,F: fun_o_int,A: $o] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
       => ( aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_o_int(F,(A))),aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))) ) ) ) ).

% cINF_insert
tff(fact_3070_cINF__insert,axiom,
    ! [A4: set_nat,F: fun_nat_int,A: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(insert_nat(A),A4))) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_nat_int(F,A)),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ) ).

% cINF_insert
tff(fact_3071_cINF__insert,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,A: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(F,A4))
       => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro8294289767544744956t_unit(F,A)),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))) ) ) ) ).

% cINF_insert
tff(fact_3072_cINF__insert,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,A: $o] :
      ( ( A4 != bot_bot_set_o )
     => ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
       => ( comple2584293577114468500t_unit(image_o_Product_unit(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_o_Product_unit(F,(A))),comple2584293577114468500t_unit(image_o_Product_unit(F,A4))) ) ) ) ).

% cINF_insert
tff(fact_3073_cINF__insert,axiom,
    ! [A4: set_nat,F: fun_nat_Product_unit,A: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,A4))
       => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_nat_Product_unit(F,A)),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))) ) ) ) ).

% cINF_insert
tff(fact_3074_cINF__insert,axiom,
    ! [A4: set_Product_unit,F: fun_Pr1628945989756059814et_nat,A: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit68592940725977398et_nat(image_8173849991827986510et_nat(F,A4))
       => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro6064029160526250199et_nat(F,A)),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A4))) ) ) ) ).

% cINF_insert
tff(fact_3075_inj__on__image__mem__iff,axiom,
    ! [F: fun_o_o,B5: set_o,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),B5)
     => ( aa_set_o_o(member_o((A)),B5)
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( aa_set_o_o(member_o(aa_o_o(F,(A))),aa_set_o_set_o(image_o_o(F),A4))
          <=> aa_set_o_o(member_o((A)),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3076_inj__on__image__mem__iff,axiom,
    ! [F: fun_o_nat,B5: set_o,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_nat(F),B5)
     => ( aa_set_o_o(member_o((A)),B5)
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( aa_set_nat_o(member_nat(aa_o_nat(F,(A))),image_o_nat(F,A4))
          <=> aa_set_o_o(member_o((A)),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3077_inj__on__image__mem__iff,axiom,
    ! [F: fun_o_int,B5: set_o,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_int(F),B5)
     => ( aa_set_o_o(member_o((A)),B5)
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( aa_set_int_o(member_int(aa_o_int(F,(A))),image_o_int(F,A4))
          <=> aa_set_o_o(member_o((A)),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3078_inj__on__image__mem__iff,axiom,
    ! [F: fun_nat_rat,B5: set_nat,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),B5)
     => ( aa_set_nat_o(member_nat(A),B5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( aa_set_rat_o(member_rat(aa_nat_rat(F,A)),aa_set_nat_set_rat(image_nat_rat(F),A4))
          <=> aa_set_nat_o(member_nat(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3079_inj__on__image__mem__iff,axiom,
    ! [F: fun_nat_o,B5: set_nat,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(F),B5)
     => ( aa_set_nat_o(member_nat(A),B5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( aa_set_o_o(member_o(aa_nat_o(F,A)),image_nat_o(F,A4))
          <=> aa_set_nat_o(member_nat(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3080_inj__on__image__mem__iff,axiom,
    ! [F: fun_nat_nat,B5: set_nat,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),B5)
     => ( aa_set_nat_o(member_nat(A),B5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( aa_set_nat_o(member_nat(aa_nat_nat(F,A)),aa_set_nat_set_nat(image_nat_nat(F),A4))
          <=> aa_set_nat_o(member_nat(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3081_inj__on__image__mem__iff,axiom,
    ! [F: fun_nat_int,B5: set_nat,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),B5)
     => ( aa_set_nat_o(member_nat(A),B5)
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( aa_set_int_o(member_int(aa_nat_int(F,A)),aa_set_nat_set_int(image_nat_int(F),A4))
          <=> aa_set_nat_o(member_nat(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3082_inj__on__image__mem__iff,axiom,
    ! [F: fun_int_o,B5: set_int,A: int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_o(F),B5)
     => ( aa_set_int_o(member_int(A),B5)
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( aa_set_o_o(member_o(aa_int_o(F,A)),image_int_o(F,A4))
          <=> aa_set_int_o(member_int(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3083_inj__on__image__mem__iff,axiom,
    ! [F: fun_int_nat,B5: set_int,A: int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),B5)
     => ( aa_set_int_o(member_int(A),B5)
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( aa_set_nat_o(member_nat(aa_int_nat(F,A)),aa_set_int_set_nat(image_int_nat(F),A4))
          <=> aa_set_int_o(member_int(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3084_inj__on__image__mem__iff,axiom,
    ! [F: fun_int_int,B5: set_int,A: int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),B5)
     => ( aa_set_int_o(member_int(A),B5)
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( aa_set_int_o(member_int(aa_int_int(F,A)),aa_set_int_set_int(image_int_int(F),A4))
          <=> aa_set_int_o(member_int(A),A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_3085_inj__on__image__eq__iff,axiom,
    ! [F: fun_nat_rat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = aa_set_nat_set_rat(image_nat_rat(F),B5) )
          <=> ( A4 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_3086_inj__on__image__eq__iff,axiom,
    ! [F: fun_nat_nat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = aa_set_nat_set_nat(image_nat_nat(F),B5) )
          <=> ( A4 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_3087_inj__on__image__eq__iff,axiom,
    ! [F: fun_nat_int,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( ( aa_set_nat_set_int(image_nat_int(F),A4) = aa_set_nat_set_int(image_nat_int(F),B5) )
          <=> ( A4 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_3088_inj__on__image__eq__iff,axiom,
    ! [F: fun_int_nat,C3: set_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C3)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C3)
         => ( ( aa_set_int_set_nat(image_int_nat(F),A4) = aa_set_int_set_nat(image_int_nat(F),B5) )
          <=> ( A4 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_3089_inj__on__image__eq__iff,axiom,
    ! [F: fun_int_int,C3: set_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),C3)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C3)
         => ( ( aa_set_int_set_int(image_int_int(F),A4) = aa_set_int_set_int(image_int_int(F),B5) )
          <=> ( A4 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_3090_subset__image__inj,axiom,
    ! [S3: set_rat,F: fun_nat_rat,T2: set_nat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(S3),aa_set_nat_set_rat(image_nat_rat(F),T2))
    <=> ? [U2: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(U2),T2)
          & aa_set_nat_o(inj_on_nat_rat(F),U2)
          & ( S3 = aa_set_nat_set_rat(image_nat_rat(F),U2) ) ) ) ).

% subset_image_inj
tff(fact_3091_subset__image__inj,axiom,
    ! [S3: set_nat,F: fun_nat_nat,T2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S3),aa_set_nat_set_nat(image_nat_nat(F),T2))
    <=> ? [U2: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(U2),T2)
          & aa_set_nat_o(inj_on_nat_nat(F),U2)
          & ( S3 = aa_set_nat_set_nat(image_nat_nat(F),U2) ) ) ) ).

% subset_image_inj
tff(fact_3092_subset__image__inj,axiom,
    ! [S3: set_int,F: fun_nat_int,T2: set_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),aa_set_nat_set_int(image_nat_int(F),T2))
    <=> ? [U2: set_nat] :
          ( aa_set_nat_o(ord_less_eq_set_nat(U2),T2)
          & aa_set_nat_o(inj_on_nat_int(F),U2)
          & ( S3 = aa_set_nat_set_int(image_nat_int(F),U2) ) ) ) ).

% subset_image_inj
tff(fact_3093_subset__image__inj,axiom,
    ! [S3: set_nat,F: fun_int_nat,T2: set_int] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S3),aa_set_int_set_nat(image_int_nat(F),T2))
    <=> ? [U2: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(U2),T2)
          & aa_set_int_o(inj_on_int_nat(F),U2)
          & ( S3 = aa_set_int_set_nat(image_int_nat(F),U2) ) ) ) ).

% subset_image_inj
tff(fact_3094_subset__image__inj,axiom,
    ! [S3: set_int,F: fun_int_int,T2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S3),aa_set_int_set_int(image_int_int(F),T2))
    <=> ? [U2: set_int] :
          ( aa_set_int_o(ord_less_eq_set_int(U2),T2)
          & aa_set_int_o(inj_on_int_int(F),U2)
          & ( S3 = aa_set_int_set_int(image_int_int(F),U2) ) ) ) ).

% subset_image_inj
tff(fact_3095_inj__img__insertE,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A4: set_Product_unit,Xa: product_unit,B5: set_Product_unit] :
      ( inj_on8151373323710067377t_unit(F,A4)
     => ( ~ member_Product_unit(Xa,B5)
       => ( ( aa_set8324764978855220220t_unit(insert_Product_unit(Xa),B5) = image_405062704495631173t_unit(F,A4) )
         => ~ ! [X6: product_unit,A7: set_Product_unit] :
                ( ~ member_Product_unit(X6,A7)
               => ( ( A4 = aa_set8324764978855220220t_unit(insert_Product_unit(X6),A7) )
                 => ( ( Xa = aa_Pro8294289767544744956t_unit(F,X6) )
                   => ( B5 != image_405062704495631173t_unit(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3096_inj__img__insertE,axiom,
    ! [F: fun_o_Product_unit,A4: set_o,Xa: product_unit,B5: set_Product_unit] :
      ( aa_set_o_o(inj_on4268752236855399398t_unit(F),A4)
     => ( ~ member_Product_unit(Xa,B5)
       => ( ( aa_set8324764978855220220t_unit(insert_Product_unit(Xa),B5) = image_o_Product_unit(F,A4) )
         => ~ ! [X6: $o,A7: set_o] :
                ( ~ aa_set_o_o(member_o((X6)),A7)
               => ( ( A4 = aa_set_o_set_o(insert_o((X6)),A7) )
                 => ( ( Xa = aa_o_Product_unit(F,(X6)) )
                   => ( B5 != image_o_Product_unit(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3097_inj__img__insertE,axiom,
    ! [F: fun_nat_rat,A4: set_nat,Xa: rat,B5: set_rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( ~ aa_set_rat_o(member_rat(Xa),B5)
       => ( ( insert_rat(Xa,B5) = aa_set_nat_set_rat(image_nat_rat(F),A4) )
         => ~ ! [X6: nat,A7: set_nat] :
                ( ~ aa_set_nat_o(member_nat(X6),A7)
               => ( ( A4 = aa_set_nat_set_nat(insert_nat(X6),A7) )
                 => ( ( Xa = aa_nat_rat(F,X6) )
                   => ( B5 != aa_set_nat_set_rat(image_nat_rat(F),A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3098_inj__img__insertE,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat,Xa: product_unit,B5: set_Product_unit] :
      ( aa_set_nat_o(inj_on7061601236592826506t_unit(F),A4)
     => ( ~ member_Product_unit(Xa,B5)
       => ( ( aa_set8324764978855220220t_unit(insert_Product_unit(Xa),B5) = image_8730104196221521654t_unit(F,A4) )
         => ~ ! [X6: nat,A7: set_nat] :
                ( ~ aa_set_nat_o(member_nat(X6),A7)
               => ( ( A4 = aa_set_nat_set_nat(insert_nat(X6),A7) )
                 => ( ( Xa = aa_nat_Product_unit(F,X6) )
                   => ( B5 != image_8730104196221521654t_unit(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3099_inj__img__insertE,axiom,
    ! [F: fun_int_Product_unit,A4: set_int,Xa: product_unit,B5: set_Product_unit] :
      ( aa_set_int_o(inj_on4178023183795006766t_unit(F),A4)
     => ( ~ member_Product_unit(Xa,B5)
       => ( ( aa_set8324764978855220220t_unit(insert_Product_unit(Xa),B5) = image_5846526143423701914t_unit(F,A4) )
         => ~ ! [X6: int,A7: set_int] :
                ( ~ aa_set_int_o(member_int(X6),A7)
               => ( ( A4 = aa_set_int_set_int(insert_int(X6),A7) )
                 => ( ( Xa = aa_int_Product_unit(F,X6) )
                   => ( B5 != image_5846526143423701914t_unit(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3100_inj__img__insertE,axiom,
    ! [F: fun_Product_unit_o,A4: set_Product_unit,Xa: $o,B5: set_o] :
      ( inj_on8854722734121536060unit_o(F,A4)
     => ( ~ aa_set_o_o(member_o((Xa)),B5)
       => ( ( aa_set_o_set_o(insert_o((Xa)),B5) = image_Product_unit_o(F,A4) )
         => ~ ! [X6: product_unit,A7: set_Product_unit] :
                ( ~ member_Product_unit(X6,A7)
               => ( ( A4 = aa_set8324764978855220220t_unit(insert_Product_unit(X6),A7) )
                 => ( ( (Xa)
                    <=> aa_Product_unit_o(F,X6) )
                   => ( B5 != image_Product_unit_o(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3101_inj__img__insertE,axiom,
    ! [F: fun_o_o,A4: set_o,Xa: $o,B5: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),A4)
     => ( ~ aa_set_o_o(member_o((Xa)),B5)
       => ( ( aa_set_o_set_o(insert_o((Xa)),B5) = aa_set_o_set_o(image_o_o(F),A4) )
         => ~ ! [X6: $o,A7: set_o] :
                ( ~ aa_set_o_o(member_o((X6)),A7)
               => ( ( A4 = aa_set_o_set_o(insert_o((X6)),A7) )
                 => ( ( (Xa)
                    <=> aa_o_o(F,(X6)) )
                   => ( B5 != aa_set_o_set_o(image_o_o(F),A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3102_inj__img__insertE,axiom,
    ! [F: fun_nat_o,A4: set_nat,Xa: $o,B5: set_o] :
      ( aa_set_nat_o(inj_on_nat_o(F),A4)
     => ( ~ aa_set_o_o(member_o((Xa)),B5)
       => ( ( aa_set_o_set_o(insert_o((Xa)),B5) = image_nat_o(F,A4) )
         => ~ ! [X6: nat,A7: set_nat] :
                ( ~ aa_set_nat_o(member_nat(X6),A7)
               => ( ( A4 = aa_set_nat_set_nat(insert_nat(X6),A7) )
                 => ( ( (Xa)
                    <=> aa_nat_o(F,X6) )
                   => ( B5 != image_nat_o(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3103_inj__img__insertE,axiom,
    ! [F: fun_int_o,A4: set_int,Xa: $o,B5: set_o] :
      ( aa_set_int_o(inj_on_int_o(F),A4)
     => ( ~ aa_set_o_o(member_o((Xa)),B5)
       => ( ( aa_set_o_set_o(insert_o((Xa)),B5) = image_int_o(F,A4) )
         => ~ ! [X6: int,A7: set_int] :
                ( ~ aa_set_int_o(member_int(X6),A7)
               => ( ( A4 = aa_set_int_set_int(insert_int(X6),A7) )
                 => ( ( (Xa)
                    <=> aa_int_o(F,X6) )
                   => ( B5 != image_int_o(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3104_inj__img__insertE,axiom,
    ! [F: fun_Product_unit_nat,A4: set_Product_unit,Xa: nat,B5: set_nat] :
      ( inj_on8430439091780834860it_nat(F,A4)
     => ( ~ aa_set_nat_o(member_nat(Xa),B5)
       => ( ( aa_set_nat_set_nat(insert_nat(Xa),B5) = image_875570014554754200it_nat(F,A4) )
         => ~ ! [X6: product_unit,A7: set_Product_unit] :
                ( ~ member_Product_unit(X6,A7)
               => ( ( A4 = aa_set8324764978855220220t_unit(insert_Product_unit(X6),A7) )
                 => ( ( Xa = aa_Product_unit_nat(F,X6) )
                   => ( B5 != image_875570014554754200it_nat(F,A7) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_3105_inj__image__mem__iff,axiom,
    ! [F: fun_o_o,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),top_top_set_o)
     => ( aa_set_o_o(member_o(aa_o_o(F,(A))),aa_set_o_set_o(image_o_o(F),A4))
      <=> aa_set_o_o(member_o((A)),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3106_inj__image__mem__iff,axiom,
    ! [F: fun_o_nat,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_nat(F),top_top_set_o)
     => ( aa_set_nat_o(member_nat(aa_o_nat(F,(A))),image_o_nat(F,A4))
      <=> aa_set_o_o(member_o((A)),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3107_inj__image__mem__iff,axiom,
    ! [F: fun_o_int,A: $o,A4: set_o] :
      ( aa_set_o_o(inj_on_o_int(F),top_top_set_o)
     => ( aa_set_int_o(member_int(aa_o_int(F,(A))),image_o_int(F,A4))
      <=> aa_set_o_o(member_o((A)),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3108_inj__image__mem__iff,axiom,
    ! [F: fun_char_o,A: char,A4: set_char] :
      ( inj_on_char_o(F,top_top_set_char)
     => ( aa_set_o_o(member_o(aa_char_o(F,A)),image_char_o(F,A4))
      <=> member_char(A,A4) ) ) ).

% inj_image_mem_iff
tff(fact_3109_inj__image__mem__iff,axiom,
    ! [F: fun_char_nat,A: char,A4: set_char] :
      ( inj_on_char_nat(F,top_top_set_char)
     => ( aa_set_nat_o(member_nat(aa_char_nat(F,A)),image_char_nat(F,A4))
      <=> member_char(A,A4) ) ) ).

% inj_image_mem_iff
tff(fact_3110_inj__image__mem__iff,axiom,
    ! [F: fun_char_int,A: char,A4: set_char] :
      ( inj_on_char_int(F,top_top_set_char)
     => ( aa_set_int_o(member_int(aa_char_int(F,A)),image_char_int(F,A4))
      <=> member_char(A,A4) ) ) ).

% inj_image_mem_iff
tff(fact_3111_inj__image__mem__iff,axiom,
    ! [F: fun_rat_o,A: rat,A4: set_rat] :
      ( aa_set_rat_o(inj_on_rat_o(F),top_top_set_rat)
     => ( aa_set_o_o(member_o(aa_rat_o(F,A)),image_rat_o(F,A4))
      <=> aa_set_rat_o(member_rat(A),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3112_inj__image__mem__iff,axiom,
    ! [F: fun_rat_nat,A: rat,A4: set_rat] :
      ( aa_set_rat_o(inj_on_rat_nat(F),top_top_set_rat)
     => ( aa_set_nat_o(member_nat(aa_rat_nat(F,A)),image_rat_nat(F,A4))
      <=> aa_set_rat_o(member_rat(A),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3113_inj__image__mem__iff,axiom,
    ! [F: fun_rat_int,A: rat,A4: set_rat] :
      ( aa_set_rat_o(inj_on_rat_int(F),top_top_set_rat)
     => ( aa_set_int_o(member_int(aa_rat_int(F,A)),image_rat_int(F,A4))
      <=> aa_set_rat_o(member_rat(A),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3114_inj__image__mem__iff,axiom,
    ! [F: fun_nat_rat,A: nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_rat_o(member_rat(aa_nat_rat(F,A)),aa_set_nat_set_rat(image_nat_rat(F),A4))
      <=> aa_set_nat_o(member_nat(A),A4) ) ) ).

% inj_image_mem_iff
tff(fact_3115_inj__image__eq__iff,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( ( aa_set_nat_set_rat(image_nat_rat(F),A4) = aa_set_nat_set_rat(image_nat_rat(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_image_eq_iff
tff(fact_3116_inj__image__eq__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( ( aa_set_nat_set_nat(image_nat_nat(F),A4) = aa_set_nat_set_nat(image_nat_nat(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_image_eq_iff
tff(fact_3117_inj__image__eq__iff,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( ( aa_set_nat_set_int(image_nat_int(F),A4) = aa_set_nat_set_int(image_nat_int(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_image_eq_iff
tff(fact_3118_inj__image__eq__iff,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( ( aa_set_int_set_nat(image_int_nat(F),A4) = aa_set_int_set_nat(image_int_nat(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_image_eq_iff
tff(fact_3119_inj__image__eq__iff,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( ( aa_set_int_set_int(image_int_int(F),A4) = aa_set_int_set_int(image_int_int(F),B5) )
      <=> ( A4 = B5 ) ) ) ).

% inj_image_eq_iff
tff(fact_3120_range__ex1__eq,axiom,
    ! [F: fun_char_o,B: $o] :
      ( inj_on_char_o(F,top_top_set_char)
     => ( aa_set_o_o(member_o((B)),image_char_o(F,top_top_set_char))
      <=> ? [X4: char] :
            ( ( (B)
            <=> aa_char_o(F,X4) )
            & ! [Y4: char] :
                ( ( (B)
                <=> aa_char_o(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3121_range__ex1__eq,axiom,
    ! [F: fun_char_nat,B: nat] :
      ( inj_on_char_nat(F,top_top_set_char)
     => ( aa_set_nat_o(member_nat(B),image_char_nat(F,top_top_set_char))
      <=> ? [X4: char] :
            ( ( B = aa_char_nat(F,X4) )
            & ! [Y4: char] :
                ( ( B = aa_char_nat(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3122_range__ex1__eq,axiom,
    ! [F: fun_char_int,B: int] :
      ( inj_on_char_int(F,top_top_set_char)
     => ( aa_set_int_o(member_int(B),image_char_int(F,top_top_set_char))
      <=> ? [X4: char] :
            ( ( B = aa_char_int(F,X4) )
            & ! [Y4: char] :
                ( ( B = aa_char_int(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3123_range__ex1__eq,axiom,
    ! [F: fun_rat_o,B: $o] :
      ( aa_set_rat_o(inj_on_rat_o(F),top_top_set_rat)
     => ( aa_set_o_o(member_o((B)),image_rat_o(F,top_top_set_rat))
      <=> ? [X4: rat] :
            ( ( (B)
            <=> aa_rat_o(F,X4) )
            & ! [Y4: rat] :
                ( ( (B)
                <=> aa_rat_o(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3124_range__ex1__eq,axiom,
    ! [F: fun_rat_nat,B: nat] :
      ( aa_set_rat_o(inj_on_rat_nat(F),top_top_set_rat)
     => ( aa_set_nat_o(member_nat(B),image_rat_nat(F,top_top_set_rat))
      <=> ? [X4: rat] :
            ( ( B = aa_rat_nat(F,X4) )
            & ! [Y4: rat] :
                ( ( B = aa_rat_nat(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3125_range__ex1__eq,axiom,
    ! [F: fun_rat_int,B: int] :
      ( aa_set_rat_o(inj_on_rat_int(F),top_top_set_rat)
     => ( aa_set_int_o(member_int(B),image_rat_int(F,top_top_set_rat))
      <=> ? [X4: rat] :
            ( ( B = aa_rat_int(F,X4) )
            & ! [Y4: rat] :
                ( ( B = aa_rat_int(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3126_range__ex1__eq,axiom,
    ! [F: fun_nat_rat,B: rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_rat_o(member_rat(B),aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat))
      <=> ? [X4: nat] :
            ( ( B = aa_nat_rat(F,X4) )
            & ! [Y4: nat] :
                ( ( B = aa_nat_rat(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3127_range__ex1__eq,axiom,
    ! [F: fun_nat_o,B: $o] :
      ( aa_set_nat_o(inj_on_nat_o(F),top_top_set_nat)
     => ( aa_set_o_o(member_o((B)),image_nat_o(F,top_top_set_nat))
      <=> ? [X4: nat] :
            ( ( (B)
            <=> aa_nat_o(F,X4) )
            & ! [Y4: nat] :
                ( ( (B)
                <=> aa_nat_o(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3128_range__ex1__eq,axiom,
    ! [F: fun_nat_nat,B: nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_o(member_nat(B),aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat))
      <=> ? [X4: nat] :
            ( ( B = aa_nat_nat(F,X4) )
            & ! [Y4: nat] :
                ( ( B = aa_nat_nat(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3129_range__ex1__eq,axiom,
    ! [F: fun_nat_int,B: int] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( aa_set_int_o(member_int(B),aa_set_nat_set_int(image_nat_int(F),top_top_set_nat))
      <=> ? [X4: nat] :
            ( ( B = aa_nat_int(F,X4) )
            & ! [Y4: nat] :
                ( ( B = aa_nat_int(F,Y4) )
               => ( Y4 = X4 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_3130_comp__inj__on,axiom,
    ! [F: fun_nat_o,A4: set_nat,G: fun_o_o] :
      ( aa_set_nat_o(inj_on_nat_o(F),A4)
     => ( aa_set_o_o(inj_on_o_o(G),image_nat_o(F,A4))
       => aa_set_nat_o(inj_on_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F)),A4) ) ) ).

% comp_inj_on
tff(fact_3131_comp__inj__on,axiom,
    ! [F: fun_Co1981900195452921617nteger,A4: set_Code_integer,G: fun_fu4800041728187187127nteger] :
      ( inj_on428699138970331399nteger(F,A4)
     => ( inj_on5093811489028909101nteger(G,image_2716445779319712667nteger(F,A4))
       => inj_on736898771723690579nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),F),A4) ) ) ).

% comp_inj_on
tff(fact_3132_comp__inj__on,axiom,
    ! [F: fun_nat_nat,A4: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(inj_on_nat_nat(G),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_set_nat_o(inj_on_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),A4) ) ) ).

% comp_inj_on
tff(fact_3133_comp__inj__on,axiom,
    ! [F: fun_int_nat,A4: set_int,G: fun_nat_num] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(inj_on_nat_num(G),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_set_int_o(inj_on_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F)),A4) ) ) ).

% comp_inj_on
tff(fact_3134_comp__inj__on,axiom,
    ! [F: fun_int_int,A4: set_int,G: fun_int_nat] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(inj_on_int_nat(G),aa_set_int_set_int(image_int_int(F),A4))
       => aa_set_int_o(inj_on_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),A4) ) ) ).

% comp_inj_on
tff(fact_3135_inj__on__imageI,axiom,
    ! [G: fun_o_o,F: fun_nat_o,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F)),A4)
     => aa_set_o_o(inj_on_o_o(G),image_nat_o(F,A4)) ) ).

% inj_on_imageI
tff(fact_3136_inj__on__imageI,axiom,
    ! [G: fun_fu4800041728187187127nteger,F: fun_Co1981900195452921617nteger,A4: set_Code_integer] :
      ( inj_on736898771723690579nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),F),A4)
     => inj_on5093811489028909101nteger(G,image_2716445779319712667nteger(F,A4)) ) ).

% inj_on_imageI
tff(fact_3137_inj__on__imageI,axiom,
    ! [G: fun_nat_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),A4)
     => aa_set_nat_o(inj_on_nat_nat(G),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% inj_on_imageI
tff(fact_3138_inj__on__imageI,axiom,
    ! [G: fun_nat_num,F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F)),A4)
     => aa_set_nat_o(inj_on_nat_num(G),aa_set_int_set_nat(image_int_nat(F),A4)) ) ).

% inj_on_imageI
tff(fact_3139_inj__on__imageI,axiom,
    ! [G: fun_int_nat,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),A4)
     => aa_set_int_o(inj_on_int_nat(G),aa_set_int_set_int(image_int_int(F),A4)) ) ).

% inj_on_imageI
tff(fact_3140_comp__inj__on__iff,axiom,
    ! [F: fun_nat_o,A4: set_nat,F2: fun_o_o] :
      ( aa_set_nat_o(inj_on_nat_o(F),A4)
     => ( aa_set_o_o(inj_on_o_o(F2),image_nat_o(F,A4))
      <=> aa_set_nat_o(inj_on_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F2),F)),A4) ) ) ).

% comp_inj_on_iff
tff(fact_3141_comp__inj__on__iff,axiom,
    ! [F: fun_Co1981900195452921617nteger,A4: set_Code_integer,F2: fun_fu4800041728187187127nteger] :
      ( inj_on428699138970331399nteger(F,A4)
     => ( inj_on5093811489028909101nteger(F2,image_2716445779319712667nteger(F,A4))
      <=> inj_on736898771723690579nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F2),F),A4) ) ) ).

% comp_inj_on_iff
tff(fact_3142_comp__inj__on__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,F2: fun_nat_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(inj_on_nat_nat(F2),aa_set_nat_set_nat(image_nat_nat(F),A4))
      <=> aa_set_nat_o(inj_on_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F2),F)),A4) ) ) ).

% comp_inj_on_iff
tff(fact_3143_comp__inj__on__iff,axiom,
    ! [F: fun_int_nat,A4: set_int,F2: fun_nat_num] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(inj_on_nat_num(F2),aa_set_int_set_nat(image_int_nat(F),A4))
      <=> aa_set_int_o(inj_on_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F2),F)),A4) ) ) ).

% comp_inj_on_iff
tff(fact_3144_comp__inj__on__iff,axiom,
    ! [F: fun_int_int,A4: set_int,F2: fun_int_nat] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(inj_on_int_nat(F2),aa_set_int_set_int(image_int_int(F),A4))
      <=> aa_set_int_o(inj_on_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F2),F)),A4) ) ) ).

% comp_inj_on_iff
tff(fact_3145_inj__compose,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger] :
      ( inj_on5093811489028909101nteger(F,top_to805621317456759325nteger)
     => ( inj_on428699138970331399nteger(G,top_to4645266643341252675nteger)
       => inj_on736898771723690579nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),top_to4645266643341252675nteger) ) ) ).

% inj_compose
tff(fact_3146_inj__compose,axiom,
    ! [F: fun_o_o,G: fun_nat_o] :
      ( aa_set_o_o(inj_on_o_o(F),top_top_set_o)
     => ( aa_set_nat_o(inj_on_nat_o(G),top_top_set_nat)
       => aa_set_nat_o(inj_on_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G)),top_top_set_nat) ) ) ).

% inj_compose
tff(fact_3147_inj__compose,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => aa_set_nat_o(inj_on_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G)),top_top_set_nat) ) ) ).

% inj_compose
tff(fact_3148_inj__compose,axiom,
    ! [F: fun_nat_num,G: fun_int_nat] :
      ( aa_set_nat_o(inj_on_nat_num(F),top_top_set_nat)
     => ( aa_set_int_o(inj_on_int_nat(G),top_top_set_int)
       => aa_set_int_o(inj_on_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G)),top_top_set_int) ) ) ).

% inj_compose
tff(fact_3149_inj__compose,axiom,
    ! [F: fun_int_nat,G: fun_int_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_int_o(inj_on_int_int(G),top_top_set_int)
       => aa_set_int_o(inj_on_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G)),top_top_set_int) ) ) ).

% inj_compose
tff(fact_3150_inf__Sup,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A),comple5685304695842803022at_nat(B5)) = comple5685304695842803022at_nat(image_3684629450409544005at_nat(inf_in2572325071724192079at_nat(A),B5)) ).

% inf_Sup
tff(fact_3151_inf__Sup,axiom,
    ! [A: set_nat,B5: set_set_nat] : aa_set_nat_set_nat(inf_inf_set_nat(A),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,B5)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(inf_inf_set_nat(A)),B5)) ).

% inf_Sup
tff(fact_3152_inf__Sup,axiom,
    ! [A: product_unit,B5: set_Product_unit] : aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(A),comple4687483117567863418t_unit(B5)) = comple4687483117567863418t_unit(image_405062704495631173t_unit(inf_inf_Product_unit(A),B5)) ).

% inf_Sup
tff(fact_3153_sup__Inf,axiom,
    ! [A: set_Pr8693737435421807431at_nat,B5: set_se6425085446898223655at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A),comple6332534001959699150at_nat(B5)) = comple6332534001959699150at_nat(image_1013614488123988933at_nat(sup_su718114333110466843at_nat(A),B5)) ).

% sup_Inf
tff(fact_3154_sup__Inf,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B5: set_se5479201105039950375at_nat] : aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A),comple6751905152567503566at_nat(B5)) = comple6751905152567503566at_nat(image_4303274092540115909at_nat(sup_su5525570899277871387at_nat(A),B5)) ).

% sup_Inf
tff(fact_3155_sup__Inf,axiom,
    ! [A: set_nat,B5: set_set_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A),comple7806235888213564991et_nat(B5)) = comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(sup_sup_set_nat(A)),B5)) ).

% sup_Inf
tff(fact_3156_sup__Inf,axiom,
    ! [A: product_unit,B5: set_Product_unit] : aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(A),comple2584293577114468500t_unit(B5)) = comple2584293577114468500t_unit(image_405062704495631173t_unit(sup_sup_Product_unit(A),B5)) ).

% sup_Inf
tff(fact_3157_bdd__below_OI2,axiom,
    ! [A4: set_o,M: nat,F: fun_o_nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_o_nat(F,(X))) )
     => aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat(F,A4)) ) ).

% bdd_below.I2
tff(fact_3158_bdd__below_OI2,axiom,
    ! [A4: set_nat,M: nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(F,X)) )
     => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% bdd_below.I2
tff(fact_3159_bdd__below_OI2,axiom,
    ! [A4: set_int,M: nat,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_int_nat(F,X)) )
     => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ).

% bdd_below.I2
tff(fact_3160_bdd__below_OI2,axiom,
    ! [A4: set_o,M: int,F: fun_o_int] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_o_int(F,(X))) )
     => aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4)) ) ).

% bdd_below.I2
tff(fact_3161_bdd__below_OI2,axiom,
    ! [A4: set_nat,M: int,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_nat_int(F,X)) )
     => aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ).

% bdd_below.I2
tff(fact_3162_bdd__below_OI2,axiom,
    ! [A4: set_int,M: int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(F,X)) )
     => aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4)) ) ).

% bdd_below.I2
tff(fact_3163_bdd__below_OI2,axiom,
    ! [A4: set_o,M: rat,F: fun_o_rat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),aa_o_rat(F,(X))) )
     => aa_set_rat_o(condit1103211067700513672ow_rat,image_o_rat(F,A4)) ) ).

% bdd_below.I2
tff(fact_3164_bdd__below_OI2,axiom,
    ! [A4: set_nat,M: rat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),aa_nat_rat(F,X)) )
     => aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ).

% bdd_below.I2
tff(fact_3165_bdd__below_OI2,axiom,
    ! [A4: set_int,M: rat,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),aa_int_rat(F,X)) )
     => aa_set_rat_o(condit1103211067700513672ow_rat,image_int_rat(F,A4)) ) ).

% bdd_below.I2
tff(fact_3166_bdd__below_OI2,axiom,
    ! [A4: set_o,M: code_integer,F: fun_o_Code_integer] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),aa_o_Code_integer(F,(X))) )
     => condit4933233095834890765nteger(image_o_Code_integer(F,A4)) ) ).

% bdd_below.I2
tff(fact_3167_bdd__belowI2,axiom,
    ! [A4: set_o,M3: nat,F: fun_o_nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_o_nat(F,(X))) )
     => aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat(F,A4)) ) ).

% bdd_belowI2
tff(fact_3168_bdd__belowI2,axiom,
    ! [A4: set_nat,M3: nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_nat_nat(F,X)) )
     => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% bdd_belowI2
tff(fact_3169_bdd__belowI2,axiom,
    ! [A4: set_int,M3: nat,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),aa_int_nat(F,X)) )
     => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ).

% bdd_belowI2
tff(fact_3170_bdd__belowI2,axiom,
    ! [A4: set_o,M3: int,F: fun_o_int] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_o_int(F,(X))) )
     => aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4)) ) ).

% bdd_belowI2
tff(fact_3171_bdd__belowI2,axiom,
    ! [A4: set_nat,M3: int,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_nat_int(F,X)) )
     => aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ).

% bdd_belowI2
tff(fact_3172_bdd__belowI2,axiom,
    ! [A4: set_int,M3: int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_int_int(F,X)) )
     => aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4)) ) ).

% bdd_belowI2
tff(fact_3173_bdd__belowI2,axiom,
    ! [A4: set_o,M3: rat,F: fun_o_rat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M3),aa_o_rat(F,(X))) )
     => aa_set_rat_o(condit1103211067700513672ow_rat,image_o_rat(F,A4)) ) ).

% bdd_belowI2
tff(fact_3174_bdd__belowI2,axiom,
    ! [A4: set_nat,M3: rat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M3),aa_nat_rat(F,X)) )
     => aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ).

% bdd_belowI2
tff(fact_3175_bdd__belowI2,axiom,
    ! [A4: set_int,M3: rat,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M3),aa_int_rat(F,X)) )
     => aa_set_rat_o(condit1103211067700513672ow_rat,image_int_rat(F,A4)) ) ).

% bdd_belowI2
tff(fact_3176_bdd__belowI2,axiom,
    ! [A4: set_o,M3: code_integer,F: fun_o_Code_integer] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M3),aa_o_Code_integer(F,(X))) )
     => condit4933233095834890765nteger(image_o_Code_integer(F,A4)) ) ).

% bdd_belowI2
tff(fact_3177_bdd__above_OI2,axiom,
    ! [A4: set_o,F: fun_o_nat,M: nat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X))),M) )
     => aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat(F,A4)) ) ).

% bdd_above.I2
tff(fact_3178_bdd__above_OI2,axiom,
    ! [A4: set_nat,F: fun_nat_nat,M: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),M) )
     => aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% bdd_above.I2
tff(fact_3179_bdd__above_OI2,axiom,
    ! [A4: set_int,F: fun_int_nat,M: nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),M) )
     => aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ).

% bdd_above.I2
tff(fact_3180_bdd__above_OI2,axiom,
    ! [A4: set_o,F: fun_o_int,M: int] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X))),M) )
     => aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4)) ) ).

% bdd_above.I2
tff(fact_3181_bdd__above_OI2,axiom,
    ! [A4: set_nat,F: fun_nat_int,M: int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X)),M) )
     => aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ).

% bdd_above.I2
tff(fact_3182_bdd__above_OI2,axiom,
    ! [A4: set_int,F: fun_int_int,M: int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X)),M) )
     => aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4)) ) ).

% bdd_above.I2
tff(fact_3183_bdd__above_OI2,axiom,
    ! [A4: set_o,F: fun_o_rat,M: rat] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F,(X))),M) )
     => aa_set_rat_o(condit1579696412822616692ve_rat,image_o_rat(F,A4)) ) ).

% bdd_above.I2
tff(fact_3184_bdd__above_OI2,axiom,
    ! [A4: set_nat,F: fun_nat_rat,M: rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),M) )
     => aa_set_rat_o(condit1579696412822616692ve_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ).

% bdd_above.I2
tff(fact_3185_bdd__above_OI2,axiom,
    ! [A4: set_int,F: fun_int_rat,M: rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),M) )
     => aa_set_rat_o(condit1579696412822616692ve_rat,image_int_rat(F,A4)) ) ).

% bdd_above.I2
tff(fact_3186_bdd__above_OI2,axiom,
    ! [A4: set_o,F: fun_o_Code_integer,M: code_integer] :
      ( ! [X: $o] :
          ( aa_set_o_o(member_o((X)),A4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_o_Code_integer(F,(X))),M) )
     => condit1118808407379112185nteger(image_o_Code_integer(F,A4)) ) ).

% bdd_above.I2
tff(fact_3187_image__Int__subset,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] : aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5))),inf_inf_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5))) ).

% image_Int_subset
tff(fact_3188_image__Int__subset,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5))),aa_set_nat_set_nat(inf_inf_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5))) ).

% image_Int_subset
tff(fact_3189_image__Int__subset,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] : aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(inf_inf_set_nat(A4),B5))),inf_inf_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5))) ).

% image_Int_subset
tff(fact_3190_image__Int__subset,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),inf_inf_set_int(A4,B5))),aa_set_nat_set_nat(inf_inf_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5))) ).

% image_Int_subset
tff(fact_3191_image__Int__subset,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),inf_inf_set_int(A4,B5))),inf_inf_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5))) ).

% image_Int_subset
tff(fact_3192_image__Int__subset,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),B5))),aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4)),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),B5))) ).

% image_Int_subset
tff(fact_3193_swap__inj__on,axiom,
    ! [A4: set_Pr7459493094073627847at_nat] : inj_on3942866938762682161at_nat(produc5456455677105495163at_nat(aTP_Lamm_ec),A4) ).

% swap_inj_on
tff(fact_3194_swap__inj__on,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] : aa_set5844364881195198285_nat_o(inj_on8430303779476321073at_nat(produc7548845685434208379at_nat(aTP_Lamm_ed)),A4) ).

% swap_inj_on
tff(fact_3195_swap__inj__on,axiom,
    ! [A4: set_Pr3280342998896498603um_nat] : inj_on2339680571444523825at_num(produc5219324331026901427at_num(aTP_Lamm_ee),A4) ).

% swap_inj_on
tff(fact_3196_swap__inj__on,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(produc2626176000494625587at_nat(aTP_Lamm_ef)),A4) ).

% swap_inj_on
tff(fact_3197_swap__inj__on,axiom,
    ! [A4: set_Pr958786334691620121nt_int] : inj_on6454360653267994289nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_eg),A4) ).

% swap_inj_on
tff(fact_3198_SUP__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_o_Product_unit(A4,(K))),comple4687483117567863418t_unit(image_o_Product_unit(A4,I3))) = comple4687483117567863418t_unit(image_o_Product_unit(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3199_SUP__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_nat_Product_unit(A4,K)),comple4687483117567863418t_unit(image_8730104196221521654t_unit(A4,I3))) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3200_SUP__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_int_Product_unit(A4,K)),comple4687483117567863418t_unit(image_5846526143423701914t_unit(A4,I3))) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3201_SUP__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_set_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(A4,(K))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3202_SUP__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(A4,K)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3203_SUP__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_int_set_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_int_set_nat(A4,K)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3204_SUP__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro676242055593665672t_unit(A4,K)),comple4687483117567863418t_unit(image_6169909464789311569t_unit(A4,I3))) = comple4687483117567863418t_unit(image_6169909464789311569t_unit(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3205_SUP__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro9123573313595863371et_nat(A4,K)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3206_SUP__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_7500966680246433010at_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(A4,(K))),comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,I3))) = comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3207_SUP__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_na317059118361677008at_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_nat1866645186517922489at_nat(A4,K)),comple2748534634209756852at_nat(image_4592875511192039216at_nat(A4,I3))) = comple2748534634209756852at_nat(image_4592875511192039216at_nat(A4,I3)) ) ) ).

% SUP_absorb
tff(fact_3208_Sup__inf,axiom,
    ! [B5: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple5685304695842803022at_nat(B5)),A) = comple5685304695842803022at_nat(image_3684629450409544005at_nat(aTP_Lamm_xa(A),B5)) ).

% Sup_inf
tff(fact_3209_Sup__inf,axiom,
    ! [B5: set_set_nat,A: set_nat] : aa_set_nat_set_nat(inf_inf_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,B5)),A) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_xb(A)),B5)) ).

% Sup_inf
tff(fact_3210_Sup__inf,axiom,
    ! [B5: set_Product_unit,A: product_unit] : aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple4687483117567863418t_unit(B5)),A) = comple4687483117567863418t_unit(image_405062704495631173t_unit(aTP_Lamm_xc(A),B5)) ).

% Sup_inf
tff(fact_3211_Inf__sup,axiom,
    ! [B5: set_se6425085446898223655at_nat,A: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple6332534001959699150at_nat(B5)),A) = comple6332534001959699150at_nat(image_1013614488123988933at_nat(aTP_Lamm_xd(A),B5)) ).

% Inf_sup
tff(fact_3212_Inf__sup,axiom,
    ! [B5: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple6751905152567503566at_nat(B5)),A) = comple6751905152567503566at_nat(image_4303274092540115909at_nat(aTP_Lamm_xe(A),B5)) ).

% Inf_sup
tff(fact_3213_Inf__sup,axiom,
    ! [B5: set_set_nat,A: set_nat] : aa_set_nat_set_nat(sup_sup_set_nat(comple7806235888213564991et_nat(B5)),A) = comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_xf(A)),B5)) ).

% Inf_sup
tff(fact_3214_Inf__sup,axiom,
    ! [B5: set_Product_unit,A: product_unit] : aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(comple2584293577114468500t_unit(B5)),A) = comple2584293577114468500t_unit(image_405062704495631173t_unit(aTP_Lamm_xg(A),B5)) ).

% Inf_sup
tff(fact_3215_INF__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_o_Product_unit(A4,(K))),comple2584293577114468500t_unit(image_o_Product_unit(A4,I3))) = comple2584293577114468500t_unit(image_o_Product_unit(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3216_INF__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_nat_Product_unit(A4,K)),comple2584293577114468500t_unit(image_8730104196221521654t_unit(A4,I3))) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3217_INF__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_int_Product_unit(A4,K)),comple2584293577114468500t_unit(image_5846526143423701914t_unit(A4,I3))) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3218_INF__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_set_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(A4,(K))),comple7806235888213564991et_nat(image_o_set_nat(A4,I3))) = comple7806235888213564991et_nat(image_o_set_nat(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3219_INF__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_nat_set_nat(A4,K)),comple7806235888213564991et_nat(image_nat_set_nat(A4,I3))) = comple7806235888213564991et_nat(image_nat_set_nat(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3220_INF__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_int_set_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_int_set_nat(A4,K)),comple7806235888213564991et_nat(image_int_set_nat(A4,I3))) = comple7806235888213564991et_nat(image_int_set_nat(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3221_INF__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr2481279687482350479t_unit] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro676242055593665672t_unit(A4,K)),comple2584293577114468500t_unit(image_6169909464789311569t_unit(A4,I3))) = comple2584293577114468500t_unit(image_6169909464789311569t_unit(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3222_INF__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_4630291819367765772at_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(A4,(K))),comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,I3))) = comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3223_INF__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_na7044673412300895210at_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_nat7062706319057040595at_nat(A4,K)),comple1930127330929219688at_nat(image_7178329752028323786at_nat(A4,I3))) = comple1930127330929219688at_nat(image_7178329752028323786at_nat(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3224_INF__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_in8804710076054006414at_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_int8822742982810151799at_nat(A4,K)),comple1930127330929219688at_nat(image_8938366415781434990at_nat(A4,I3))) = comple1930127330929219688at_nat(image_8938366415781434990at_nat(A4,I3)) ) ) ).

% INF_absorb
tff(fact_3225_translation__subtract__Int,axiom,
    ! [A: int,S: set_int,T: set_int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),inf_inf_set_int(S,T)) = inf_inf_set_int(aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),S),aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_qh,A)),T)) ).

% translation_subtract_Int
tff(fact_3226_translation__subtract__Int,axiom,
    ! [A: rat,S: set_rat,T: set_rat] : image_rat_rat(aTP_Lamm_qi(A),inf_inf_set_rat(S,T)) = inf_inf_set_rat(image_rat_rat(aTP_Lamm_qi(A),S),image_rat_rat(aTP_Lamm_qi(A),T)) ).

% translation_subtract_Int
tff(fact_3227_translation__subtract__Int,axiom,
    ! [A: code_integer,S: set_Code_integer,T: set_Code_integer] : image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),inf_in1364745209274528805nteger(S,T)) = inf_in1364745209274528805nteger(image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),S),image_4470545334726330049nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,A),T)) ).

% translation_subtract_Int
tff(fact_3228_UN__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_set_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(A4,(K))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3229_UN__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(A4,K)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3230_UN__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_int_set_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_int_set_nat(A4,K)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3231_UN__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro9123573313595863371et_nat(A4,K)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,I3))) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3232_UN__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_7500966680246433010at_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(A4,(K))),comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,I3))) = comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3233_UN__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_na317059118361677008at_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_nat1866645186517922489at_nat(A4,K)),comple2748534634209756852at_nat(image_4592875511192039216at_nat(A4,I3))) = comple2748534634209756852at_nat(image_4592875511192039216at_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3234_UN__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_in5473042712587599220at_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_int7022628780743844701at_nat(A4,K)),comple2748534634209756852at_nat(image_525487068563185620at_nat(A4,I3))) = comple2748534634209756852at_nat(image_525487068563185620at_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3235_UN__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr6108405906935189705at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro1853492592809407170at_nat(A4,K)),comple2748534634209756852at_nat(image_5803077947706746251at_nat(A4,I3))) = comple2748534634209756852at_nat(image_5803077947706746251at_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3236_UN__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_7855627885593154802at_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_o_s4772616771316612843at_nat(A4,(K))),comple4364388969068068532at_nat(image_2496312346330513076at_nat(A4,I3))) = comple4364388969068068532at_nat(image_2496312346330513076at_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3237_UN__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_na5670658592214072528at_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_nat8843416860741599929at_nat(A4,K)),comple4364388969068068532at_nat(image_1087905878993176368at_nat(A4,I3))) = comple4364388969068068532at_nat(image_1087905878993176368at_nat(A4,I3)) ) ) ).

% UN_absorb
tff(fact_3238_INT__absorb,axiom,
    ! [K: produc3843707927480180839at_nat,I3: set_Pr4329608150637261639at_nat,A4: fun_Pr6212429452034653949at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(K),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro441628658106787830at_nat(A4,K)),comple1930127330929219688at_nat(image_7979982168039674559at_nat(A4,I3))) = comple1930127330929219688at_nat(image_7979982168039674559at_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3239_INT__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr1469247711798767971at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro2745879313366468188at_nat(A4,K)),comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(A4),I3))) = comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(A4),I3)) ) ) ).

% INT_absorb
tff(fact_3240_INT__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_4630291819367765772at_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(A4,(K))),comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,I3))) = comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3241_INT__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_na7044673412300895210at_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_nat7062706319057040595at_nat(A4,K)),comple1930127330929219688at_nat(image_7178329752028323786at_nat(A4,I3))) = comple1930127330929219688at_nat(image_7178329752028323786at_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3242_INT__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_in8804710076054006414at_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_int8822742982810151799at_nat(A4,K)),comple1930127330929219688at_nat(image_8938366415781434990at_nat(A4,I3))) = comple1930127330929219688at_nat(image_8938366415781434990at_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3243_INT__absorb,axiom,
    ! [K: produc3843707927480180839at_nat,I3: set_Pr4329608150637261639at_nat,A4: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(K),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro1692590229789857841et_nat(A4,K)),comple7806235888213564991et_nat(image_3160451284896210088et_nat(A4,I3))) = comple7806235888213564991et_nat(image_3160451284896210088et_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3244_INT__absorb,axiom,
    ! [K: product_prod_nat_nat,I3: set_Pr1261947904930325089at_nat,A4: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(K),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro9123573313595863371et_nat(A4,K)),comple7806235888213564991et_nat(image_15824709712370754et_nat(A4,I3))) = comple7806235888213564991et_nat(image_15824709712370754et_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3245_INT__absorb,axiom,
    ! [K: $o,I3: set_o,A4: fun_o_set_nat] :
      ( aa_set_o_o(member_o((K)),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(A4,(K))),comple7806235888213564991et_nat(image_o_set_nat(A4,I3))) = comple7806235888213564991et_nat(image_o_set_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3246_INT__absorb,axiom,
    ! [K: nat,I3: set_nat,A4: fun_nat_set_nat] :
      ( aa_set_nat_o(member_nat(K),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_nat_set_nat(A4,K)),comple7806235888213564991et_nat(image_nat_set_nat(A4,I3))) = comple7806235888213564991et_nat(image_nat_set_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3247_INT__absorb,axiom,
    ! [K: int,I3: set_int,A4: fun_int_set_nat] :
      ( aa_set_int_o(member_int(K),I3)
     => ( aa_set_nat_set_nat(inf_inf_set_nat(aa_int_set_nat(A4,K)),comple7806235888213564991et_nat(image_int_set_nat(A4,I3))) = comple7806235888213564991et_nat(image_int_set_nat(A4,I3)) ) ) ).

% INT_absorb
tff(fact_3248_Int__Union2,axiom,
    ! [B5: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple5685304695842803022at_nat(B5)),A4) = comple5685304695842803022at_nat(image_3684629450409544005at_nat(aTP_Lamm_xa(A4),B5)) ).

% Int_Union2
tff(fact_3249_Int__Union2,axiom,
    ! [B5: set_set_nat,A4: set_nat] : aa_set_nat_set_nat(inf_inf_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,B5)),A4) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_xb(A4)),B5)) ).

% Int_Union2
tff(fact_3250_Int__Union,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple5685304695842803022at_nat(B5)) = comple5685304695842803022at_nat(image_3684629450409544005at_nat(inf_in2572325071724192079at_nat(A4),B5)) ).

% Int_Union
tff(fact_3251_Int__Union,axiom,
    ! [A4: set_nat,B5: set_set_nat] : aa_set_nat_set_nat(inf_inf_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,B5)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(inf_inf_set_nat(A4)),B5)) ).

% Int_Union
tff(fact_3252_Un__Inter,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B5: set_se6425085446898223655at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple6332534001959699150at_nat(B5)) = comple6332534001959699150at_nat(image_1013614488123988933at_nat(sup_su718114333110466843at_nat(A4),B5)) ).

% Un_Inter
tff(fact_3253_Un__Inter,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: set_se5479201105039950375at_nat] : aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),comple6751905152567503566at_nat(B5)) = comple6751905152567503566at_nat(image_4303274092540115909at_nat(sup_su5525570899277871387at_nat(A4),B5)) ).

% Un_Inter
tff(fact_3254_Un__Inter,axiom,
    ! [A4: set_nat,B5: set_set_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A4),comple7806235888213564991et_nat(B5)) = comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(sup_sup_set_nat(A4)),B5)) ).

% Un_Inter
tff(fact_3255_inj__on__iff__surj,axiom,
    ! [A4: set_rat,A6: set_nat] :
      ( ( A4 != bot_bot_set_rat )
     => ( ? [F5: fun_rat_nat] :
            ( aa_set_rat_o(inj_on_rat_nat(F5),A4)
            & aa_set_nat_o(ord_less_eq_set_nat(image_rat_nat(F5,A4)),A6) )
      <=> ? [G3: fun_nat_rat] : aa_set_nat_set_rat(image_nat_rat(G3),A6) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_3256_inj__on__iff__surj,axiom,
    ! [A4: set_int,A6: set_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( ? [F5: fun_int_nat] :
            ( aa_set_int_o(inj_on_int_nat(F5),A4)
            & aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F5),A4)),A6) )
      <=> ? [G3: fun_nat_int] : aa_set_nat_set_int(image_nat_int(G3),A6) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_3257_inj__on__iff__surj,axiom,
    ! [A4: set_int,A6: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ? [F5: fun_int_int] :
            ( aa_set_int_o(inj_on_int_int(F5),A4)
            & aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F5),A4)),A6) )
      <=> ? [G3: fun_int_int] : aa_set_int_set_int(image_int_int(G3),A6) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_3258_inj__on__iff__surj,axiom,
    ! [A4: set_nat,A6: set_rat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ? [F5: fun_nat_rat] :
            ( aa_set_nat_o(inj_on_nat_rat(F5),A4)
            & aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F5),A4)),A6) )
      <=> ? [G3: fun_rat_nat] : image_rat_nat(G3,A6) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_3259_inj__on__iff__surj,axiom,
    ! [A4: set_nat,A6: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ? [F5: fun_nat_nat] :
            ( aa_set_nat_o(inj_on_nat_nat(F5),A4)
            & aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F5),A4)),A6) )
      <=> ? [G3: fun_nat_nat] : aa_set_nat_set_nat(image_nat_nat(G3),A6) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_3260_inj__on__iff__surj,axiom,
    ! [A4: set_nat,A6: set_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( ? [F5: fun_nat_int] :
            ( aa_set_nat_o(inj_on_nat_int(F5),A4)
            & aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F5),A4)),A6) )
      <=> ? [G3: fun_int_nat] : aa_set_int_set_nat(image_int_nat(G3),A6) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_3261_Int__Inter__eq_I1_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B8: set_se7855581050983116737at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple1930127330929219688at_nat(B8)) = $ite(B8 = bot_bo3083307316010499117at_nat,A4,comple1930127330929219688at_nat(image_3684629450409544005at_nat(inf_in2572325071724192079at_nat(A4),B8))) ).

% Int_Inter_eq(1)
tff(fact_3262_Int__Inter__eq_I1_J,axiom,
    ! [A4: set_nat,B8: set_set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(B8)) = $ite(B8 = bot_bot_set_set_nat,A4,comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(inf_inf_set_nat(A4)),B8))) ).

% Int_Inter_eq(1)
tff(fact_3263_Int__Inter__eq_I2_J,axiom,
    ! [B8: set_se7855581050983116737at_nat,A4: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(B8)),A4) = $ite(B8 = bot_bo3083307316010499117at_nat,A4,comple1930127330929219688at_nat(image_3684629450409544005at_nat(aTP_Lamm_xa(A4),B8))) ).

% Int_Inter_eq(2)
tff(fact_3264_Int__Inter__eq_I2_J,axiom,
    ! [B8: set_set_nat,A4: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(B8)),A4) = $ite(B8 = bot_bot_set_set_nat,A4,comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(aTP_Lamm_xb(A4)),B8))) ).

% Int_Inter_eq(2)
tff(fact_3265_inj__image__subset__iff,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),aa_set_nat_set_rat(image_nat_rat(F),B5))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B5) ) ) ).

% inj_image_subset_iff
tff(fact_3266_inj__image__subset__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B5) ) ) ).

% inj_image_subset_iff
tff(fact_3267_inj__image__subset__iff,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),aa_set_nat_set_int(image_nat_int(F),B5))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A4),B5) ) ) ).

% inj_image_subset_iff
tff(fact_3268_inj__image__subset__iff,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5))
      <=> aa_set_int_o(ord_less_eq_set_int(A4),B5) ) ) ).

% inj_image_subset_iff
tff(fact_3269_inj__image__subset__iff,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),aa_set_int_set_int(image_int_int(F),B5))
      <=> aa_set_int_o(ord_less_eq_set_int(A4),B5) ) ) ).

% inj_image_subset_iff
tff(fact_3270_inj__on__image__set__diff,axiom,
    ! [F: fun_nat_rat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)) = minus_minus_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_3271_inj__on__image__set__diff,axiom,
    ! [F: fun_nat_nat,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_3272_inj__on__image__set__diff,axiom,
    ! [F: fun_nat_int,C3: set_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)),C3)
       => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C3)
         => ( aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)) = minus_minus_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_3273_inj__on__image__set__diff,axiom,
    ! [F: fun_int_nat,C3: set_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,B5)),C3)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C3)
         => ( aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,B5)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_3274_inj__on__image__set__diff,axiom,
    ! [F: fun_int_int,C3: set_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A4,B5)),C3)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C3)
         => ( aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,B5)) = minus_minus_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_3275_image__set__diff,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)) = minus_minus_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set_nat_set_rat(image_nat_rat(F),B5)) ) ) ).

% image_set_diff
tff(fact_3276_image__set__diff,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),aa_set_nat_set_nat(image_nat_nat(F),B5)) ) ) ).

% image_set_diff
tff(fact_3277_image__set__diff,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( aa_set_nat_set_int(image_nat_int(F),aa_set_nat_set_nat(minus_minus_set_nat(A4),B5)) = minus_minus_set_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set_nat_set_int(image_nat_int(F),B5)) ) ) ).

% image_set_diff
tff(fact_3278_image__set__diff,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,B5)) = aa_set_nat_set_nat(minus_minus_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),aa_set_int_set_nat(image_int_nat(F),B5)) ) ) ).

% image_set_diff
tff(fact_3279_image__set__diff,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,B5)) = minus_minus_set_int(aa_set_int_set_int(image_int_int(F),A4),aa_set_int_set_int(image_int_int(F),B5)) ) ) ).

% image_set_diff
tff(fact_3280_cINF__lower2,axiom,
    ! [F: fun_o_nat,A4: set_o,Xa: $o,U: nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Xa))),U)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(F,A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3281_cINF__lower2,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Xa: nat,U: nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Xa)),U)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3282_cINF__lower2,axiom,
    ! [F: fun_int_nat,A4: set_int,Xa: int,U: nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Xa)),U)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3283_cINF__lower2,axiom,
    ! [F: fun_o_int,A4: set_o,Xa: $o,U: int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(Xa))),U)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3284_cINF__lower2,axiom,
    ! [F: fun_nat_int,A4: set_nat,Xa: nat,U: int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Xa)),U)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3285_cINF__lower2,axiom,
    ! [F: fun_int_int,A4: set_int,Xa: int,U: int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,Xa)),U)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3286_cINF__lower2,axiom,
    ! [F: fun_o_Product_unit,A4: set_o,Xa: $o,U: product_unit] :
      ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(Xa))),U)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3287_cINF__lower2,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat,Xa: nat,U: product_unit] :
      ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,Xa)),U)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3288_cINF__lower2,axiom,
    ! [F: fun_int_Product_unit,A4: set_int,Xa: int,U: product_unit] :
      ( condit9161856309968323677t_unit(image_5846526143423701914t_unit(F,A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,Xa)),U)
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3289_cINF__lower2,axiom,
    ! [F: fun_o_set_nat,A4: set_o,Xa: $o,U: set_nat] :
      ( condit68592940725977398et_nat(image_o_set_nat(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(Xa))),U)
         => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,A4))),U) ) ) ) ).

% cINF_lower2
tff(fact_3290_cINF__lower,axiom,
    ! [F: fun_o_nat,A4: set_o,Xa: $o] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(F,A4))),aa_o_nat(F,(Xa))) ) ) ).

% cINF_lower
tff(fact_3291_cINF__lower,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Xa: nat] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_nat_nat(F,Xa)) ) ) ).

% cINF_lower
tff(fact_3292_cINF__lower,axiom,
    ! [F: fun_int_nat,A4: set_int,Xa: int] :
      ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_int_nat(F,Xa)) ) ) ).

% cINF_lower
tff(fact_3293_cINF__lower,axiom,
    ! [F: fun_o_int,A4: set_o,Xa: $o] :
      ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))),aa_o_int(F,(Xa))) ) ) ).

% cINF_lower
tff(fact_3294_cINF__lower,axiom,
    ! [F: fun_nat_int,A4: set_nat,Xa: nat] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_nat_int(F,Xa)) ) ) ).

% cINF_lower
tff(fact_3295_cINF__lower,axiom,
    ! [F: fun_int_int,A4: set_int,Xa: int] :
      ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))),aa_int_int(F,Xa)) ) ) ).

% cINF_lower
tff(fact_3296_cINF__lower,axiom,
    ! [F: fun_o_Product_unit,A4: set_o,Xa: $o] :
      ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,A4))),aa_o_Product_unit(F,(Xa))) ) ) ).

% cINF_lower
tff(fact_3297_cINF__lower,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat,Xa: nat] :
      ( condit9161856309968323677t_unit(image_8730104196221521654t_unit(F,A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))),aa_nat_Product_unit(F,Xa)) ) ) ).

% cINF_lower
tff(fact_3298_cINF__lower,axiom,
    ! [F: fun_int_Product_unit,A4: set_int,Xa: int] :
      ( condit9161856309968323677t_unit(image_5846526143423701914t_unit(F,A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))),aa_int_Product_unit(F,Xa)) ) ) ).

% cINF_lower
tff(fact_3299_cINF__lower,axiom,
    ! [F: fun_o_set_nat,A4: set_o,Xa: $o] :
      ( condit68592940725977398et_nat(image_o_set_nat(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => aa_set_nat_o(ord_less_eq_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,A4))),aa_o_set_nat(F,(Xa))) ) ) ).

% cINF_lower
tff(fact_3300_cSUP__upper2,axiom,
    ! [F: fun_o_nat,A4: set_o,Xa: $o,U: nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_o_nat(F,(Xa)))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3301_cSUP__upper2,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Xa: nat,U: nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_nat_nat(F,Xa))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3302_cSUP__upper2,axiom,
    ! [F: fun_int_nat,A4: set_int,Xa: int,U: nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_int_nat(F,Xa))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3303_cSUP__upper2,axiom,
    ! [F: fun_o_int,A4: set_o,Xa: $o,U: int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_o_int(F,(Xa)))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3304_cSUP__upper2,axiom,
    ! [F: fun_nat_int,A4: set_nat,Xa: nat,U: int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_nat_int(F,Xa))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3305_cSUP__upper2,axiom,
    ! [F: fun_int_int,A4: set_int,Xa: int,U: int] :
      ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_int_int(F,Xa))
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3306_cSUP__upper2,axiom,
    ! [F: fun_o_Product_unit,A4: set_o,Xa: $o,U: product_unit] :
      ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_o_Product_unit(F,(Xa)))
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3307_cSUP__upper2,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat,Xa: nat,U: product_unit] :
      ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,A4))
     => ( aa_set_nat_o(member_nat(Xa),A4)
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_nat_Product_unit(F,Xa))
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3308_cSUP__upper2,axiom,
    ! [F: fun_int_Product_unit,A4: set_int,Xa: int,U: product_unit] :
      ( condit2080475194861589617t_unit(image_5846526143423701914t_unit(F,A4))
     => ( aa_set_int_o(member_int(Xa),A4)
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_int_Product_unit(F,Xa))
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3309_cSUP__upper2,axiom,
    ! [F: fun_o_set_nat,A4: set_o,Xa: $o,U: set_nat] :
      ( condit5477540289124974626et_nat(image_o_set_nat(F,A4))
     => ( aa_set_o_o(member_o((Xa)),A4)
       => ( aa_set_nat_o(ord_less_eq_set_nat(U),aa_o_set_nat(F,(Xa)))
         => aa_set_nat_o(ord_less_eq_set_nat(U),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))) ) ) ) ).

% cSUP_upper2
tff(fact_3310_cSUP__upper,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_nat] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat(F,A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(Xa))),aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))) ) ) ).

% cSUP_upper
tff(fact_3311_cSUP__upper,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Xa)),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ).

% cSUP_upper
tff(fact_3312_cSUP__upper,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Xa)),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ).

% cSUP_upper
tff(fact_3313_cSUP__upper,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_int] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(Xa))),aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))) ) ) ).

% cSUP_upper
tff(fact_3314_cSUP__upper,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Xa)),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ).

% cSUP_upper
tff(fact_3315_cSUP__upper,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,Xa)),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ).

% cSUP_upper
tff(fact_3316_cSUP__upper,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_Product_unit] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(Xa))),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ) ) ).

% cSUP_upper
tff(fact_3317_cSUP__upper,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_Product_unit] :
      ( aa_set_nat_o(member_nat(Xa),A4)
     => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(F,A4))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,Xa)),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ) ) ).

% cSUP_upper
tff(fact_3318_cSUP__upper,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_Product_unit] :
      ( aa_set_int_o(member_int(Xa),A4)
     => ( condit2080475194861589617t_unit(image_5846526143423701914t_unit(F,A4))
       => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,Xa)),comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))) ) ) ).

% cSUP_upper
tff(fact_3319_cSUP__upper,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_set_nat] :
      ( aa_set_o_o(member_o((Xa)),A4)
     => ( condit5477540289124974626et_nat(image_o_set_nat(F,A4))
       => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(F,(Xa))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))) ) ) ).

% cSUP_upper
tff(fact_3320_Union__image__empty,axiom,
    ! [A4: set_nat,F: fun_Pr1628945989756059814et_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,bot_bo3957492148770167129t_unit))) = A4 ).

% Union_image_empty
tff(fact_3321_Union__image__empty,axiom,
    ! [A4: set_nat,F: fun_o_set_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,bot_bot_set_o))) = A4 ).

% Union_image_empty
tff(fact_3322_Union__image__empty,axiom,
    ! [A4: set_nat,F: fun_nat_set_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,bot_bot_set_nat))) = A4 ).

% Union_image_empty
tff(fact_3323_Union__image__empty,axiom,
    ! [A4: set_nat,F: fun_Pr9038118721296872730et_nat] : aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,bot_bo2099793752762293965at_nat))) = A4 ).

% Union_image_empty
tff(fact_3324_Union__image__empty,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,F: fun_Pr53831763054929981at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_7288820656315755647at_nat(F,bot_bo3957492148770167129t_unit))) = A4 ).

% Union_image_empty
tff(fact_3325_Union__image__empty,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,F: fun_o_7500966680246433010at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_8493167801588156084at_nat(F,bot_bot_set_o))) = A4 ).

% Union_image_empty
tff(fact_3326_Union__image__empty,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,F: fun_na317059118361677008at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_4592875511192039216at_nat(F,bot_bot_set_nat))) = A4 ).

% Union_image_empty
tff(fact_3327_Union__image__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_Pr1686037257031381053at_nat] : aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),comple4364388969068068532at_nat(image_5172358348707908735at_nat(F,bot_bo3957492148770167129t_unit))) = A4 ).

% Union_image_empty
tff(fact_3328_Union__image__empty,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,F: fun_Pr6108405906935189705at_nat] : aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_5803077947706746251at_nat(F,bot_bo2099793752762293965at_nat))) = A4 ).

% Union_image_empty
tff(fact_3329_Union__image__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,F: fun_o_7855627885593154802at_nat] : aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),comple4364388969068068532at_nat(image_2496312346330513076at_nat(F,bot_bot_set_o))) = A4 ).

% Union_image_empty
tff(fact_3330_Union__image__insert,axiom,
    ! [F: fun_nat_set_nat,A: nat,B5: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,aa_set_nat_set_nat(insert_nat(A),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,B5))) ).

% Union_image_insert
tff(fact_3331_Union__image__insert,axiom,
    ! [F: fun_Pr1628945989756059814et_nat,A: product_unit,B5: set_Product_unit] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro6064029160526250199et_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,B5))) ).

% Union_image_insert
tff(fact_3332_Union__image__insert,axiom,
    ! [F: fun_o_set_nat,A: $o,B5: set_o] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,aa_set_o_set_o(insert_o((A)),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(F,(A))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,B5))) ).

% Union_image_insert
tff(fact_3333_Union__image__insert,axiom,
    ! [F: fun_Pr9038118721296872730et_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),B5))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro9123573313595863371et_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,B5))) ).

% Union_image_insert
tff(fact_3334_Union__image__insert,axiom,
    ! [F: fun_na317059118361677008at_nat,A: nat,B5: set_nat] : comple2748534634209756852at_nat(image_4592875511192039216at_nat(F,aa_set_nat_set_nat(insert_nat(A),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_nat1866645186517922489at_nat(F,A)),comple2748534634209756852at_nat(image_4592875511192039216at_nat(F,B5))) ).

% Union_image_insert
tff(fact_3335_Union__image__insert,axiom,
    ! [F: fun_Pr53831763054929981at_nat,A: product_unit,B5: set_Product_unit] : comple2748534634209756852at_nat(image_7288820656315755647at_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro2353044888477550134at_nat(F,A)),comple2748534634209756852at_nat(image_7288820656315755647at_nat(F,B5))) ).

% Union_image_insert
tff(fact_3336_Union__image__insert,axiom,
    ! [F: fun_o_7500966680246433010at_nat,A: $o,B5: set_o] : comple2748534634209756852at_nat(image_8493167801588156084at_nat(F,aa_set_o_set_o(insert_o((A)),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(F,(A))),comple2748534634209756852at_nat(image_8493167801588156084at_nat(F,B5))) ).

% Union_image_insert
tff(fact_3337_Union__image__insert,axiom,
    ! [F: fun_Pr6108405906935189705at_nat,A: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] : comple2748534634209756852at_nat(image_5803077947706746251at_nat(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),B5))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro1853492592809407170at_nat(F,A)),comple2748534634209756852at_nat(image_5803077947706746251at_nat(F,B5))) ).

% Union_image_insert
tff(fact_3338_Union__image__insert,axiom,
    ! [F: fun_na5670658592214072528at_nat,A: nat,B5: set_nat] : comple4364388969068068532at_nat(image_1087905878993176368at_nat(F,aa_set_nat_set_nat(insert_nat(A),B5))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_nat8843416860741599929at_nat(F,A)),comple4364388969068068532at_nat(image_1087905878993176368at_nat(F,B5))) ).

% Union_image_insert
tff(fact_3339_Union__image__insert,axiom,
    ! [F: fun_Pr1686037257031381053at_nat,A: product_unit,B5: set_Product_unit] : comple4364388969068068532at_nat(image_5172358348707908735at_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),B5))) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_Pro2875264725485350454at_nat(F,A)),comple4364388969068068532at_nat(image_5172358348707908735at_nat(F,B5))) ).

% Union_image_insert
tff(fact_3340_SUP__insert,axiom,
    ! [F: fun_nat_Product_unit,A: nat,A4: set_nat] : comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_nat_Product_unit(F,A)),comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))) ).

% SUP_insert
tff(fact_3341_SUP__insert,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A: product_unit,A4: set_Product_unit] : comple4687483117567863418t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro8294289767544744956t_unit(F,A)),comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))) ).

% SUP_insert
tff(fact_3342_SUP__insert,axiom,
    ! [F: fun_o_Product_unit,A: $o,A4: set_o] : comple4687483117567863418t_unit(image_o_Product_unit(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_o_Product_unit(F,(A))),comple4687483117567863418t_unit(image_o_Product_unit(F,A4))) ).

% SUP_insert
tff(fact_3343_SUP__insert,axiom,
    ! [F: fun_nat_set_nat,A: nat,A4: set_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(F,A4))) ).

% SUP_insert
tff(fact_3344_SUP__insert,axiom,
    ! [F: fun_Pr1628945989756059814et_nat,A: product_unit,A4: set_Product_unit] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro6064029160526250199et_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,A4))) ).

% SUP_insert
tff(fact_3345_SUP__insert,axiom,
    ! [F: fun_o_set_nat,A: $o,A4: set_o] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(F,(A))),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(F,A4))) ).

% SUP_insert
tff(fact_3346_SUP__insert,axiom,
    ! [F: fun_Pr2481279687482350479t_unit,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro676242055593665672t_unit(F,A)),comple4687483117567863418t_unit(image_6169909464789311569t_unit(F,A4))) ).

% SUP_insert
tff(fact_3347_SUP__insert,axiom,
    ! [F: fun_Pr9038118721296872730et_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro9123573313595863371et_nat(F,A)),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(F,A4))) ).

% SUP_insert
tff(fact_3348_SUP__insert,axiom,
    ! [F: fun_na317059118361677008at_nat,A: nat,A4: set_nat] : comple2748534634209756852at_nat(image_4592875511192039216at_nat(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_nat1866645186517922489at_nat(F,A)),comple2748534634209756852at_nat(image_4592875511192039216at_nat(F,A4))) ).

% SUP_insert
tff(fact_3349_SUP__insert,axiom,
    ! [F: fun_Pr53831763054929981at_nat,A: product_unit,A4: set_Product_unit] : comple2748534634209756852at_nat(image_7288820656315755647at_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro2353044888477550134at_nat(F,A)),comple2748534634209756852at_nat(image_7288820656315755647at_nat(F,A4))) ).

% SUP_insert
tff(fact_3350_INF__inf__const2,axiom,
    ! [I3: set_Product_unit,F: fun_Pr2191293291710260995t_unit,Xa: product_unit] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(aa_Pro1375803928533059540t_unit(aTP_Lamm_xh(F),Xa),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_405062704495631173t_unit(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3351_INF__inf__const2,axiom,
    ! [I3: set_o,F: fun_o_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_o )
     => ( comple2584293577114468500t_unit(image_o_Product_unit(aa_Pro1198011010171541257t_unit(aTP_Lamm_xi(F),Xa),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_o_Product_unit(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3352_INF__inf__const2,axiom,
    ! [I3: set_nat,F: fun_nat_Product_unit,Xa: product_unit] :
      ( ( I3 != bot_bot_set_nat )
     => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_Pro6532773420561294981t_unit(aTP_Lamm_xj(F),Xa),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3353_INF__inf__const2,axiom,
    ! [I3: set_Product_unit,F: fun_Pr1628945989756059814et_nat,Xa: set_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_ss(F),Xa),I3)) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3354_INF__inf__const2,axiom,
    ! [I3: set_o,F: fun_o_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( comple7806235888213564991et_nat(image_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_xk(F),Xa),I3)) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_o_set_nat(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3355_INF__inf__const2,axiom,
    ! [I3: set_nat,F: fun_nat_set_nat,Xa: set_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( comple7806235888213564991et_nat(image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_xl(F),Xa),I3)) = aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3356_INF__inf__const2,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,F: fun_Pr2481279687482350479t_unit,Xa: product_unit] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( comple2584293577114468500t_unit(image_6169909464789311569t_unit(aa_Pro4251916131025800416t_unit(aTP_Lamm_xm(F),Xa),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3357_INF__inf__const2,axiom,
    ! [I3: set_Product_unit,F: fun_Pr3332065415989118679at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( comple1930127330929219688at_nat(image_3617315916088684441at_nat(aa_set9196459572129487356at_nat(aTP_Lamm_xn(F),Xa),I3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_3617315916088684441at_nat(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3358_INF__inf__const2,axiom,
    ! [I3: set_o,F: fun_o_4630291819367765772at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( comple1930127330929219688at_nat(image_7077428525161401678at_nat(aa_set7523300704893535665at_nat(aTP_Lamm_xo(F),Xa),I3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7077428525161401678at_nat(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3359_INF__inf__const2,axiom,
    ! [I3: set_nat,F: fun_na7044673412300895210at_nat,Xa: set_Pr1261947904930325089at_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( comple1930127330929219688at_nat(image_7178329752028323786at_nat(aa_set340465950006982789at_nat(aTP_Lamm_xp(F),Xa),I3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7178329752028323786at_nat(F,I3))),Xa) ) ) ).

% INF_inf_const2
tff(fact_3360_INF__inf__const1,axiom,
    ! [I3: set_Product_unit,Xa: product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( comple2584293577114468500t_unit(image_405062704495631173t_unit(aa_fun6153996563283759940t_unit(aTP_Lamm_xq(Xa),F),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Xa),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3361_INF__inf__const1,axiom,
    ! [I3: set_o,Xa: product_unit,F: fun_o_Product_unit] :
      ( ( I3 != bot_bot_set_o )
     => ( comple2584293577114468500t_unit(image_o_Product_unit(aa_fun4820891174006887598t_unit(aTP_Lamm_xr(Xa),F),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Xa),comple2584293577114468500t_unit(image_o_Product_unit(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3362_INF__inf__const1,axiom,
    ! [I3: set_nat,Xa: product_unit,F: fun_nat_Product_unit] :
      ( ( I3 != bot_bot_set_nat )
     => ( comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_fun4858661924651670332t_unit(aTP_Lamm_xs(Xa),F),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Xa),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3363_INF__inf__const1,axiom,
    ! [I3: set_Product_unit,Xa: set_nat,F: fun_Pr1628945989756059814et_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_si(Xa),F),I3)) = aa_set_nat_set_nat(inf_inf_set_nat(Xa),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3364_INF__inf__const1,axiom,
    ! [I3: set_o,Xa: set_nat,F: fun_o_set_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( comple7806235888213564991et_nat(image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_xt(Xa),F),I3)) = aa_set_nat_set_nat(inf_inf_set_nat(Xa),comple7806235888213564991et_nat(image_o_set_nat(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3365_INF__inf__const1,axiom,
    ! [I3: set_nat,Xa: set_nat,F: fun_nat_set_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( comple7806235888213564991et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_xu(Xa),F),I3)) = aa_set_nat_set_nat(inf_inf_set_nat(Xa),comple7806235888213564991et_nat(image_nat_set_nat(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3366_INF__inf__const1,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,Xa: product_unit,F: fun_Pr2481279687482350479t_unit] :
      ( ( I3 != bot_bo2099793752762293965at_nat )
     => ( comple2584293577114468500t_unit(image_6169909464789311569t_unit(aa_fun7695145498385251164t_unit(aTP_Lamm_xv(Xa),F),I3)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Xa),comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3367_INF__inf__const1,axiom,
    ! [I3: set_Product_unit,Xa: set_Pr1261947904930325089at_nat,F: fun_Pr3332065415989118679at_nat] :
      ( ( I3 != bot_bo3957492148770167129t_unit )
     => ( comple1930127330929219688at_nat(image_3617315916088684441at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_xw(Xa),F),I3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Xa),comple1930127330929219688at_nat(image_3617315916088684441at_nat(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3368_INF__inf__const1,axiom,
    ! [I3: set_o,Xa: set_Pr1261947904930325089at_nat,F: fun_o_4630291819367765772at_nat] :
      ( ( I3 != bot_bot_set_o )
     => ( comple1930127330929219688at_nat(image_7077428525161401678at_nat(aa_fun213375806635181270at_nat(aTP_Lamm_xx(Xa),F),I3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Xa),comple1930127330929219688at_nat(image_7077428525161401678at_nat(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3369_INF__inf__const1,axiom,
    ! [I3: set_nat,Xa: set_Pr1261947904930325089at_nat,F: fun_na7044673412300895210at_nat] :
      ( ( I3 != bot_bot_set_nat )
     => ( comple1930127330929219688at_nat(image_7178329752028323786at_nat(aa_fun5076269500744484796at_nat(aTP_Lamm_xy(Xa),F),I3)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Xa),comple1930127330929219688at_nat(image_7178329752028323786at_nat(F,I3))) ) ) ).

% INF_inf_const1
tff(fact_3370_INF__insert,axiom,
    ! [F: fun_nat_Product_unit,A: nat,A4: set_nat] : comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_nat_Product_unit(F,A)),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))) ).

% INF_insert
tff(fact_3371_INF__insert,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A: product_unit,A4: set_Product_unit] : comple2584293577114468500t_unit(image_405062704495631173t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro8294289767544744956t_unit(F,A)),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))) ).

% INF_insert
tff(fact_3372_INF__insert,axiom,
    ! [F: fun_o_Product_unit,A: $o,A4: set_o] : comple2584293577114468500t_unit(image_o_Product_unit(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_o_Product_unit(F,(A))),comple2584293577114468500t_unit(image_o_Product_unit(F,A4))) ).

% INF_insert
tff(fact_3373_INF__insert,axiom,
    ! [F: fun_nat_set_nat,A: nat,A4: set_nat] : comple7806235888213564991et_nat(image_nat_set_nat(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_nat_set_nat(F,A)),comple7806235888213564991et_nat(image_nat_set_nat(F,A4))) ).

% INF_insert
tff(fact_3374_INF__insert,axiom,
    ! [F: fun_Pr1628945989756059814et_nat,A: product_unit,A4: set_Product_unit] : comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro6064029160526250199et_nat(F,A)),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A4))) ).

% INF_insert
tff(fact_3375_INF__insert,axiom,
    ! [F: fun_o_set_nat,A: $o,A4: set_o] : comple7806235888213564991et_nat(image_o_set_nat(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(F,(A))),comple7806235888213564991et_nat(image_o_set_nat(F,A4))) ).

% INF_insert
tff(fact_3376_INF__insert,axiom,
    ! [F: fun_Pr2481279687482350479t_unit,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4))) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro676242055593665672t_unit(F,A)),comple2584293577114468500t_unit(image_6169909464789311569t_unit(F,A4))) ).

% INF_insert
tff(fact_3377_INF__insert,axiom,
    ! [F: fun_na7044673412300895210at_nat,A: nat,A4: set_nat] : comple1930127330929219688at_nat(image_7178329752028323786at_nat(F,aa_set_nat_set_nat(insert_nat(A),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_nat7062706319057040595at_nat(F,A)),comple1930127330929219688at_nat(image_7178329752028323786at_nat(F,A4))) ).

% INF_insert
tff(fact_3378_INF__insert,axiom,
    ! [F: fun_Pr3332065415989118679at_nat,A: product_unit,A4: set_Product_unit] : comple1930127330929219688at_nat(image_3617315916088684441at_nat(F,aa_set8324764978855220220t_unit(insert_Product_unit(A),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro5062749757247690704at_nat(F,A)),comple1930127330929219688at_nat(image_3617315916088684441at_nat(F,A4))) ).

% INF_insert
tff(fact_3379_INF__insert,axiom,
    ! [F: fun_o_4630291819367765772at_nat,A: $o,A4: set_o] : comple1930127330929219688at_nat(image_7077428525161401678at_nat(F,aa_set_o_set_o(insert_o((A)),A4))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(F,(A))),comple1930127330929219688at_nat(image_7077428525161401678at_nat(F,A4))) ).

% INF_insert
tff(fact_3380_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,C3: set_Product_unit,B5: set_nat] :
      aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(A4,C3))),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,B5,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_vl(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3381_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_o_set_nat,C3: set_o,B5: set_nat] :
      aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_o,B5,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_vm(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3382_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_nat_set_nat,C3: set_nat,B5: set_nat] :
      aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_nat,B5,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_vn(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3383_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_Pr9038118721296872730et_nat,C3: set_Pr1261947904930325089at_nat,B5: set_nat] :
      aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,C3))),B5) = $ite(C3 = bot_bo2099793752762293965at_nat,B5,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(aa_set3599674627694894166et_nat(aTP_Lamm_xz(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3384_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_Pr53831763054929981at_nat,C3: set_Product_unit,B5: set_Pr8693737435421807431at_nat] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_7288820656315755647at_nat(A4,C3))),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,B5,comple2748534634209756852at_nat(image_7288820656315755647at_nat(aa_set3561981400713889608at_nat(aTP_Lamm_ya(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3385_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_o_7500966680246433010at_nat,C3: set_o,B5: set_Pr8693737435421807431at_nat] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_8493167801588156084at_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_o,B5,comple2748534634209756852at_nat(image_8493167801588156084at_nat(aa_set4424263627741053053at_nat(aTP_Lamm_yb(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3386_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_na317059118361677008at_nat,C3: set_nat,B5: set_Pr8693737435421807431at_nat] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_4592875511192039216at_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_nat,B5,comple2748534634209756852at_nat(image_4592875511192039216at_nat(aa_set911474328896584197at_nat(aTP_Lamm_yc(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3387_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_Pr1686037257031381053at_nat,C3: set_Product_unit,B5: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple4364388969068068532at_nat(image_5172358348707908735at_nat(A4,C3))),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,B5,comple4364388969068068532at_nat(image_5172358348707908735at_nat(aa_set6512053633918173000at_nat(aTP_Lamm_yd(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3388_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_Pr6108405906935189705at_nat,C3: set_Pr1261947904930325089at_nat,B5: set_Pr8693737435421807431at_nat] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(comple2748534634209756852at_nat(image_5803077947706746251at_nat(A4,C3))),B5) = $ite(C3 = bot_bo2099793752762293965at_nat,B5,comple2748534634209756852at_nat(image_5803077947706746251at_nat(aa_set2180152308823829844at_nat(aTP_Lamm_ye(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3389_UN__extend__simps_I2_J,axiom,
    ! [A4: fun_o_7855627885593154802at_nat,C3: set_o,B5: set_Pr4329608150637261639at_nat] :
      aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(comple4364388969068068532at_nat(image_2496312346330513076at_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_o,B5,comple4364388969068068532at_nat(image_2496312346330513076at_nat(aa_set7164036131237892221at_nat(aTP_Lamm_yf(A4),B5),C3))) ).

% UN_extend_simps(2)
tff(fact_3390_UN__extend__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,A4,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_vb(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3391_UN__extend__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_o_set_nat,C3: set_o] :
      aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,A4,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_vc(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3392_UN__extend__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat,C3: set_nat] :
      aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_nat,A4,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_vd(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3393_UN__extend__simps_I3_J,axiom,
    ! [A4: set_nat,B5: fun_Pr9038118721296872730et_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set_nat_set_nat(sup_sup_set_nat(A4),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,C3))) = $ite(C3 = bot_bo2099793752762293965at_nat,A4,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(aa_fun994922058233850812et_nat(aTP_Lamm_yg(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3394_UN__extend__simps_I3_J,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B5: fun_Pr53831763054929981at_nat,C3: set_Product_unit] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_7288820656315755647at_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,A4,comple2748534634209756852at_nat(image_7288820656315755647at_nat(aa_fun3999407453037076664at_nat(aTP_Lamm_yh(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3395_UN__extend__simps_I3_J,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B5: fun_o_7500966680246433010at_nat,C3: set_o] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_8493167801588156084at_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,A4,comple2748534634209756852at_nat(image_8493167801588156084at_nat(aa_fun1001176853729534498at_nat(aTP_Lamm_yi(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3396_UN__extend__simps_I3_J,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B5: fun_na317059118361677008at_nat,C3: set_nat] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_4592875511192039216at_nat(B5,C3))) = $ite(C3 = bot_bot_set_nat,A4,comple2748534634209756852at_nat(image_4592875511192039216at_nat(aa_fun379109724618789564at_nat(aTP_Lamm_yj(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3397_UN__extend__simps_I3_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: fun_Pr1686037257031381053at_nat,C3: set_Product_unit] :
      aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),comple4364388969068068532at_nat(image_5172358348707908735at_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,A4,comple4364388969068068532at_nat(image_5172358348707908735at_nat(aa_fun3072769712689967288at_nat(aTP_Lamm_yk(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3398_UN__extend__simps_I3_J,axiom,
    ! [A4: set_Pr8693737435421807431at_nat,B5: fun_Pr6108405906935189705at_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(A4),comple2748534634209756852at_nat(image_5803077947706746251at_nat(B5,C3))) = $ite(C3 = bot_bo2099793752762293965at_nat,A4,comple2748534634209756852at_nat(image_5803077947706746251at_nat(aa_fun393720513254445520at_nat(aTP_Lamm_yl(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3399_UN__extend__simps_I3_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: fun_o_7855627885593154802at_nat,C3: set_o] :
      aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(A4),comple4364388969068068532at_nat(image_2496312346330513076at_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,A4,comple4364388969068068532at_nat(image_2496312346330513076at_nat(aa_fun4159517961099164194at_nat(aTP_Lamm_ym(A4),B5),C3))) ).

% UN_extend_simps(3)
tff(fact_3400_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_Pr3332065415989118679at_nat,C3: set_Product_unit,B5: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_3617315916088684441at_nat(A4,C3))),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,B5,comple1930127330929219688at_nat(image_3617315916088684441at_nat(aa_set9196459572129487356at_nat(aTP_Lamm_xn(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3401_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_Pr6212429452034653949at_nat,C3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7979982168039674559at_nat(A4,C3))),B5) = $ite(C3 = bot_bo228742789529271731at_nat,B5,comple1930127330929219688at_nat(image_7979982168039674559at_nat(aa_set2412372282518133026at_nat(aTP_Lamm_yn(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3402_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_Pr1469247711798767971at_nat,C3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(A4),C3))),B5) = $ite(C3 = bot_bo2099793752762293965at_nat,B5,comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(aa_set6004322782020247304at_nat(aTP_Lamm_yo(A4),B5)),C3))) ).

% INT_extend_simps(1)
tff(fact_3403_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_o_4630291819367765772at_nat,C3: set_o,B5: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7077428525161401678at_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_o,B5,comple1930127330929219688at_nat(image_7077428525161401678at_nat(aa_set7523300704893535665at_nat(aTP_Lamm_xo(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3404_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_na7044673412300895210at_nat,C3: set_nat,B5: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(comple1930127330929219688at_nat(image_7178329752028323786at_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_nat,B5,comple1930127330929219688at_nat(image_7178329752028323786at_nat(aa_set340465950006982789at_nat(aTP_Lamm_xp(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3405_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,C3: set_Product_unit,B5: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_8173849991827986510et_nat(A4,C3))),B5) = $ite(C3 = bot_bo3957492148770167129t_unit,B5,comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_ss(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3406_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_Pr6764980218223570944et_nat,C3: set_Pr4329608150637261639at_nat,B5: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_3160451284896210088et_nat(A4,C3))),B5) = $ite(C3 = bot_bo228742789529271731at_nat,B5,comple7806235888213564991et_nat(image_3160451284896210088et_nat(aa_set3215376554459393340et_nat(aTP_Lamm_yp(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3407_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_Pr9038118721296872730et_nat,C3: set_Pr1261947904930325089at_nat,B5: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_15824709712370754et_nat(A4,C3))),B5) = $ite(C3 = bot_bo2099793752762293965at_nat,B5,comple7806235888213564991et_nat(image_15824709712370754et_nat(aa_set3599674627694894166et_nat(aTP_Lamm_yq(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3408_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_o_set_nat,C3: set_o,B5: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_o_set_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_o,B5,comple7806235888213564991et_nat(image_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_xk(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3409_INT__extend__simps_I1_J,axiom,
    ! [A4: fun_nat_set_nat,C3: set_nat,B5: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(comple7806235888213564991et_nat(image_nat_set_nat(A4,C3))),B5) = $ite(C3 = bot_bot_set_nat,B5,comple7806235888213564991et_nat(image_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_xl(A4),B5),C3))) ).

% INT_extend_simps(1)
tff(fact_3410_INT__extend__simps_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr3332065415989118679at_nat,C3: set_Product_unit] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple1930127330929219688at_nat(image_3617315916088684441at_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,A4,comple1930127330929219688at_nat(image_3617315916088684441at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_xw(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3411_INT__extend__simps_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr6212429452034653949at_nat,C3: set_Pr4329608150637261639at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple1930127330929219688at_nat(image_7979982168039674559at_nat(B5,C3))) = $ite(C3 = bot_bo228742789529271731at_nat,A4,comple1930127330929219688at_nat(image_7979982168039674559at_nat(aa_fun3339055349710477880at_nat(aTP_Lamm_yr(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3412_INT__extend__simps_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr1469247711798767971at_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(B5),C3))) = $ite(C3 = bot_bo2099793752762293965at_nat,A4,comple1930127330929219688at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(aa_fun5344049784256257540at_nat(aTP_Lamm_ys(A4),B5)),C3))) ).

% INT_extend_simps(2)
tff(fact_3413_INT__extend__simps_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_o_4630291819367765772at_nat,C3: set_o] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple1930127330929219688at_nat(image_7077428525161401678at_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,A4,comple1930127330929219688at_nat(image_7077428525161401678at_nat(aa_fun213375806635181270at_nat(aTP_Lamm_xx(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3414_INT__extend__simps_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_na7044673412300895210at_nat,C3: set_nat] :
      aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),comple1930127330929219688at_nat(image_7178329752028323786at_nat(B5,C3))) = $ite(C3 = bot_bot_set_nat,A4,comple1930127330929219688at_nat(image_7178329752028323786at_nat(aa_fun5076269500744484796at_nat(aTP_Lamm_xy(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3415_INT__extend__simps_I2_J,axiom,
    ! [A4: set_nat,B5: fun_Pr1628945989756059814et_nat,C3: set_Product_unit] :
      aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(image_8173849991827986510et_nat(B5,C3))) = $ite(C3 = bot_bo3957492148770167129t_unit,A4,comple7806235888213564991et_nat(image_8173849991827986510et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_si(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3416_INT__extend__simps_I2_J,axiom,
    ! [A4: set_nat,B5: fun_Pr6764980218223570944et_nat,C3: set_Pr4329608150637261639at_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(image_3160451284896210088et_nat(B5,C3))) = $ite(C3 = bot_bo228742789529271731at_nat,A4,comple7806235888213564991et_nat(image_3160451284896210088et_nat(aa_fun2592980488468313788et_nat(aTP_Lamm_yt(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3417_INT__extend__simps_I2_J,axiom,
    ! [A4: set_nat,B5: fun_Pr9038118721296872730et_nat,C3: set_Pr1261947904930325089at_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(image_15824709712370754et_nat(B5,C3))) = $ite(C3 = bot_bo2099793752762293965at_nat,A4,comple7806235888213564991et_nat(image_15824709712370754et_nat(aa_fun994922058233850812et_nat(aTP_Lamm_yu(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3418_INT__extend__simps_I2_J,axiom,
    ! [A4: set_nat,B5: fun_o_set_nat,C3: set_o] :
      aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(image_o_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_o,A4,comple7806235888213564991et_nat(image_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_xt(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3419_INT__extend__simps_I2_J,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat,C3: set_nat] :
      aa_set_nat_set_nat(inf_inf_set_nat(A4),comple7806235888213564991et_nat(image_nat_set_nat(B5,C3))) = $ite(C3 = bot_bot_set_nat,A4,comple7806235888213564991et_nat(image_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_xu(A4),B5),C3))) ).

% INT_extend_simps(2)
tff(fact_3420_le__cINF__iff,axiom,
    ! [A4: set_int,F: fun_int_nat,U: nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4)))
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_int_nat(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3421_le__cINF__iff,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_nat,U: nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(F,A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A4)))
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_Product_unit_nat(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3422_le__cINF__iff,axiom,
    ! [A4: set_o,F: fun_o_nat,U: nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat(F,A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(F,A4)))
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_o_nat(F,(X4))) ) ) ) ) ).

% le_cINF_iff
tff(fact_3423_le__cINF__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat,U: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)))
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),aa_nat_nat(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3424_le__cINF__iff,axiom,
    ! [A4: set_int,F: fun_int_int,U: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4)))
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_int_int(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3425_le__cINF__iff,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,U: int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(F,A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,A4)))
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_Product_unit_int(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3426_le__cINF__iff,axiom,
    ! [A4: set_o,F: fun_o_int,U: int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(F,A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4)))
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_o_int(F,(X4))) ) ) ) ) ).

% le_cINF_iff
tff(fact_3427_le__cINF__iff,axiom,
    ! [A4: set_nat,F: fun_nat_int,U: int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4)))
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,U),aa_nat_int(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3428_le__cINF__iff,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,U: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(F,A4))
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4)))
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,A4)
             => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_Pro8294289767544744956t_unit(F,X4)) ) ) ) ) ).

% le_cINF_iff
tff(fact_3429_le__cINF__iff,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,U: product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( condit9161856309968323677t_unit(image_o_Product_unit(F,A4))
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(U),comple2584293577114468500t_unit(image_o_Product_unit(F,A4)))
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => aa_Product_unit_o(ord_le3221252021190050221t_unit(U),aa_o_Product_unit(F,(X4))) ) ) ) ) ).

% le_cINF_iff
tff(fact_3430_cINF__mono,axiom,
    ! [B5: set_int,F: fun_nat_nat,A4: set_nat,G: fun_int_nat] :
      ( ( B5 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( ! [M4: int] :
              ( aa_set_int_o(member_int(M4),B5)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X5)),aa_int_nat(G,M4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ).

% cINF_mono
tff(fact_3431_cINF__mono,axiom,
    ! [B5: set_int,F: fun_int_nat,A4: set_int,G: fun_int_nat] :
      ( ( B5 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( ! [M4: int] :
              ( aa_set_int_o(member_int(M4),B5)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X5)),aa_int_nat(G,M4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ).

% cINF_mono
tff(fact_3432_cINF__mono,axiom,
    ! [B5: set_Product_unit,F: fun_nat_nat,A4: set_nat,G: fun_Product_unit_nat] :
      ( ( B5 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( ! [M4: product_unit] :
              ( member_Product_unit(M4,B5)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X5)),aa_Product_unit_nat(G,M4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(G,B5))) ) ) ) ).

% cINF_mono
tff(fact_3433_cINF__mono,axiom,
    ! [B5: set_Product_unit,F: fun_int_nat,A4: set_int,G: fun_Product_unit_nat] :
      ( ( B5 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( ! [M4: product_unit] :
              ( member_Product_unit(M4,B5)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X5)),aa_Product_unit_nat(G,M4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(G,B5))) ) ) ) ).

% cINF_mono
tff(fact_3434_cINF__mono,axiom,
    ! [B5: set_o,F: fun_nat_nat,A4: set_nat,G: fun_o_nat] :
      ( ( B5 != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( ! [M4: $o] :
              ( aa_set_o_o(member_o((M4)),B5)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X5)),aa_o_nat(G,(M4))) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(G,B5))) ) ) ) ).

% cINF_mono
tff(fact_3435_cINF__mono,axiom,
    ! [B5: set_o,F: fun_int_nat,A4: set_int,G: fun_o_nat] :
      ( ( B5 != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( ! [M4: $o] :
              ( aa_set_o_o(member_o((M4)),B5)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X5)),aa_o_nat(G,(M4))) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(G,B5))) ) ) ) ).

% cINF_mono
tff(fact_3436_cINF__mono,axiom,
    ! [B5: set_nat,F: fun_nat_nat,A4: set_nat,G: fun_nat_nat] :
      ( ( B5 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( ! [M4: nat] :
              ( aa_set_nat_o(member_nat(M4),B5)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X5)),aa_nat_nat(G,M4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ).

% cINF_mono
tff(fact_3437_cINF__mono,axiom,
    ! [B5: set_nat,F: fun_int_nat,A4: set_int,G: fun_nat_nat] :
      ( ( B5 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( ! [M4: nat] :
              ( aa_set_nat_o(member_nat(M4),B5)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),A4)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X5)),aa_nat_nat(G,M4)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ).

% cINF_mono
tff(fact_3438_cINF__mono,axiom,
    ! [B5: set_int,F: fun_nat_int,A4: set_nat,G: fun_int_int] :
      ( ( B5 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( ! [M4: int] :
              ( aa_set_int_o(member_int(M4),B5)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),A4)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X5)),aa_int_int(G,M4)) ) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(G),B5))) ) ) ) ).

% cINF_mono
tff(fact_3439_cINF__mono,axiom,
    ! [B5: set_int,F: fun_int_int,A4: set_int,G: fun_int_int] :
      ( ( B5 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( ! [M4: int] :
              ( aa_set_int_o(member_int(M4),B5)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),A4)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X5)),aa_int_int(G,M4)) ) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(G),B5))) ) ) ) ).

% cINF_mono
tff(fact_3440_cSUP__le__iff,axiom,
    ! [A4: set_int,F: fun_int_nat,U: nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),U)
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3441_cSUP__le__iff,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_nat,U: nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(F,A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A4))),U)
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3442_cSUP__le__iff,axiom,
    ! [A4: set_o,F: fun_o_nat,U: nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat(F,A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))),U)
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X4))),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3443_cSUP__le__iff,axiom,
    ! [A4: set_nat,F: fun_nat_nat,U: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),U)
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3444_cSUP__le__iff,axiom,
    ! [A4: set_int,F: fun_int_int,U: int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),U)
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3445_cSUP__le__iff,axiom,
    ! [A4: set_Product_unit,F: fun_Product_unit_int,U: int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(F,A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))),U)
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3446_cSUP__le__iff,axiom,
    ! [A4: set_o,F: fun_o_int,U: int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(F,A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))),U)
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X4))),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3447_cSUP__le__iff,axiom,
    ! [A4: set_nat,F: fun_nat_int,U: int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))),U)
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3448_cSUP__le__iff,axiom,
    ! [A4: set_Product_unit,F: fun_Pr2191293291710260995t_unit,U: product_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(F,A4))
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))),U)
        <=> ! [X4: product_unit] :
              ( member_Product_unit(X4,A4)
             => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(F,X4)),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3449_cSUP__le__iff,axiom,
    ! [A4: set_o,F: fun_o_Product_unit,U: product_unit] :
      ( ( A4 != bot_bot_set_o )
     => ( condit2080475194861589617t_unit(image_o_Product_unit(F,A4))
       => ( aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_o_Product_unit(F,A4))),U)
        <=> ! [X4: $o] :
              ( aa_set_o_o(member_o((X4)),A4)
             => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,(X4))),U) ) ) ) ) ).

% cSUP_le_iff
tff(fact_3450_cSUP__mono,axiom,
    ! [A4: set_int,G: fun_nat_nat,B5: set_nat,F: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))
       => ( ! [N2: int] :
              ( aa_set_int_o(member_int(N2),A4)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,N2)),aa_nat_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3451_cSUP__mono,axiom,
    ! [A4: set_int,G: fun_int_nat,B5: set_int,F: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),B5))
       => ( ! [N2: int] :
              ( aa_set_int_o(member_int(N2),A4)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,N2)),aa_int_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3452_cSUP__mono,axiom,
    ! [A4: set_Product_unit,G: fun_nat_nat,B5: set_nat,F: fun_Product_unit_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))
       => ( ! [N2: product_unit] :
              ( member_Product_unit(N2,A4)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,N2)),aa_nat_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3453_cSUP__mono,axiom,
    ! [A4: set_Product_unit,G: fun_int_nat,B5: set_int,F: fun_Product_unit_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),B5))
       => ( ! [N2: product_unit] :
              ( member_Product_unit(N2,A4)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,N2)),aa_int_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3454_cSUP__mono,axiom,
    ! [A4: set_o,G: fun_nat_nat,B5: set_nat,F: fun_o_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))
       => ( ! [N2: $o] :
              ( aa_set_o_o(member_o((N2)),A4)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(N2))),aa_nat_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3455_cSUP__mono,axiom,
    ! [A4: set_o,G: fun_int_nat,B5: set_int,F: fun_o_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),B5))
       => ( ! [N2: $o] :
              ( aa_set_o_o(member_o((N2)),A4)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(N2))),aa_int_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3456_cSUP__mono,axiom,
    ! [A4: set_nat,G: fun_nat_nat,B5: set_nat,F: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))
       => ( ! [N2: nat] :
              ( aa_set_nat_o(member_nat(N2),A4)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3457_cSUP__mono,axiom,
    ! [A4: set_nat,G: fun_int_nat,B5: set_int,F: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),B5))
       => ( ! [N2: nat] :
              ( aa_set_nat_o(member_nat(N2),A4)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_int_nat(G,X5)) ) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3458_cSUP__mono,axiom,
    ! [A4: set_int,G: fun_nat_int,B5: set_nat,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(G),B5))
       => ( ! [N2: int] :
              ( aa_set_int_o(member_int(N2),A4)
             => ? [X5: nat] :
                  ( aa_set_nat_o(member_nat(X5),B5)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,N2)),aa_nat_int(G,X5)) ) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3459_cSUP__mono,axiom,
    ! [A4: set_int,G: fun_int_int,B5: set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(G),B5))
       => ( ! [N2: int] :
              ( aa_set_int_o(member_int(N2),A4)
             => ? [X5: int] :
                  ( aa_set_int_o(member_int(X5),B5)
                  & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,N2)),aa_int_int(G,X5)) ) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(G),B5))) ) ) ) ).

% cSUP_mono
tff(fact_3460_prod__set__defs_I1_J,axiom,
    ! [X5: produc7812382424243267045t_unit] : basic_685971451221398055t_unit(X5) = insert_ref_a(aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,X5),bot_bot_set_ref_a) ).

% prod_set_defs(1)
tff(fact_3461_image__INT,axiom,
    ! [F: fun_int_int,C3: set_int,A4: set_o,B5: fun_o_set_int,J4: $o] :
      ( aa_set_int_o(inj_on_int_int(F),C3)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_int_o(ord_less_eq_set_int(aa_o_set_int(B5,(X))),C3) )
       => ( aa_set_o_o(member_o((J4)),A4)
         => ( aa_set_int_set_int(image_int_int(F),comple3628384868704368283et_int(image_o_set_int(B5,A4))) = comple3628384868704368283et_int(image_o_set_int(aa_fun1981618354277381628et_int(aTP_Lamm_yv(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3462_image__INT,axiom,
    ! [F: fun_int_int,C3: set_int,A4: set_nat,B5: fun_nat_set_int,J4: nat] :
      ( aa_set_int_o(inj_on_int_int(F),C3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(B5,X)),C3) )
       => ( aa_set_nat_o(member_nat(J4),A4)
         => ( aa_set_int_set_int(image_int_int(F),comple3628384868704368283et_int(image_nat_set_int(B5,A4))) = comple3628384868704368283et_int(image_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_yw(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3463_image__INT,axiom,
    ! [F: fun_int_int,C3: set_int,A4: set_int,B5: fun_int_set_int,J4: int] :
      ( aa_set_int_o(inj_on_int_int(F),C3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(B5,X)),C3) )
       => ( aa_set_int_o(member_int(J4),A4)
         => ( aa_set_int_set_int(image_int_int(F),comple3628384868704368283et_int(image_int_set_int(B5,A4))) = comple3628384868704368283et_int(image_int_set_int(aa_fun2302365292583850068et_int(aTP_Lamm_yx(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3464_image__INT,axiom,
    ! [F: fun_int_nat,C3: set_int,A4: set_o,B5: fun_o_set_int,J4: $o] :
      ( aa_set_int_o(inj_on_int_nat(F),C3)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_int_o(ord_less_eq_set_int(aa_o_set_int(B5,(X))),C3) )
       => ( aa_set_o_o(member_o((J4)),A4)
         => ( aa_set_int_set_nat(image_int_nat(F),comple3628384868704368283et_int(image_o_set_int(B5,A4))) = comple7806235888213564991et_nat(image_o_set_nat(aa_fun4471701499209108384et_nat(aTP_Lamm_yy(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3465_image__INT,axiom,
    ! [F: fun_int_nat,C3: set_int,A4: set_nat,B5: fun_nat_set_int,J4: nat] :
      ( aa_set_int_o(inj_on_int_nat(F),C3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(B5,X)),C3) )
       => ( aa_set_nat_o(member_nat(J4),A4)
         => ( aa_set_int_set_nat(image_int_nat(F),comple3628384868704368283et_int(image_nat_set_int(B5,A4))) = comple7806235888213564991et_nat(image_nat_set_nat(aa_fun77027002720559424et_nat(aTP_Lamm_yz(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3466_image__INT,axiom,
    ! [F: fun_int_nat,C3: set_int,A4: set_int,B5: fun_int_set_int,J4: int] :
      ( aa_set_int_o(inj_on_int_nat(F),C3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A4)
           => aa_set_int_o(ord_less_eq_set_int(aa_int_set_int(B5,X)),C3) )
       => ( aa_set_int_o(member_int(J4),A4)
         => ( aa_set_int_set_nat(image_int_nat(F),comple3628384868704368283et_int(image_int_set_int(B5,A4))) = comple7806235888213564991et_nat(image_int_set_nat(aa_fun4792448437515576824et_nat(aTP_Lamm_za(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3467_image__INT,axiom,
    ! [F: fun_nat_rat,C3: set_nat,A4: set_o,B5: fun_o_set_nat,J4: $o] :
      ( aa_set_nat_o(inj_on_nat_rat(F),C3)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(B5,(X))),C3) )
       => ( aa_set_o_o(member_o((J4)),A4)
         => ( aa_set_nat_set_rat(image_nat_rat(F),comple7806235888213564991et_nat(image_o_set_nat(B5,A4))) = comple4298007329820168263et_rat(image_o_set_rat(aa_fun7077510056935631620et_rat(aTP_Lamm_zb(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3468_image__INT,axiom,
    ! [F: fun_nat_int,C3: set_nat,A4: set_o,B5: fun_o_set_nat,J4: $o] :
      ( aa_set_nat_o(inj_on_nat_int(F),C3)
     => ( ! [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_o_set_nat(B5,(X))),C3) )
       => ( aa_set_o_o(member_o((J4)),A4)
         => ( aa_set_nat_set_int(image_nat_int(F),comple7806235888213564991et_nat(image_o_set_nat(B5,A4))) = comple3628384868704368283et_int(image_o_set_int(aa_fun1744041212626806616et_int(aTP_Lamm_zc(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3469_image__INT,axiom,
    ! [F: fun_nat_rat,C3: set_nat,A4: set_nat,B5: fun_nat_set_nat,J4: nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),C3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(B5,X)),C3) )
       => ( aa_set_nat_o(member_nat(J4),A4)
         => ( aa_set_nat_set_rat(image_nat_rat(F),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))) = comple4298007329820168263et_rat(image_nat_set_rat(aa_fun3693742223670881516et_rat(aTP_Lamm_zd(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3470_image__INT,axiom,
    ! [F: fun_nat_int,C3: set_nat,A4: set_nat,B5: fun_nat_set_nat,J4: nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),C3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
           => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(B5,X)),C3) )
       => ( aa_set_nat_o(member_nat(J4),A4)
         => ( aa_set_nat_set_int(image_nat_int(F),comple7806235888213564991et_nat(image_nat_set_nat(B5,A4))) = comple3628384868704368283et_int(image_nat_set_int(aa_fun7583645416216832320et_int(aTP_Lamm_ze(F),B5),A4)) ) ) ) ) ).

% image_INT
tff(fact_3471_cINF__superset__mono,axiom,
    ! [A4: set_int,G: fun_int_nat,B5: set_int,F: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(G),B5))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),B5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X)),aa_int_nat(F,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(G),B5))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3472_cINF__superset__mono,axiom,
    ! [A4: set_Product_unit,G: fun_Product_unit_nat,B5: set_Product_unit,F: fun_Product_unit_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_875570014554754200it_nat(G,B5))
       => ( ord_le3507040750410214029t_unit(A4,B5)
         => ( ! [X: product_unit] :
                ( member_Product_unit(X,B5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(G,X)),aa_Product_unit_nat(F,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(G,B5))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3473_cINF__superset__mono,axiom,
    ! [A4: set_o,G: fun_o_nat,B5: set_o,F: fun_o_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,image_o_nat(G,B5))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( ! [X: $o] :
                ( aa_set_o_o(member_o((X)),B5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(G,(X))),aa_o_nat(F,(X))) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(G,B5))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(F,A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3474_cINF__superset__mono,axiom,
    ! [A4: set_nat,G: fun_nat_nat,B5: set_nat,F: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),B5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X)),aa_nat_nat(F,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3475_cINF__superset__mono,axiom,
    ! [A4: set_int,G: fun_int_int,B5: set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(G),B5))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),B5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(G,X)),aa_int_int(F,X)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(G),B5))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3476_cINF__superset__mono,axiom,
    ! [A4: set_Product_unit,G: fun_Product_unit_int,B5: set_Product_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(G,B5))
       => ( ord_le3507040750410214029t_unit(A4,B5)
         => ( ! [X: product_unit] :
                ( member_Product_unit(X,B5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(G,X)),aa_Product_unit_int(F,X)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(G,B5))),aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(F,A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3477_cINF__superset__mono,axiom,
    ! [A4: set_o,G: fun_o_int,B5: set_o,F: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(G,B5))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( ! [X: $o] :
                ( aa_set_o_o(member_o((X)),B5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(G,(X))),aa_o_int(F,(X))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,image_o_int(G,B5))),aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3478_cINF__superset__mono,axiom,
    ! [A4: set_nat,G: fun_nat_int,B5: set_nat,F: fun_nat_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(G),B5))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),B5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(G,X)),aa_nat_int(F,X)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(G),B5))),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3479_cINF__superset__mono,axiom,
    ! [A4: set_int,G: fun_int_Product_unit,B5: set_int,F: fun_int_Product_unit] :
      ( ( A4 != bot_bot_set_int )
     => ( condit9161856309968323677t_unit(image_5846526143423701914t_unit(G,B5))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),B5)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(G,X)),aa_int_Product_unit(F,X)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_5846526143423701914t_unit(G,B5))),comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3480_cINF__superset__mono,axiom,
    ! [A4: set_Product_unit,G: fun_Pr2191293291710260995t_unit,B5: set_Product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit9161856309968323677t_unit(image_405062704495631173t_unit(G,B5))
       => ( ord_le3507040750410214029t_unit(A4,B5)
         => ( ! [X: product_unit] :
                ( member_Product_unit(X,B5)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(G,X)),aa_Pro8294289767544744956t_unit(F,X)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple2584293577114468500t_unit(image_405062704495631173t_unit(G,B5))),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))) ) ) ) ) ).

% cINF_superset_mono
tff(fact_3481_cSUP__subset__mono,axiom,
    ! [A4: set_int,G: fun_int_nat,B5: set_int,F: fun_int_nat] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(G),B5))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(G),B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3482_cSUP__subset__mono,axiom,
    ! [A4: set_Product_unit,G: fun_Product_unit_nat,B5: set_Product_unit,F: fun_Product_unit_nat] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_875570014554754200it_nat(G,B5))
       => ( ord_le3507040750410214029t_unit(A4,B5)
         => ( ! [X: product_unit] :
                ( member_Product_unit(X,A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,X)),aa_Product_unit_nat(G,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A4))),aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(G,B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3483_cSUP__subset__mono,axiom,
    ! [A4: set_o,G: fun_o_nat,B5: set_o,F: fun_o_nat] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,image_o_nat(G,B5))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( ! [X: $o] :
                ( aa_set_o_o(member_o((X)),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,(X))),aa_o_nat(G,(X))) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))),aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(G,B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3484_cSUP__subset__mono,axiom,
    ! [A4: set_nat,G: fun_nat_nat,B5: set_nat,F: fun_nat_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),A4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),aa_nat_nat(G,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(G),B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3485_cSUP__subset__mono,axiom,
    ! [A4: set_int,G: fun_int_int,B5: set_int,F: fun_int_int] :
      ( ( A4 != bot_bot_set_int )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(G),B5))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X)),aa_int_int(G,X)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(G),B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3486_cSUP__subset__mono,axiom,
    ! [A4: set_Product_unit,G: fun_Product_unit_int,B5: set_Product_unit,F: fun_Product_unit_int] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(G,B5))
       => ( ord_le3507040750410214029t_unit(A4,B5)
         => ( ! [X: product_unit] :
                ( member_Product_unit(X,A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Product_unit_int(F,X)),aa_Product_unit_int(G,X)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))),aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(G,B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3487_cSUP__subset__mono,axiom,
    ! [A4: set_o,G: fun_o_int,B5: set_o,F: fun_o_int] :
      ( ( A4 != bot_bot_set_o )
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(G,B5))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => ( ! [X: $o] :
                ( aa_set_o_o(member_o((X)),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,(X))),aa_o_int(G,(X))) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))),aa_set_int_int(complete_Sup_Sup_int,image_o_int(G,B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3488_cSUP__subset__mono,axiom,
    ! [A4: set_nat,G: fun_nat_int,B5: set_nat,F: fun_nat_int] :
      ( ( A4 != bot_bot_set_nat )
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(G),B5))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),A4)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(G),B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3489_cSUP__subset__mono,axiom,
    ! [A4: set_int,G: fun_int_Product_unit,B5: set_int,F: fun_int_Product_unit] :
      ( ( A4 != bot_bot_set_int )
     => ( condit2080475194861589617t_unit(image_5846526143423701914t_unit(G,B5))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,X)),aa_int_Product_unit(G,X)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_5846526143423701914t_unit(F,A4))),comple4687483117567863418t_unit(image_5846526143423701914t_unit(G,B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3490_cSUP__subset__mono,axiom,
    ! [A4: set_Product_unit,G: fun_Pr2191293291710260995t_unit,B5: set_Product_unit,F: fun_Pr2191293291710260995t_unit] :
      ( ( A4 != bot_bo3957492148770167129t_unit )
     => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(G,B5))
       => ( ord_le3507040750410214029t_unit(A4,B5)
         => ( ! [X: product_unit] :
                ( member_Product_unit(X,A4)
               => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(F,X)),aa_Pro8294289767544744956t_unit(G,X)) )
           => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))),comple4687483117567863418t_unit(image_405062704495631173t_unit(G,B5))) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_3491_UNION__fun__upd,axiom,
    ! [A4: fun_in4731230917697954842t_unit,I: int,B5: set_Product_unit,J3: set_int] :
      comple3349607820504768346t_unit(image_8658272795163236474t_unit(fun_up5054257004045450754t_unit(A4,I,B5),J3)) = sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_8658272795163236474t_unit(A4,minus_minus_set_int(J3,aa_set_int_set_int(insert_int(I),bot_bot_set_int)))),
        $ite(aa_set_int_o(member_int(I),J3),B5,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3492_UNION__fun__upd,axiom,
    ! [A4: fun_int_set_o,I: int,B5: set_o,J3: set_int] :
      comple90263536869209701_set_o(image_int_set_o(fun_upd_int_set_o(A4,I,B5),J3)) = sup_sup_set_o(comple90263536869209701_set_o(image_int_set_o(A4,minus_minus_set_int(J3,aa_set_int_set_int(insert_int(I),bot_bot_set_int)))),
        $ite(aa_set_int_o(member_int(I),J3),B5,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3493_UNION__fun__upd,axiom,
    ! [A4: fun_Pr3013102738158121955t_unit,I: product_unit,B5: set_Product_unit,J3: set_Product_unit] :
      comple3349607820504768346t_unit(image_7778121751606633509t_unit(fun_up488859312337516701t_unit(A4,I,B5),J3)) = sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_7778121751606633509t_unit(A4,minus_6452836326544984404t_unit(J3,aa_set8324764978855220220t_unit(insert_Product_unit(I),bot_bo3957492148770167129t_unit)))),
        $ite(member_Product_unit(I,J3),B5,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3494_UNION__fun__upd,axiom,
    ! [A4: fun_Pr6739565440344758894_set_o,I: product_unit,B5: set_o,J3: set_Product_unit] :
      comple90263536869209701_set_o(image_6930536745178679216_set_o(fun_up1762901246852280360_set_o(A4,I,B5),J3)) = sup_sup_set_o(comple90263536869209701_set_o(image_6930536745178679216_set_o(A4,minus_6452836326544984404t_unit(J3,aa_set8324764978855220220t_unit(insert_Product_unit(I),bot_bo3957492148770167129t_unit)))),
        $ite(member_Product_unit(I,J3),B5,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3495_UNION__fun__upd,axiom,
    ! [A4: fun_o_1833192751255691672t_unit,I: $o,B5: set_Product_unit,J3: set_o] :
      comple3349607820504768346t_unit(image_663195431845683034t_unit(fun_up4718931970374059986t_unit(A4,(I),B5),J3)) = sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_663195431845683034t_unit(A4,minus_minus_set_o(J3,aa_set_o_set_o(insert_o((I)),bot_bot_set_o)))),
        $ite(aa_set_o_o(member_o((I)),J3),B5,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3496_UNION__fun__upd,axiom,
    ! [A4: fun_o_set_o,I: $o,B5: set_o,J3: set_o] :
      comple90263536869209701_set_o(image_o_set_o(fun_upd_o_set_o(A4,(I),B5),J3)) = sup_sup_set_o(comple90263536869209701_set_o(image_o_set_o(A4,minus_minus_set_o(J3,aa_set_o_set_o(insert_o((I)),bot_bot_set_o)))),
        $ite(aa_set_o_o(member_o((I)),J3),B5,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3497_UNION__fun__upd,axiom,
    ! [A4: fun_na896991816813928310t_unit,I: nat,B5: set_Product_unit,J3: set_nat] :
      comple3349607820504768346t_unit(image_4824033694279209942t_unit(fun_up1220017903161424222t_unit(A4,I,B5),J3)) = sup_su793286257634532545t_unit(comple3349607820504768346t_unit(image_4824033694279209942t_unit(A4,aa_set_nat_set_nat(minus_minus_set_nat(J3),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)))),
        $ite(aa_set_nat_o(member_nat(I),J3),B5,bot_bo3957492148770167129t_unit)) ).

% UNION_fun_upd
tff(fact_3498_UNION__fun__upd,axiom,
    ! [A4: fun_nat_set_o,I: nat,B5: set_o,J3: set_nat] :
      comple90263536869209701_set_o(image_nat_set_o(fun_upd_nat_set_o(A4,I,B5),J3)) = sup_sup_set_o(comple90263536869209701_set_o(image_nat_set_o(A4,aa_set_nat_set_nat(minus_minus_set_nat(J3),aa_set_nat_set_nat(insert_nat(I),bot_bot_set_nat)))),
        $ite(aa_set_nat_o(member_nat(I),J3),B5,bot_bot_set_o)) ).

% UNION_fun_upd
tff(fact_3499_UNION__fun__upd,axiom,
    ! [A4: fun_int_set_nat,I: int,B5: set_nat,J3: set_int] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(fun_upd_int_set_nat(A4,I,B5),J3)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(A4,minus_minus_set_int(J3,aa_set_int_set_int(insert_int(I),bot_bot_set_int))))),
        $ite(aa_set_int_o(member_int(I),J3),B5,bot_bot_set_nat)) ).

% UNION_fun_upd
tff(fact_3500_UNION__fun__upd,axiom,
    ! [A4: fun_Pr1628945989756059814et_nat,I: product_unit,B5: set_nat,J3: set_Product_unit] :
      aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(fun_up4569834200710200790et_nat(A4,I,B5),J3)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(A4,minus_6452836326544984404t_unit(J3,aa_set8324764978855220220t_unit(insert_Product_unit(I),bot_bo3957492148770167129t_unit))))),
        $ite(member_Product_unit(I,J3),B5,bot_bot_set_nat)) ).

% UNION_fun_upd
tff(fact_3501_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_rat,G: fun_nat_rat,A4: set_nat] : image_2021340605491722874at_rat(aa_fun8989050709201395191at_rat(aTP_Lamm_zf(F),G),A4) = produc8728714639637146746at_rat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set4177764166417342247et_rat(aa_fun4037799688509761353et_rat(aTP_Lamm_zg(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3502_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_rat,G: fun_nat_nat,A4: set_nat] : image_5529569163885119602at_nat(aa_fun3500786184992586743at_nat(aTP_Lamm_zh(F),G),A4) = produc140472662868866674at_nat(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set1334378467040243999et_nat(aa_fun2282505651424660281et_nat(aTP_Lamm_zi(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3503_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_rat,G: fun_nat_int,A4: set_nat] : image_1351718144375922894at_int(aa_fun8155003486069834487at_int(aTP_Lamm_zj(F),G),A4) = produc137982192359816398at_int(aa_set_nat_set_rat(image_nat_rat(F),A4),aa_set8067667358963293051et_int(aa_fun8515194321480695793et_int(aTP_Lamm_zk(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3504_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_int,G: fun_nat_rat,A4: set_nat] : image_3337248961327643342nt_rat(aa_fun6353284437548643403nt_rat(aTP_Lamm_zl(F),G),A4) = produc821250958618291406nt_rat(aa_set_nat_set_int(image_nat_int(F),A4),aa_set3777344917092667259et_rat(aa_fun1292496876412202397et_rat(aTP_Lamm_zm(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3505_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_int,G: fun_nat_nat,A4: set_nat] : image_6845477519721040070nt_nat(aa_fun865019913339834955nt_nat(aTP_Lamm_zn(F),G),A4) = produc1456381018704787142nt_nat(aa_set_nat_set_int(image_nat_int(F),A4),aa_set933959217715569011et_nat(aa_fun8760574876181877133et_nat(aTP_Lamm_zo(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3506_image__split__eq__Sigma,axiom,
    ! [F: fun_int_int,G: fun_int_nat,A4: set_int] : image_659947567330398058nt_nat(aa_fun1008949613753415571nt_nat(aTP_Lamm_zp(F),G),A4) = produc1456381018704787142nt_nat(aa_set_int_set_int(image_int_int(F),A4),aa_set5326422216746463695et_nat(aa_fun7229874773247449157et_nat(aTP_Lamm_zq(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3507_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,A4: set_nat] : image_2667626500211843362nt_int(aa_fun5519237214417082699nt_int(aTP_Lamm_zr(F),G),A4) = produc1453890548195736866nt_int(aa_set_nat_set_int(image_nat_int(F),A4),aa_set7667248109638618063et_int(aa_fun5769891509383136837et_int(aTP_Lamm_zs(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3508_image__split__eq__Sigma,axiom,
    ! [F: fun_int_int,G: fun_int_int,A4: set_int] : image_5705468584675977158nt_int(aa_fun5663166914830663315nt_int(aTP_Lamm_zt(F),G),A4) = produc1453890548195736866nt_int(aa_set_int_set_int(image_int_int(F),A4),aa_set2836339071814736939et_int(aa_fun4239191406448708861et_int(aTP_Lamm_zu(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3509_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_Product_unit,G: fun_nat_rat,A4: set_nat] : image_469905907365485675it_rat(aa_fun7557576995580850244it_rat(aTP_Lamm_zv(F),G),A4) = produc6240246942209516957it_rat(image_8730104196221521654t_unit(F,A4),aa_set717647588170897386et_rat(aa_fun1928299002622888286et_rat(aTP_Lamm_zw(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3510_image__split__eq__Sigma,axiom,
    ! [F: fun_nat_Product_unit,G: fun_nat_nat,A4: set_nat] : image_3978134465758882403it_nat(aa_fun2380121109533724724it_nat(aTP_Lamm_zx(F),G),A4) = produc6875377002296012693it_nat(image_8730104196221521654t_unit(F,A4),aa_set7097633925648574946et_nat(aa_fun5826417969294899038et_nat(aTP_Lamm_zy(F),G),A4)) ).

% image_split_eq_Sigma
tff(fact_3511_If__the__inv__into__f__f,axiom,
    ! [I: num,C3: set_num,G: fun_num_o,Xa: num] :
      ( member_num(I,C3)
     => ( inj_on_num_o(G,C3)
       => ( aa_num_num(comp_o_num_num(aa_num_fun_o_num(aa_fun4126176751244095145_o_num(aTP_Lamm_zz(C3),G),Xa),G),I) = aa_num_num(id_num,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3512_If__the__inv__into__f__f,axiom,
    ! [I: num,C3: set_num,G: fun_num_nat,Xa: num] :
      ( member_num(I,C3)
     => ( inj_on_num_nat(G,C3)
       => ( aa_num_num(comp_nat_num_num(aa_num_fun_nat_num(aa_fun5532158766709364899at_num(aTP_Lamm_aaa(C3),G),Xa),G),I) = aa_num_num(id_num,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3513_If__the__inv__into__f__f,axiom,
    ! [I: num,C3: set_num,G: fun_num_int,Xa: num] :
      ( member_num(I,C3)
     => ( inj_on_num_int(G,C3)
       => ( aa_num_num(comp_int_num_num(aa_num_fun_int_num(aa_fun463138793435060571nt_num(aTP_Lamm_aab(C3),G),Xa),G),I) = aa_num_num(id_num,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3514_If__the__inv__into__f__f,axiom,
    ! [I: $o,C3: set_o,G: fun_o_o,Xa: $o] :
      ( aa_set_o_o(member_o((I)),C3)
     => ( aa_set_o_o(inj_on_o_o(G),C3)
       => ( aa_o_o(comp_o_o_o(aa_o_fun_o_o(aa_fun7163513353746205997un_o_o(aTP_Lamm_aac(C3),G),(Xa)),G),(I))
        <=> aa_o_o(id_o,(I)) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3515_If__the__inv__into__f__f,axiom,
    ! [I: $o,C3: set_o,G: fun_o_nat,Xa: $o] :
      ( aa_set_o_o(member_o((I)),C3)
     => ( aa_set_o_o(inj_on_o_nat(G),C3)
       => ( aa_o_o(comp_nat_o_o(aa_o_fun_nat_o(aa_fun336398880946066315_nat_o(aTP_Lamm_aad(C3),G),(Xa)),G),(I))
        <=> aa_o_o(id_o,(I)) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3516_If__the__inv__into__f__f,axiom,
    ! [I: $o,C3: set_o,G: fun_o_int,Xa: $o] :
      ( aa_set_o_o(member_o((I)),C3)
     => ( aa_set_o_o(inj_on_o_int(G),C3)
       => ( aa_o_o(comp_int_o_o(aa_o_fun_int_o(aa_fun5004808018037445259_int_o(aTP_Lamm_aae(C3),G),(Xa)),G),(I))
        <=> aa_o_o(id_o,(I)) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3517_If__the__inv__into__f__f,axiom,
    ! [I: nat,C3: set_nat,G: fun_nat_rat,Xa: nat] :
      ( aa_set_nat_o(member_nat(I),C3)
     => ( aa_set_nat_o(inj_on_nat_rat(G),C3)
       => ( aa_nat_nat(comp_rat_nat_nat(aa_nat_fun_rat_nat(aa_fun4643191702187387861at_nat(aTP_Lamm_aaf(C3),G),Xa),G),I) = aa_nat_nat(id_nat,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3518_If__the__inv__into__f__f,axiom,
    ! [I: nat,C3: set_nat,G: fun_nat_o,Xa: nat] :
      ( aa_set_nat_o(member_nat(I),C3)
     => ( aa_set_nat_o(inj_on_nat_o(G),C3)
       => ( aa_nat_nat(comp_o_nat_nat(aa_nat_fun_o_nat(aa_fun3953650679475262431_o_nat(aTP_Lamm_aag(C3),G),Xa),G),I) = aa_nat_nat(id_nat,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3519_If__the__inv__into__f__f,axiom,
    ! [I: nat,C3: set_nat,G: fun_nat_nat,Xa: nat] :
      ( aa_set_nat_o(member_nat(I),C3)
     => ( aa_set_nat_o(inj_on_nat_nat(G),C3)
       => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aah(C3),G),Xa)),G),I) = aa_nat_nat(id_nat,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3520_If__the__inv__into__f__f,axiom,
    ! [I: nat,C3: set_nat,G: fun_nat_int,Xa: nat] :
      ( aa_set_nat_o(member_nat(I),C3)
     => ( aa_set_nat_o(inj_on_nat_int(G),C3)
       => ( aa_nat_nat(comp_int_nat_nat(aa_nat_fun_int_nat(aa_fun1581638925303081085nt_nat(aTP_Lamm_aai(C3),G),Xa),G),I) = aa_nat_nat(id_nat,I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_3521_Field__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,Ra: set_Pr7459493094073627847at_nat] : field_1428494348145443988at_nat(insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),Ra)) = sup_su474284620922681339at_nat(insert2290795007606087959at_nat(A,insert2290795007606087959at_nat(B,bot_bo1534825418459847827at_nat)),field_1428494348145443988at_nat(Ra)) ).

% Field_insert
tff(fact_3522_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Ra: set_Pr4329608150637261639at_nat] : field_3145993188331922222at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),Ra)) = sup_su3642409539654194069at_nat(insert9200635055090092081at_nat(A,insert9200635055090092081at_nat(B,bot_bo3083307316010499117at_nat)),field_3145993188331922222at_nat(Ra)) ).

% Field_insert
tff(fact_3523_Field__insert,axiom,
    ! [A: int,B: int,Ra: set_Pr958786334691620121nt_int] : field_int(insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),Ra)) = sup_sup_set_int(aa_set_int_set_int(insert_int(A),aa_set_int_set_int(insert_int(B),bot_bot_set_int)),field_int(Ra)) ).

% Field_insert
tff(fact_3524_Field__insert,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,Ra: set_Pr553994874890374343at_nat] : field_3044959742847878580at_nat(insert5475735164933346999at_nat(produc4662710985925991255at_nat(A,B),Ra)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(insert5050368324300391991at_nat(A,insert5050368324300391991at_nat(B,bot_bo5327735625951526323at_nat))),field_3044959742847878580at_nat(Ra)) ).

% Field_insert
tff(fact_3525_Field__insert,axiom,
    ! [A: product_unit,B: product_unit,Ra: set_Pr5094982260447487303t_unit] : field_Product_unit(insert4616891024742550839t_unit(produc5692694829384537815t_unit(A,B),Ra)) = sup_su793286257634532545t_unit(aa_set8324764978855220220t_unit(insert_Product_unit(A),aa_set8324764978855220220t_unit(insert_Product_unit(B),bot_bo3957492148770167129t_unit)),field_Product_unit(Ra)) ).

% Field_insert
tff(fact_3526_Field__insert,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,Ra: set_Pr5564308138774400199at_nat] : field_2862844103579723188at_nat(insert398606708087465655at_nat(produc2335810615602371415at_nat(A,B),Ra)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(B),bot_bo228742789529271731at_nat))),field_2862844103579723188at_nat(Ra)) ).

% Field_insert
tff(fact_3527_Field__insert,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Ra: set_Pr8693737435421807431at_nat] : field_4706345496283612622at_nat(insert5050368324300391991at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),Ra)) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(B),bot_bo2099793752762293965at_nat))),field_4706345496283612622at_nat(Ra)) ).

% Field_insert
tff(fact_3528_Field__insert,axiom,
    ! [A: $o,B: $o,Ra: set_Product_prod_o_o] : field_o(insert6201435330877294327od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),Ra)) = sup_sup_set_o(aa_set_o_set_o(insert_o((A)),aa_set_o_set_o(insert_o((B)),bot_bot_set_o)),field_o(Ra)) ).

% Field_insert
tff(fact_3529_Field__insert,axiom,
    ! [A: nat,B: nat,Ra: set_Pr1261947904930325089at_nat] : aa_set736839985058952875et_nat(field_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),Ra)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_set_nat_set_nat(insert_nat(A),aa_set_nat_set_nat(insert_nat(B),bot_bot_set_nat))),aa_set736839985058952875et_nat(field_nat,Ra)) ).

% Field_insert
tff(fact_3530_bdd__below__image__antimono,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( order_2045666486212196060at_rat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% bdd_below_image_antimono
tff(fact_3531_bdd__below__image__antimono,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( order_2680796546298691796at_nat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% bdd_below_image_antimono
tff(fact_3532_bdd__below__image__antimono,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( order_2678306075789641520at_int(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% bdd_below_image_antimono
tff(fact_3533_bdd__below__image__antimono,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( order_3680150258199746352nt_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,A4)
       => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ).

% bdd_below_image_antimono
tff(fact_3534_bdd__below__image__antimono,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( order_3677659787690696076nt_int(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,A4)
       => aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4)) ) ) ).

% bdd_below_image_antimono
tff(fact_3535_bdd__above__image__antimono,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( order_2045666486212196060at_rat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => aa_set_rat_o(condit1579696412822616692ve_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% bdd_above_image_antimono
tff(fact_3536_bdd__above__image__antimono,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( order_2680796546298691796at_nat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% bdd_above_image_antimono
tff(fact_3537_bdd__above__image__antimono,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( order_2678306075789641520at_int(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% bdd_above_image_antimono
tff(fact_3538_bdd__above__image__antimono,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( order_3680150258199746352nt_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ).

% bdd_above_image_antimono
tff(fact_3539_bdd__above__image__antimono,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( order_3677659787690696076nt_int(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4)) ) ) ).

% bdd_above_image_antimono
tff(fact_3540_the__inv__f__o__f__id,axiom,
    ! [F: fun_nat_nat,Z: nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,the_inv_into_nat_nat(top_top_set_nat,F)),F),Z) = aa_nat_nat(id_nat,Z) ) ) ).

% the_inv_f_o_f_id
tff(fact_3541_If__the__inv__into__in__Func,axiom,
    ! [G: fun_Product_unit_o,C3: set_Product_unit,B5: set_Product_unit,Xa: product_unit] :
      ( inj_on8854722734121536060unit_o(G,C3)
     => ( ord_le3507040750410214029t_unit(C3,sup_su793286257634532545t_unit(B5,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))
       => member2508651315732621391t_unit(aa_Pro1198011010171541257t_unit(aa_set4357366655728970727t_unit(aTP_Lamm_aaj(G),C3),Xa),bNF_We8730861560085513481t_unit(top_top_set_o,sup_su793286257634532545t_unit(B5,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3542_If__the__inv__into__in__Func,axiom,
    ! [G: fun_o_o,C3: set_o,B5: set_o,Xa: $o] :
      ( aa_set_o_o(inj_on_o_o(G),C3)
     => ( aa_set_o_o(ord_less_eq_set_o(C3),sup_sup_set_o(B5,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))
       => member_fun_o_o(aa_o_fun_o_o(aa_set4838170466384450098un_o_o(aTP_Lamm_aak(G),C3),(Xa)),bNF_We7690717809131995284nc_o_o(top_top_set_o,sup_sup_set_o(B5,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3543_If__the__inv__into__in__Func,axiom,
    ! [G: fun_nat_o,C3: set_nat,B5: set_nat,Xa: nat] :
      ( aa_set_nat_o(inj_on_nat_o(G),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C3),aa_set_nat_set_nat(sup_sup_set_nat(B5),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))
       => member_fun_o_nat(aa_nat_fun_o_nat(aa_set7102390759555960768_o_nat(aTP_Lamm_aal(G),C3),Xa),bNF_We6830391624356636116_o_nat(top_top_set_o,aa_set_nat_set_nat(sup_sup_set_nat(B5),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3544_If__the__inv__into__in__Func,axiom,
    ! [G: fun_Pr4311113161502339697t_char,C3: set_Product_unit,B5: set_Product_unit,Xa: product_unit] :
      ( inj_on1959294024837303853t_char(G,C3)
     => ( ord_le3507040750410214029t_unit(C3,sup_su793286257634532545t_unit(B5,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))
       => member7491614633282302462t_unit(aa_Pro30065888502685636t_unit(aa_set5681920794977454460t_unit(aTP_Lamm_aam(G),C3),Xa),bNF_We7396736742984212070t_unit(top_top_set_char,sup_su793286257634532545t_unit(B5,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3545_If__the__inv__into__in__Func,axiom,
    ! [G: fun_o_char,C3: set_o,B5: set_o,Xa: $o] :
      ( aa_set_o_o(inj_on_o_char(G),C3)
     => ( aa_set_o_o(ord_less_eq_set_o(C3),sup_sup_set_o(B5,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))
       => member_fun_char_o(aa_o_fun_char_o(aa_set1568417695978065863char_o(aTP_Lamm_aan(G),C3),(Xa)),bNF_We6174688282833725809char_o(top_top_set_char,sup_sup_set_o(B5,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3546_If__the__inv__into__in__Func,axiom,
    ! [G: fun_nat_char,C3: set_nat,B5: set_nat,Xa: nat] :
      ( aa_set_nat_o(inj_on_nat_char(G),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C3),aa_set_nat_set_nat(sup_sup_set_nat(B5),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))
       => member_fun_char_nat(aa_nat_fun_char_nat(aa_set484384187446105903ar_nat(aTP_Lamm_aao(G),C3),Xa),bNF_We935846424581991991ar_nat(top_top_set_char,aa_set_nat_set_nat(sup_sup_set_nat(B5),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3547_If__the__inv__into__in__Func,axiom,
    ! [G: fun_Product_unit_rat,C3: set_Product_unit,B5: set_Product_unit,Xa: product_unit] :
      ( inj_on7795309031694339124it_rat(G,C3)
     => ( ord_le3507040750410214029t_unit(C3,sup_su793286257634532545t_unit(B5,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))
       => member1122892197520815799t_unit(aa_Pro130249013586881149t_unit(aa_set475620726854874293t_unit(aTP_Lamm_aap(G),C3),Xa),bNF_We8564238559827753503t_unit(top_top_set_rat,sup_su793286257634532545t_unit(B5,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3548_If__the__inv__into__in__Func,axiom,
    ! [G: fun_o_rat,C3: set_o,B5: set_o,Xa: $o] :
      ( aa_set_o_o(inj_on_o_rat(G),C3)
     => ( aa_set_o_o(ord_less_eq_set_o(C3),sup_sup_set_o(B5,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))
       => member_fun_rat_o(aa_o_fun_rat_o(aa_set4692966247657309312_rat_o(aTP_Lamm_aaq(G),C3),(Xa)),bNF_We6195730058148253354_rat_o(top_top_set_rat,sup_sup_set_o(B5,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3549_If__the__inv__into__in__Func,axiom,
    ! [G: fun_nat_rat,C3: set_nat,B5: set_nat,Xa: nat] :
      ( aa_set_nat_o(inj_on_nat_rat(G),C3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C3),aa_set_nat_set_nat(sup_sup_set_nat(B5),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))
       => member_fun_rat_nat(aa_nat_fun_rat_nat(aa_set58180079379113142at_nat(aTP_Lamm_aar(G),C3),Xa),bNF_We7223962090624190782at_nat(top_top_set_rat,aa_set_nat_set_nat(sup_sup_set_nat(B5),aa_set_nat_set_nat(insert_nat(Xa),bot_bot_set_nat)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3550_If__the__inv__into__in__Func,axiom,
    ! [G: fun_int_nat,C3: set_int,B5: set_int,Xa: int] :
      ( aa_set_int_o(inj_on_int_nat(G),C3)
     => ( aa_set_int_o(ord_less_eq_set_int(C3),sup_sup_set_int(B5,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))
       => member_fun_nat_int(aa_int_fun_nat_int(aa_set123921758692524994at_int(aTP_Lamm_aas(G),C3),Xa),bNF_We7538026264050006418at_int(top_top_set_nat,sup_sup_set_int(B5,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_3551_mono__cInf,axiom,
    ! [F: fun_o_nat,A4: set_o] :
      ( order_mono_o_nat(F)
     => ( aa_set_o_o(condit5413489452508810728elow_o,A4)
       => ( ( A4 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F,aa_set_o_o(complete_Inf_Inf_o,A4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(F,A4))) ) ) ) ).

% mono_cInf
tff(fact_3552_mono__cInf,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => ( ( A4 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_set_nat_nat(complete_Inf_Inf_nat,A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))) ) ) ) ).

% mono_cInf
tff(fact_3553_mono__cInf,axiom,
    ! [F: fun_o_int,A4: set_o] :
      ( order_mono_o_int(F)
     => ( aa_set_o_o(condit5413489452508810728elow_o,A4)
       => ( ( A4 != bot_bot_set_o )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(F,aa_set_o_o(complete_Inf_Inf_o,A4))),aa_set_int_int(complete_Inf_Inf_int,image_o_int(F,A4))) ) ) ) ).

% mono_cInf
tff(fact_3554_mono__cInf,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( order_mono_nat_int(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => ( ( A4 != bot_bot_set_nat )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_set_nat_nat(complete_Inf_Inf_nat,A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(F),A4))) ) ) ) ).

% mono_cInf
tff(fact_3555_mono__cInf,axiom,
    ! [F: fun_o_Product_unit,A4: set_o] :
      ( order_7782127260464396005t_unit(F)
     => ( aa_set_o_o(condit5413489452508810728elow_o,A4)
       => ( ( A4 != bot_bot_set_o )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_o_Product_unit(F,aa_set_o_o(complete_Inf_Inf_o,A4))),comple2584293577114468500t_unit(image_o_Product_unit(F,A4))) ) ) ) ).

% mono_cInf
tff(fact_3556_mono__cInf,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( order_5727276828720722763t_unit(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => ( ( A4 != bot_bot_set_nat )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,aa_set_nat_nat(complete_Inf_Inf_nat,A4))),comple2584293577114468500t_unit(image_8730104196221521654t_unit(F,A4))) ) ) ) ).

% mono_cInf
tff(fact_3557_mono__cInf,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => ( ( A4 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,aa_set_int_int(complete_Inf_Inf_int,A4))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(F),A4))) ) ) ) ).

% mono_cInf
tff(fact_3558_mono__cInf,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( order_mono_int_int(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => ( ( A4 != bot_bot_set_int )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,aa_set_int_int(complete_Inf_Inf_int,A4))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(F),A4))) ) ) ) ).

% mono_cInf
tff(fact_3559_mono__cInf,axiom,
    ! [F: fun_int_Product_unit,A4: set_int] :
      ( order_2843698775922903023t_unit(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => ( ( A4 != bot_bot_set_int )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_int_Product_unit(F,aa_set_int_int(complete_Inf_Inf_int,A4))),comple2584293577114468500t_unit(image_5846526143423701914t_unit(F,A4))) ) ) ) ).

% mono_cInf
tff(fact_3560_mono__cInf,axiom,
    ! [F: fun_Product_unit_nat,A4: set_Product_unit] :
      ( order_7096114683908731117it_nat(F)
     => ( condit9161856309968323677t_unit(A4)
       => ( ( A4 != bot_bo3957492148770167129t_unit )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Product_unit_nat(F,comple2584293577114468500t_unit(A4))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(F,A4))) ) ) ) ).

% mono_cInf
tff(fact_3561_mono__cINF,axiom,
    ! [F: fun_nat_nat,A4: fun_int_nat,I3: set_int] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(A4),I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(A4),I3)))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_be(F),A4)),I3))) ) ) ) ).

% mono_cINF
tff(fact_3562_mono__cINF,axiom,
    ! [F: fun_nat_nat,A4: fun_nat_nat,I3: set_nat] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3)))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_bd(F),A4)),I3))) ) ) ) ).

% mono_cINF
tff(fact_3563_mono__cINF,axiom,
    ! [F: fun_nat_int,A4: fun_int_nat,I3: set_int] :
      ( order_mono_nat_int(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(A4),I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(A4),I3)))),aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_bg(F),A4)),I3))) ) ) ) ).

% mono_cINF
tff(fact_3564_mono__cINF,axiom,
    ! [F: fun_nat_int,A4: fun_nat_nat,I3: set_nat] :
      ( order_mono_nat_int(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3)))),aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_bf(F),A4)),I3))) ) ) ) ).

% mono_cINF
tff(fact_3565_mono__cINF,axiom,
    ! [F: fun_nat_Product_unit,A4: fun_int_nat,I3: set_int] :
      ( order_5727276828720722763t_unit(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(A4),I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(A4),I3)))),comple2584293577114468500t_unit(image_5846526143423701914t_unit(aa_fun8867120635469333699t_unit(aTP_Lamm_aat(F),A4),I3))) ) ) ) ).

% mono_cINF
tff(fact_3566_mono__cINF,axiom,
    ! [F: fun_nat_Product_unit,A4: fun_nat_nat,I3: set_nat] :
      ( order_5727276828720722763t_unit(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_nat_Product_unit(F,aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3)))),comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_fun3939150890428253051t_unit(aTP_Lamm_aau(F),A4),I3))) ) ) ) ).

% mono_cINF
tff(fact_3567_mono__cINF,axiom,
    ! [F: fun_int_nat,A4: fun_int_int,I3: set_int] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(A4),I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(A4),I3)))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_aav(F),A4)),I3))) ) ) ) ).

% mono_cINF
tff(fact_3568_mono__cINF,axiom,
    ! [F: fun_int_nat,A4: fun_Product_unit_int,I3: set_Product_unit] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_873079544045703924it_int(A4,I3))
       => ( ( I3 != bot_bo3957492148770167129t_unit )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(A4,I3)))),aa_set_nat_nat(complete_Inf_Inf_nat,image_875570014554754200it_nat(aa_fun7819058828246937184it_nat(aTP_Lamm_aaw(F),A4),I3))) ) ) ) ).

% mono_cINF
tff(fact_3569_mono__cINF,axiom,
    ! [F: fun_int_nat,A4: fun_o_int,I3: set_o] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,image_o_int(A4,I3))
       => ( ( I3 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,aa_set_int_int(complete_Inf_Inf_int,image_o_int(A4,I3)))),aa_set_nat_nat(complete_Inf_Inf_nat,image_o_nat(aa_fun8905617046332959520_o_nat(aTP_Lamm_aax(F),A4),I3))) ) ) ) ).

% mono_cINF
tff(fact_3570_mono__cINF,axiom,
    ! [F: fun_int_nat,A4: fun_nat_int,I3: set_nat] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(A4),I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,aa_set_int_int(complete_Inf_Inf_int,aa_set_nat_set_int(image_nat_int(A4),I3)))),aa_set_nat_nat(complete_Inf_Inf_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_aay(F),A4)),I3))) ) ) ) ).

% mono_cINF
tff(fact_3571_relcompp__distrib2,axiom,
    ! [S3: fun_Pr2028404973800837934_nat_o,T2: fun_Pr2028404973800837934_nat_o,R2: fun_Pr4738963024209204295_int_o] : relcom8911704206367525917at_int(sup_su52541223548345602_nat_o(S3,T2),R2) = sup_su556009932677587827_int_o(relcom8911704206367525917at_int(S3,R2),relcom8911704206367525917at_int(T2,R2)) ).

% relcompp_distrib2
tff(fact_3572_relcompp__distrib2,axiom,
    ! [S3: fun_int_fun_int_o,T2: fun_int_fun_int_o,R2: fun_in4891666191172872833eger_o] : relcom967497705283160040nteger(sup_su7393609867835711046_int_o(S3,T2),R2) = sup_su3989259347931230805eger_o(relcom967497705283160040nteger(S3,R2),relcom967497705283160040nteger(T2,R2)) ).

% relcompp_distrib2
tff(fact_3573_relcompp__distrib,axiom,
    ! [R2: fun_Pr2028404973800837934_nat_o,S3: fun_Pr4738963024209204295_int_o,T2: fun_Pr4738963024209204295_int_o] : relcom8911704206367525917at_int(R2,sup_su556009932677587827_int_o(S3,T2)) = sup_su556009932677587827_int_o(relcom8911704206367525917at_int(R2,S3),relcom8911704206367525917at_int(R2,T2)) ).

% relcompp_distrib
tff(fact_3574_relcompp__distrib,axiom,
    ! [R2: fun_int_fun_int_o,S3: fun_in4891666191172872833eger_o,T2: fun_in4891666191172872833eger_o] : relcom967497705283160040nteger(R2,sup_su3989259347931230805eger_o(S3,T2)) = sup_su3989259347931230805eger_o(relcom967497705283160040nteger(R2,S3),relcom967497705283160040nteger(R2,T2)) ).

% relcompp_distrib
tff(fact_3575_Field__Union,axiom,
    ! [R2: set_se7855581050983116737at_nat] : aa_set736839985058952875et_nat(field_nat,comple5685304695842803022at_nat(R2)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_5853224409075402146et_nat(field_nat,R2)) ).

% Field_Union
tff(fact_3576_the__inv__into__onto,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( image_rat_nat(the_inv_into_nat_rat(A4,F),aa_set_nat_set_rat(image_nat_rat(F),A4)) = A4 ) ) ).

% the_inv_into_onto
tff(fact_3577_the__inv__into__onto,axiom,
    ! [F: fun_rat_nat,A4: set_rat] :
      ( aa_set_rat_o(inj_on_rat_nat(F),A4)
     => ( aa_set_nat_set_rat(image_nat_rat(the_inv_into_rat_nat(A4,F)),image_rat_nat(F,A4)) = A4 ) ) ).

% the_inv_into_onto
tff(fact_3578_the__inv__into__onto,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_set_nat(image_nat_nat(the_inv_into_nat_nat(A4,F)),aa_set_nat_set_nat(image_nat_nat(F),A4)) = A4 ) ) ).

% the_inv_into_onto
tff(fact_3579_the__inv__into__onto,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_set_int(image_nat_int(the_inv_into_int_nat(A4,F)),aa_set_int_set_nat(image_int_nat(F),A4)) = A4 ) ) ).

% the_inv_into_onto
tff(fact_3580_the__inv__into__onto,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_set_nat(image_int_nat(the_inv_into_nat_int(A4,F)),aa_set_nat_set_int(image_nat_int(F),A4)) = A4 ) ) ).

% the_inv_into_onto
tff(fact_3581_the__inv__into__onto,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_set_int(image_int_int(the_inv_into_int_int(A4,F)),aa_set_int_set_int(image_int_int(F),A4)) = A4 ) ) ).

% the_inv_into_onto
tff(fact_3582_image__vimage__eq,axiom,
    ! [F: fun_ch7922468903353698889at_nat,A4: set_Pr1261947904930325089at_nat] : image_8382823969816519465at_nat(F,vimage8942442433166402803at_nat(F,A4)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),image_8382823969816519465at_nat(F,top_top_set_char)) ).

% image_vimage_eq
tff(fact_3583_image__vimage__eq,axiom,
    ! [F: fun_li4460313466401262074at_nat,A4: set_Pr1261947904930325089at_nat] : image_2795870212791037530at_nat(F,vimage8527374361057053988at_nat(F,A4)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),image_2795870212791037530at_nat(F,top_top_set_list_nat)) ).

% image_vimage_eq
tff(fact_3584_image__vimage__eq,axiom,
    ! [F: fun_ra4806459883200776834at_nat,A4: set_Pr1261947904930325089at_nat] : image_9175953278516507234at_nat(F,vimage2119786153496215084at_nat(F,A4)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),image_9175953278516507234at_nat(F,top_top_set_rat)) ).

% image_vimage_eq
tff(fact_3585_image__vimage__eq,axiom,
    ! [F: fun_nat_rat,A4: set_rat] : aa_set_nat_set_rat(image_nat_rat(F),vimage_nat_rat(F,A4)) = inf_inf_set_rat(A4,aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3586_image__vimage__eq,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(vimage_nat_nat(F),A4)) = aa_set_nat_set_nat(inf_inf_set_nat(A4),aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3587_image__vimage__eq,axiom,
    ! [F: fun_nat_int,A4: set_int] : aa_set_nat_set_int(image_nat_int(F),vimage_nat_int(F,A4)) = inf_inf_set_int(A4,aa_set_nat_set_int(image_nat_int(F),top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3588_image__vimage__eq,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_Pr1261947904930325089at_nat] : image_5846123807819985514at_nat(F,vimage8013328719654469172at_nat(F,A4)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),image_5846123807819985514at_nat(F,top_top_set_nat)) ).

% image_vimage_eq
tff(fact_3589_image__vimage__eq,axiom,
    ! [F: fun_int_nat,A4: set_nat] : aa_set_int_set_nat(image_int_nat(F),aa_set_nat_set_int(vimage_int_nat(F),A4)) = aa_set_nat_set_nat(inf_inf_set_nat(A4),aa_set_int_set_nat(image_int_nat(F),top_top_set_int)) ).

% image_vimage_eq
tff(fact_3590_image__vimage__eq,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(vimage_int_int(F),A4)) = inf_inf_set_int(A4,aa_set_int_set_int(image_int_int(F),top_top_set_int)) ).

% image_vimage_eq
tff(fact_3591_image__vimage__eq,axiom,
    ! [F: fun_in4514472496968388910at_nat,A4: set_Pr1261947904930325089at_nat] : image_8883965892284119310at_nat(F,vimage1827798767263827160at_nat(F,A4)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(A4),image_8883965892284119310at_nat(F,top_top_set_int)) ).

% image_vimage_eq
tff(fact_3592_vimage__comp,axiom,
    ! [F: fun_nat_o,G: fun_o_o,Xa: set_o] : aa_set_o_set_nat(vimage_nat_o(F),aa_set_o_set_o(vimage_o_o(G),Xa)) = aa_set_o_set_nat(vimage_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F)),Xa) ).

% vimage_comp
tff(fact_3593_vimage__comp,axiom,
    ! [F: fun_int_nat,G: fun_nat_num,Xa: set_num] : aa_set_nat_set_int(vimage_int_nat(F),aa_set_num_set_nat(vimage_nat_num(G),Xa)) = aa_set_num_set_int(vimage_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F)),Xa) ).

% vimage_comp
tff(fact_3594_vimage__comp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(vimage_nat_nat(G),Xa)) = aa_set_nat_set_nat(vimage_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),Xa) ).

% vimage_comp
tff(fact_3595_vimage__comp,axiom,
    ! [F: fun_int_int,G: fun_int_nat,Xa: set_nat] : aa_set_int_set_int(vimage_int_int(F),aa_set_nat_set_int(vimage_int_nat(G),Xa)) = aa_set_nat_set_int(vimage_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),Xa) ).

% vimage_comp
tff(fact_3596_vimage__comp,axiom,
    ! [F: fun_Co1981900195452921617nteger,G: fun_fu4800041728187187127nteger,Xa: set_fu6062072743142154809nteger] : aa_set5924712930599384894nteger(vimage5717976381819148369nteger(F),aa_set7755499389519020644nteger(vimage1838862866765761655nteger(G),Xa)) = aa_set1297487899767428746nteger(vimage2869426828964670109nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),F)),Xa) ).

% vimage_comp
tff(fact_3597_set_Ocompositionality,axiom,
    ! [F: fun_nat_o,G: fun_o_o,Set: set_o] : aa_set_o_set_nat(vimage_nat_o(F),aa_set_o_set_o(vimage_o_o(G),Set)) = aa_set_o_set_nat(vimage_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F)),Set) ).

% set.compositionality
tff(fact_3598_set_Ocompositionality,axiom,
    ! [F: fun_int_nat,G: fun_nat_num,Set: set_num] : aa_set_nat_set_int(vimage_int_nat(F),aa_set_num_set_nat(vimage_nat_num(G),Set)) = aa_set_num_set_int(vimage_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F)),Set) ).

% set.compositionality
tff(fact_3599_set_Ocompositionality,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Set: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(vimage_nat_nat(G),Set)) = aa_set_nat_set_nat(vimage_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)),Set) ).

% set.compositionality
tff(fact_3600_set_Ocompositionality,axiom,
    ! [F: fun_int_int,G: fun_int_nat,Set: set_nat] : aa_set_int_set_int(vimage_int_int(F),aa_set_nat_set_int(vimage_int_nat(G),Set)) = aa_set_nat_set_int(vimage_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)),Set) ).

% set.compositionality
tff(fact_3601_set_Ocompositionality,axiom,
    ! [F: fun_Co1981900195452921617nteger,G: fun_fu4800041728187187127nteger,Set: set_fu6062072743142154809nteger] : aa_set5924712930599384894nteger(vimage5717976381819148369nteger(F),aa_set7755499389519020644nteger(vimage1838862866765761655nteger(G),Set)) = aa_set1297487899767428746nteger(vimage2869426828964670109nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),F)),Set) ).

% set.compositionality
tff(fact_3602_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J4: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( member6009668390874684176at_nat(produc2335810615602371415at_nat(I,J4),R2)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(J4),field_2862844103579723188at_nat(R2)) ) ).

% FieldI2
tff(fact_3603_FieldI2,axiom,
    ! [I: product_prod_nat_nat,J4: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(I),J4),R2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(J4),field_4706345496283612622at_nat(R2)) ) ).

% FieldI2
tff(fact_3604_FieldI2,axiom,
    ! [I: $o,J4: $o,R2: set_Product_prod_o_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((I)),(J4)),R2)
     => aa_set_o_o(member_o((J4)),field_o(R2)) ) ).

% FieldI2
tff(fact_3605_FieldI2,axiom,
    ! [I: set_Pr4329608150637261639at_nat,J4: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,I),J4),R2)
     => member6273874620674182256at_nat(J4,field_1428494348145443988at_nat(R2)) ) ).

% FieldI2
tff(fact_3606_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J4: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,I),J4)),R2)
     => member2643936169264416010at_nat(J4,field_3145993188331922222at_nat(R2)) ) ).

% FieldI2
tff(fact_3607_FieldI2,axiom,
    ! [I: nat,J4: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,I),J4)),R2)
     => aa_set_nat_o(member_nat(J4),aa_set736839985058952875et_nat(field_nat,R2)) ) ).

% FieldI2
tff(fact_3608_FieldI2,axiom,
    ! [I: int,J4: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J4)),R2)
     => aa_set_int_o(member_int(J4),field_int(R2)) ) ).

% FieldI2
tff(fact_3609_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J4: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( member6009668390874684176at_nat(produc2335810615602371415at_nat(I,J4),R2)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(I),field_2862844103579723188at_nat(R2)) ) ).

% FieldI1
tff(fact_3610_FieldI1,axiom,
    ! [I: product_prod_nat_nat,J4: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(I),J4),R2)
     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(I),field_4706345496283612622at_nat(R2)) ) ).

% FieldI1
tff(fact_3611_FieldI1,axiom,
    ! [I: $o,J4: $o,R2: set_Product_prod_o_o] :
      ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((I)),(J4)),R2)
     => aa_set_o_o(member_o((I)),field_o(R2)) ) ).

% FieldI1
tff(fact_3612_FieldI1,axiom,
    ! [I: set_Pr4329608150637261639at_nat,J4: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,I),J4),R2)
     => member6273874620674182256at_nat(I,field_1428494348145443988at_nat(R2)) ) ).

% FieldI1
tff(fact_3613_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J4: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,I),J4)),R2)
     => member2643936169264416010at_nat(I,field_3145993188331922222at_nat(R2)) ) ).

% FieldI1
tff(fact_3614_FieldI1,axiom,
    ! [I: nat,J4: nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,I),J4)),R2)
     => aa_set_nat_o(member_nat(I),aa_set736839985058952875et_nat(field_nat,R2)) ) ).

% FieldI1
tff(fact_3615_FieldI1,axiom,
    ! [I: int,J4: int,R2: set_Pr958786334691620121nt_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J4)),R2)
     => aa_set_int_o(member_int(I),field_int(R2)) ) ).

% FieldI1
tff(fact_3616_inf__Int__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat,X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(inf_in2372659476514261276_nat_o(aa_set4399942904220837859_nat_o(aTP_Lamm_ep,R2),aa_set4399942904220837859_nat_o(aTP_Lamm_ep,S3)),X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),inf_in2456762090751560885at_nat(R2,S3)) ) ).

% inf_Int_eq2
tff(fact_3617_inf__Int__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat,X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(inf_in484816107290947356_nat_o(aa_set5605188905385065315_nat_o(aTP_Lamm_eq,R2),aa_set5605188905385065315_nat_o(aTP_Lamm_eq,S3)),X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),aa_set4883462369968598652at_nat(inf_in7913087082777306421at_nat(R2),S3)) ) ).

% inf_Int_eq2
tff(fact_3618_inf__Int__eq2,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num,X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(inf_in1133183212291220558_num_o(aa_set108017733112466993_num_o(aTP_Lamm_er,R2),aa_set108017733112466993_num_o(aTP_Lamm_er,S3)),X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),inf_in7510916698018314649at_num(R2,S3)) ) ).

% inf_Int_eq2
tff(fact_3619_inf__Int__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int,X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(inf_in8908015913983109984_int_o(aa_set742265667465749333_int_o(aTP_Lamm_et,R2),aa_set742265667465749333_int_o(aTP_Lamm_et,S3)),X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),inf_in2269163501485487111nt_int(R2,S3)) ) ).

% inf_Int_eq2
tff(fact_3620_inf__Int__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(inf_in375002649321709592_nat_o(aa_set2634951030311766213_nat_o(aTP_Lamm_es,R2),aa_set2634951030311766213_nat_o(aTP_Lamm_es,S3)),X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(R2),S3)) ) ).

% inf_Int_eq2
tff(fact_3621_fun__upd__comp,axiom,
    ! [F: fun_o_o,G: fun_nat_o,Xa: nat,Y2: $o] : aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),aa_o_fun_nat_o(fun_upd_nat_o(G,Xa),(Y2))) = aa_o_fun_nat_o(fun_upd_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G),Xa),aa_o_o(F,(Y2))) ).

% fun_upd_comp
tff(fact_3622_fun__upd__comp,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,Xa: int,Y2: nat] : aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),fun_upd_int_nat(G,Xa,Y2)) = fun_upd_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G),Xa,aa_nat_num(F,Y2)) ).

% fun_upd_comp
tff(fact_3623_fun__upd__comp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,Xa: nat,Y2: nat] : aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),aa_nat_fun_nat_nat(fun_upd_nat_nat(G,Xa),Y2)) = aa_nat_fun_nat_nat(fun_upd_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,F),G),Xa),aa_nat_nat(F,Y2)) ).

% fun_upd_comp
tff(fact_3624_fun__upd__comp,axiom,
    ! [F: fun_int_nat,G: fun_int_int,Xa: int,Y2: int] : aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),aa_int_fun_int_int(fun_upd_int_int(G,Xa),Y2)) = fun_upd_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,F),G),Xa,aa_int_nat(F,Y2)) ).

% fun_upd_comp
tff(fact_3625_fun__upd__comp,axiom,
    ! [F: fun_fu4800041728187187127nteger,G: fun_Co1981900195452921617nteger,Xa: code_integer,Y2: fun_Co8499238673658658103nteger] : aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),fun_up7364656672652868627nteger(G,Xa,Y2)) = fun_up1280136195588363103nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(F),G),Xa,aa_fun2509385170507305080nteger(F,Y2)) ).

% fun_upd_comp
tff(fact_3626_image__vimage__subset,axiom,
    ! [F: fun_nat_rat,A4: set_rat] : aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),vimage_nat_rat(F,A4))),A4) ).

% image_vimage_subset
tff(fact_3627_image__vimage__subset,axiom,
    ! [F: fun_nat_nat,A4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),aa_set_nat_set_nat(vimage_nat_nat(F),A4))),A4) ).

% image_vimage_subset
tff(fact_3628_image__vimage__subset,axiom,
    ! [F: fun_nat_int,A4: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),vimage_nat_int(F,A4))),A4) ).

% image_vimage_subset
tff(fact_3629_image__vimage__subset,axiom,
    ! [F: fun_int_nat,A4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),aa_set_nat_set_int(vimage_int_nat(F),A4))),A4) ).

% image_vimage_subset
tff(fact_3630_image__vimage__subset,axiom,
    ! [F: fun_int_int,A4: set_int] : aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),aa_set_int_set_int(vimage_int_int(F),A4))),A4) ).

% image_vimage_subset
tff(fact_3631_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),vimage_nat_rat(F,B5)) ) ).

% image_subset_iff_subset_vimage
tff(fact_3632_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(vimage_nat_nat(F),B5)) ) ).

% image_subset_iff_subset_vimage
tff(fact_3633_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
    <=> aa_set_nat_o(ord_less_eq_set_nat(A4),vimage_nat_int(F,B5)) ) ).

% image_subset_iff_subset_vimage
tff(fact_3634_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
    <=> aa_set_int_o(ord_less_eq_set_int(A4),aa_set_nat_set_int(vimage_int_nat(F),B5)) ) ).

% image_subset_iff_subset_vimage
tff(fact_3635_image__subset__iff__subset__vimage,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
    <=> aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(vimage_int_int(F),B5)) ) ).

% image_subset_iff_subset_vimage
tff(fact_3636_surj__image__vimage__eq,axiom,
    ! [F: fun_char_char,A4: set_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( image_char_char(F,vimage_char_char(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3637_surj__image__vimage__eq,axiom,
    ! [F: fun_char_rat,A4: set_rat] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ( image_char_rat(F,vimage_char_rat(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3638_surj__image__vimage__eq,axiom,
    ! [F: fun_char_nat,A4: set_nat] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
     => ( image_char_nat(F,vimage_char_nat(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3639_surj__image__vimage__eq,axiom,
    ! [F: fun_char_int,A4: set_int] :
      ( ( image_char_int(F,top_top_set_char) = top_top_set_int )
     => ( image_char_int(F,vimage_char_int(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3640_surj__image__vimage__eq,axiom,
    ! [F: fun_rat_char,A4: set_char] :
      ( ( image_rat_char(F,top_top_set_rat) = top_top_set_char )
     => ( image_rat_char(F,vimage_rat_char(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3641_surj__image__vimage__eq,axiom,
    ! [F: fun_rat_rat,A4: set_rat] :
      ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
     => ( image_rat_rat(F,vimage_rat_rat(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3642_surj__image__vimage__eq,axiom,
    ! [F: fun_rat_nat,A4: set_nat] :
      ( ( image_rat_nat(F,top_top_set_rat) = top_top_set_nat )
     => ( image_rat_nat(F,vimage_rat_nat(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3643_surj__image__vimage__eq,axiom,
    ! [F: fun_rat_int,A4: set_int] :
      ( ( image_rat_int(F,top_top_set_rat) = top_top_set_int )
     => ( image_rat_int(F,vimage_rat_int(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3644_surj__image__vimage__eq,axiom,
    ! [F: fun_nat_char,A4: set_char] :
      ( ( image_nat_char(F,top_top_set_nat) = top_top_set_char )
     => ( image_nat_char(F,vimage_nat_char(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3645_surj__image__vimage__eq,axiom,
    ! [F: fun_nat_rat,A4: set_rat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = top_top_set_rat )
     => ( aa_set_nat_set_rat(image_nat_rat(F),vimage_nat_rat(F,A4)) = A4 ) ) ).

% surj_image_vimage_eq
tff(fact_3646_bdd__below__image__mono,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( order_mono_nat_rat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => aa_set_rat_o(condit1103211067700513672ow_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% bdd_below_image_mono
tff(fact_3647_bdd__below__image__mono,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% bdd_below_image_mono
tff(fact_3648_bdd__below__image__mono,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( order_mono_nat_int(F)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,A4)
       => aa_set_int_o(condit1735850657277959132ow_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% bdd_below_image_mono
tff(fact_3649_bdd__below__image__mono,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => aa_set_nat_o(condit1738341127787009408ow_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ).

% bdd_below_image_mono
tff(fact_3650_bdd__below__image__mono,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( order_mono_int_int(F)
     => ( aa_set_int_o(condit1735850657277959132ow_int,A4)
       => aa_set_int_o(condit1735850657277959132ow_int,aa_set_int_set_int(image_int_int(F),A4)) ) ) ).

% bdd_below_image_mono
tff(fact_3651_bdd__above__image__mono,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( order_mono_nat_rat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => aa_set_rat_o(condit1579696412822616692ve_rat,aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% bdd_above_image_mono
tff(fact_3652_bdd__above__image__mono,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ) ).

% bdd_above_image_mono
tff(fact_3653_bdd__above__image__mono,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( order_mono_nat_int(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(F),A4)) ) ) ).

% bdd_above_image_mono
tff(fact_3654_bdd__above__image__mono,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,A4)
       => aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(F),A4)) ) ) ).

% bdd_above_image_mono
tff(fact_3655_bdd__above__image__mono,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( order_mono_int_int(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,A4)
       => aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(F),A4)) ) ) ).

% bdd_above_image_mono
tff(fact_3656_sup__Un__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat,X5: set_Pr4329608150637261639at_nat,Xa4: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(sup_su4361989354738342146_nat_o(aa_set4399942904220837859_nat_o(aTP_Lamm_ep,R2),aa_set4399942904220837859_nat_o(aTP_Lamm_ep,S3)),X5),Xa4)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X5),Xa4),sup_su5395584912770621083at_nat(R2,S3)) ) ).

% sup_Un_eq2
tff(fact_3657_sup__Un__eq2,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num,X5: nat,Xa4: num] :
      ( aa_num_o(aa_nat_fun_num_o(sup_su8842149202998597428_num_o(aa_set108017733112466993_num_o(aTP_Lamm_er,R2),aa_set108017733112466993_num_o(aTP_Lamm_er,S3)),X5),Xa4)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X5),Xa4),sup_su2042722026077122175at_num(R2,S3)) ) ).

% sup_Un_eq2
tff(fact_3658_sup__Un__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,X5: nat,Xa4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(sup_su8083968640029086462_nat_o(aa_set2634951030311766213_nat_o(aTP_Lamm_es,R2),aa_set2634951030311766213_nat_o(aTP_Lamm_es,S3)),X5),Xa4)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X5),Xa4)),aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(R2),S3)) ) ).

% sup_Un_eq2
tff(fact_3659_sup__Un__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int,X5: int,Xa4: int] :
      ( aa_int_o(aa_int_fun_int_o(sup_su7393609867835711046_int_o(aa_set742265667465749333_int_o(aTP_Lamm_et,R2),aa_set742265667465749333_int_o(aTP_Lamm_et,S3)),X5),Xa4)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X5),Xa4)),sup_su6024340866399070445nt_int(R2,S3)) ) ).

% sup_Un_eq2
tff(fact_3660_sup__Un__eq2,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,Xa4: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(sup_su52541223548345602_nat_o(aTP_Lamm_aaz(R2),aTP_Lamm_aaz(S3)),X5),Xa4)
    <=> member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X5),Xa4),aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(R2),S3)) ) ).

% sup_Un_eq2
tff(fact_3661_sup__Un__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat,X5: set_Pr1261947904930325089at_nat,Xa4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(sup_su5424986631308460290_nat_o(aa_set5605188905385065315_nat_o(aTP_Lamm_eq,R2),aa_set5605188905385065315_nat_o(aTP_Lamm_eq,S3)),X5),Xa4)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X5),Xa4)),aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(R2),S3)) ) ).

% sup_Un_eq2
tff(fact_3662_vimage__Union,axiom,
    ! [F: fun_nat_nat,A4: set_set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4)) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(vimage_nat_nat(F)),A4)) ).

% vimage_Union
tff(fact_3663_vimage__Times,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_nat,B5: set_nat] : aa_set8868280134176360700at_nat(vimage2449269961533847803at_nat(F),produc457027306803732586at_nat(A4,aTP_Lamm_gm(B5))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(vimage4653281326611754070at_nat(comp_P1567445206330693457at_nat(product_fst_nat_nat,F),A4)),vimage4653281326611754070at_nat(comp_P1567445206330693457at_nat(product_snd_nat_nat,F),B5)) ).

% vimage_Times
tff(fact_3664_vimage__Times,axiom,
    ! [F: fun_Pr6075555455866405753t_unit,A4: set_ref_a,B5: set_he8644174446132254637t_unit] : vimage4308919176130087787t_unit(F,produc3946373962484743846t_unit(A4,aTP_Lamm_ju(B5))) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(vimage7322659125625010723_ref_a(comp_P7986581389989349338at_nat(produc473321116828791851t_unit,F),A4)),vimage1024995831485383111t_unit(comp_P6390991081591698364at_nat(produc1126150082675451753t_unit,F),B5)) ).

% vimage_Times
tff(fact_3665_inj__on__fun__updI,axiom,
    ! [F: fun_nat_rat,A4: set_nat,Y2: rat,Xa: nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( ~ aa_set_rat_o(member_rat(Y2),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => aa_set_nat_o(inj_on_nat_rat(fun_upd_nat_rat(F,Xa,Y2)),A4) ) ) ).

% inj_on_fun_updI
tff(fact_3666_inj__on__fun__updI,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Y2: nat,Xa: nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( ~ aa_set_nat_o(member_nat(Y2),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(fun_upd_nat_nat(F,Xa),Y2)),A4) ) ) ).

% inj_on_fun_updI
tff(fact_3667_inj__on__fun__updI,axiom,
    ! [F: fun_int_nat,A4: set_int,Y2: nat,Xa: int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( ~ aa_set_nat_o(member_nat(Y2),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_set_int_o(inj_on_int_nat(fun_upd_int_nat(F,Xa,Y2)),A4) ) ) ).

% inj_on_fun_updI
tff(fact_3668_inj__on__fun__updI,axiom,
    ! [F: fun_nat_int,A4: set_nat,Y2: int,Xa: nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( ~ aa_set_int_o(member_int(Y2),aa_set_nat_set_int(image_nat_int(F),A4))
       => aa_set_nat_o(inj_on_nat_int(fun_upd_nat_int(F,Xa,Y2)),A4) ) ) ).

% inj_on_fun_updI
tff(fact_3669_inj__on__fun__updI,axiom,
    ! [F: fun_int_int,A4: set_int,Y2: int,Xa: int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( ~ aa_set_int_o(member_int(Y2),aa_set_int_set_int(image_int_int(F),A4))
       => aa_set_int_o(inj_on_int_int(aa_int_fun_int_int(fun_upd_int_int(F,Xa),Y2)),A4) ) ) ).

% inj_on_fun_updI
tff(fact_3670_set_Ocomp,axiom,
    ! [F: fun_nat_o,G: fun_o_o] : comp_s9189197539837705563_set_o(vimage_nat_o(F),vimage_o_o(G)) = vimage_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,G),F)) ).

% set.comp
tff(fact_3671_set_Ocomp,axiom,
    ! [F: fun_int_nat,G: fun_nat_num] : comp_s6458263553158497821et_num(vimage_int_nat(F),vimage_nat_num(G)) = vimage_int_num(aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,G),F)) ).

% set.comp
tff(fact_3672_set_Ocomp,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] : comp_s8964582002068861047et_nat(vimage_nat_nat(F),vimage_nat_nat(G)) = vimage_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,G),F)) ).

% set.comp
tff(fact_3673_set_Ocomp,axiom,
    ! [F: fun_int_int,G: fun_int_nat] : comp_s7717450105097109807et_nat(vimage_int_int(F),vimage_int_nat(G)) = vimage_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,G),F)) ).

% set.comp
tff(fact_3674_set_Ocomp,axiom,
    ! [F: fun_Co1981900195452921617nteger,G: fun_fu4800041728187187127nteger] : comp_s2335681566234270110nteger(vimage5717976381819148369nteger(F),vimage1838862866765761655nteger(G)) = vimage2869426828964670109nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(G),F)) ).

% set.comp
tff(fact_3675_surj__vimage__empty,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A4: set_Product_unit] :
      ( ( image_405062704495631173t_unit(F,top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit )
     => ( ( vimage7995052115951654139t_unit(F,A4) = bot_bo3957492148770167129t_unit )
      <=> ( A4 = bot_bo3957492148770167129t_unit ) ) ) ).

% surj_vimage_empty
tff(fact_3676_surj__vimage__empty,axiom,
    ! [F: fun_Product_unit_o,A4: set_o] :
      ( ( image_Product_unit_o(F,top_to1996260823553986621t_unit) = top_top_set_o )
     => ( ( vimage8808348054653608838unit_o(F,A4) = bot_bo3957492148770167129t_unit )
      <=> ( A4 = bot_bot_set_o ) ) ) ).

% surj_vimage_empty
tff(fact_3677_surj__vimage__empty,axiom,
    ! [F: fun_o_Product_unit,A4: set_Product_unit] :
      ( ( image_o_Product_unit(F,top_top_set_o) = top_to1996260823553986621t_unit )
     => ( ( vimage4222377557387472176t_unit(F,A4) = bot_bot_set_o )
      <=> ( A4 = bot_bo3957492148770167129t_unit ) ) ) ).

% surj_vimage_empty
tff(fact_3678_surj__vimage__empty,axiom,
    ! [F: fun_o_o,A4: set_o] :
      ( ( aa_set_o_set_o(image_o_o(F),top_top_set_o) = top_top_set_o )
     => ( ( aa_set_o_set_o(vimage_o_o(F),A4) = bot_bot_set_o )
      <=> ( A4 = bot_bot_set_o ) ) ) ).

% surj_vimage_empty
tff(fact_3679_surj__vimage__empty,axiom,
    ! [F: fun_Pr4311113161502339697t_char,A4: set_char] :
      ( ( image_7248553556944771225t_char(F,top_to1996260823553986621t_unit) = top_top_set_char )
     => ( ( vimage2648087219933717859t_char(F,A4) = bot_bo3957492148770167129t_unit )
      <=> ( A4 = bot_bot_set_char ) ) ) ).

% surj_vimage_empty
tff(fact_3680_surj__vimage__empty,axiom,
    ! [F: fun_o_char,A4: set_char] :
      ( ( image_o_char(F,top_top_set_o) = top_top_set_char )
     => ( ( vimage_o_char(F,A4) = bot_bot_set_o )
      <=> ( A4 = bot_bot_set_char ) ) ) ).

% surj_vimage_empty
tff(fact_3681_surj__vimage__empty,axiom,
    ! [F: fun_Product_unit_rat,A4: set_rat] :
      ( ( image_240439954468258464it_rat(F,top_to1996260823553986621t_unit) = top_top_set_rat )
     => ( ( vimage5618198413390092650it_rat(F,A4) = bot_bo3957492148770167129t_unit )
      <=> ( A4 = bot_bot_set_rat ) ) ) ).

% surj_vimage_empty
tff(fact_3682_surj__vimage__empty,axiom,
    ! [F: fun_o_rat,A4: set_rat] :
      ( ( image_o_rat(F,top_top_set_o) = top_top_set_rat )
     => ( ( vimage_o_rat(F,A4) = bot_bot_set_o )
      <=> ( A4 = bot_bot_set_rat ) ) ) ).

% surj_vimage_empty
tff(fact_3683_surj__vimage__empty,axiom,
    ! [F: fun_Product_unit_nat,A4: set_nat] :
      ( ( image_875570014554754200it_nat(F,top_to1996260823553986621t_unit) = top_top_set_nat )
     => ( ( vimage6253328473476588386it_nat(F,A4) = bot_bo3957492148770167129t_unit )
      <=> ( A4 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
tff(fact_3684_surj__vimage__empty,axiom,
    ! [F: fun_o_nat,A4: set_nat] :
      ( ( image_o_nat(F,top_top_set_o) = top_top_set_nat )
     => ( ( vimage_o_nat(F,A4) = bot_bot_set_o )
      <=> ( A4 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
tff(fact_3685_vimage__subsetD,axiom,
    ! [F: fun_char_char,B5: set_char,A4: set_char] :
      ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
     => ( ord_less_eq_set_char(vimage_char_char(F,B5),A4)
       => ord_less_eq_set_char(B5,image_char_char(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3686_vimage__subsetD,axiom,
    ! [F: fun_char_rat,B5: set_rat,A4: set_char] :
      ( ( image_char_rat(F,top_top_set_char) = top_top_set_rat )
     => ( ord_less_eq_set_char(vimage_char_rat(F,B5),A4)
       => aa_set_rat_o(ord_less_eq_set_rat(B5),image_char_rat(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3687_vimage__subsetD,axiom,
    ! [F: fun_char_nat,B5: set_nat,A4: set_char] :
      ( ( image_char_nat(F,top_top_set_char) = top_top_set_nat )
     => ( ord_less_eq_set_char(vimage_char_nat(F,B5),A4)
       => aa_set_nat_o(ord_less_eq_set_nat(B5),image_char_nat(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3688_vimage__subsetD,axiom,
    ! [F: fun_char_int,B5: set_int,A4: set_char] :
      ( ( image_char_int(F,top_top_set_char) = top_top_set_int )
     => ( ord_less_eq_set_char(vimage_char_int(F,B5),A4)
       => aa_set_int_o(ord_less_eq_set_int(B5),image_char_int(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3689_vimage__subsetD,axiom,
    ! [F: fun_rat_char,B5: set_char,A4: set_rat] :
      ( ( image_rat_char(F,top_top_set_rat) = top_top_set_char )
     => ( aa_set_rat_o(ord_less_eq_set_rat(vimage_rat_char(F,B5)),A4)
       => ord_less_eq_set_char(B5,image_rat_char(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3690_vimage__subsetD,axiom,
    ! [F: fun_rat_rat,B5: set_rat,A4: set_rat] :
      ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
     => ( aa_set_rat_o(ord_less_eq_set_rat(vimage_rat_rat(F,B5)),A4)
       => aa_set_rat_o(ord_less_eq_set_rat(B5),image_rat_rat(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3691_vimage__subsetD,axiom,
    ! [F: fun_rat_nat,B5: set_nat,A4: set_rat] :
      ( ( image_rat_nat(F,top_top_set_rat) = top_top_set_nat )
     => ( aa_set_rat_o(ord_less_eq_set_rat(vimage_rat_nat(F,B5)),A4)
       => aa_set_nat_o(ord_less_eq_set_nat(B5),image_rat_nat(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3692_vimage__subsetD,axiom,
    ! [F: fun_rat_int,B5: set_int,A4: set_rat] :
      ( ( image_rat_int(F,top_top_set_rat) = top_top_set_int )
     => ( aa_set_rat_o(ord_less_eq_set_rat(vimage_rat_int(F,B5)),A4)
       => aa_set_int_o(ord_less_eq_set_int(B5),image_rat_int(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3693_vimage__subsetD,axiom,
    ! [F: fun_nat_char,B5: set_char,A4: set_nat] :
      ( ( image_nat_char(F,top_top_set_nat) = top_top_set_char )
     => ( aa_set_nat_o(ord_less_eq_set_nat(vimage_nat_char(F,B5)),A4)
       => ord_less_eq_set_char(B5,image_nat_char(F,A4)) ) ) ).

% vimage_subsetD
tff(fact_3694_vimage__subsetD,axiom,
    ! [F: fun_nat_rat,B5: set_rat,A4: set_nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat) = top_top_set_rat )
     => ( aa_set_nat_o(ord_less_eq_set_nat(vimage_nat_rat(F,B5)),A4)
       => aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ) ).

% vimage_subsetD
tff(fact_3695_Pair__vimage__Sigma,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_set_num] :
      vimage5436458796434496692at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),produc6237731129814287028at_num(A4,F)) = $ite(aa_set_nat_o(member_nat(Xa),A4),aa_nat_set_num(F,Xa),bot_bot_set_num) ).

% Pair_vimage_Sigma
tff(fact_3696_Pair__vimage__Sigma,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_set_int] :
      vimage7872673496510460816nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),produc1453890548195736866nt_int(A4,F)) = $ite(aa_set_int_o(member_int(Xa),A4),aa_int_set_int(F,Xa),bot_bot_set_int) ).

% Pair_vimage_Sigma
tff(fact_3697_Pair__vimage__Sigma,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_1833192751255691672t_unit] :
      vimage508909887551888138t_unit(produc3901946521647900940t_unit((Xa)),produc411009820408249917t_unit(A4,F)) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_o_s3492205133957624465t_unit(F,(Xa)),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_3698_Pair__vimage__Sigma,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_na896991816813928310t_unit] :
      vimage7305857734363866308t_unit(produc3434450474625521892t_unit(Xa),produc5506539147108004339t_unit(A4,F)) = $ite(aa_set_nat_o(member_nat(Xa),A4),aa_nat2714212862977473631t_unit(F,Xa),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_3699_Pair__vimage__Sigma,axiom,
    ! [Xa: int,A4: set_int,F: fun_in4731230917697954842t_unit] :
      vimage4422279681566046568t_unit(produc550872421827702152t_unit(Xa),produc2622961094310184599t_unit(A4,F)) = $ite(aa_set_int_o(member_int(Xa),A4),aa_int6548451963861500163t_unit(F,Xa),bot_bo3957492148770167129t_unit) ).

% Pair_vimage_Sigma
tff(fact_3700_Pair__vimage__Sigma,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_set_o] :
      vimage8945963521958007626od_o_o(product_Pair_o_o((Xa)),product_Sigma_o_o(A4,F)) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_o_set_o(F,(Xa)),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_3701_Pair__vimage__Sigma,axiom,
    ! [Xa: nat,A4: set_nat,F: fun_nat_set_o] :
      vimage6125388955539849434_nat_o(product_Pair_nat_o(Xa),product_Sigma_nat_o(A4,F)) = $ite(aa_set_nat_o(member_nat(Xa),A4),aa_nat_set_o(F,Xa),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_3702_Pair__vimage__Sigma,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_set_o] :
      vimage4304346610107161470_int_o(product_Pair_int_o(Xa),product_Sigma_int_o(A4,F)) = $ite(aa_set_int_o(member_int(Xa),A4),aa_int_set_o(F,Xa),bot_bot_set_o) ).

% Pair_vimage_Sigma
tff(fact_3703_Pair__vimage__Sigma,axiom,
    ! [Xa: $o,A4: set_o,F: fun_o_set_nat] :
      vimage5656379182745576100_o_nat(product_Pair_o_nat((Xa)),product_Sigma_o_nat(A4,F)) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_o_set_nat(F,(Xa)),bot_bot_set_nat) ).

% Pair_vimage_Sigma
tff(fact_3704_Pair__vimage__Sigma,axiom,
    ! [Xa: int,A4: set_int,F: fun_int_set_nat] :
      vimage9012682431555523728nt_nat(product_Pair_int_nat(Xa),produc1456381018704787142nt_nat(A4,F)) = $ite(aa_set_int_o(member_int(Xa),A4),aa_int_set_nat(F,Xa),bot_bot_set_nat) ).

% Pair_vimage_Sigma
tff(fact_3705_inj__vimage__image__eq,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( vimage_nat_rat(F,aa_set_nat_set_rat(image_nat_rat(F),A4)) = A4 ) ) ).

% inj_vimage_image_eq
tff(fact_3706_inj__vimage__image__eq,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(image_nat_nat(F),A4)) = A4 ) ) ).

% inj_vimage_image_eq
tff(fact_3707_inj__vimage__image__eq,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( vimage_nat_int(F,aa_set_nat_set_int(image_nat_int(F),A4)) = A4 ) ) ).

% inj_vimage_image_eq
tff(fact_3708_inj__vimage__image__eq,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_nat_set_int(vimage_int_nat(F),aa_set_int_set_nat(image_int_nat(F),A4)) = A4 ) ) ).

% inj_vimage_image_eq
tff(fact_3709_inj__vimage__image__eq,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( aa_set_int_set_int(vimage_int_int(F),aa_set_int_set_int(image_int_int(F),A4)) = A4 ) ) ).

% inj_vimage_image_eq
tff(fact_3710_inj__graph,axiom,
    inj_on1557069173596624977at_nat(aTP_Lamm_abb,top_to6864348034779713667at_nat) ).

% inj_graph
tff(fact_3711_inj__graph,axiom,
    inj_on5183350185949902417nt_int(aTP_Lamm_abd,top_to6561186464541008699nt_int) ).

% inj_graph
tff(fact_3712_f__the__inv__into__f,axiom,
    ! [F: fun_nat_rat,A4: set_nat,Y2: rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_rat_o(member_rat(Y2),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => ( aa_nat_rat(F,aa_rat_nat(the_inv_into_nat_rat(A4,F),Y2)) = Y2 ) ) ) ).

% f_the_inv_into_f
tff(fact_3713_f__the__inv__into__f,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Y2: nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(member_nat(Y2),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_nat_nat(F,aa_nat_nat(the_inv_into_nat_nat(A4,F),Y2)) = Y2 ) ) ) ).

% f_the_inv_into_f
tff(fact_3714_f__the__inv__into__f,axiom,
    ! [F: fun_int_nat,A4: set_int,Y2: nat] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(member_nat(Y2),aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_int_nat(F,aa_nat_int(the_inv_into_int_nat(A4,F),Y2)) = Y2 ) ) ) ).

% f_the_inv_into_f
tff(fact_3715_f__the__inv__into__f,axiom,
    ! [F: fun_nat_int,A4: set_nat,Y2: int] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_o(member_int(Y2),aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_nat_int(F,aa_int_nat(the_inv_into_nat_int(A4,F),Y2)) = Y2 ) ) ) ).

% f_the_inv_into_f
tff(fact_3716_f__the__inv__into__f,axiom,
    ! [F: fun_int_int,A4: set_int,Y2: int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(member_int(Y2),aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_int_int(F,aa_int_int(the_inv_into_int_int(A4,F),Y2)) = Y2 ) ) ) ).

% f_the_inv_into_f
tff(fact_3717_inj__on__the__inv__into,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => aa_set_rat_o(inj_on_rat_nat(the_inv_into_nat_rat(A4,F)),aa_set_nat_set_rat(image_nat_rat(F),A4)) ) ).

% inj_on_the_inv_into
tff(fact_3718_inj__on__the__inv__into,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => aa_set_nat_o(inj_on_nat_nat(the_inv_into_nat_nat(A4,F)),aa_set_nat_set_nat(image_nat_nat(F),A4)) ) ).

% inj_on_the_inv_into
tff(fact_3719_inj__on__the__inv__into,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => aa_set_int_o(inj_on_int_nat(the_inv_into_nat_int(A4,F)),aa_set_nat_set_int(image_nat_int(F),A4)) ) ).

% inj_on_the_inv_into
tff(fact_3720_inj__on__the__inv__into,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => aa_set_nat_o(inj_on_nat_int(the_inv_into_int_nat(A4,F)),aa_set_int_set_nat(image_int_nat(F),A4)) ) ).

% inj_on_the_inv_into
tff(fact_3721_inj__on__the__inv__into,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => aa_set_int_o(inj_on_int_int(the_inv_into_int_int(A4,F)),aa_set_int_set_int(image_int_int(F),A4)) ) ).

% inj_on_the_inv_into
tff(fact_3722_vimage__fst,axiom,
    ! [A4: set_ref_a] : vimage2910454108108208703_ref_a(produc473321116828791851t_unit,A4) = produc3946373962484743846t_unit(A4,aTP_Lamm_abe) ).

% vimage_fst
tff(fact_3723_vimage__fst,axiom,
    ! [A4: set_nat] : vimage4653281326611754070at_nat(product_fst_nat_nat,A4) = produc457027306803732586at_nat(A4,aTP_Lamm_abf) ).

% vimage_fst
tff(fact_3724_vimage__snd,axiom,
    ! [A4: set_he8644174446132254637t_unit] : vimage6116673589273818795t_unit(produc1126150082675451753t_unit,A4) = produc3946373962484743846t_unit(top_top_set_ref_a,aTP_Lamm_ju(A4)) ).

% vimage_snd
tff(fact_3725_vimage__snd,axiom,
    ! [A4: set_nat] : vimage4653281326611754070at_nat(product_snd_nat_nat,A4) = produc457027306803732586at_nat(top_top_set_nat,aTP_Lamm_gm(A4)) ).

% vimage_snd
tff(fact_3726_mono__Sup,axiom,
    ! [F: fun_set_nat_set_nat,A4: set_set_nat] :
      ( order_1005506198798982270et_nat(F)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(F),A4))),aa_set_nat_set_nat(F,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))) ) ).

% mono_Sup
tff(fact_3727_mono__Sup,axiom,
    ! [F: fun_Pr1628945989756059814et_nat,A4: set_Product_unit] :
      ( order_509894322932672675et_nat(F)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(F,A4))),aa_Pro6064029160526250199et_nat(F,comple4687483117567863418t_unit(A4))) ) ).

% mono_Sup
tff(fact_3728_mono__Sup,axiom,
    ! [F: fun_se552304175655956064t_unit,A4: set_set_nat] :
      ( order_7928700340835332885t_unit(F)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_6369283972875870912t_unit(F,A4))),aa_set4259463141574134601t_unit(F,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,A4))) ) ).

% mono_Sup
tff(fact_3729_mono__Sup,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A4: set_Product_unit] :
      ( order_4454465170704313136t_unit(F)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_405062704495631173t_unit(F,A4))),aa_Pro8294289767544744956t_unit(F,comple4687483117567863418t_unit(A4))) ) ).

% mono_Sup
tff(fact_3730_mono__Inf,axiom,
    ! [F: fun_set_nat_set_nat,A4: set_set_nat] :
      ( order_1005506198798982270et_nat(F)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(F,comple7806235888213564991et_nat(A4))),comple7806235888213564991et_nat(aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(F),A4))) ) ).

% mono_Inf
tff(fact_3731_mono__Inf,axiom,
    ! [F: fun_se552304175655956064t_unit,A4: set_set_nat] :
      ( order_7928700340835332885t_unit(F)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_set4259463141574134601t_unit(F,comple7806235888213564991et_nat(A4))),comple2584293577114468500t_unit(image_6369283972875870912t_unit(F,A4))) ) ).

% mono_Inf
tff(fact_3732_mono__Inf,axiom,
    ! [F: fun_Pr1628945989756059814et_nat,A4: set_Product_unit] :
      ( order_509894322932672675et_nat(F)
     => aa_set_nat_o(ord_less_eq_set_nat(aa_Pro6064029160526250199et_nat(F,comple2584293577114468500t_unit(A4))),comple7806235888213564991et_nat(image_8173849991827986510et_nat(F,A4))) ) ).

% mono_Inf
tff(fact_3733_mono__Inf,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,A4: set_Product_unit] :
      ( order_4454465170704313136t_unit(F)
     => aa_Product_unit_o(ord_le3221252021190050221t_unit(aa_Pro8294289767544744956t_unit(F,comple2584293577114468500t_unit(A4))),comple2584293577114468500t_unit(image_405062704495631173t_unit(F,A4))) ) ).

% mono_Inf
tff(fact_3734_Pow__insert,axiom,
    ! [A: nat,A4: set_nat] : pow_nat(aa_set_nat_set_nat(insert_nat(A),A4)) = sup_sup_set_set_nat(pow_nat(A4),aa_set1013257548857649334et_nat(image_7916887816326733075et_nat(insert_nat(A)),pow_nat(A4))) ).

% Pow_insert
tff(fact_3735_Pow__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] : aa_set6242432332095302236at_nat(pow_Pr3466873192417945886at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(A),A4)) = sup_su3642409539654194069at_nat(aa_set6242432332095302236at_nat(pow_Pr3466873192417945886at_nat,A4),image_3684629450409544005at_nat(insert8211810215607154385at_nat(A),aa_set6242432332095302236at_nat(pow_Pr3466873192417945886at_nat,A4))) ).

% Pow_insert
tff(fact_3736_Pow__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] : aa_set7518756442812098908at_nat(pow_Pr1023649790151954692at_nat,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(A),A4)) = sup_su474284620922681339at_nat(aa_set7518756442812098908at_nat(pow_Pr1023649790151954692at_nat,A4),image_4303274092540115909at_nat(insert9069300056098147895at_nat(A),aa_set7518756442812098908at_nat(pow_Pr1023649790151954692at_nat,A4))) ).

% Pow_insert
tff(fact_3737_Pow__insert,axiom,
    ! [A: product_unit,A4: set_Product_unit] : pow_Product_unit(aa_set8324764978855220220t_unit(insert_Product_unit(A),A4)) = sup_su5975557051335651745t_unit(pow_Product_unit(A4),image_4019945301284682565t_unit(insert_Product_unit(A),pow_Product_unit(A4))) ).

% Pow_insert
tff(fact_3738_Pow__insert,axiom,
    ! [A: $o,A4: set_o] : pow_o(aa_set_o_set_o(insert_o((A)),A4)) = sup_sup_set_set_o(pow_o(A4),image_set_o_set_o(insert_o((A)),pow_o(A4))) ).

% Pow_insert
tff(fact_3739_vimage__subsetI,axiom,
    ! [F: fun_nat_rat,B5: set_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => aa_set_nat_o(ord_less_eq_set_nat(vimage_nat_rat(F,B5)),A4) ) ) ).

% vimage_subsetI
tff(fact_3740_vimage__subsetI,axiom,
    ! [F: fun_nat_nat,B5: set_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(vimage_nat_nat(F),B5)),A4) ) ) ).

% vimage_subsetI
tff(fact_3741_vimage__subsetI,axiom,
    ! [F: fun_nat_int,B5: set_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_nat_set_int(image_nat_int(F),A4))
       => aa_set_nat_o(ord_less_eq_set_nat(vimage_nat_int(F,B5)),A4) ) ) ).

% vimage_subsetI
tff(fact_3742_vimage__subsetI,axiom,
    ! [F: fun_int_nat,B5: set_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(vimage_int_nat(F),B5)),A4) ) ) ).

% vimage_subsetI
tff(fact_3743_vimage__subsetI,axiom,
    ! [F: fun_int_int,B5: set_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_int_set_int(image_int_int(F),A4))
       => aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(vimage_int_int(F),B5)),A4) ) ) ).

% vimage_subsetI
tff(fact_3744_vimage__eq__UN,axiom,
    ! [F: fun_nat_Product_unit,B5: set_Product_unit] : vimage4884490618288580032t_unit(F,B5) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_8173849991827986510et_nat(aTP_Lamm_abg(F),B5)) ).

% vimage_eq_UN
tff(fact_3745_vimage__eq__UN,axiom,
    ! [F: fun_na367238361360896496at_nat,B5: set_Pr4329608150637261639at_nat] : vimage6435164912253009178at_nat(F,B5) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(aTP_Lamm_abh(F),B5)) ).

% vimage_eq_UN
tff(fact_3746_vimage__eq__UN,axiom,
    ! [F: fun_na1476630412504255114at_nat,B5: set_Pr1261947904930325089at_nat] : vimage8013328719654469172at_nat(F,B5) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(aTP_Lamm_abi(F),B5)) ).

% vimage_eq_UN
tff(fact_3747_vimage__eq__UN,axiom,
    ! [F: fun_nat_o,B5: set_o] : aa_set_o_set_nat(vimage_nat_o(F),B5) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(aTP_Lamm_abj(F),B5)) ).

% vimage_eq_UN
tff(fact_3748_vimage__eq__UN,axiom,
    ! [F: fun_nat_nat,B5: set_nat] : aa_set_nat_set_nat(vimage_nat_nat(F),B5) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(aTP_Lamm_abk(F),B5)) ).

% vimage_eq_UN
tff(fact_3749_the__inv__into__into,axiom,
    ! [F: fun_nat_rat,A4: set_nat,Xa: rat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_rat_o(member_rat(Xa),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => aa_set_nat_o(member_nat(aa_rat_nat(the_inv_into_nat_rat(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3750_the__inv__into__into,axiom,
    ! [F: fun_o_o,A4: set_o,Xa: $o,B5: set_o] :
      ( aa_set_o_o(inj_on_o_o(F),A4)
     => ( aa_set_o_o(member_o((Xa)),aa_set_o_set_o(image_o_o(F),A4))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => aa_set_o_o(member_o(aa_o_o(the_inv_into_o_o(A4,F),(Xa))),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3751_the__inv__into__into,axiom,
    ! [F: fun_nat_o,A4: set_nat,Xa: $o,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_o(F),A4)
     => ( aa_set_o_o(member_o((Xa)),image_nat_o(F,A4))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => aa_set_nat_o(member_nat(aa_o_nat(the_inv_into_nat_o(A4,F),(Xa))),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3752_the__inv__into__into,axiom,
    ! [F: fun_int_o,A4: set_int,Xa: $o,B5: set_int] :
      ( aa_set_int_o(inj_on_int_o(F),A4)
     => ( aa_set_o_o(member_o((Xa)),image_int_o(F,A4))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => aa_set_int_o(member_int(aa_o_int(the_inv_into_int_o(A4,F),(Xa))),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3753_the__inv__into__into,axiom,
    ! [F: fun_o_nat,A4: set_o,Xa: nat,B5: set_o] :
      ( aa_set_o_o(inj_on_o_nat(F),A4)
     => ( aa_set_nat_o(member_nat(Xa),image_o_nat(F,A4))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => aa_set_o_o(member_o(aa_nat_o(the_inv_into_o_nat(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3754_the__inv__into__into,axiom,
    ! [F: fun_nat_nat,A4: set_nat,Xa: nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(member_nat(Xa),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => aa_set_nat_o(member_nat(aa_nat_nat(the_inv_into_nat_nat(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3755_the__inv__into__into,axiom,
    ! [F: fun_int_nat,A4: set_int,Xa: nat,B5: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(member_nat(Xa),aa_set_int_set_nat(image_int_nat(F),A4))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => aa_set_int_o(member_int(aa_nat_int(the_inv_into_int_nat(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3756_the__inv__into__into,axiom,
    ! [F: fun_o_int,A4: set_o,Xa: int,B5: set_o] :
      ( aa_set_o_o(inj_on_o_int(F),A4)
     => ( aa_set_int_o(member_int(Xa),image_o_int(F,A4))
       => ( aa_set_o_o(ord_less_eq_set_o(A4),B5)
         => aa_set_o_o(member_o(aa_int_o(the_inv_into_o_int(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3757_the__inv__into__into,axiom,
    ! [F: fun_nat_int,A4: set_nat,Xa: int,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_o(member_int(Xa),aa_set_nat_set_int(image_nat_int(F),A4))
       => ( aa_set_nat_o(ord_less_eq_set_nat(A4),B5)
         => aa_set_nat_o(member_nat(aa_int_nat(the_inv_into_nat_int(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3758_the__inv__into__into,axiom,
    ! [F: fun_int_int,A4: set_int,Xa: int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(member_int(Xa),aa_set_int_set_int(image_int_int(F),A4))
       => ( aa_set_int_o(ord_less_eq_set_int(A4),B5)
         => aa_set_int_o(member_int(aa_int_int(the_inv_into_int_int(A4,F),Xa)),B5) ) ) ) ).

% the_inv_into_into
tff(fact_3759_the__inv__into__comp,axiom,
    ! [F: fun_nat_rat,G: fun_nat_nat,A4: set_nat,Xa: rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),aa_set_nat_set_nat(image_nat_nat(G),A4))
     => ( aa_set_nat_o(inj_on_nat_nat(G),A4)
       => ( aa_set_rat_o(member_rat(Xa),aa_set_nat_set_rat(image_nat_rat(F),aa_set_nat_set_nat(image_nat_nat(G),A4)))
         => ( aa_rat_nat(the_inv_into_nat_rat(A4,comp_nat_rat_nat(F,G)),Xa) = aa_rat_nat(comp_nat_nat_rat(the_inv_into_nat_nat(A4,G),the_inv_into_nat_rat(aa_set_nat_set_nat(image_nat_nat(G),A4),F)),Xa) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3760_the__inv__into__comp,axiom,
    ! [F: fun_nat_num,G: fun_int_nat,A4: set_int,Xa: num] :
      ( aa_set_nat_o(inj_on_nat_num(F),aa_set_int_set_nat(image_int_nat(G),A4))
     => ( aa_set_int_o(inj_on_int_nat(G),A4)
       => ( member_num(Xa,image_nat_num(F,aa_set_int_set_nat(image_int_nat(G),A4)))
         => ( aa_num_int(the_inv_into_int_num(A4,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,F),G)),Xa) = aa_num_int(comp_nat_int_num(the_inv_into_int_nat(A4,G),the_inv_into_nat_num(aa_set_int_set_nat(image_int_nat(G),A4),F)),Xa) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3761_the__inv__into__comp,axiom,
    ! [F: fun_nat_rat,G: fun_int_nat,A4: set_int,Xa: rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),aa_set_int_set_nat(image_int_nat(G),A4))
     => ( aa_set_int_o(inj_on_int_nat(G),A4)
       => ( aa_set_rat_o(member_rat(Xa),aa_set_nat_set_rat(image_nat_rat(F),aa_set_int_set_nat(image_int_nat(G),A4)))
         => ( aa_rat_int(the_inv_into_int_rat(A4,comp_nat_rat_int(F,G)),Xa) = aa_rat_int(comp_nat_int_rat(the_inv_into_int_nat(A4,G),the_inv_into_nat_rat(aa_set_int_set_nat(image_int_nat(G),A4),F)),Xa) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3762_the__inv__into__comp,axiom,
    ! [F: fun_o_o,G: fun_nat_o,A4: set_nat,Xa: $o] :
      ( aa_set_o_o(inj_on_o_o(F),image_nat_o(G,A4))
     => ( aa_set_nat_o(inj_on_nat_o(G),A4)
       => ( aa_set_o_o(member_o((Xa)),aa_set_o_set_o(image_o_o(F),image_nat_o(G,A4)))
         => ( aa_o_nat(the_inv_into_nat_o(A4,aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,F),G)),(Xa)) = aa_o_nat(comp_o_nat_o(the_inv_into_nat_o(A4,G),the_inv_into_o_o(image_nat_o(G,A4),F)),(Xa)) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3763_the__inv__into__comp,axiom,
    ! [F: fun_rat_o,G: fun_nat_rat,A4: set_nat,Xa: $o] :
      ( aa_set_rat_o(inj_on_rat_o(F),aa_set_nat_set_rat(image_nat_rat(G),A4))
     => ( aa_set_nat_o(inj_on_nat_rat(G),A4)
       => ( aa_set_o_o(member_o((Xa)),image_rat_o(F,aa_set_nat_set_rat(image_nat_rat(G),A4)))
         => ( aa_o_nat(the_inv_into_nat_o(A4,comp_rat_o_nat(F,G)),(Xa)) = aa_o_nat(comp_rat_nat_o(the_inv_into_nat_rat(A4,G),the_inv_into_rat_o(aa_set_nat_set_rat(image_nat_rat(G),A4),F)),(Xa)) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3764_the__inv__into__comp,axiom,
    ! [F: fun_nat_o,G: fun_nat_nat,A4: set_nat,Xa: $o] :
      ( aa_set_nat_o(inj_on_nat_o(F),aa_set_nat_set_nat(image_nat_nat(G),A4))
     => ( aa_set_nat_o(inj_on_nat_nat(G),A4)
       => ( aa_set_o_o(member_o((Xa)),image_nat_o(F,aa_set_nat_set_nat(image_nat_nat(G),A4)))
         => ( aa_o_nat(the_inv_into_nat_o(A4,comp_nat_o_nat(F,G)),(Xa)) = aa_o_nat(comp_nat_nat_o(the_inv_into_nat_nat(A4,G),the_inv_into_nat_o(aa_set_nat_set_nat(image_nat_nat(G),A4),F)),(Xa)) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3765_the__inv__into__comp,axiom,
    ! [F: fun_int_o,G: fun_nat_int,A4: set_nat,Xa: $o] :
      ( aa_set_int_o(inj_on_int_o(F),aa_set_nat_set_int(image_nat_int(G),A4))
     => ( aa_set_nat_o(inj_on_nat_int(G),A4)
       => ( aa_set_o_o(member_o((Xa)),image_int_o(F,aa_set_nat_set_int(image_nat_int(G),A4)))
         => ( aa_o_nat(the_inv_into_nat_o(A4,comp_int_o_nat(F,G)),(Xa)) = aa_o_nat(comp_int_nat_o(the_inv_into_nat_int(A4,G),the_inv_into_int_o(aa_set_nat_set_int(image_nat_int(G),A4),F)),(Xa)) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3766_the__inv__into__comp,axiom,
    ! [F: fun_nat_o,G: fun_int_nat,A4: set_int,Xa: $o] :
      ( aa_set_nat_o(inj_on_nat_o(F),aa_set_int_set_nat(image_int_nat(G),A4))
     => ( aa_set_int_o(inj_on_int_nat(G),A4)
       => ( aa_set_o_o(member_o((Xa)),image_nat_o(F,aa_set_int_set_nat(image_int_nat(G),A4)))
         => ( aa_o_int(the_inv_into_int_o(A4,comp_nat_o_int(F,G)),(Xa)) = aa_o_int(comp_nat_int_o(the_inv_into_int_nat(A4,G),the_inv_into_nat_o(aa_set_int_set_nat(image_int_nat(G),A4),F)),(Xa)) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3767_the__inv__into__comp,axiom,
    ! [F: fun_int_o,G: fun_int_int,A4: set_int,Xa: $o] :
      ( aa_set_int_o(inj_on_int_o(F),aa_set_int_set_int(image_int_int(G),A4))
     => ( aa_set_int_o(inj_on_int_int(G),A4)
       => ( aa_set_o_o(member_o((Xa)),image_int_o(F,aa_set_int_set_int(image_int_int(G),A4)))
         => ( aa_o_int(the_inv_into_int_o(A4,comp_int_o_int(F,G)),(Xa)) = aa_o_int(comp_int_int_o(the_inv_into_int_int(A4,G),the_inv_into_int_o(aa_set_int_set_int(image_int_int(G),A4),F)),(Xa)) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3768_the__inv__into__comp,axiom,
    ! [F: fun_o_nat,G: fun_o_o,A4: set_o,Xa: nat] :
      ( aa_set_o_o(inj_on_o_nat(F),aa_set_o_set_o(image_o_o(G),A4))
     => ( aa_set_o_o(inj_on_o_o(G),A4)
       => ( aa_set_nat_o(member_nat(Xa),image_o_nat(F,aa_set_o_set_o(image_o_o(G),A4)))
         => ( aa_nat_o(the_inv_into_o_nat(A4,comp_o_nat_o(F,G)),Xa)
          <=> aa_nat_o(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,the_inv_into_o_o(A4,G)),the_inv_into_o_nat(aa_set_o_set_o(image_o_o(G),A4),F)),Xa) ) ) ) ) ).

% the_inv_into_comp
tff(fact_3769_fun__upd__image,axiom,
    ! [F: fun_int_int,Xa: int,Y2: int,A4: set_int] :
      aa_set_int_set_int(image_int_int(aa_int_fun_int_int(fun_upd_int_int(F,Xa),Y2)),A4) = $ite(aa_set_int_o(member_int(Xa),A4),aa_set_int_set_int(insert_int(Y2),aa_set_int_set_int(image_int_int(F),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),aa_set_int_set_int(image_int_int(F),A4)) ).

% fun_upd_image
tff(fact_3770_fun__upd__image,axiom,
    ! [F: fun_int_nat,Xa: int,Y2: nat,A4: set_int] :
      aa_set_int_set_nat(image_int_nat(fun_upd_int_nat(F,Xa,Y2)),A4) = $ite(aa_set_int_o(member_int(Xa),A4),aa_set_nat_set_nat(insert_nat(Y2),aa_set_int_set_nat(image_int_nat(F),minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),aa_set_int_set_nat(image_int_nat(F),A4)) ).

% fun_upd_image
tff(fact_3771_fun__upd__image,axiom,
    ! [F: fun_int_Product_unit,Xa: int,Y2: product_unit,A4: set_int] :
      image_5846526143423701914t_unit(fun_up2220634978854545698t_unit(F,Xa,Y2),A4) = $ite(aa_set_int_o(member_int(Xa),A4),aa_set8324764978855220220t_unit(insert_Product_unit(Y2),image_5846526143423701914t_unit(F,minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),image_5846526143423701914t_unit(F,A4)) ).

% fun_upd_image
tff(fact_3772_fun__upd__image,axiom,
    ! [F: fun_int_o,Xa: int,Y2: $o,A4: set_int] :
      image_int_o(aa_o_fun_int_o(fun_upd_int_o(F,Xa),(Y2)),A4) = $ite(aa_set_int_o(member_int(Xa),A4),aa_set_o_set_o(insert_o((Y2)),image_int_o(F,minus_minus_set_int(A4,aa_set_int_set_int(insert_int(Xa),bot_bot_set_int)))),image_int_o(F,A4)) ).

% fun_upd_image
tff(fact_3773_fun__upd__image,axiom,
    ! [F: fun_Product_unit_nat,Xa: product_unit,Y2: nat,A4: set_Product_unit] :
      image_875570014554754200it_nat(fun_up6473050886840373792it_nat(F,Xa,Y2),A4) = $ite(member_Product_unit(Xa,A4),aa_set_nat_set_nat(insert_nat(Y2),image_875570014554754200it_nat(F,minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))),image_875570014554754200it_nat(F,A4)) ).

% fun_upd_image
tff(fact_3774_fun__upd__image,axiom,
    ! [F: fun_Pr2191293291710260995t_unit,Xa: product_unit,Y2: product_unit,A4: set_Product_unit] :
      image_405062704495631173t_unit(fun_up6875000483049295805t_unit(F,Xa,Y2),A4) = $ite(member_Product_unit(Xa,A4),aa_set8324764978855220220t_unit(insert_Product_unit(Y2),image_405062704495631173t_unit(F,minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))),image_405062704495631173t_unit(F,A4)) ).

% fun_upd_image
tff(fact_3775_fun__upd__image,axiom,
    ! [F: fun_Product_unit_o,Xa: product_unit,Y2: $o,A4: set_Product_unit] :
      image_Product_unit_o(fun_up5432024876597553480unit_o(F,Xa,(Y2)),A4) = $ite(member_Product_unit(Xa,A4),aa_set_o_set_o(insert_o((Y2)),image_Product_unit_o(F,minus_6452836326544984404t_unit(A4,aa_set8324764978855220220t_unit(insert_Product_unit(Xa),bot_bo3957492148770167129t_unit)))),image_Product_unit_o(F,A4)) ).

% fun_upd_image
tff(fact_3776_fun__upd__image,axiom,
    ! [F: fun_o_nat,Xa: $o,Y2: nat,A4: set_o] :
      image_o_nat(fun_upd_o_nat(F,(Xa),Y2),A4) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_set_nat_set_nat(insert_nat(Y2),image_o_nat(F,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))),image_o_nat(F,A4)) ).

% fun_upd_image
tff(fact_3777_fun__upd__image,axiom,
    ! [F: fun_o_Product_unit,Xa: $o,Y2: product_unit,A4: set_o] :
      image_o_Product_unit(fun_up846054379331416818t_unit(F,(Xa),Y2),A4) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_set8324764978855220220t_unit(insert_Product_unit(Y2),image_o_Product_unit(F,minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))),image_o_Product_unit(F,A4)) ).

% fun_upd_image
tff(fact_3778_fun__upd__image,axiom,
    ! [F: fun_o_o,Xa: $o,Y2: $o,A4: set_o] :
      aa_set_o_set_o(image_o_o(aa_o_fun_o_o(fun_upd_o_o(F,(Xa)),(Y2))),A4) = $ite(aa_set_o_o(member_o((Xa)),A4),aa_set_o_set_o(insert_o((Y2)),aa_set_o_set_o(image_o_o(F),minus_minus_set_o(A4,aa_set_o_set_o(insert_o((Xa)),bot_bot_set_o)))),aa_set_o_set_o(image_o_o(F),A4)) ).

% fun_upd_image
tff(fact_3779_mono__cSUP,axiom,
    ! [F: fun_nat_nat,A4: fun_int_nat,I3: set_int] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_int_set_nat(image_int_nat(A4),I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_be(F),A4)),I3))),aa_nat_nat(F,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(A4),I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3780_mono__cSUP,axiom,
    ! [F: fun_nat_nat,A4: fun_nat_nat,I3: set_nat] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_bd(F),A4)),I3))),aa_nat_nat(F,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(A4),I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3781_mono__cSUP,axiom,
    ! [F: fun_int_nat,A4: fun_int_int,I3: set_int] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_int_set_int(image_int_int(A4),I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_aav(F),A4)),I3))),aa_int_nat(F,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(A4),I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3782_mono__cSUP,axiom,
    ! [F: fun_int_nat,A4: fun_Product_unit_int,I3: set_Product_unit] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_873079544045703924it_int(A4,I3))
       => ( ( I3 != bot_bo3957492148770167129t_unit )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(aa_fun7819058828246937184it_nat(aTP_Lamm_aaw(F),A4),I3))),aa_int_nat(F,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(A4,I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3783_mono__cSUP,axiom,
    ! [F: fun_int_nat,A4: fun_o_int,I3: set_o] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,image_o_int(A4,I3))
       => ( ( I3 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(aa_fun8905617046332959520_o_nat(aTP_Lamm_aax(F),A4),I3))),aa_int_nat(F,aa_set_int_int(complete_Sup_Sup_int,image_o_int(A4,I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3784_mono__cSUP,axiom,
    ! [F: fun_int_nat,A4: fun_nat_int,I3: set_nat] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,aa_set_nat_set_int(image_nat_int(A4),I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_aay(F),A4)),I3))),aa_int_nat(F,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(A4),I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3785_mono__cSUP,axiom,
    ! [F: fun_Product_unit_nat,A4: fun_int_Product_unit,I3: set_int] :
      ( order_7096114683908731117it_nat(F)
     => ( condit2080475194861589617t_unit(image_5846526143423701914t_unit(A4,I3))
       => ( ( I3 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(aa_fun1692700469306394999nt_nat(aTP_Lamm_abl(F),A4)),I3))),aa_Product_unit_nat(F,comple4687483117567863418t_unit(image_5846526143423701914t_unit(A4,I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3786_mono__cSUP,axiom,
    ! [F: fun_Product_unit_nat,A4: fun_Pr2191293291710260995t_unit,I3: set_Product_unit] :
      ( order_7096114683908731117it_nat(F)
     => ( condit2080475194861589617t_unit(image_405062704495631173t_unit(A4,I3))
       => ( ( I3 != bot_bo3957492148770167129t_unit )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(aa_fun7205615774095635567it_nat(aTP_Lamm_abm(F),A4),I3))),aa_Product_unit_nat(F,comple4687483117567863418t_unit(image_405062704495631173t_unit(A4,I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3787_mono__cSUP,axiom,
    ! [F: fun_Product_unit_nat,A4: fun_o_Product_unit,I3: set_o] :
      ( order_7096114683908731117it_nat(F)
     => ( condit2080475194861589617t_unit(image_o_Product_unit(A4,I3))
       => ( ( I3 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(aa_fun6071669926149765765_o_nat(aTP_Lamm_abn(F),A4),I3))),aa_Product_unit_nat(F,comple4687483117567863418t_unit(image_o_Product_unit(A4,I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3788_mono__cSUP,axiom,
    ! [F: fun_Product_unit_nat,A4: fun_nat_Product_unit,I3: set_nat] :
      ( order_7096114683908731117it_nat(F)
     => ( condit2080475194861589617t_unit(image_8730104196221521654t_unit(A4,I3))
       => ( ( I3 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun2105170996421215863at_nat(aTP_Lamm_abo(F),A4)),I3))),aa_Product_unit_nat(F,comple4687483117567863418t_unit(image_8730104196221521654t_unit(A4,I3)))) ) ) ) ).

% mono_cSUP
tff(fact_3789_mono__cSup,axiom,
    ! [F: fun_o_nat,A4: set_o] :
      ( order_mono_o_nat(F)
     => ( aa_set_o_o(condit5488710616941104124bove_o,A4)
       => ( ( A4 != bot_bot_set_o )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_o_nat(F,A4))),aa_o_nat(F,aa_set_o_o(complete_Sup_Sup_o,A4))) ) ) ) ).

% mono_cSup
tff(fact_3790_mono__cSup,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( order_mono_nat_nat(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => ( ( A4 != bot_bot_set_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_nat_nat(F,aa_set_nat_nat(complete_Sup_Sup_nat,A4))) ) ) ) ).

% mono_cSup
tff(fact_3791_mono__cSup,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( order_mono_int_nat(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,A4)
       => ( ( A4 != bot_bot_set_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,aa_set_int_set_nat(image_int_nat(F),A4))),aa_int_nat(F,aa_set_int_int(complete_Sup_Sup_int,A4))) ) ) ) ).

% mono_cSup
tff(fact_3792_mono__cSup,axiom,
    ! [F: fun_Product_unit_nat,A4: set_Product_unit] :
      ( order_7096114683908731117it_nat(F)
     => ( condit2080475194861589617t_unit(A4)
       => ( ( A4 != bot_bo3957492148770167129t_unit )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(complete_Sup_Sup_nat,image_875570014554754200it_nat(F,A4))),aa_Product_unit_nat(F,comple4687483117567863418t_unit(A4))) ) ) ) ).

% mono_cSup
tff(fact_3793_mono__cSup,axiom,
    ! [F: fun_o_int,A4: set_o] :
      ( order_mono_o_int(F)
     => ( aa_set_o_o(condit5488710616941104124bove_o,A4)
       => ( ( A4 != bot_bot_set_o )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_o_int(F,A4))),aa_o_int(F,aa_set_o_o(complete_Sup_Sup_o,A4))) ) ) ) ).

% mono_cSup
tff(fact_3794_mono__cSup,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( order_mono_nat_int(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => ( ( A4 != bot_bot_set_nat )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_nat_set_int(image_nat_int(F),A4))),aa_nat_int(F,aa_set_nat_nat(complete_Sup_Sup_nat,A4))) ) ) ) ).

% mono_cSup
tff(fact_3795_mono__cSup,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( order_mono_int_int(F)
     => ( aa_set_int_o(condit2212336002400062152ve_int,A4)
       => ( ( A4 != bot_bot_set_int )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(F),A4))),aa_int_int(F,aa_set_int_int(complete_Sup_Sup_int,A4))) ) ) ) ).

% mono_cSup
tff(fact_3796_mono__cSup,axiom,
    ! [F: fun_Product_unit_int,A4: set_Product_unit] :
      ( order_7093624213399680841it_int(F)
     => ( condit2080475194861589617t_unit(A4)
       => ( ( A4 != bot_bo3957492148770167129t_unit )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(F,A4))),aa_Product_unit_int(F,comple4687483117567863418t_unit(A4))) ) ) ) ).

% mono_cSup
tff(fact_3797_mono__cSup,axiom,
    ! [F: fun_o_Product_unit,A4: set_o] :
      ( order_7782127260464396005t_unit(F)
     => ( aa_set_o_o(condit5488710616941104124bove_o,A4)
       => ( ( A4 != bot_bot_set_o )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_o_Product_unit(F,A4))),aa_o_Product_unit(F,aa_set_o_o(complete_Sup_Sup_o,A4))) ) ) ) ).

% mono_cSup
tff(fact_3798_mono__cSup,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( order_5727276828720722763t_unit(F)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,A4)
       => ( ( A4 != bot_bot_set_nat )
         => aa_Product_unit_o(ord_le3221252021190050221t_unit(comple4687483117567863418t_unit(image_8730104196221521654t_unit(F,A4))),aa_nat_Product_unit(F,aa_set_nat_nat(complete_Sup_Sup_nat,A4))) ) ) ) ).

% mono_cSup
tff(fact_3799_Func__map__surj,axiom,
    ! [F1: fun_nat_rat,A12: set_nat,B1: set_rat,F22: fun_int_int,B22: set_int,A23: set_int] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_int(F22),B22)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_int ) )
           => ( bNF_We7904740386373615514nt_rat(B22,B1) = image_2476731758076941825nt_rat(bNF_We576767280138975660at_int(B22,F1,F22),bNF_We8539870446460111250nt_nat(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3800_Func__map__surj,axiom,
    ! [F1: fun_nat_nat,A12: set_nat,B1: set_nat,F22: fun_int_int,B22: set_int,A23: set_int] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_int(F22),B22)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_int ) )
           => ( bNF_We8539870446460111250nt_nat(B22,B1) = image_5984960316470338553nt_nat(bNF_We893321924073841572at_int(B22,F1,F22),bNF_We8539870446460111250nt_nat(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3801_Func__map__surj,axiom,
    ! [F1: fun_nat_int,A12: set_nat,B1: set_int,F22: fun_int_int,B22: set_int,A23: set_int] :
      ( ( aa_set_nat_set_int(image_nat_int(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_int(F22),B22)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_int ) )
           => ( bNF_We8537379975951060974nt_int(B22,B1) = image_1807109296961141845nt_int(bNF_We1892675635974896128nt_int(B22,F1,F22),bNF_We8539870446460111250nt_nat(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3802_Func__map__surj,axiom,
    ! [F1: fun_int_nat,A12: set_int,B1: set_nat,F22: fun_int_int,B22: set_int,A23: set_int] :
      ( ( aa_set_int_set_nat(image_int_nat(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_int(F22),B22)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_int ) )
           => ( bNF_We8539870446460111250nt_nat(B22,B1) = image_713032908031272021nt_nat(bNF_We1264975484310686208at_int(B22,F1,F22),bNF_We8537379975951060974nt_int(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3803_Func__map__surj,axiom,
    ! [F1: fun_int_int,A12: set_int,B1: set_int,F22: fun_int_int,B22: set_int,A23: set_int] :
      ( ( aa_set_int_set_int(image_int_int(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_int(F22),B22)
       => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_int ) )
           => ( bNF_We8537379975951060974nt_int(B22,B1) = image_5758553925376851121nt_int(bNF_We2264329196211740764nt_int(B22,F1,F22),bNF_We8537379975951060974nt_int(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3804_Func__map__surj,axiom,
    ! [F1: fun_nat_rat,A12: set_nat,B1: set_rat,F22: fun_int_nat,B22: set_int,A23: set_nat] :
      ( ( aa_set_nat_set_rat(image_nat_rat(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_nat(F22),B22)
       => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_nat ) )
           => ( bNF_We7904740386373615514nt_rat(B22,B1) = image_2760782832447051685nt_rat(bNF_We579257750648025936at_nat(B22,F1,F22),bNF_We7540516734559056694at_nat(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3805_Func__map__surj,axiom,
    ! [F1: fun_nat_nat,A12: set_nat,B1: set_nat,F22: fun_int_nat,B22: set_int,A23: set_nat] :
      ( ( aa_set_nat_set_nat(image_nat_nat(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_nat(F22),B22)
       => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_nat ) )
           => ( bNF_We8539870446460111250nt_nat(B22,B1) = image_6269011390840448413nt_nat(bNF_We895812394582891848at_nat(B22,F1,F22),bNF_We7540516734559056694at_nat(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3806_Func__map__surj,axiom,
    ! [F1: fun_nat_int,A12: set_nat,B1: set_int,F22: fun_int_nat,B22: set_int,A23: set_nat] :
      ( ( aa_set_nat_set_int(image_nat_int(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_nat(F22),B22)
       => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_nat ) )
           => ( bNF_We8537379975951060974nt_int(B22,B1) = image_2091160371331251705nt_int(bNF_We1895166106483946404nt_nat(B22,F1,F22),bNF_We7540516734559056694at_nat(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3807_Func__map__surj,axiom,
    ! [F1: fun_int_nat,A12: set_int,B1: set_nat,F22: fun_int_nat,B22: set_int,A23: set_nat] :
      ( ( aa_set_int_set_nat(image_int_nat(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_nat(F22),B22)
       => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_nat ) )
           => ( bNF_We8539870446460111250nt_nat(B22,B1) = image_997083982401381881nt_nat(bNF_We1267465954819736484at_nat(B22,F1,F22),bNF_We7538026264050006418at_int(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3808_Func__map__surj,axiom,
    ! [F1: fun_int_int,A12: set_int,B1: set_int,F22: fun_int_nat,B22: set_int,A23: set_nat] :
      ( ( aa_set_int_set_int(image_int_int(F1),A12) = B1 )
     => ( aa_set_int_o(inj_on_int_nat(F22),B22)
       => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F22),B22)),A23)
         => ( ( ( B22 = bot_bot_set_int )
             => ( A23 = bot_bot_set_nat ) )
           => ( bNF_We8537379975951060974nt_int(B22,B1) = image_6042604999746960981nt_int(bNF_We2266819666720791040nt_nat(B22,F1,F22),bNF_We7538026264050006418at_int(A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_3809_cofinal__def,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat] :
      ( bNF_Ca6037544080215331608at_nat(A4,Ra)
    <=> ! [X4: set_Pr4329608150637261639at_nat] :
          ( member6273874620674182256at_nat(X4,field_1428494348145443988at_nat(Ra))
         => ? [Xa3: set_Pr4329608150637261639at_nat] :
              ( member6273874620674182256at_nat(Xa3,A4)
              & ( X4 != Xa3 )
              & member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X4),Xa3),Ra) ) ) ) ).

% cofinal_def
tff(fact_3810_cofinal__def,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(bNF_Ca8448034394003515314at_nat(A4),Ra)
    <=> ! [X4: set_Pr1261947904930325089at_nat] :
          ( member2643936169264416010at_nat(X4,field_3145993188331922222at_nat(Ra))
         => ? [Xa3: set_Pr1261947904930325089at_nat] :
              ( member2643936169264416010at_nat(Xa3,A4)
              & ( X4 != Xa3 )
              & aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X4),Xa3)),Ra) ) ) ) ).

% cofinal_def
tff(fact_3811_cofinal__def,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(bNF_Ca5159621140400407487al_nat(A4),Ra)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_set736839985058952875et_nat(field_nat,Ra))
         => ? [Xa3: nat] :
              ( aa_set_nat_o(member_nat(Xa3),A4)
              & ( X4 != Xa3 )
              & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa3)),Ra) ) ) ) ).

% cofinal_def
tff(fact_3812_cofinal__def,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int] :
      ( bNF_Ca5157130669891357211al_int(A4,Ra)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),field_int(Ra))
         => ? [Xa3: int] :
              ( aa_set_int_o(member_int(Xa3),A4)
              & ( X4 != Xa3 )
              & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),Ra) ) ) ) ).

% cofinal_def
tff(fact_3813_Total__subset__Id,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat] :
      ( total_5117575981256876747at_nat(field_1428494348145443988at_nat(Ra),Ra)
     => ( ord_le5997549366648089703at_nat(Ra,id_set4888139568991089827at_nat)
       => ( ( Ra = bot_bo4948859079157340979at_nat )
          | ? [A3: set_Pr4329608150637261639at_nat] : Ra = insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A3),A3),bot_bo4948859079157340979at_nat) ) ) ) ).

% Total_subset_Id
tff(fact_3814_Total__subset__Id,axiom,
    ! [Ra: set_Pr958786334691620121nt_int] :
      ( total_on_int(field_int(Ra),Ra)
     => ( ord_le2843351958646193337nt_int(Ra,id_int2)
       => ( ( Ra = bot_bo1796632182523588997nt_int )
          | ? [A3: int] : Ra = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),A3),bot_bo1796632182523588997nt_int) ) ) ) ).

% Total_subset_Id
tff(fact_3815_Total__subset__Id,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat] :
      ( total_3592101749530773125at_nat(field_4706345496283612622at_nat(Ra),Ra)
     => ( ord_le3000389064537975527at_nat(Ra,id_Pro2258643101195443293at_nat)
       => ( ( Ra = bot_bo5327735625951526323at_nat )
          | ? [A3: product_prod_nat_nat] : Ra = insert5050368324300391991at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A3),A3),bot_bo5327735625951526323at_nat) ) ) ) ).

% Total_subset_Id
tff(fact_3816_Total__subset__Id,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(total_7212417256332265445at_nat(field_3145993188331922222at_nat(Ra)),Ra)
     => ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,Ra),id_set1796276052702428605at_nat)
       => ( ( Ra = bot_bo228742789529271731at_nat )
          | ? [A3: set_Pr1261947904930325089at_nat] : Ra = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A3),A3)),bot_bo228742789529271731at_nat) ) ) ) ).

% Total_subset_Id
tff(fact_3817_Total__subset__Id,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(total_on_nat(aa_set736839985058952875et_nat(field_nat,Ra)),Ra)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Ra),id_nat2)
       => ( ( Ra = bot_bo2099793752762293965at_nat )
          | ? [A3: nat] : Ra = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),A3)),bot_bo2099793752762293965at_nat) ) ) ) ).

% Total_subset_Id
tff(fact_3818_inf__img__fin__dom_H,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(F),A4))
     => ( ~ aa_set_int_o(finite_finite_int,A4)
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),aa_set_int_set_int(image_int_int(F),A4))
            & ~ aa_set_int_o(finite_finite_int,inf_inf_set_int(aa_set_int_set_int(vimage_int_int(F),aa_set_int_set_int(insert_int(X),bot_bot_set_int)),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3819_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: rat] :
            ( aa_set_rat_o(member_rat(X),aa_set_nat_set_rat(image_nat_rat(F),A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_rat(F,insert_rat(X,bot_bot_set_rat))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3820_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),aa_set_nat_set_int(image_nat_int(F),A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_int(F,aa_set_int_set_int(insert_int(X),bot_bot_set_int))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3821_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( finite4290736615968046902t_unit(image_8730104196221521654t_unit(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: product_unit] :
            ( member_Product_unit(X,image_8730104196221521654t_unit(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(X),bot_bo3957492148770167129t_unit))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3822_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),image_nat_o(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(aa_set_o_set_nat(vimage_nat_o(F),aa_set_o_set_o(insert_o((X)),bot_bot_set_o))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3823_inf__img__fin__dom_H,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( ~ aa_set_int_o(finite_finite_int,A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),aa_set_int_set_nat(image_int_nat(F),A4))
            & ~ aa_set_int_o(finite_finite_int,inf_inf_set_int(aa_set_nat_set_int(vimage_int_nat(F),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat)),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3824_inf__img__fin__dom_H,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(F),A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3825_inf__img__fin__dom_H,axiom,
    ! [F: fun_Pr2481279687482350479t_unit,A4: set_Pr1261947904930325089at_nat] :
      ( finite4290736615968046902t_unit(image_6169909464789311569t_unit(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ? [X: product_unit] :
            ( member_Product_unit(X,image_6169909464789311569t_unit(F,A4))
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(vimage3104651464403166471t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(X),bot_bo3957492148770167129t_unit))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3826_inf__img__fin__dom_H,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_5846123807819985514at_nat(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),image_5846123807819985514at_nat(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage8013328719654469172at_nat(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3827_inf__img__fin__dom_H,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set_o_o(finite_finite_o,image_3693632289388996572_nat_o(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),image_3693632289388996572_nat_o(F,A4))
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(vimage2444243245536687506_nat_o(F,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_3828_inf__img__fin__domE_H,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(F),A4))
     => ( ~ aa_set_int_o(finite_finite_int,A4)
       => ~ ! [Y: int] :
              ( aa_set_int_o(member_int(Y),aa_set_int_set_int(image_int_int(F),A4))
             => aa_set_int_o(finite_finite_int,inf_inf_set_int(aa_set_int_set_int(vimage_int_int(F),aa_set_int_set_int(insert_int(Y),bot_bot_set_int)),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3829_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ~ ! [Y: rat] :
              ( aa_set_rat_o(member_rat(Y),aa_set_nat_set_rat(image_nat_rat(F),A4))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_rat(F,insert_rat(Y,bot_bot_set_rat))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3830_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ~ ! [Y: int] :
              ( aa_set_int_o(member_int(Y),aa_set_nat_set_int(image_nat_int(F),A4))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_int(F,aa_set_int_set_int(insert_int(Y),bot_bot_set_int))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3831_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( finite4290736615968046902t_unit(image_8730104196221521654t_unit(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ~ ! [Y: product_unit] :
              ( member_Product_unit(Y,image_8730104196221521654t_unit(F,A4))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(Y),bot_bo3957492148770167129t_unit))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3832_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ~ ! [Y: $o] :
              ( aa_set_o_o(member_o((Y)),image_nat_o(F,A4))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(aa_set_o_set_nat(vimage_nat_o(F),aa_set_o_set_o(insert_o((Y)),bot_bot_set_o))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3833_inf__img__fin__domE_H,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( ~ aa_set_int_o(finite_finite_int,A4)
       => ~ ! [Y: nat] :
              ( aa_set_nat_o(member_nat(Y),aa_set_int_set_nat(image_int_nat(F),A4))
             => aa_set_int_o(finite_finite_int,inf_inf_set_int(aa_set_nat_set_int(vimage_int_nat(F),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat)),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3834_inf__img__fin__domE_H,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ~ ! [Y: nat] :
              ( aa_set_nat_o(member_nat(Y),aa_set_nat_set_nat(image_nat_nat(F),A4))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(insert_nat(Y),bot_bot_set_nat))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3835_inf__img__fin__domE_H,axiom,
    ! [F: fun_Pr7647638413817135296at_int,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set_int_o(finite_finite_int,image_2483585944268220136at_int(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ~ ! [Y: int] :
              ( aa_set_int_o(member_int(Y),image_2483585944268220136at_int(F,A4))
             => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(vimage4650790856102703794at_int(F,aa_set_int_set_int(insert_int(Y),bot_bot_set_int))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3836_inf__img__fin__domE_H,axiom,
    ! [F: fun_Pr2481279687482350479t_unit,A4: set_Pr1261947904930325089at_nat] :
      ( finite4290736615968046902t_unit(image_6169909464789311569t_unit(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ~ ! [Y: product_unit] :
              ( member_Product_unit(Y,image_6169909464789311569t_unit(F,A4))
             => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(vimage3104651464403166471t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(Y),bot_bo3957492148770167129t_unit))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3837_inf__img__fin__domE_H,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_5846123807819985514at_nat(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ~ ! [Y: product_prod_nat_nat] :
              ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y),image_5846123807819985514at_nat(F,A4))
             => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(inf_inf_set_nat(vimage8013328719654469172at_nat(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y),bot_bo2099793752762293965at_nat))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_3838_prod__set__simps_I2_J,axiom,
    ! [Xa: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] : basic_5713354656523905413at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Xa),Y2)) = insert2290795007606087959at_nat(Y2,bot_bo1534825418459847827at_nat) ).

% prod_set_simps(2)
tff(fact_3839_prod__set__simps_I2_J,axiom,
    ! [Xa: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : basic_484852565060450053at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Xa),Y2)) = insert9200635055090092081at_nat(Y2,bot_bo3083307316010499117at_nat) ).

% prod_set_simps(2)
tff(fact_3840_prod__set__simps_I2_J,axiom,
    ! [Xa: nat,Y2: num] : basic_snds_nat_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xa),Y2)) = insert_num(Y2,bot_bot_set_num) ).

% prod_set_simps(2)
tff(fact_3841_prod__set__simps_I2_J,axiom,
    ! [Xa: int,Y2: int] : basic_snds_int_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xa),Y2)) = aa_set_int_set_int(insert_int(Y2),bot_bot_set_int) ).

% prod_set_simps(2)
tff(fact_3842_prod__set__simps_I2_J,axiom,
    ! [Xa: nat,Y2: nat] : aa_Pro9123573313595863371et_nat(basic_snds_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)) = aa_set_nat_set_nat(insert_nat(Y2),bot_bot_set_nat) ).

% prod_set_simps(2)
tff(fact_3843_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( order_2711500226769404828at_nat(field_2862844103579723188at_nat(Ra),Ra)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),field_2862844103579723188at_nat(Ra))
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),field_2862844103579723188at_nat(Ra))
         => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(A,B),Ra)
          <=> ~ member6009668390874684176at_nat(produc2335810615602371415at_nat(B,A),minus_8235382265540049710at_nat(Ra,id_Pro5428700290092274115at_nat)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3844_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Product_prod_o_o,A: $o,B: $o] :
      ( order_6238756238976269133r_on_o(field_o(Ra),Ra)
     => ( aa_set_o_o(member_o((A)),field_o(Ra))
       => ( aa_set_o_o(member_o((B)),field_o(Ra))
         => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),Ra)
          <=> ~ member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((B)),(A)),minus_7052513330764642158od_o_o(Ra,id_o2)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3845_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( order_7480980593162297270at_nat(field_4706345496283612622at_nat(Ra),Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),field_4706345496283612622at_nat(Ra))
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),field_4706345496283612622at_nat(Ra))
         => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),Ra)
          <=> ~ member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(B),A),minus_8321449233255521966at_nat(Ra,id_Pro2258643101195443293at_nat)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3846_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( order_1878278401035043452at_nat(field_1428494348145443988at_nat(Ra),Ra)
     => ( member6273874620674182256at_nat(A,field_1428494348145443988at_nat(Ra))
       => ( member6273874620674182256at_nat(B,field_1428494348145443988at_nat(Ra))
         => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),Ra)
          <=> ~ member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,B),A),minus_8674369445256047406at_nat(Ra,id_set4888139568991089827at_nat)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3847_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(order_5942792088423767318at_nat(field_3145993188331922222at_nat(Ra)),Ra)
     => ( member2643936169264416010at_nat(A,field_3145993188331922222at_nat(Ra))
       => ( member2643936169264416010at_nat(B,field_3145993188331922222at_nat(Ra))
         => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),Ra)
          <=> ~ aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,B),A)),aa_set4883462369968598652at_nat(minus_3314409938677909166at_nat(Ra),id_set1796276052702428605at_nat)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3848_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(order_4473980167227706203on_nat(aa_set736839985058952875et_nat(field_nat,Ra)),Ra)
     => ( aa_set_nat_o(member_nat(A),aa_set736839985058952875et_nat(field_nat,Ra))
       => ( aa_set_nat_o(member_nat(B),aa_set736839985058952875et_nat(field_nat,Ra))
         => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),Ra)
          <=> ~ aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),A)),aa_set8868280134176360700at_nat(minus_1356011639430497352at_nat(Ra),id_nat2)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3849_Linear__order__in__diff__Id,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,A: int,B: int] :
      ( order_4471489696718655927on_int(field_int(Ra),Ra)
     => ( aa_set_int_o(member_int(A),field_int(Ra))
       => ( aa_set_int_o(member_int(B),field_int(Ra))
         => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),Ra)
          <=> ~ aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,B),A)),minus_1052850069191792384nt_int(Ra,id_int2)) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_3850_aboveS__def,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(order_4940263126897443436at_nat(Ra),A) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_Pro989595756709598047_nat_o(aTP_Lamm_abp(Ra),A)) ).

% aboveS_def
tff(fact_3851_aboveS__def,axiom,
    ! [Ra: set_Pr2560585780119916871nt_int,A: product_prod_int_int] : order_1761765819289301284nt_int(Ra,A) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_Pro3952998155787061599_int_o(aTP_Lamm_abq(Ra),A)) ).

% aboveS_def
tff(fact_3852_aboveS__def,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] : aa_set7518756442812098908at_nat(order_2442354982230848562at_nat(Ra),A) = collec4573900263915890866at_nat(aa_set7096950279330728543_nat_o(aTP_Lamm_abr(Ra),A)) ).

% aboveS_def
tff(fact_3853_aboveS__def,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] : aa_set6242432332095302236at_nat(order_4440095836999870412at_nat(Ra),A) = collec5514110066124741708at_nat(aa_set7675215768510388063_nat_o(aTP_Lamm_abs(Ra),A)) ).

% aboveS_def
tff(fact_3854_aboveS__def,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat,A: nat] : aa_nat_set_nat(order_aboveS_nat(Ra),A) = aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_abt(Ra),A)) ).

% aboveS_def
tff(fact_3855_aboveS__def,axiom,
    ! [Ra: set_Pr958786334691620121nt_int,A: int] : aa_int_set_int(order_aboveS_int(Ra),A) = aa_fun_int_o_set_int(collect_int,aa_int_fun_int_o(aTP_Lamm_abu(Ra),A)) ).

% aboveS_def
tff(fact_3856_card__vimage__inj,axiom,
    ! [F: fun_nat_rat,A4: set_rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),top_top_set_nat)
     => ( aa_set_rat_o(ord_less_eq_set_rat(A4),aa_set_nat_set_rat(image_nat_rat(F),top_top_set_nat))
       => ( aa_set_nat_nat(finite_card_nat,vimage_nat_rat(F,A4)) = finite_card_rat(A4) ) ) ) ).

% card_vimage_inj
tff(fact_3857_card__vimage__inj,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat))
       => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(vimage_nat_nat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% card_vimage_inj
tff(fact_3858_card__vimage__inj,axiom,
    ! [F: fun_nat_int,A4: set_int] :
      ( aa_set_nat_o(inj_on_nat_int(F),top_top_set_nat)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_nat_set_int(image_nat_int(F),top_top_set_nat))
       => ( aa_set_nat_nat(finite_card_nat,vimage_nat_int(F,A4)) = finite_card_int(A4) ) ) ) ).

% card_vimage_inj
tff(fact_3859_card__vimage__inj,axiom,
    ! [F: fun_int_nat,A4: set_nat] :
      ( aa_set_int_o(inj_on_int_nat(F),top_top_set_int)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_int_set_nat(image_int_nat(F),top_top_set_int))
       => ( finite_card_int(aa_set_nat_set_int(vimage_int_nat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% card_vimage_inj
tff(fact_3860_card__vimage__inj,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(image_int_int(F),top_top_set_int))
       => ( finite_card_int(aa_set_int_set_int(vimage_int_int(F),A4)) = finite_card_int(A4) ) ) ) ).

% card_vimage_inj
tff(fact_3861_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr7459493094073627847at_nat] :
      ( order_1878278401035043452at_nat(field_1428494348145443988at_nat(Ra),Ra)
     => ( order_423636163458052581at_nat(field_1428494348145443988at_nat(Ra),Ra)
      <=> ! [A8: set_se5479201105039950375at_nat] :
            ( ord_le7966958417815640519at_nat(A8,field_1428494348145443988at_nat(Ra))
           => ( ( A8 != bot_bo1534825418459847827at_nat )
             => ? [X4: set_Pr4329608150637261639at_nat] :
                  ( member6273874620674182256at_nat(X4,A8)
                  & ! [Xa3: set_Pr4329608150637261639at_nat] :
                      ( member6273874620674182256at_nat(Xa3,A8)
                     => member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X4),Xa3),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3862_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(order_5942792088423767318at_nat(field_3145993188331922222at_nat(Ra)),Ra)
     => ( aa_set5844364881195198285_nat_o(order_1968183844437341695at_nat(field_3145993188331922222at_nat(Ra)),Ra)
      <=> ! [A8: set_se7855581050983116737at_nat] :
            ( ord_le2077887516847798113at_nat(A8,field_3145993188331922222at_nat(Ra))
           => ( ( A8 != bot_bo3083307316010499117at_nat )
             => ? [X4: set_Pr1261947904930325089at_nat] :
                  ( member2643936169264416010at_nat(X4,A8)
                  & ! [Xa3: set_Pr1261947904930325089at_nat] :
                      ( member2643936169264416010at_nat(Xa3,A8)
                     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X4),Xa3)),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3863_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr958786334691620121nt_int] :
      ( order_4471489696718655927on_int(field_int(Ra),Ra)
     => ( order_2886507596567047182on_int(field_int(Ra),Ra)
      <=> ! [A8: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(A8),field_int(Ra))
           => ( ( A8 != bot_bot_set_int )
             => ? [X4: int] :
                  ( aa_set_int_o(member_int(X4),A8)
                  & ! [Xa3: int] :
                      ( aa_set_int_o(member_int(Xa3),A8)
                     => aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3)),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3864_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr5094982260447487303t_unit] :
      ( order_888282469287196226t_unit(field_Product_unit(Ra),Ra)
     => ( order_3164716736300062379t_unit(field_Product_unit(Ra),Ra)
      <=> ! [A8: set_Product_unit] :
            ( ord_le3507040750410214029t_unit(A8,field_Product_unit(Ra))
           => ( ( A8 != bot_bo3957492148770167129t_unit )
             => ? [X4: product_unit] :
                  ( member_Product_unit(X4,A8)
                  & ! [Xa3: product_unit] :
                      ( member_Product_unit(Xa3,A8)
                     => member7821138191923116944t_unit(produc5692694829384537815t_unit(X4,Xa3),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3865_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr5564308138774400199at_nat] :
      ( order_2711500226769404828at_nat(field_2862844103579723188at_nat(Ra),Ra)
     => ( order_2339240034117818117at_nat(field_2862844103579723188at_nat(Ra),Ra)
      <=> ! [A8: set_Pr4329608150637261639at_nat] :
            ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(ord_le1268244103169919719at_nat,A8),field_2862844103579723188at_nat(Ra))
           => ( ( A8 != bot_bo228742789529271731at_nat )
             => ? [X4: produc3843707927480180839at_nat] :
                  ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X4),A8)
                  & ! [Xa3: produc3843707927480180839at_nat] :
                      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Xa3),A8)
                     => member6009668390874684176at_nat(produc2335810615602371415at_nat(X4,Xa3),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3866_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr8693737435421807431at_nat] :
      ( order_7480980593162297270at_nat(field_4706345496283612622at_nat(Ra),Ra)
     => ( order_8331247224601271455at_nat(field_4706345496283612622at_nat(Ra),Ra)
      <=> ! [A8: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A8),field_4706345496283612622at_nat(Ra))
           => ( ( A8 != bot_bo2099793752762293965at_nat )
             => ? [X4: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A8)
                  & ! [Xa3: product_prod_nat_nat] :
                      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),A8)
                     => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X4),Xa3),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3867_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Product_prod_o_o] :
      ( order_6238756238976269133r_on_o(field_o(Ra),Ra)
     => ( order_2615775362358406454r_on_o(field_o(Ra),Ra)
      <=> ! [A8: set_o] :
            ( aa_set_o_o(ord_less_eq_set_o(A8),field_o(Ra))
           => ( ( A8 != bot_bot_set_o )
             => ? [X4: $o] :
                  ( aa_set_o_o(member_o((X4)),A8)
                  & ! [Xa3: $o] :
                      ( aa_set_o_o(member_o((Xa3)),A8)
                     => member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((X4)),(Xa3)),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3868_Linear__order__Well__order__iff,axiom,
    ! [Ra: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(order_4473980167227706203on_nat(aa_set736839985058952875et_nat(field_nat,Ra)),Ra)
     => ( aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(aa_set736839985058952875et_nat(field_nat,Ra)),Ra)
      <=> ! [A8: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(A8),aa_set736839985058952875et_nat(field_nat,Ra))
           => ( ( A8 != bot_bot_set_nat )
             => ? [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),A8)
                  & ! [Xa3: nat] :
                      ( aa_set_nat_o(member_nat(Xa3),A8)
                     => aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Xa3)),Ra) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_3869_finite__imageI,axiom,
    ! [F3: set_int,Ha: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,F3)
     => aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(Ha),F3)) ) ).

% finite_imageI
tff(fact_3870_finite__imageI,axiom,
    ! [F3: set_int,Ha: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(Ha),F3)) ) ).

% finite_imageI
tff(fact_3871_finite__imageI,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,Ha: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(Ha),F3)) ) ).

% finite_imageI
tff(fact_3872_finite__imageI,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,Ha: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,F3)
     => aa_set_nat_o(finite_finite_nat,image_2486076414777270412at_nat(Ha,F3)) ) ).

% finite_imageI
tff(fact_3873_finite__imageI,axiom,
    ! [F3: set_nat,Ha: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(Ha),F3)) ) ).

% finite_imageI
tff(fact_3874_finite__imageI,axiom,
    ! [F3: set_nat,Ha: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(Ha),F3)) ) ).

% finite_imageI
tff(fact_3875_finite__imageI,axiom,
    ! [F3: set_nat,Ha: fun_na1476630412504255114at_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_5846123807819985514at_nat(Ha,F3)) ) ).

% finite_imageI
tff(fact_3876_finite__imageI,axiom,
    ! [F3: set_nat,Ha: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(Ha),F3)) ) ).

% finite_imageI
tff(fact_3877_IdI,axiom,
    ! [A: product_prod_nat_nat] : member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),A),id_Pro2258643101195443293at_nat) ).

% IdI
tff(fact_3878_IdI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),A),id_set4888139568991089827at_nat) ).

% IdI
tff(fact_3879_IdI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),A)),id_set1796276052702428605at_nat) ).

% IdI
tff(fact_3880_IdI,axiom,
    ! [A: nat] : aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),A)),id_nat2) ).

% IdI
tff(fact_3881_IdI,axiom,
    ! [A: int] : aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),A)),id_int2) ).

% IdI
tff(fact_3882_pair__in__Id__conv,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),id_Pro2258643101195443293at_nat)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_3883_pair__in__Id__conv,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),id_set4888139568991089827at_nat)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_3884_pair__in__Id__conv,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),id_set1796276052702428605at_nat)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_3885_pair__in__Id__conv,axiom,
    ! [A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),id_nat2)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_3886_pair__in__Id__conv,axiom,
    ! [A: int,B: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),id_int2)
    <=> ( A = B ) ) ).

% pair_in_Id_conv
tff(fact_3887_finite__UN,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr1469247711798767971at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(B5),A4)))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_Pro2745879313366468188at_nat(B5,X4)) ) ) ) ).

% finite_UN
tff(fact_3888_finite__UN,axiom,
    ! [A4: set_nat,B5: fun_na7044673412300895210at_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(image_7178329752028323786at_nat(B5,A4)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_nat7062706319057040595at_nat(B5,X4)) ) ) ) ).

% finite_UN
tff(fact_3889_finite__UN,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,A4)))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),A4)
           => aa_set_nat_o(finite_finite_nat,aa_Pro9123573313595863371et_nat(B5,X4)) ) ) ) ).

% finite_UN
tff(fact_3890_finite__UN,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A4)
           => aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(B5,X4)) ) ) ) ).

% finite_UN
tff(fact_3891_finite__UN__I,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: fun_Pr6212429452034653949at_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( ! [A3: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_Pro441628658106787830at_nat(B5,A3)) )
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(image_7979982168039674559at_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3892_finite__UN__I,axiom,
    ! [A4: set_o,B5: fun_o_4630291819367765772at_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_o_s3322794332960822277at_nat(B5,(A3))) )
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(image_7077428525161401678at_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3893_finite__UN__I,axiom,
    ! [A4: set_int,B5: fun_in8804710076054006414at_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_int8822742982810151799at_nat(B5,A3)) )
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(image_8938366415781434990at_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3894_finite__UN__I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr1469247711798767971at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [A3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_Pro2745879313366468188at_nat(B5,A3)) )
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(aa_set6242432332095302236at_nat(image_5423882813909314213at_nat(B5),A4))) ) ) ).

% finite_UN_I
tff(fact_3895_finite__UN__I,axiom,
    ! [A4: set_nat,B5: fun_na7044673412300895210at_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_nat7062706319057040595at_nat(B5,A3)) )
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,comple5685304695842803022at_nat(image_7178329752028323786at_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3896_finite__UN__I,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B5: fun_Pr6764980218223570944et_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,A4)
     => ( ! [A3: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A3),A4)
           => aa_set_nat_o(finite_finite_nat,aa_Pro1692590229789857841et_nat(B5,A3)) )
       => aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_3160451284896210088et_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3897_finite__UN__I,axiom,
    ! [A4: set_o,B5: fun_o_set_nat] :
      ( aa_set_o_o(finite_finite_o,A4)
     => ( ! [A3: $o] :
            ( aa_set_o_o(member_o((A3)),A4)
           => aa_set_nat_o(finite_finite_nat,aa_o_set_nat(B5,(A3))) )
       => aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_o_set_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3898_finite__UN__I,axiom,
    ! [A4: set_int,B5: fun_int_set_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ! [A3: int] :
            ( aa_set_int_o(member_int(A3),A4)
           => aa_set_nat_o(finite_finite_nat,aa_int_set_nat(B5,A3)) )
       => aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_int_set_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3899_finite__UN__I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( ! [A3: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A3),A4)
           => aa_set_nat_o(finite_finite_nat,aa_Pro9123573313595863371et_nat(B5,A3)) )
       => aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3900_finite__UN__I,axiom,
    ! [A4: set_nat,B5: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ! [A3: nat] :
            ( aa_set_nat_o(member_nat(A3),A4)
           => aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(B5,A3)) )
       => aa_set_nat_o(finite_finite_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(B5,A4))) ) ) ).

% finite_UN_I
tff(fact_3901_bdd__below__UN,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,A4: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I3)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,I3)))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),I3)
           => aa_set_nat_o(condit1738341127787009408ow_nat,aa_Pro9123573313595863371et_nat(A4,X4)) ) ) ) ).

% bdd_below_UN
tff(fact_3902_bdd__below__UN,axiom,
    ! [I3: set_nat,A4: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,I3)
     => ( aa_set_nat_o(condit1738341127787009408ow_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),I3)
           => aa_set_nat_o(condit1738341127787009408ow_nat,aa_nat_set_nat(A4,X4)) ) ) ) ).

% bdd_below_UN
tff(fact_3903_bdd__above__UN,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,A4: fun_Pr9038118721296872730et_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,I3)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_15824709712370754et_nat(A4,I3)))
      <=> ! [X4: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),I3)
           => aa_set_nat_o(condit2214826472909112428ve_nat,aa_Pro9123573313595863371et_nat(A4,X4)) ) ) ) ).

% bdd_above_UN
tff(fact_3904_bdd__above__UN,axiom,
    ! [I3: set_nat,A4: fun_nat_set_nat] :
      ( aa_set_nat_o(finite_finite_nat,I3)
     => ( aa_set_nat_o(condit2214826472909112428ve_nat,aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(A4,I3)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),I3)
           => aa_set_nat_o(condit2214826472909112428ve_nat,aa_nat_set_nat(A4,X4)) ) ) ) ).

% bdd_above_UN
tff(fact_3905_card__image__le,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_int_set_nat(image_int_nat(F),A4))),finite_card_int(A4)) ) ).

% card_image_le
tff(fact_3906_card__image__le,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_int_set_int(image_int_int(F),A4))),finite_card_int(A4)) ) ).

% card_image_le
tff(fact_3907_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_rat(aa_set_nat_set_rat(image_nat_rat(F),A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_3908_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_3909_card__image__le,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(aa_set_nat_set_int(image_nat_int(F),A4))),aa_set_nat_nat(finite_card_nat,A4)) ) ).

% card_image_le
tff(fact_3910_surj__card__le,axiom,
    ! [A4: set_int,B5: set_nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B5)),finite_card_int(A4)) ) ) ).

% surj_card_le
tff(fact_3911_surj__card__le,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_int_set_int(image_int_int(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B5)),finite_card_int(A4)) ) ) ).

% surj_card_le
tff(fact_3912_surj__card__le,axiom,
    ! [A4: set_nat,B5: set_rat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_rat(B5)),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% surj_card_le
tff(fact_3913_surj__card__le,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,B5)),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% surj_card_le
tff(fact_3914_surj__card__le,axiom,
    ! [A4: set_nat,B5: set_int,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_nat_set_int(image_nat_int(F),A4))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(B5)),aa_set_nat_nat(finite_card_nat,A4)) ) ) ).

% surj_card_le
tff(fact_3915_eq__card__imp__inj__on,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( aa_set_nat_nat(finite_card_nat,aa_set_int_set_nat(image_int_nat(F),A4)) = finite_card_int(A4) )
       => aa_set_int_o(inj_on_int_nat(F),A4) ) ) ).

% eq_card_imp_inj_on
tff(fact_3916_eq__card__imp__inj__on,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( ( finite_card_int(aa_set_int_set_int(image_int_int(F),A4)) = finite_card_int(A4) )
       => aa_set_int_o(inj_on_int_int(F),A4) ) ) ).

% eq_card_imp_inj_on
tff(fact_3917_eq__card__imp__inj__on,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( finite_card_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) )
       => aa_set_nat_o(inj_on_nat_rat(F),A4) ) ) ).

% eq_card_imp_inj_on
tff(fact_3918_eq__card__imp__inj__on,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) )
       => aa_set_nat_o(inj_on_nat_nat(F),A4) ) ) ).

% eq_card_imp_inj_on
tff(fact_3919_eq__card__imp__inj__on,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( ( finite_card_int(aa_set_nat_set_int(image_nat_int(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) )
       => aa_set_nat_o(inj_on_nat_int(F),A4) ) ) ).

% eq_card_imp_inj_on
tff(fact_3920_inj__on__iff__eq__card,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(inj_on_int_nat(F),A4)
      <=> ( aa_set_nat_nat(finite_card_nat,aa_set_int_set_nat(image_int_nat(F),A4)) = finite_card_int(A4) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_3921_inj__on__iff__eq__card,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(inj_on_int_int(F),A4)
      <=> ( finite_card_int(aa_set_int_set_int(image_int_int(F),A4)) = finite_card_int(A4) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_3922_inj__on__iff__eq__card,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(inj_on_nat_rat(F),A4)
      <=> ( finite_card_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_3923_inj__on__iff__eq__card,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(inj_on_nat_nat(F),A4)
      <=> ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_3924_inj__on__iff__eq__card,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(inj_on_nat_int(F),A4)
      <=> ( finite_card_int(aa_set_nat_set_int(image_nat_int(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_3925_well__order__on__domain,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,Ra: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( order_2339240034117818117at_nat(A4,Ra)
     => ( member6009668390874684176at_nat(produc2335810615602371415at_nat(A,B),Ra)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(A),A4)
          & aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(B),A4) ) ) ) ).

% well_order_on_domain
tff(fact_3926_well__order__on__domain,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,Ra: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( order_8331247224601271455at_nat(A4,Ra)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),Ra)
       => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(A),A4)
          & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(B),A4) ) ) ) ).

% well_order_on_domain
tff(fact_3927_well__order__on__domain,axiom,
    ! [A4: set_o,Ra: set_Product_prod_o_o,A: $o,B: $o] :
      ( order_2615775362358406454r_on_o(A4,Ra)
     => ( member7466972457876170832od_o_o(aa_o_P8186867570850176203od_o_o(product_Pair_o_o((A)),(B)),Ra)
       => ( aa_set_o_o(member_o((A)),A4)
          & aa_set_o_o(member_o((B)),A4) ) ) ) ).

% well_order_on_domain
tff(fact_3928_well__order__on__domain,axiom,
    ! [A4: set_se5479201105039950375at_nat,Ra: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( order_423636163458052581at_nat(A4,Ra)
     => ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),Ra)
       => ( member6273874620674182256at_nat(A,A4)
          & member6273874620674182256at_nat(B,A4) ) ) ) ).

% well_order_on_domain
tff(fact_3929_well__order__on__domain,axiom,
    ! [A4: set_se7855581050983116737at_nat,Ra: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(order_1968183844437341695at_nat(A4),Ra)
     => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),Ra)
       => ( member2643936169264416010at_nat(A,A4)
          & member2643936169264416010at_nat(B,A4) ) ) ) ).

% well_order_on_domain
tff(fact_3930_well__order__on__domain,axiom,
    ! [A4: set_int,Ra: set_Pr958786334691620121nt_int,A: int,B: int] :
      ( order_2886507596567047182on_int(A4,Ra)
     => ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),Ra)
       => ( aa_set_int_o(member_int(A),A4)
          & aa_set_int_o(member_int(B),A4) ) ) ) ).

% well_order_on_domain
tff(fact_3931_well__order__on__domain,axiom,
    ! [A4: set_nat,Ra: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(A4),Ra)
     => ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),Ra)
       => ( aa_set_nat_o(member_nat(A),A4)
          & aa_set_nat_o(member_nat(B),A4) ) ) ) ).

% well_order_on_domain
tff(fact_3932_infinite__countable__subset,axiom,
    ! [S3: set_rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,S3)
     => ? [F4: fun_nat_rat] :
          ( aa_set_nat_o(inj_on_nat_rat(F4),top_top_set_nat)
          & aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F4),top_top_set_nat)),S3) ) ) ).

% infinite_countable_subset
tff(fact_3933_infinite__countable__subset,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
     => ? [F4: fun_nat_int] :
          ( aa_set_nat_o(inj_on_nat_int(F4),top_top_set_nat)
          & aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F4),top_top_set_nat)),S3) ) ) ).

% infinite_countable_subset
tff(fact_3934_infinite__countable__subset,axiom,
    ! [S3: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
     => ? [F4: fun_na1476630412504255114at_nat] :
          ( aa_set_nat_o(inj_on5538052773655684606at_nat(F4),top_top_set_nat)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F4,top_top_set_nat)),S3) ) ) ).

% infinite_countable_subset
tff(fact_3935_infinite__countable__subset,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
     => ? [F4: fun_nat_nat] :
          ( aa_set_nat_o(inj_on_nat_nat(F4),top_top_set_nat)
          & aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F4),top_top_set_nat)),S3) ) ) ).

% infinite_countable_subset
tff(fact_3936_infinite__iff__countable__subset,axiom,
    ! [S3: set_rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,S3)
    <=> ? [F5: fun_nat_rat] :
          ( aa_set_nat_o(inj_on_nat_rat(F5),top_top_set_nat)
          & aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F5),top_top_set_nat)),S3) ) ) ).

% infinite_iff_countable_subset
tff(fact_3937_infinite__iff__countable__subset,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ? [F5: fun_nat_int] :
          ( aa_set_nat_o(inj_on_nat_int(F5),top_top_set_nat)
          & aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F5),top_top_set_nat)),S3) ) ) ).

% infinite_iff_countable_subset
tff(fact_3938_infinite__iff__countable__subset,axiom,
    ! [S3: set_Pr1261947904930325089at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
    <=> ? [F5: fun_na1476630412504255114at_nat] :
          ( aa_set_nat_o(inj_on5538052773655684606at_nat(F5),top_top_set_nat)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F5,top_top_set_nat)),S3) ) ) ).

% infinite_iff_countable_subset
tff(fact_3939_infinite__iff__countable__subset,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [F5: fun_nat_nat] :
          ( aa_set_nat_o(inj_on_nat_nat(F5),top_top_set_nat)
          & aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F5),top_top_set_nat)),S3) ) ) ).

% infinite_iff_countable_subset
tff(fact_3940_IdE,axiom,
    ! [P: produc859450856879609959at_nat] :
      ( member8206827879206165904at_nat(P,id_Pro2258643101195443293at_nat)
     => ~ ! [X: product_prod_nat_nat] : P != aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(X),X) ) ).

% IdE
tff(fact_3941_IdE,axiom,
    ! [P: produc1319942482725812455at_nat] :
      ( member1466754251312161552at_nat(P,id_set4888139568991089827at_nat)
     => ~ ! [X: set_Pr4329608150637261639at_nat] : P != aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,X),X) ) ).

% IdE
tff(fact_3942_IdE,axiom,
    ! [P: produc3843707927480180839at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(P),id_set1796276052702428605at_nat)
     => ~ ! [X: set_Pr1261947904930325089at_nat] : P != aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X),X) ) ).

% IdE
tff(fact_3943_IdE,axiom,
    ! [P: product_prod_nat_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(P),id_nat2)
     => ~ ! [X: nat] : P != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),X) ) ).

% IdE
tff(fact_3944_IdE,axiom,
    ! [P: product_prod_int_int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(P),id_int2)
     => ~ ! [X: int] : P != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),X) ) ).

% IdE
tff(fact_3945_IdD,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(A),B),id_Pro2258643101195443293at_nat)
     => ( A = B ) ) ).

% IdD
tff(fact_3946_IdD,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,A),B),id_set4888139568991089827at_nat)
     => ( A = B ) ) ).

% IdD
tff(fact_3947_IdD,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,A),B)),id_set1796276052702428605at_nat)
     => ( A = B ) ) ).

% IdD
tff(fact_3948_IdD,axiom,
    ! [A: nat,B: nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),id_nat2)
     => ( A = B ) ) ).

% IdD
tff(fact_3949_IdD,axiom,
    ! [A: int,B: int] :
      ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)),id_int2)
     => ( A = B ) ) ).

% IdD
tff(fact_3950_card__bij__eq,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int,G: fun_int_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
       => ( aa_set_int_o(inj_on_int_int(G),B5)
         => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(G),B5)),A4)
           => ( aa_set_int_o(finite_finite_int,A4)
             => ( aa_set_int_o(finite_finite_int,B5)
               => ( finite_card_int(A4) = finite_card_int(B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3951_card__bij__eq,axiom,
    ! [F: fun_rat_nat,A4: set_rat,B5: set_nat,G: fun_nat_rat] :
      ( aa_set_rat_o(inj_on_rat_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(image_rat_nat(F,A4)),B5)
       => ( aa_set_nat_o(inj_on_nat_rat(G),B5)
         => ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(G),B5)),A4)
           => ( aa_set_rat_o(finite_finite_rat,A4)
             => ( aa_set_nat_o(finite_finite_nat,B5)
               => ( finite_card_rat(A4) = aa_set_nat_nat(finite_card_nat,B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3952_card__bij__eq,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat,G: fun_nat_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
       => ( aa_set_nat_o(inj_on_nat_int(G),B5)
         => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(G),B5)),A4)
           => ( aa_set_int_o(finite_finite_int,A4)
             => ( aa_set_nat_o(finite_finite_nat,B5)
               => ( finite_card_int(A4) = aa_set_nat_nat(finite_card_nat,B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3953_card__bij__eq,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,G: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4)),B5)
       => ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(G),B5)
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(G),B5)),A4)
           => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
             => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
               => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3954_card__bij__eq,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_nat,G: fun_na1476630412504255114at_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(F,A4)),B5)
       => ( aa_set_nat_o(inj_on5538052773655684606at_nat(G),B5)
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(G,B5)),A4)
           => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
             => ( aa_set_nat_o(finite_finite_nat,B5)
               => ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4) = aa_set_nat_nat(finite_card_nat,B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3955_card__bij__eq,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat,G: fun_rat_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
       => ( aa_set_rat_o(inj_on_rat_nat(G),B5)
         => ( aa_set_nat_o(ord_less_eq_set_nat(image_rat_nat(G,B5)),A4)
           => ( aa_set_nat_o(finite_finite_nat,A4)
             => ( aa_set_rat_o(finite_finite_rat,B5)
               => ( aa_set_nat_nat(finite_card_nat,A4) = finite_card_rat(B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3956_card__bij__eq,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int,G: fun_int_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
       => ( aa_set_int_o(inj_on_int_nat(G),B5)
         => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(G),B5)),A4)
           => ( aa_set_nat_o(finite_finite_nat,A4)
             => ( aa_set_int_o(finite_finite_int,B5)
               => ( aa_set_nat_nat(finite_card_nat,A4) = finite_card_int(B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3957_card__bij__eq,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat,B5: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat] :
      ( aa_set_nat_o(inj_on5538052773655684606at_nat(F),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F,A4)),B5)
       => ( aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(G),B5)
         => ( aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(G,B5)),A4)
           => ( aa_set_nat_o(finite_finite_nat,A4)
             => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
               => ( aa_set_nat_nat(finite_card_nat,A4) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3958_card__bij__eq,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
       => ( aa_set_nat_o(inj_on_nat_nat(G),B5)
         => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(G),B5)),A4)
           => ( aa_set_nat_o(finite_finite_nat,A4)
             => ( aa_set_nat_o(finite_finite_nat,B5)
               => ( aa_set_nat_nat(finite_card_nat,A4) = aa_set_nat_nat(finite_card_nat,B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_3959_card__le__inj,axiom,
    ! [A4: set_int,B5: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B5))
         => ? [F4: fun_int_int] :
              ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F4),A4)),B5)
              & aa_set_int_o(inj_on_int_int(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3960_card__le__inj,axiom,
    ! [A4: set_int,B5: set_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),aa_set_nat_nat(finite_card_nat,B5))
         => ? [F4: fun_int_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F4),A4)),B5)
              & aa_set_int_o(inj_on_int_nat(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3961_card__le__inj,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5))
         => ? [F4: fun_Pr2118491707810114563at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F4),A4)),B5)
              & aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3962_card__le__inj,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4)),aa_set_nat_nat(finite_card_nat,B5))
         => ? [F4: fun_Pr2602117396471556196at_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(F4,A4)),B5)
              & aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3963_card__le__inj,axiom,
    ! [A4: set_nat,B5: set_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_rat_o(finite_finite_rat,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_rat(B5))
         => ? [F4: fun_nat_rat] :
              ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F4),A4)),B5)
              & aa_set_nat_o(inj_on_nat_rat(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3964_card__le__inj,axiom,
    ! [A4: set_nat,B5: set_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_int(B5))
         => ? [F4: fun_nat_int] :
              ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F4),A4)),B5)
              & aa_set_nat_o(inj_on_nat_int(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3965_card__le__inj,axiom,
    ! [A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5))
         => ? [F4: fun_na1476630412504255114at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F4,A4)),B5)
              & aa_set_nat_o(inj_on5538052773655684606at_nat(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3966_card__le__inj,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5))
         => ? [F4: fun_nat_nat] :
              ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F4),A4)),B5)
              & aa_set_nat_o(inj_on_nat_nat(F4),A4) ) ) ) ) ).

% card_le_inj
tff(fact_3967_card__inj__on__le,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
       => ( aa_set_rat_o(finite_finite_rat,B5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_rat(B5)) ) ) ) ).

% card_inj_on_le
tff(fact_3968_card__inj__on__le,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
       => ( aa_set_int_o(finite_finite_int,B5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_int(B5)) ) ) ) ).

% card_inj_on_le
tff(fact_3969_card__inj__on__le,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
       => ( aa_set_int_o(finite_finite_int,B5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B5)) ) ) ) ).

% card_inj_on_le
tff(fact_3970_card__inj__on__le,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
       => ( aa_set_nat_o(finite_finite_nat,B5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5)) ) ) ) ).

% card_inj_on_le
tff(fact_3971_card__inj__on__le,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
       => ( aa_set_nat_o(finite_finite_nat,B5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),aa_set_nat_nat(finite_card_nat,B5)) ) ) ) ).

% card_inj_on_le
tff(fact_3972_inj__on__iff__card__le,axiom,
    ! [A4: set_int,B5: set_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( ? [F5: fun_int_int] :
              ( aa_set_int_o(inj_on_int_int(F5),A4)
              & aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F5),A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),finite_card_int(B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3973_inj__on__iff__card__le,axiom,
    ! [A4: set_int,B5: set_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( ? [F5: fun_int_nat] :
              ( aa_set_int_o(inj_on_int_nat(F5),A4)
              & aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F5),A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_int(A4)),aa_set_nat_nat(finite_card_nat,B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3974_inj__on__iff__card__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
       => ( ? [F5: fun_Pr2118491707810114563at_nat] :
              ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F5),A4)
              & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F5),A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3975_inj__on__iff__card__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( ? [F5: fun_Pr2602117396471556196at_nat] :
              ( aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F5),A4)
              & aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(F5,A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set4749074711808745205at_nat(finite711546835091564841at_nat,A4)),aa_set_nat_nat(finite_card_nat,B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3976_inj__on__iff__card__le,axiom,
    ! [A4: set_nat,B5: set_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_rat_o(finite_finite_rat,B5)
       => ( ? [F5: fun_nat_rat] :
              ( aa_set_nat_o(inj_on_nat_rat(F5),A4)
              & aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F5),A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_rat(B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3977_inj__on__iff__card__le,axiom,
    ! [A4: set_nat,B5: set_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( ? [F5: fun_nat_int] :
              ( aa_set_nat_o(inj_on_nat_int(F5),A4)
              & aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F5),A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),finite_card_int(B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3978_inj__on__iff__card__le,axiom,
    ! [A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
       => ( ? [F5: fun_na1476630412504255114at_nat] :
              ( aa_set_nat_o(inj_on5538052773655684606at_nat(F5),A4)
              & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F5,A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set4749074711808745205at_nat(finite711546835091564841at_nat,B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3979_inj__on__iff__card__le,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( ? [F5: fun_nat_nat] :
              ( aa_set_nat_o(inj_on_nat_nat(F5),A4)
              & aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F5),A4)),B5) )
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,A4)),aa_set_nat_nat(finite_card_nat,B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_3980_surjective__iff__injective__gen,axiom,
    ! [S3: set_int,T2: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,T2)
       => ( ( finite_card_int(S3) = finite_card_int(T2) )
         => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),S3)),T2)
           => ( ! [X4: int] :
                  ( aa_set_int_o(member_int(X4),T2)
                 => ? [Xa3: int] :
                      ( aa_set_int_o(member_int(Xa3),S3)
                      & ( aa_int_int(F,Xa3) = X4 ) ) )
            <=> aa_set_int_o(inj_on_int_int(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3981_surjective__iff__injective__gen,axiom,
    ! [S3: set_int,T2: set_nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ( finite_card_int(S3) = aa_set_nat_nat(finite_card_nat,T2) )
         => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),S3)),T2)
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),T2)
                 => ? [Xa3: int] :
                      ( aa_set_int_o(member_int(Xa3),S3)
                      & ( aa_int_nat(F,Xa3) = X4 ) ) )
            <=> aa_set_int_o(inj_on_int_nat(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3982_surjective__iff__injective__gen,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat,F: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T2)
       => ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,S3) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,T2) )
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),S3)),T2)
           => ( ! [X4: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),T2)
                 => ? [Xa3: product_prod_nat_nat] :
                      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),S3)
                      & ( aa_Pro262675715343573244at_nat(F,Xa3) = X4 ) ) )
            <=> aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3983_surjective__iff__injective__gen,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T2: set_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,S3)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ( aa_set4749074711808745205at_nat(finite711546835091564841at_nat,S3) = aa_set_nat_nat(finite_card_nat,T2) )
         => ( aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(F,S3)),T2)
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),T2)
                 => ? [Xa3: product_prod_nat_nat] :
                      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa3),S3)
                      & ( aa_Pro4243173770044397973at_nat(F,Xa3) = X4 ) ) )
            <=> aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3984_surjective__iff__injective__gen,axiom,
    ! [S3: set_nat,T2: set_rat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_rat_o(finite_finite_rat,T2)
       => ( ( aa_set_nat_nat(finite_card_nat,S3) = finite_card_rat(T2) )
         => ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),S3)),T2)
           => ( ! [X4: rat] :
                  ( aa_set_rat_o(member_rat(X4),T2)
                 => ? [Xa3: nat] :
                      ( aa_set_nat_o(member_nat(Xa3),S3)
                      & ( aa_nat_rat(F,Xa3) = X4 ) ) )
            <=> aa_set_nat_o(inj_on_nat_rat(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3985_surjective__iff__injective__gen,axiom,
    ! [S3: set_nat,T2: set_int,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_int_o(finite_finite_int,T2)
       => ( ( aa_set_nat_nat(finite_card_nat,S3) = finite_card_int(T2) )
         => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),S3)),T2)
           => ( ! [X4: int] :
                  ( aa_set_int_o(member_int(X4),T2)
                 => ? [Xa3: nat] :
                      ( aa_set_nat_o(member_nat(Xa3),S3)
                      & ( aa_nat_int(F,Xa3) = X4 ) ) )
            <=> aa_set_nat_o(inj_on_nat_int(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3986_surjective__iff__injective__gen,axiom,
    ! [S3: set_nat,T2: set_Pr1261947904930325089at_nat,F: fun_na1476630412504255114at_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,T2)
       => ( ( aa_set_nat_nat(finite_card_nat,S3) = aa_set4749074711808745205at_nat(finite711546835091564841at_nat,T2) )
         => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F,S3)),T2)
           => ( ! [X4: product_prod_nat_nat] :
                  ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X4),T2)
                 => ? [Xa3: nat] :
                      ( aa_set_nat_o(member_nat(Xa3),S3)
                      & ( aa_nat7603221163087113075at_nat(F,Xa3) = X4 ) ) )
            <=> aa_set_nat_o(inj_on5538052773655684606at_nat(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3987_surjective__iff__injective__gen,axiom,
    ! [S3: set_nat,T2: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(finite_finite_nat,T2)
       => ( ( aa_set_nat_nat(finite_card_nat,S3) = aa_set_nat_nat(finite_card_nat,T2) )
         => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),S3)),T2)
           => ( ! [X4: nat] :
                  ( aa_set_nat_o(member_nat(X4),T2)
                 => ? [Xa3: nat] :
                      ( aa_set_nat_o(member_nat(Xa3),S3)
                      & ( aa_nat_nat(F,Xa3) = X4 ) ) )
            <=> aa_set_nat_o(inj_on_nat_nat(F),S3) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_3988_pigeonhole__infinite,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(F),A4))
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ~ aa_set_int_o(finite_finite_int,aa_fun_int_o_set_int(collect_int,aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_abv(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3989_pigeonhole__infinite,axiom,
    ! [A4: set_o,F: fun_o_nat] :
      ( ~ aa_set_o_o(finite_finite_o,A4)
     => ( aa_set_nat_o(finite_finite_nat,image_o_nat(F,A4))
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
            & ~ aa_set_o_o(finite_finite_o,aa_fun_o_o_set_o(collect_o,aa_o_fun_o_o(aa_fun8960405049393213551un_o_o(aTP_Lamm_abw(A4),F),(X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3990_pigeonhole__infinite,axiom,
    ! [A4: set_int,F: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(F),A4))
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ~ aa_set_int_o(finite_finite_int,aa_fun_int_o_set_int(collect_int,aa_int_fun_int_o(aa_fun927631379751134075_int_o(aTP_Lamm_abx(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3991_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ~ aa_set_nat_o(finite_finite_nat,aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aa_fun2969884873976723847_nat_o(aTP_Lamm_aby(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3992_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(F),A4))
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ~ aa_set_nat_o(finite_finite_nat,aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_abz(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3993_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A4)
            & ~ aa_set_nat_o(finite_finite_nat,aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_aca(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3994_pigeonhole__infinite,axiom,
    ! [A4: set_o,F: fun_o_67843798866261420at_nat] :
      ( ~ aa_set_o_o(finite_finite_o,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_3855930084881510382at_nat(F,A4))
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),A4)
            & ~ aa_set_o_o(finite_finite_o,aa_fun_o_o_set_o(collect_o,aa_o_fun_o_o(aa_fun4676553822554939524un_o_o(aTP_Lamm_acb(A4),F),(X)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3995_pigeonhole__infinite,axiom,
    ! [A4: set_int,F: fun_in4514472496968388910at_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_8883965892284119310at_nat(F,A4))
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A4)
            & ~ aa_set_int_o(finite_finite_int,aa_fun_int_o_set_int(collect_int,aa_int_fun_int_o(aa_fun1661645502662368768_int_o(aTP_Lamm_acc(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3996_pigeonhole__infinite,axiom,
    ! [A4: set_Pr958786334691620121nt_int,F: fun_Pr5160692531401422620nt_nat] :
      ( ~ finite2998713641127702882nt_int(A4)
     => ( aa_set_nat_o(finite_finite_nat,image_5044651549707136836nt_nat(F,A4))
       => ? [X: product_prod_int_int] :
            ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(X),A4)
            & ~ finite2998713641127702882nt_int(aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_Pro3952998155787061599_int_o(aa_fun4260993738209906510_int_o(aTP_Lamm_acd(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3997_pigeonhole__infinite,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set_nat_o(finite_finite_nat,image_2486076414777270412at_nat(F,A4))
       => ? [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),A4)
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_Pro989595756709598047_nat_o(aa_fun2769474628653446662_nat_o(aTP_Lamm_ace(A4),F),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3998_snds_Ointros,axiom,
    ! [P: produc7812382424243267045t_unit] : member242389114213480694t_unit(aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P),basic_1496934909540232809t_unit(P)) ).

% snds.intros
tff(fact_3999_snds_Osimps,axiom,
    ! [A: heap_e7401611519738050253t_unit,P: produc7812382424243267045t_unit] :
      ( member242389114213480694t_unit(A,basic_1496934909540232809t_unit(P))
    <=> ( A = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P) ) ) ).

% snds.simps
tff(fact_4000_snds_Ocases,axiom,
    ! [A: heap_e7401611519738050253t_unit,P: produc7812382424243267045t_unit] :
      ( member242389114213480694t_unit(A,basic_1496934909540232809t_unit(P))
     => ( A = aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,P) ) ) ).

% snds.cases
tff(fact_4001_card__image,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( finite_card_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ).

% card_image
tff(fact_4002_card__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_nat_set_nat(image_nat_nat(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ).

% card_image
tff(fact_4003_card__image,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( finite_card_int(aa_set_nat_set_int(image_nat_int(F),A4)) = aa_set_nat_nat(finite_card_nat,A4) ) ) ).

% card_image
tff(fact_4004_card__image,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_nat(finite_card_nat,aa_set_int_set_nat(image_int_nat(F),A4)) = finite_card_int(A4) ) ) ).

% card_image
tff(fact_4005_card__image,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( finite_card_int(aa_set_int_set_int(image_int_int(F),A4)) = finite_card_int(A4) ) ) ).

% card_image
tff(fact_4006_all__finite__subset__image,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(ord_less_eq_set_int(B6),aa_set_int_set_int(image_int_int(F),A4)) )
         => aa_set_int_o(P2,B6) )
    <=> ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(ord_less_eq_set_int(B6),A4) )
         => aa_set_int_o(P2,aa_set_int_set_int(image_int_int(F),B6)) ) ) ).

% all_finite_subset_image
tff(fact_4007_all__finite__subset__image,axiom,
    ! [F: fun_nat_rat,A4: set_nat,P2: fun_set_rat_o] :
      ( ! [B6: set_rat] :
          ( ( aa_set_rat_o(finite_finite_rat,B6)
            & aa_set_rat_o(ord_less_eq_set_rat(B6),aa_set_nat_set_rat(image_nat_rat(F),A4)) )
         => aa_set_rat_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),A4) )
         => aa_set_rat_o(P2,aa_set_nat_set_rat(image_nat_rat(F),B6)) ) ) ).

% all_finite_subset_image
tff(fact_4008_all__finite__subset__image,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_set_int_o] :
      ( ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(ord_less_eq_set_int(B6),aa_set_nat_set_int(image_nat_int(F),A4)) )
         => aa_set_int_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),A4) )
         => aa_set_int_o(P2,aa_set_nat_set_int(image_nat_int(F),B6)) ) ) ).

% all_finite_subset_image
tff(fact_4009_all__finite__subset__image,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_se4782206882984618170_nat_o] :
      ( ! [B6: set_Pr1261947904930325089at_nat] :
          ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
            & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4)) )
         => aa_set3702741264012733747_nat_o(P2,B6) )
    <=> ! [B6: set_Pr1261947904930325089at_nat] :
          ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
            & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),A4) )
         => aa_set3702741264012733747_nat_o(P2,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),B6)) ) ) ).

% all_finite_subset_image
tff(fact_4010_all__finite__subset__image,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat,P2: fun_se4782206882984618170_nat_o] :
      ( ! [B6: set_Pr1261947904930325089at_nat] :
          ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
            & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),image_5846123807819985514at_nat(F,A4)) )
         => aa_set3702741264012733747_nat_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),A4) )
         => aa_set3702741264012733747_nat_o(P2,image_5846123807819985514at_nat(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_4011_all__finite__subset__image,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),aa_set_int_set_nat(image_int_nat(F),A4)) )
         => aa_set_nat_o(P2,B6) )
    <=> ! [B6: set_int] :
          ( ( aa_set_int_o(finite_finite_int,B6)
            & aa_set_int_o(ord_less_eq_set_int(B6),A4) )
         => aa_set_nat_o(P2,aa_set_int_set_nat(image_int_nat(F),B6)) ) ) ).

% all_finite_subset_image
tff(fact_4012_all__finite__subset__image,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),image_2486076414777270412at_nat(F,A4)) )
         => aa_set_nat_o(P2,B6) )
    <=> ! [B6: set_Pr1261947904930325089at_nat] :
          ( ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
            & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),A4) )
         => aa_set_nat_o(P2,image_2486076414777270412at_nat(F,B6)) ) ) ).

% all_finite_subset_image
tff(fact_4013_all__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),aa_set_nat_set_nat(image_nat_nat(F),A4)) )
         => aa_set_nat_o(P2,B6) )
    <=> ! [B6: set_nat] :
          ( ( aa_set_nat_o(finite_finite_nat,B6)
            & aa_set_nat_o(ord_less_eq_set_nat(B6),A4) )
         => aa_set_nat_o(P2,aa_set_nat_set_nat(image_nat_nat(F),B6)) ) ) ).

% all_finite_subset_image
tff(fact_4014_ex__finite__subset__image,axiom,
    ! [F: fun_int_int,A4: set_int,P2: fun_set_int_o] :
      ( ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(ord_less_eq_set_int(B6),aa_set_int_set_int(image_int_int(F),A4))
          & aa_set_int_o(P2,B6) )
    <=> ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(ord_less_eq_set_int(B6),A4)
          & aa_set_int_o(P2,aa_set_int_set_int(image_int_int(F),B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4015_ex__finite__subset__image,axiom,
    ! [F: fun_nat_rat,A4: set_nat,P2: fun_set_rat_o] :
      ( ? [B6: set_rat] :
          ( aa_set_rat_o(finite_finite_rat,B6)
          & aa_set_rat_o(ord_less_eq_set_rat(B6),aa_set_nat_set_rat(image_nat_rat(F),A4))
          & aa_set_rat_o(P2,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
          & aa_set_rat_o(P2,aa_set_nat_set_rat(image_nat_rat(F),B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4016_ex__finite__subset__image,axiom,
    ! [F: fun_nat_int,A4: set_nat,P2: fun_set_int_o] :
      ( ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(ord_less_eq_set_int(B6),aa_set_nat_set_int(image_nat_int(F),A4))
          & aa_set_int_o(P2,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
          & aa_set_int_o(P2,aa_set_nat_set_int(image_nat_int(F),B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4017_ex__finite__subset__image,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_se4782206882984618170_nat_o] :
      ( ? [B6: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
          & aa_set3702741264012733747_nat_o(P2,B6) )
    <=> ? [B6: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),A4)
          & aa_set3702741264012733747_nat_o(P2,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4018_ex__finite__subset__image,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat,P2: fun_se4782206882984618170_nat_o] :
      ( ? [B6: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),image_5846123807819985514at_nat(F,A4))
          & aa_set3702741264012733747_nat_o(P2,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
          & aa_set3702741264012733747_nat_o(P2,image_5846123807819985514at_nat(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4019_ex__finite__subset__image,axiom,
    ! [F: fun_int_nat,A4: set_int,P2: fun_set_nat_o] :
      ( ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),aa_set_int_set_nat(image_int_nat(F),A4))
          & aa_set_nat_o(P2,B6) )
    <=> ? [B6: set_int] :
          ( aa_set_int_o(finite_finite_int,B6)
          & aa_set_int_o(ord_less_eq_set_int(B6),A4)
          & aa_set_nat_o(P2,aa_set_int_set_nat(image_int_nat(F),B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4020_ex__finite__subset__image,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat,P2: fun_set_nat_o] :
      ( ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),image_2486076414777270412at_nat(F,A4))
          & aa_set_nat_o(P2,B6) )
    <=> ? [B6: set_Pr1261947904930325089at_nat] :
          ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B6)
          & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B6),A4)
          & aa_set_nat_o(P2,image_2486076414777270412at_nat(F,B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4021_ex__finite__subset__image,axiom,
    ! [F: fun_nat_nat,A4: set_nat,P2: fun_set_nat_o] :
      ( ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),aa_set_nat_set_nat(image_nat_nat(F),A4))
          & aa_set_nat_o(P2,B6) )
    <=> ? [B6: set_nat] :
          ( aa_set_nat_o(finite_finite_nat,B6)
          & aa_set_nat_o(ord_less_eq_set_nat(B6),A4)
          & aa_set_nat_o(P2,aa_set_nat_set_nat(image_nat_nat(F),B6)) ) ) ).

% ex_finite_subset_image
tff(fact_4022_finite__subset__image,axiom,
    ! [B5: set_int,F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,B5)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_int_set_int(image_int_int(F),A4))
       => ? [C2: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C2),A4)
            & aa_set_int_o(finite_finite_int,C2)
            & ( B5 = aa_set_int_set_int(image_int_int(F),C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4023_finite__subset__image,axiom,
    ! [B5: set_rat,F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(finite_finite_rat,B5)
     => ( aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => ? [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
            & aa_set_nat_o(finite_finite_nat,C2)
            & ( B5 = aa_set_nat_set_rat(image_nat_rat(F),C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4024_finite__subset__image,axiom,
    ! [B5: set_int,F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(finite_finite_int,B5)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_nat_set_int(image_nat_int(F),A4))
       => ? [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
            & aa_set_nat_o(finite_finite_nat,C2)
            & ( B5 = aa_set_nat_set_int(image_nat_int(F),C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4025_finite__subset__image,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
       => ? [C2: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A4)
            & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,C2)
            & ( B5 = aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4026_finite__subset__image,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,F: fun_na1476630412504255114at_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),image_5846123807819985514at_nat(F,A4))
       => ? [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
            & aa_set_nat_o(finite_finite_nat,C2)
            & ( B5 = image_5846123807819985514at_nat(F,C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4027_finite__subset__image,axiom,
    ! [B5: set_nat,F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(finite_finite_nat,B5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_int_set_nat(image_int_nat(F),A4))
       => ? [C2: set_int] :
            ( aa_set_int_o(ord_less_eq_set_int(C2),A4)
            & aa_set_int_o(finite_finite_int,C2)
            & ( B5 = aa_set_int_set_nat(image_int_nat(F),C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4028_finite__subset__image,axiom,
    ! [B5: set_nat,F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,B5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),image_2486076414777270412at_nat(F,A4))
       => ? [C2: set_Pr1261947904930325089at_nat] :
            ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A4)
            & aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,C2)
            & ( B5 = image_2486076414777270412at_nat(F,C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4029_finite__subset__image,axiom,
    ! [B5: set_nat,F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => ? [C2: set_nat] :
            ( aa_set_nat_o(ord_less_eq_set_nat(C2),A4)
            & aa_set_nat_o(finite_finite_nat,C2)
            & ( B5 = aa_set_nat_set_nat(image_nat_nat(F),C2) ) ) ) ) ).

% finite_subset_image
tff(fact_4030_finite__surj,axiom,
    ! [A4: set_int,B5: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_int_set_int(image_int_int(F),A4))
       => aa_set_int_o(finite_finite_int,B5) ) ) ).

% finite_surj
tff(fact_4031_finite__surj,axiom,
    ! [A4: set_int,B5: set_nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_int_set_nat(image_int_nat(F),A4))
       => aa_set_nat_o(finite_finite_nat,B5) ) ) ).

% finite_surj
tff(fact_4032_finite__surj,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5) ) ) ).

% finite_surj
tff(fact_4033_finite__surj,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_nat,F: fun_Pr2602117396471556196at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),image_2486076414777270412at_nat(F,A4))
       => aa_set_nat_o(finite_finite_nat,B5) ) ) ).

% finite_surj
tff(fact_4034_finite__surj,axiom,
    ! [A4: set_nat,B5: set_rat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_rat_o(ord_less_eq_set_rat(B5),aa_set_nat_set_rat(image_nat_rat(F),A4))
       => aa_set_rat_o(finite_finite_rat,B5) ) ) ).

% finite_surj
tff(fact_4035_finite__surj,axiom,
    ! [A4: set_nat,B5: set_int,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(B5),aa_set_nat_set_int(image_nat_int(F),A4))
       => aa_set_int_o(finite_finite_int,B5) ) ) ).

% finite_surj
tff(fact_4036_finite__surj,axiom,
    ! [A4: set_nat,B5: set_Pr1261947904930325089at_nat,F: fun_na1476630412504255114at_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B5),image_5846123807819985514at_nat(F,A4))
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5) ) ) ).

% finite_surj
tff(fact_4037_finite__surj,axiom,
    ! [A4: set_nat,B5: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_set_nat_o(finite_finite_nat,B5) ) ) ).

% finite_surj
tff(fact_4038_finite__image__iff,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(F),A4))
      <=> aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_image_iff
tff(fact_4039_finite__image__iff,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_image_iff
tff(fact_4040_finite__image__iff,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(F),A4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_image_iff
tff(fact_4041_finite__image__iff,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
      <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_image_iff
tff(fact_4042_finite__image__iff,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on5538052773655684606at_nat(F),A4)
     => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_5846123807819985514at_nat(F,A4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_image_iff
tff(fact_4043_finite__image__iff,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(F),A4))
      <=> aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_image_iff
tff(fact_4044_finite__image__iff,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F),A4)
     => ( aa_set_nat_o(finite_finite_nat,image_2486076414777270412at_nat(F,A4))
      <=> aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_image_iff
tff(fact_4045_finite__image__iff,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
      <=> aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_image_iff
tff(fact_4046_finite__imageD,axiom,
    ! [F: fun_int_int,A4: set_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_int_set_int(image_int_int(F),A4))
     => ( aa_set_int_o(inj_on_int_int(F),A4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_imageD
tff(fact_4047_finite__imageD,axiom,
    ! [F: fun_nat_rat,A4: set_nat] :
      ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(F),A4))
     => ( aa_set_nat_o(inj_on_nat_rat(F),A4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_imageD
tff(fact_4048_finite__imageD,axiom,
    ! [F: fun_nat_int,A4: set_nat] :
      ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(F),A4))
     => ( aa_set_nat_o(inj_on_nat_int(F),A4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_imageD
tff(fact_4049_finite__imageD,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
     => ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),A4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_imageD
tff(fact_4050_finite__imageD,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_5846123807819985514at_nat(F,A4))
     => ( aa_set_nat_o(inj_on5538052773655684606at_nat(F),A4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_imageD
tff(fact_4051_finite__imageD,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( aa_set_int_o(inj_on_int_nat(F),A4)
       => aa_set_int_o(finite_finite_int,A4) ) ) ).

% finite_imageD
tff(fact_4052_finite__imageD,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,image_2486076414777270412at_nat(F,A4))
     => ( aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F),A4)
       => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ).

% finite_imageD
tff(fact_4053_finite__imageD,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( aa_set_nat_o(inj_on_nat_nat(F),A4)
       => aa_set_nat_o(finite_finite_nat,A4) ) ) ).

% finite_imageD
tff(fact_4054_finite__range__imageI,axiom,
    ! [G: fun_char_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,image_char_nat(G,top_top_set_char))
     => aa_set_nat_o(finite_finite_nat,image_char_nat(aa_fun8194089762075812217ar_nat(aTP_Lamm_acf(G),F),top_top_set_char)) ) ).

% finite_range_imageI
tff(fact_4055_finite__range__imageI,axiom,
    ! [G: fun_rat_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,image_rat_nat(G,top_top_set_rat))
     => aa_set_nat_o(finite_finite_nat,image_rat_nat(aa_fun5474894869278726016at_nat(aTP_Lamm_acg(G),F),top_top_set_rat)) ) ).

% finite_range_imageI
tff(fact_4056_finite__range__imageI,axiom,
    ! [G: fun_nat_rat,F: fun_rat_rat] :
      ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(G),top_top_set_nat))
     => aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(aa_fun4974911892022385552at_rat(aTP_Lamm_ach(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4057_finite__range__imageI,axiom,
    ! [G: fun_nat_rat,F: fun_rat_int] :
      ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(G),top_top_set_nat))
     => aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(aa_fun550824187084660792at_int(aTP_Lamm_aci(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4058_finite__range__imageI,axiom,
    ! [G: fun_nat_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(G),top_top_set_nat))
     => aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(aa_fun481707715832943588at_rat(aTP_Lamm_acj(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4059_finite__range__imageI,axiom,
    ! [G: fun_nat_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(G),top_top_set_nat))
     => aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(aa_fun5280992047749994636at_int(aTP_Lamm_ack(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4060_finite__range__imageI,axiom,
    ! [G: fun_nat_rat,F: fun_rat_nat] :
      ( aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(G),top_top_set_nat))
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun777230578178148224at_nat(aTP_Lamm_acl(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4061_finite__range__imageI,axiom,
    ! [G: fun_nat_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(G),top_top_set_nat))
     => aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(aa_fun5507398438843482068at_nat(aTP_Lamm_acm(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4062_finite__range__imageI,axiom,
    ! [G: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(G),top_top_set_nat))
     => aa_set_rat_o(finite_finite_rat,aa_set_nat_set_rat(image_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_acn(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4063_finite__range__imageI,axiom,
    ! [G: fun_nat_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(G),top_top_set_nat))
     => aa_set_int_o(finite_finite_int,aa_set_nat_set_int(image_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_aco(G),F)),top_top_set_nat)) ) ).

% finite_range_imageI
tff(fact_4064_Id__fstsnd__eq,axiom,
    id_Pro2258643101195443293at_nat = collec7088162979684241874at_nat(aTP_Lamm_acp) ).

% Id_fstsnd_eq
tff(fact_4065_Id__fstsnd__eq,axiom,
    id_nat2 = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aTP_Lamm_acq) ).

% Id_fstsnd_eq
tff(fact_4066_Id__fstsnd__eq,axiom,
    id_int2 = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aTP_Lamm_acr) ).

% Id_fstsnd_eq
tff(fact_4067_finite__surj__inj,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(A4),aa_set_int_set_int(image_int_int(F),A4))
       => aa_set_int_o(inj_on_int_int(F),A4) ) ) ).

% finite_surj_inj
tff(fact_4068_finite__surj__inj,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A4),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
       => aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),A4) ) ) ).

% finite_surj_inj
tff(fact_4069_finite__surj__inj,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A4),aa_set_nat_set_nat(image_nat_nat(F),A4))
       => aa_set_nat_o(inj_on_nat_nat(F),A4) ) ) ).

% finite_surj_inj
tff(fact_4070_inj__on__finite,axiom,
    ! [F: fun_int_int,A4: set_int,B5: set_int] :
      ( aa_set_int_o(inj_on_int_int(F),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),B5)
       => ( aa_set_int_o(finite_finite_int,B5)
         => aa_set_int_o(finite_finite_int,A4) ) ) ) ).

% inj_on_finite
tff(fact_4071_inj__on__finite,axiom,
    ! [F: fun_nat_rat,A4: set_nat,B5: set_rat] :
      ( aa_set_nat_o(inj_on_nat_rat(F),A4)
     => ( aa_set_rat_o(ord_less_eq_set_rat(aa_set_nat_set_rat(image_nat_rat(F),A4)),B5)
       => ( aa_set_rat_o(finite_finite_rat,B5)
         => aa_set_nat_o(finite_finite_nat,A4) ) ) ) ).

% inj_on_finite
tff(fact_4072_inj__on__finite,axiom,
    ! [F: fun_nat_int,A4: set_nat,B5: set_int] :
      ( aa_set_nat_o(inj_on_nat_int(F),A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_nat_set_int(image_nat_int(F),A4)),B5)
       => ( aa_set_int_o(finite_finite_int,B5)
         => aa_set_nat_o(finite_finite_nat,A4) ) ) ) ).

% inj_on_finite
tff(fact_4073_inj__on__finite,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4)),B5)
       => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
         => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ) ).

% inj_on_finite
tff(fact_4074_inj__on__finite,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat,B5: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(inj_on5538052773655684606at_nat(F),A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(F,A4)),B5)
       => ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,B5)
         => aa_set_nat_o(finite_finite_nat,A4) ) ) ) ).

% inj_on_finite
tff(fact_4075_inj__on__finite,axiom,
    ! [F: fun_int_nat,A4: set_int,B5: set_nat] :
      ( aa_set_int_o(inj_on_int_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(F),A4)),B5)
       => ( aa_set_nat_o(finite_finite_nat,B5)
         => aa_set_int_o(finite_finite_int,A4) ) ) ) ).

% inj_on_finite
tff(fact_4076_inj__on__finite,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat,B5: set_nat] :
      ( aa_set3702741264012733747_nat_o(inj_on2178005380612969504at_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(image_2486076414777270412at_nat(F,A4)),B5)
       => ( aa_set_nat_o(finite_finite_nat,B5)
         => aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4) ) ) ) ).

% inj_on_finite
tff(fact_4077_inj__on__finite,axiom,
    ! [F: fun_nat_nat,A4: set_nat,B5: set_nat] :
      ( aa_set_nat_o(inj_on_nat_nat(F),A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),B5)
       => ( aa_set_nat_o(finite_finite_nat,B5)
         => aa_set_nat_o(finite_finite_nat,A4) ) ) ) ).

% inj_on_finite
tff(fact_4078_endo__inj__surj,axiom,
    ! [A4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A4)
     => ( aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(F),A4)),A4)
       => ( aa_set_int_o(inj_on_int_int(F),A4)
         => ( aa_set_int_set_int(image_int_int(F),A4) = A4 ) ) ) ) ).

% endo_inj_surj
tff(fact_4079_endo__inj__surj,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4)),A4)
       => ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),A4)
         => ( aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4) = A4 ) ) ) ) ).

% endo_inj_surj
tff(fact_4080_endo__inj__surj,axiom,
    ! [A4: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(F),A4)),A4)
       => ( aa_set_nat_o(inj_on_nat_nat(F),A4)
         => ( aa_set_nat_set_nat(image_nat_nat(F),A4) = A4 ) ) ) ) ).

% endo_inj_surj
tff(fact_4081_finite__UNIV__surj__inj,axiom,
    ! [F: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat)
     => ( ( aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),top_to4669805908274784177at_nat) = top_to4669805908274784177at_nat )
       => aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),top_to4669805908274784177at_nat) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4082_finite__UNIV__surj__inj,axiom,
    ! [F: fun_char_char] :
      ( finite_finite_char(top_top_set_char)
     => ( ( image_char_char(F,top_top_set_char) = top_top_set_char )
       => inj_on_char_char(F,top_top_set_char) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4083_finite__UNIV__surj__inj,axiom,
    ! [F: fun_li7731370096534215997st_nat] :
      ( finite8100373058378681591st_nat(top_top_set_list_nat)
     => ( ( image_7976474329151083847st_nat(F,top_top_set_list_nat) = top_top_set_list_nat )
       => inj_on3049792774292151987st_nat(F,top_top_set_list_nat) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4084_finite__UNIV__surj__inj,axiom,
    ! [F: fun_rat_rat] :
      ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => ( ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat )
       => aa_set_rat_o(inj_on_rat_rat(F),top_top_set_rat) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4085_finite__UNIV__surj__inj,axiom,
    ! [F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
     => ( ( aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat) = top_top_set_nat )
       => aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4086_finite__UNIV__surj__inj,axiom,
    ! [F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,top_top_set_int)
     => ( ( aa_set_int_set_int(image_int_int(F),top_top_set_int) = top_top_set_int )
       => aa_set_int_o(inj_on_int_int(F),top_top_set_int) ) ) ).

% finite_UNIV_surj_inj
tff(fact_4087_finite__UNIV__inj__surj,axiom,
    ! [F: fun_Pr2118491707810114563at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,top_to4669805908274784177at_nat)
     => ( aa_set3702741264012733747_nat_o(inj_on8969904277767023793at_nat(F),top_to4669805908274784177at_nat)
       => ( aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),top_to4669805908274784177at_nat) = top_to4669805908274784177at_nat ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4088_finite__UNIV__inj__surj,axiom,
    ! [F: fun_char_char] :
      ( finite_finite_char(top_top_set_char)
     => ( inj_on_char_char(F,top_top_set_char)
       => ( image_char_char(F,top_top_set_char) = top_top_set_char ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4089_finite__UNIV__inj__surj,axiom,
    ! [F: fun_li7731370096534215997st_nat] :
      ( finite8100373058378681591st_nat(top_top_set_list_nat)
     => ( inj_on3049792774292151987st_nat(F,top_top_set_list_nat)
       => ( image_7976474329151083847st_nat(F,top_top_set_list_nat) = top_top_set_list_nat ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4090_finite__UNIV__inj__surj,axiom,
    ! [F: fun_rat_rat] :
      ( aa_set_rat_o(finite_finite_rat,top_top_set_rat)
     => ( aa_set_rat_o(inj_on_rat_rat(F),top_top_set_rat)
       => ( image_rat_rat(F,top_top_set_rat) = top_top_set_rat ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4091_finite__UNIV__inj__surj,axiom,
    ! [F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,top_top_set_nat)
     => ( aa_set_nat_o(inj_on_nat_nat(F),top_top_set_nat)
       => ( aa_set_nat_set_nat(image_nat_nat(F),top_top_set_nat) = top_top_set_nat ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4092_finite__UNIV__inj__surj,axiom,
    ! [F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,top_top_set_int)
     => ( aa_set_int_o(inj_on_int_int(F),top_top_set_int)
       => ( aa_set_int_set_int(image_int_int(F),top_top_set_int) = top_top_set_int ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_4093_finite__vimageD,axiom,
    ! [Ha: fun_char_char,F3: set_char] :
      ( finite_finite_char(vimage_char_char(Ha,F3))
     => ( ( image_char_char(Ha,top_top_set_char) = top_top_set_char )
       => finite_finite_char(F3) ) ) ).

% finite_vimageD
tff(fact_4094_finite__vimageD,axiom,
    ! [Ha: fun_char_rat,F3: set_rat] :
      ( finite_finite_char(vimage_char_rat(Ha,F3))
     => ( ( image_char_rat(Ha,top_top_set_char) = top_top_set_rat )
       => aa_set_rat_o(finite_finite_rat,F3) ) ) ).

% finite_vimageD
tff(fact_4095_finite__vimageD,axiom,
    ! [Ha: fun_char_nat,F3: set_nat] :
      ( finite_finite_char(vimage_char_nat(Ha,F3))
     => ( ( image_char_nat(Ha,top_top_set_char) = top_top_set_nat )
       => aa_set_nat_o(finite_finite_nat,F3) ) ) ).

% finite_vimageD
tff(fact_4096_finite__vimageD,axiom,
    ! [Ha: fun_char_int,F3: set_int] :
      ( finite_finite_char(vimage_char_int(Ha,F3))
     => ( ( image_char_int(Ha,top_top_set_char) = top_top_set_int )
       => aa_set_int_o(finite_finite_int,F3) ) ) ).

% finite_vimageD
tff(fact_4097_finite__vimageD,axiom,
    ! [Ha: fun_rat_char,F3: set_char] :
      ( aa_set_rat_o(finite_finite_rat,vimage_rat_char(Ha,F3))
     => ( ( image_rat_char(Ha,top_top_set_rat) = top_top_set_char )
       => finite_finite_char(F3) ) ) ).

% finite_vimageD
tff(fact_4098_finite__vimageD,axiom,
    ! [Ha: fun_rat_rat,F3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,vimage_rat_rat(Ha,F3))
     => ( ( image_rat_rat(Ha,top_top_set_rat) = top_top_set_rat )
       => aa_set_rat_o(finite_finite_rat,F3) ) ) ).

% finite_vimageD
tff(fact_4099_finite__vimageD,axiom,
    ! [Ha: fun_rat_nat,F3: set_nat] :
      ( aa_set_rat_o(finite_finite_rat,vimage_rat_nat(Ha,F3))
     => ( ( image_rat_nat(Ha,top_top_set_rat) = top_top_set_nat )
       => aa_set_nat_o(finite_finite_nat,F3) ) ) ).

% finite_vimageD
tff(fact_4100_finite__vimageD,axiom,
    ! [Ha: fun_rat_int,F3: set_int] :
      ( aa_set_rat_o(finite_finite_rat,vimage_rat_int(Ha,F3))
     => ( ( image_rat_int(Ha,top_top_set_rat) = top_top_set_int )
       => aa_set_int_o(finite_finite_int,F3) ) ) ).

% finite_vimageD
tff(fact_4101_finite__vimageD,axiom,
    ! [Ha: fun_nat_char,F3: set_char] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_char(Ha,F3))
     => ( ( image_nat_char(Ha,top_top_set_nat) = top_top_set_char )
       => finite_finite_char(F3) ) ) ).

% finite_vimageD
tff(fact_4102_finite__vimageD,axiom,
    ! [Ha: fun_nat_rat,F3: set_rat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_rat(Ha,F3))
     => ( ( aa_set_nat_set_rat(image_nat_rat(Ha),top_top_set_nat) = top_top_set_rat )
       => aa_set_rat_o(finite_finite_rat,F3) ) ) ).

% finite_vimageD
tff(fact_4103_prod__set__defs_I2_J,axiom,
    ! [X5: produc7812382424243267045t_unit] : basic_1496934909540232809t_unit(X5) = insert8563697753282147485t_unit(aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,X5),bot_bo3686245532977126937t_unit) ).

% prod_set_defs(2)
tff(fact_4104_inf__img__fin__dom,axiom,
    ! [F: fun_nat_Product_unit,A4: set_nat] :
      ( finite4290736615968046902t_unit(image_8730104196221521654t_unit(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: product_unit] :
            ( member_Product_unit(X,image_8730104196221521654t_unit(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,vimage4884490618288580032t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(X),bot_bo3957492148770167129t_unit))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4105_inf__img__fin__dom,axiom,
    ! [F: fun_nat_o,A4: set_nat] :
      ( aa_set_o_o(finite_finite_o,image_nat_o(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),image_nat_o(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_o_set_nat(vimage_nat_o(F),aa_set_o_set_o(insert_o((X)),bot_bot_set_o))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4106_inf__img__fin__dom,axiom,
    ! [F: fun_int_nat,A4: set_int] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_int_set_nat(image_int_nat(F),A4))
     => ( ~ aa_set_int_o(finite_finite_int,A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),aa_set_int_set_nat(image_int_nat(F),A4))
            & ~ aa_set_int_o(finite_finite_int,aa_set_nat_set_int(vimage_int_nat(F),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4107_inf__img__fin__dom,axiom,
    ! [F: fun_nat_nat,A4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(image_nat_nat(F),A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),aa_set_nat_set_nat(image_nat_nat(F),A4))
            & ~ aa_set_nat_o(finite_finite_nat,aa_set_nat_set_nat(vimage_nat_nat(F),aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4108_inf__img__fin__dom,axiom,
    ! [F: fun_Pr2481279687482350479t_unit,A4: set_Pr1261947904930325089at_nat] :
      ( finite4290736615968046902t_unit(image_6169909464789311569t_unit(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ? [X: product_unit] :
            ( member_Product_unit(X,image_6169909464789311569t_unit(F,A4))
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,vimage3104651464403166471t_unit(F,aa_set8324764978855220220t_unit(insert_Product_unit(X),bot_bo3957492148770167129t_unit))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4109_inf__img__fin__dom,axiom,
    ! [F: fun_na1476630412504255114at_nat,A4: set_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,image_5846123807819985514at_nat(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),image_5846123807819985514at_nat(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,vimage8013328719654469172at_nat(F,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4110_inf__img__fin__dom,axiom,
    ! [F: fun_Pr4796943543331071514_nat_o,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set_o_o(finite_finite_o,image_3693632289388996572_nat_o(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ? [X: $o] :
            ( aa_set_o_o(member_o((X)),image_3693632289388996572_nat_o(F,A4))
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,vimage2444243245536687506_nat_o(F,aa_set_o_set_o(insert_o((X)),bot_bot_set_o))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4111_inf__img__fin__dom,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set_nat_o(finite_finite_nat,image_2486076414777270412at_nat(F,A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),image_2486076414777270412at_nat(F,A4))
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,vimage4653281326611754070at_nat(F,aa_set_nat_set_nat(insert_nat(X),bot_bot_set_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4112_inf__img__fin__dom,axiom,
    ! [F: fun_Pr2118491707810114563at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
     => ( ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,A4)
       => ? [X: product_prod_nat_nat] :
            ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(X),aa_set8868280134176360700at_nat(image_5168914502847457605at_nat(F),A4))
            & ~ aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,aa_set8868280134176360700at_nat(vimage2449269961533847803at_nat(F),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(X),bot_bo2099793752762293965at_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4113_inf__img__fin__dom,axiom,
    ! [F: fun_na367238361360896496at_nat,A4: set_nat] :
      ( aa_set5844364881195198285_nat_o(finite4343798906461161616at_nat,image_2499013659892586576at_nat(F,A4))
     => ( ~ aa_set_nat_o(finite_finite_nat,A4)
       => ? [X: produc3843707927480180839at_nat] :
            ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(X),image_2499013659892586576at_nat(F,A4))
            & ~ aa_set_nat_o(finite_finite_nat,vimage6435164912253009178at_nat(F,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(X),bot_bo228742789529271731at_nat))) ) ) ) ).

% inf_img_fin_dom
tff(fact_4114_nat__descend__induct,axiom,
    ! [N3: nat,P2: fun_nat_o,M3: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),K2)
         => aa_nat_o(P2,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N3)
           => ( ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I5)
                 => aa_nat_o(P2,I5) )
             => aa_nat_o(P2,K2) ) )
       => aa_nat_o(P2,M3) ) ) ).

% nat_descend_induct
tff(fact_4115_Field__natLeq__on,axiom,
    ! [N3: nat] : aa_set736839985058952875et_nat(field_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3)))) = aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_act,N3)) ).

% Field_natLeq_on
tff(fact_4116_natLeq__on__well__order__on,axiom,
    ! [N3: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_act,N3))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3)))) ).

% natLeq_on_well_order_on
tff(fact_4117_natLeq__on__Well__order,axiom,
    ! [N3: nat] : aa_set3702741264012733747_nat_o(order_2888998067076097458on_nat(aa_set736839985058952875et_nat(field_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3))))),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3)))) ).

% natLeq_on_Well_order
tff(fact_4118_natLeq__on__wo__rel,axiom,
    ! [N3: nat] : aa_set3702741264012733747_nat_o(bNF_We3818239936649020644el_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3)))) ).

% natLeq_on_wo_rel
tff(fact_4119_wf__less,axiom,
    aa_set3702741264012733747_nat_o(wf_nat,aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,ord_less_nat))) ).

% wf_less
tff(fact_4120_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,ord_less_nat)) ).

% natLess_def
tff(fact_4121_card_Ocomp__fun__commute__on,axiom,
    aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,suc),suc) = aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(comp_nat_nat_nat,suc),suc) ).

% card.comp_fun_commute_on
tff(fact_4122_pred__nat__def,axiom,
    pred_nat = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acu)) ).

% pred_nat_def
tff(fact_4123_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),A4)) ).

% zero_notin_Suc_image
tff(fact_4124_Restr__natLeq2,axiom,
    ! [N3: nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N3),aa_nat1818925681583257441et_nat(aTP_Lamm_acv,N3))) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3))) ).

% Restr_natLeq2
tff(fact_4125_Restr__natLeq,axiom,
    ! [N3: nat] : aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_act,N3)),aa_nat1818925681583257441et_nat(aTP_Lamm_acw,N3))) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_acs(N3))) ).

% Restr_natLeq
tff(fact_4126_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,ord_less_eq_nat)) ).

% natLeq_def
tff(fact_4127_rat__denum,axiom,
    ? [F4: fun_nat_rat] : aa_set_nat_set_rat(image_nat_rat(F4),top_top_set_nat) = top_top_set_rat ).

% rat_denum
tff(fact_4128_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = aa_set_nat_set_nat(image_nat_nat(suc),top_top_set_nat) ).

% greaterThan_0
tff(fact_4129_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J4: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,I),J4)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,I)),aa_nat_nat(suc,J4)) ).

% image_Suc_atLeastLessThan
tff(fact_4130_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_4131_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N3: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N3)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N3))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4132_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xa: nat,Y2: nat] :
      aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acx,C)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Xa),Y2)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y2),
        aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y2),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),Y2),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4133_surj__nat__to__rat__surj,axiom,
    aa_set_nat_set_rat(image_nat_rat(nat_to_rat_surj),top_top_set_nat) = top_top_set_rat ).

% surj_nat_to_rat_surj
tff(fact_4134_range__mod,axiom,
    ! [N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
     => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acy,N3)),top_top_set_nat) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N3) ) ) ).

% range_mod
tff(fact_4135_divmod__nat__if,axiom,
    ! [M3: nat,N3: nat] :
      aa_nat7603221163087113075at_nat(divmod_nat(M3),N3) = $ite(
        ( ( N3 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N3) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),M3),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_acz),aa_nat7603221163087113075at_nat(divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M3),N3)),N3)) ) ).

% divmod_nat_if
tff(fact_4136_Rats__eq__range__nat__to__rat__surj,axiom,
    field_6020823756834552118ts_rat = aa_set_nat_set_rat(image_nat_rat(nat_to_rat_surj),top_top_set_nat) ).

% Rats_eq_range_nat_to_rat_surj
tff(fact_4137_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J4: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I),J4)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,I)),aa_nat_nat(suc,J4)) ).

% image_Suc_atLeastAtMost
tff(fact_4138_UN__atMost__UNIV,axiom,
    aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_4139_atMost__Suc__eq__insert__0,axiom,
    ! [N3: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N3)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,N3))) ).

% atMost_Suc_eq_insert_0
tff(fact_4140_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N3: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N3)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N3))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4141_image__Suc__lessThan,axiom,
    ! [N3: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,N3)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N3) ).

% image_Suc_lessThan
tff(fact_4142_UN__lessThan__UNIV,axiom,
    aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_4143_lessThan__Suc__eq__insert__0,axiom,
    ! [N3: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N3)) = aa_set_nat_set_nat(insert_nat(zero_zero_nat),aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,N3))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4144_image__Suc__atMost,axiom,
    ! [N3: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,N3)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),aa_nat_nat(suc,N3)) ).

% image_Suc_atMost
tff(fact_4145_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] : aa_set_nat_set_int(image_nat_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A),B)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or1266510415728281911st_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastAtMost
tff(fact_4146_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] : aa_set_nat_set_int(image_nat_int(semiri1314217659103216013at_int),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,A),B)) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% image_int_atLeastLessThan
tff(fact_4147_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa_set_int_set_int(image_int_int(aa_int_fun_int_int(aTP_Lamm_ada,L)),aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,L),U) ).

% image_add_int_atLeastLessThan
tff(fact_4148_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),T)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_4149_pair__less__iff1,axiom,
    ! [Xa: nat,Y2: nat,Z: nat] :
      ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Y2)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),Z) ) ).

% pair_less_iff1
tff(fact_4150_total__pair__less,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : total_3592101749530773125at_nat(A4,fun_pair_less) ).

% total_pair_less
tff(fact_4151_Inf__int__def,axiom,
    ! [X3: set_int] : aa_set_int_int(complete_Inf_Inf_int,X3) = aa_int_int(uminus_uminus_int,aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(uminus_uminus_int),X3))) ).

% Inf_int_def
tff(fact_4152_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat(fun_pair_less) ).

% wf_pair_less
tff(fact_4153_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),T)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_4154_bij__betw__Suc,axiom,
    ! [M: set_nat,N: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M),N)
    <=> ( aa_set_nat_set_nat(image_nat_nat(suc),M) = N ) ) ).

% bij_betw_Suc
tff(fact_4155_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),T)
       => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),T)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_4156_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),S)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),T)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_4157_pair__leq__def,axiom,
    fun_pair_leq = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(fun_pair_less),id_Pro2258643101195443293at_nat) ).

% pair_leq_def
tff(fact_4158_wmax__insertI,axiom,
    ! [Y2: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,Xa: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),YS)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,XS),YS)),fun_max_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),XS)),YS)),fun_max_weak) ) ) ) ).

% wmax_insertI
tff(fact_4159_wmin__insertI,axiom,
    ! [Xa: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y2: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),XS)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),fun_pair_leq)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,XS),YS)),fun_min_weak)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,XS),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y2),YS))),fun_min_weak) ) ) ) ).

% wmin_insertI
tff(fact_4160_smax__insertI,axiom,
    ! [Y2: product_prod_nat_nat,Y3: set_Pr1261947904930325089at_nat,Xa: product_prod_nat_nat,X3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Y2),Y3)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X3),Y3)),fun_max_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Xa),X3)),Y3)),fun_max_strict) ) ) ) ).

% smax_insertI
tff(fact_4161_smin__insertI,axiom,
    ! [Xa: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y2: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Xa),XS)
     => ( member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Xa),Y2),fun_pair_less)
       => ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,XS),YS)),fun_min_strict)
         => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,XS),aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Y2),YS))),fun_min_strict) ) ) ) ).

% smin_insertI
tff(fact_4162_max__rpair__set,axiom,
    aa_Pro4518350200472837805_nat_o(fun_re2478310338295953701at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,fun_max_strict),fun_max_weak)) ).

% max_rpair_set
tff(fact_4163_min__rpair__set,axiom,
    aa_Pro4518350200472837805_nat_o(fun_re2478310338295953701at_nat,aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,fun_min_strict),fun_min_weak)) ).

% min_rpair_set
tff(fact_4164_wmax__emptyI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,X3)
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,bot_bo2099793752762293965at_nat),X3)),fun_max_weak) ) ).

% wmax_emptyI
tff(fact_4165_smax__emptyI,axiom,
    ! [Y3: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(finite6177210948735845034at_nat,Y3)
     => ( ( Y3 != bot_bo2099793752762293965at_nat )
       => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,bot_bo2099793752762293965at_nat),Y3)),fun_max_strict) ) ) ).

% smax_emptyI
tff(fact_4166_smin__emptyI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( X3 != bot_bo2099793752762293965at_nat )
     => aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X3),bot_bo2099793752762293965at_nat)),fun_min_strict) ) ).

% smin_emptyI
tff(fact_4167_wmin__emptyI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] : aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,X3),bot_bo2099793752762293965at_nat)),fun_min_weak) ).

% wmin_emptyI
tff(fact_4168_max__strict__def,axiom,
    fun_max_strict = max_ex8135407076693332796at_nat(fun_pair_less) ).

% max_strict_def
tff(fact_4169_min__strict__def,axiom,
    fun_min_strict = min_ex6901939911449802026at_nat(fun_pair_less) ).

% min_strict_def
tff(fact_4170_min__weak__def,axiom,
    fun_min_weak = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(min_ex6901939911449802026at_nat(fun_pair_leq)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat)),bot_bo228742789529271731at_nat)) ).

% min_weak_def
tff(fact_4171_max__weak__def,axiom,
    fun_max_weak = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(max_ex8135407076693332796at_nat(fun_pair_leq)),aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,bot_bo2099793752762293965at_nat),bot_bo2099793752762293965at_nat)),bot_bo228742789529271731at_nat)) ).

% max_weak_def
tff(fact_4172_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( aa_int_set_int(aa_int8639817059373041269et_int(set_or4662586982721622107an_int,zero_zero_int),U) = aa_set_nat_set_int(image_nat_int(semiri1314217659103216013at_int),aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_4173_UN__atLeast__UNIV,axiom,
    aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_4174_trans__pair__less,axiom,
    trans_4347625901269045472at_nat(fun_pair_less) ).

% trans_pair_less
tff(fact_4175_less__eq__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adc),aa_int1417691210696471063at_nat(rep_Integ,Xa)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_4176_nat_Orep__eq,axiom,
    ! [Xa: int] : aa_int_nat(nat2,Xa) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ).

% nat.rep_eq
tff(fact_4177_less__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ade),aa_int1417691210696471063at_nat(rep_Integ,Xa)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_4178_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adc),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_4179_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adc),Xa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_4180_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_adf(D2))) ).

% int_ge_less_than2_def
tff(fact_4181_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_adg(D2))) ).

% int_ge_less_than_def
tff(fact_4182_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ade),ord_less_int) ).

% less_int.transfer
tff(fact_4183_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adi),minus_minus_int) ).

% minus_int.transfer
tff(fact_4184_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adk),plus_plus_int) ).

% plus_int.transfer
tff(fact_4185_nat_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xa) ).

% nat.abs_eq
tff(fact_4186_nat_Otransfer,axiom,
    aa_fun_int_nat_o(aa_fun1998507904093502224_nat_o(bNF_re4555766996558763186at_nat(pcr_int,fequal_nat),produc6842872674320459806at_nat(minus_minus_nat)),nat2) ).

% nat.transfer
tff(fact_4187_uminus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ef),Xa)) ).

% uminus_int.abs_eq
tff(fact_4188_uminus__int_Otransfer,axiom,
    aa_fun_int_int_o(aa_fun856177901615811547_int_o(bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc2626176000494625587at_nat(aTP_Lamm_ef)),uminus_uminus_int) ).

% uminus_int.transfer
tff(fact_4189_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adm),Xa),Xb)) ).

% times_int.abs_eq
tff(fact_4190_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adk),Xa),Xb)) ).

% plus_int.abs_eq
tff(fact_4191_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adi),Xa),Xb)) ).

% minus_int.abs_eq
tff(fact_4192_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adm),times_times_int) ).

% times_int.transfer
tff(fact_4193_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ade),Xa),Xb) ) ).

% less_int.abs_eq
tff(fact_4194_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_ef)) ).

% uminus_int_def
tff(fact_4195_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adm),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adm)) ).

% times_int.rsp
tff(fact_4196_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_4197_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adi),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adi)) ).

% minus_int.rsp
tff(fact_4198_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adk),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adk)) ).

% plus_int.rsp
tff(fact_4199_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adc),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adc)) ).

% less_eq_int.rsp
tff(fact_4200_uminus__int_Orsp,axiom,
    aa_fun3538214411680374423_nat_o(aa_fun6920661532540581549_nat_o(bNF_re2241393799969408733at_nat(intrel,intrel),produc2626176000494625587at_nat(aTP_Lamm_ef)),produc2626176000494625587at_nat(aTP_Lamm_ef)) ).

% uminus_int.rsp
tff(fact_4201_nat_Orsp,axiom,
    aa_fun5720009856248676976_nat_o(aa_fun2062416747992930591_nat_o(bNF_re8246922863344978751at_nat(intrel,fequal_nat),produc6842872674320459806at_nat(minus_minus_nat)),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_4202_intrel__def,axiom,
    intrel = aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ado) ).

% intrel_def
tff(fact_4203_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ade),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ade)) ).

% less_int.rsp
tff(fact_4204_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adm)) ).

% times_int_def
tff(fact_4205_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adi)) ).

% minus_int_def
tff(fact_4206_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),aa_fun6875833325056119164at_nat(produc4528391901023682385at_nat,aTP_Lamm_adk)) ).

% plus_int_def
tff(fact_4207_rat__less__code,axiom,
    ! [P: rat,Q: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P),Q)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_adq(Q)),aa_rat7754553326175492643nt_int(quotient_of,P)) ) ).

% rat_less_code
tff(fact_4208_rat__less__eq__code,axiom,
    ! [P: rat,Q: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P),Q)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_ads(Q)),aa_rat7754553326175492643nt_int(quotient_of,P)) ) ).

% rat_less_eq_code
tff(fact_4209_rat__uminus__code,axiom,
    ! [P: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(uminus_uminus_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_adt),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_uminus_code
tff(fact_4210_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_ade)) ).

% less_int_def
tff(fact_4211_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),aa_fun5912894404062470332_nat_o(produc6395135316500328296_nat_o,aTP_Lamm_adc)) ).

% less_eq_int_def
tff(fact_4212_rat__minus__code,axiom,
    ! [P: rat,Q: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(minus_minus_rat(P),Q)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_adv(Q)),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_minus_code
tff(fact_4213_rat__plus__code,axiom,
    ! [P: rat,Q: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(plus_plus_rat(P),Q)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_adx(Q)),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_plus_code
tff(fact_4214_rat__divide__code,axiom,
    ! [P: rat,Q: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(divide_divide_rat(P),Q)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_adz(Q)),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_divide_code
tff(fact_4215_rat__times__code,axiom,
    ! [P: rat,Q: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(times_times_rat(P),Q)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aeb(Q)),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_times_code
tff(fact_4216_pcr__int__def,axiom,
    pcr_int = relcom8911704206367525917at_int(basic_5328504652464829177at_nat(fequal_nat,fequal_nat),cr_int) ).

% pcr_int_def
tff(fact_4217_UNIV__unit,axiom,
    top_to1996260823553986621t_unit = aa_set8324764978855220220t_unit(insert_Product_unit(product_Unity),bot_bo3957492148770167129t_unit) ).

% UNIV_unit
tff(fact_4218_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
tff(fact_4219_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa_Pro8294289767544744956t_unit(uminus2952777764628376836t_unit,Uu) = product_Unity ).

% uminus_unit_def
tff(fact_4220_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uu),Uv) = product_Unity ).

% inf_unit_def
tff(fact_4221_old_Ounit_Oexhaust,axiom,
    ! [Y2: product_unit] : Y2 = product_Unity ).

% old.unit.exhaust
tff(fact_4222_Inf__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple2584293577114468500t_unit(Uu) = product_Unity ).

% Inf_unit_def
tff(fact_4223_Sup__unit__def,axiom,
    ! [Uu: set_Product_unit] : comple4687483117567863418t_unit(Uu) = product_Unity ).

% Sup_unit_def
tff(fact_4224_sup__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(Uu),Uv) = product_Unity ).

% sup_unit_def
tff(fact_4225_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
tff(fact_4226_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
tff(fact_4227_Gcd__eq__Max,axiom,
    ! [M: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M)
     => ( ( M != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_aed,M))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_4228_Gcd__int__eq,axiom,
    ! [N: set_nat] : aa_set_int_int(gcd_Gcd_int,aa_set_nat_set_int(image_nat_int(semiri1314217659103216013at_int),N)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,N)) ).

% Gcd_int_eq
tff(fact_4229_Gcd__int__def,axiom,
    ! [K3: set_int] : aa_set_int_int(gcd_Gcd_int,K3) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Gcd_nat,aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,nat2),abs_abs_int)),K3))) ).

% Gcd_int_def
tff(fact_4230_Gcd__nat__abs__eq,axiom,
    ! [K3: set_int] : aa_set_nat_nat(gcd_Gcd_nat,aa_set_int_set_nat(image_int_nat(aTP_Lamm_aee),K3)) = aa_int_nat(nat2,aa_set_int_int(gcd_Gcd_int,K3)) ).

% Gcd_nat_abs_eq
tff(fact_4231_range__abs__Nats,axiom,
    aa_set_int_set_int(image_int_int(abs_abs_int),top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_4232_Gcd__abs__eq,axiom,
    ! [K3: set_int] : aa_set_int_int(gcd_Gcd_int,aa_set_int_set_int(image_int_int(abs_abs_int),K3)) = aa_set_int_int(gcd_Gcd_int,K3) ).

% Gcd_abs_eq
tff(fact_4233_int__in__range__abs,axiom,
    ! [N3: nat] : aa_set_int_o(member_int(aa_nat_int(semiri1314217659103216013at_int,N3)),aa_set_int_set_int(image_int_int(abs_abs_int),top_top_set_int)) ).

% int_in_range_abs
tff(fact_4234_measure__function__int,axiom,
    fun_is_measure_int(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,nat2),abs_abs_int)) ).

% measure_function_int
tff(fact_4235_rat__abs__code,axiom,
    ! [P: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(abs_abs_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aef),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_abs_code
tff(fact_4236_surj__int__encode,axiom,
    aa_set_int_set_nat(image_int_nat(nat_int_encode),top_top_set_int) = top_top_set_nat ).

% surj_int_encode
tff(fact_4237_surj__int__decode,axiom,
    aa_set_nat_set_int(image_nat_int(nat_int_decode),top_top_set_nat) = top_top_set_int ).

% surj_int_decode
tff(fact_4238_Lcm__abs__eq,axiom,
    ! [K3: set_int] : aa_set_int_int(gcd_Lcm_int,aa_set_int_set_int(image_int_int(abs_abs_int),K3)) = aa_set_int_int(gcd_Lcm_int,K3) ).

% Lcm_abs_eq
tff(fact_4239_Lcm__int__def,axiom,
    ! [K3: set_int] : aa_set_int_int(gcd_Lcm_int,K3) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,aa_set_int_set_nat(image_int_nat(aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,nat2),abs_abs_int)),K3))) ).

% Lcm_int_def
tff(fact_4240_Lcm__int__eq,axiom,
    ! [N: set_nat] : aa_set_int_int(gcd_Lcm_int,aa_set_nat_set_int(image_nat_int(semiri1314217659103216013at_int),N)) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(gcd_Lcm_nat,N)) ).

% Lcm_int_eq
tff(fact_4241_Lcm__nat__abs__eq,axiom,
    ! [K3: set_int] : aa_set_nat_nat(gcd_Lcm_nat,aa_set_int_set_nat(image_int_nat(aTP_Lamm_aee),K3)) = aa_int_nat(nat2,aa_set_int_int(gcd_Lcm_int,K3)) ).

% Lcm_nat_abs_eq
tff(fact_4242_rat__inverse__code,axiom,
    ! [P: rat] : aa_rat7754553326175492643nt_int(quotient_of,aa_rat_rat(inverse_inverse_rat,P)) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aeg),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_inverse_code
tff(fact_4243_euclidean__size__int__def,axiom,
    euclid4774559944035922753ze_int = aa_fun1234824742305470092nt_nat(aa_fun135663990063072770nt_nat(comp_int_nat_int,nat2),abs_abs_int) ).

% euclidean_size_int_def
tff(fact_4244_Unity__def,axiom,
    product_Unity = aa_o_Product_unit(product_Abs_unit,$true) ).

% Unity_def
tff(fact_4245_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_aeh) ).

% prod_encode_def
tff(fact_4246_Abs__unit__cases,axiom,
    ! [Xa: product_unit] :
      ~ ! [Y: $o] :
          ( ( Xa = aa_o_Product_unit(product_Abs_unit,(Y)) )
         => ~ aa_set_o_o(member_o((Y)),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ) ).

% Abs_unit_cases
tff(fact_4247_Abs__unit__induct,axiom,
    ! [P2: fun_Product_unit_o,Xa: product_unit] :
      ( ! [Y: $o] :
          ( aa_set_o_o(member_o((Y)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
         => aa_Product_unit_o(P2,aa_o_Product_unit(product_Abs_unit,(Y))) )
     => aa_Product_unit_o(P2,Xa) ) ).

% Abs_unit_induct
tff(fact_4248_Abs__unit__inject,axiom,
    ! [Xa: $o,Y2: $o] :
      ( aa_set_o_o(member_o((Xa)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
     => ( aa_set_o_o(member_o((Y2)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
       => ( ( aa_o_Product_unit(product_Abs_unit,(Xa)) = aa_o_Product_unit(product_Abs_unit,(Y2)) )
        <=> ( (Xa)
          <=> (Y2) ) ) ) ) ).

% Abs_unit_inject
tff(fact_4249_surj__prod__encode,axiom,
    image_2486076414777270412at_nat(nat_prod_encode,top_to4669805908274784177at_nat) = top_top_set_nat ).

% surj_prod_encode
tff(fact_4250_type__definition__unit,axiom,
    aa_set_o_o(type_d6188575255521822967unit_o(product_Rep_unit,product_Abs_unit),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ).

% type_definition_unit
tff(fact_4251_Rep__unit__inject,axiom,
    ! [Xa: product_unit,Y2: product_unit] :
      ( ( aa_Product_unit_o(product_Rep_unit,Xa)
      <=> aa_Product_unit_o(product_Rep_unit,Y2) )
    <=> ( Xa = Y2 ) ) ).

% Rep_unit_inject
tff(fact_4252_Rep__unit__induct,axiom,
    ! [Y2: $o,P2: fun_o_o] :
      ( aa_set_o_o(member_o((Y2)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
     => ( ! [X: product_unit] : aa_o_o(P2,aa_Product_unit_o(product_Rep_unit,X))
       => aa_o_o(P2,(Y2)) ) ) ).

% Rep_unit_induct
tff(fact_4253_Rep__unit__cases,axiom,
    ! [Y2: $o] :
      ( aa_set_o_o(member_o((Y2)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
     => ~ ! [X: product_unit] :
            ( (Y2)
          <=> ~ aa_Product_unit_o(product_Rep_unit,X) ) ) ).

% Rep_unit_cases
tff(fact_4254_Rep__unit,axiom,
    ! [Xa: product_unit] : aa_set_o_o(member_o(aa_Product_unit_o(product_Rep_unit,Xa)),aa_set_o_set_o(insert_o($true),bot_bot_set_o)) ).

% Rep_unit
tff(fact_4255_Rep__unit__inverse,axiom,
    ! [Xa: product_unit] : aa_o_Product_unit(product_Abs_unit,aa_Product_unit_o(product_Rep_unit,Xa)) = Xa ).

% Rep_unit_inverse
tff(fact_4256_Abs__unit__inverse,axiom,
    ! [Y2: $o] :
      ( aa_set_o_o(member_o((Y2)),aa_set_o_set_o(insert_o($true),bot_bot_set_o))
     => ( aa_Product_unit_o(product_Rep_unit,aa_o_Product_unit(product_Abs_unit,(Y2)))
      <=> (Y2) ) ) ).

% Abs_unit_inverse
tff(fact_4257_surj__prod__decode,axiom,
    image_5846123807819985514at_nat(nat_prod_decode,top_top_set_nat) = top_to4669805908274784177at_nat ).

% surj_prod_decode
tff(fact_4258_pair__less__def,axiom,
    fun_pair_less = lex_prod_nat_nat(less_than,less_than) ).

% pair_less_def
tff(fact_4259_list__decode_Oelims,axiom,
    ! [Xa: nat,Y2: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,Xa) = Y2 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y2 != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( Xa = aa_nat_nat(suc,N2) )
             => ( Y2 != produc2761476792215241774st_nat(aTP_Lamm_aei,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_4260_nat__to__rat__surj__def,axiom,
    ! [N3: nat] : aa_nat_rat(nat_to_rat_surj,N3) = produc6207742614233964070at_rat(aTP_Lamm_aej,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) ).

% nat_to_rat_surj_def
tff(fact_4261_list__decode_Osimps_I2_J,axiom,
    ! [N3: nat] : aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N3)) = produc2761476792215241774st_nat(aTP_Lamm_aei,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) ).

% list_decode.simps(2)
tff(fact_4262_list__decode_Opelims,axiom,
    ! [Xa: nat,Y2: list_nat] :
      ( ( aa_nat_list_nat(nat_list_decode,Xa) = Y2 )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),Xa)
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y2 = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N2: nat] :
                ( ( Xa = aa_nat_nat(suc,N2) )
               => ( ( Y2 = produc2761476792215241774st_nat(aTP_Lamm_aei,aa_nat7603221163087113075at_nat(nat_prod_decode,N2)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_4263_surj__list__decode,axiom,
    image_nat_list_nat(nat_list_decode,top_top_set_nat) = top_top_set_list_nat ).

% surj_list_decode
tff(fact_4264_list__decode_Opsimps_I2_J,axiom,
    ! [N3: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N3))
     => ( aa_nat_list_nat(nat_list_decode,aa_nat_nat(suc,N3)) = produc2761476792215241774st_nat(aTP_Lamm_aei,aa_nat7603221163087113075at_nat(nat_prod_decode,N3)) ) ) ).

% list_decode.psimps(2)
tff(fact_4265_surj__sum__encode,axiom,
    image_1320371278474632150at_nat(nat_sum_encode,top_to6661820994512907621at_nat) = top_top_set_nat ).

% surj_sum_encode
tff(fact_4266_surj__list__encode,axiom,
    image_list_nat_nat(nat_list_encode,top_top_set_list_nat) = top_top_set_nat ).

% surj_list_encode
tff(fact_4267_surj__sum__decode,axiom,
    image_678696785212003926at_nat(nat_sum_decode,top_top_set_nat) = top_to6661820994512907621at_nat ).

% surj_sum_decode
tff(fact_4268_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aTP_Lamm_aek(L)),Qr) ).

% divmod_step_int_def
tff(fact_4269_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ael(L)),Qr) ).

% divmod_step_nat_def
tff(fact_4270_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : aa_Pro4693618923521437820nteger(unique4921790084139445826nteger(L),Qr) = aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_aem(L)),Qr) ).

% divmod_step_integer_def
tff(fact_4271_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_aen,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_4272_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_num(code_num_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_aeo,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_4273_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_aep(L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_aeq(L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_4274_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_aer,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_4275_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_4276_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_aes(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_4277_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(aa_Cod1978467581589928094nteger(comp_C7848664941760151970nteger(aa_fun811764697647618220nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger),times_3573771949741848930nteger),sgn_sgn_Code_integer),L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),aa_Pro4693618923521437820nteger(produc6916734918728496179nteger(aTP_Lamm_aet(L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_4278_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_4279_num__of__integer__def,axiom,
    code_num_of_integer = map_fu1227494855608507351um_num(code_int_of_integer,id_num,aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,num_of_nat),nat2)) ).

% num_of_integer_def
tff(fact_4280_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_4281_num__of__integer_Orsp,axiom,
    aa_fun_int_num_o(aa_fun7722276931801436589_num_o(bNF_re7626690874201225453um_num(fequal_int,fequal_num),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,num_of_nat),nat2)),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,num_of_nat),nat2)) ).

% num_of_integer.rsp
tff(fact_4282_num__of__integer_Otransfer,axiom,
    aa_fun9207913800938841190_num_o(aa_fun5465311938146857276_num_o(bNF_re6718328864250387230um_num(code_pcr_integer,fequal_num),aa_fun7214996087693374330nt_num(aa_fun6849364718390456542nt_num(comp_nat_num_int,num_of_nat),nat2)),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_4283_pcr__integer__def,axiom,
    code_pcr_integer = relcom967497705283160040nteger(fequal_int,code_cr_integer) ).

% pcr_integer_def
tff(fact_4284_Random__Pred_Onot__randompred__def,axiom,
    ! [P2: fun_Pr160617923510306311atural,X5: produc7822875418678951345atural] : aa_Pro6676232280479689400atural(random6974930770145893639ompred(P2),X5) = produc6665183775751917029atural(aTP_Lamm_aeu,aa_Pro6676232280479689400atural(P2,X5)) ).

% Random_Pred.not_randompred_def
tff(fact_4285_rat__floor__code,axiom,
    ! [P: rat] : aa_rat_int(archim3151403230148437115or_rat,P) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,divide_divide_int),aa_rat7754553326175492643nt_int(quotient_of,P)) ).

% rat_floor_code
tff(fact_4286_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(aa_fun914008879417524284nt_int(produc8211389475949308722nt_int,aTP_Lamm_aev),Qr) ).

% Divides.adjust_div_def
tff(fact_4287_take__bit__num__code,axiom,
    ! [N3: nat,M3: num] : bit_take_bit_num(N3,M3) = produc478579273971653890on_num(aTP_Lamm_afa,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,N3),M3)) ).

% take_bit_num_code
tff(fact_4288_list__ex__iff__not__all__inverval__nat,axiom,
    ! [P2: fun_nat_o,I: nat,J4: nat] :
      ( list_ex_nat(P2,upt(I,J4))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(aa_fun5937193402907943164_nat_o(aa_fun5662400728881950724_nat_o(comp_o_o_nat,fNot),P2)),I),J4) ) ).

% list_ex_iff_not_all_inverval_nat
tff(fact_4289_ATP_Olambda__1,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat] : aa_set7135727717115484412at_nat(aTP_Lamm_mt,Uu) = insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uu),Uu),bot_bo4948859079157340979at_nat) ).

% ATP.lambda_1
tff(fact_4290_ATP_Olambda__2,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat] : aa_set1427696295806717282at_nat(aTP_Lamm_mv,Uu) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uu),Uu)),bot_bo228742789529271731at_nat) ).

% ATP.lambda_2
tff(fact_4291_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_mw,Uu) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),Uu)),bot_bo2099793752762293965at_nat) ).

% ATP.lambda_3
tff(fact_4292_ATP_Olambda__4,axiom,
    ! [Uu: int] : aa_int8519581412571446831nt_int(aTP_Lamm_mu,Uu) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),Uu),bot_bo1796632182523588997nt_int) ).

% ATP.lambda_4
tff(fact_4293_ATP_Olambda__5,axiom,
    ! [Uu: produc859450856879609959at_nat] :
      ( aa_Pro602569247023830573_nat_o(aTP_Lamm_acp,Uu)
    <=> ( aa_Pro5901169740204579478at_nat(produc3213797794245857475at_nat,Uu) = aa_Pro5901169740204579478at_nat(produc6408287024330202629at_nat,Uu) ) ) ).

% ATP.lambda_5
tff(fact_4294_ATP_Olambda__6,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_acq,Uu)
    <=> ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uu) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uu) ) ) ).

% ATP.lambda_6
tff(fact_4295_ATP_Olambda__7,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_acr,Uu)
    <=> ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu) ) ) ).

% ATP.lambda_7
tff(fact_4296_ATP_Olambda__8,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea8848428470362602134t_unit(aTP_Lamm_fe,Uu) = produc8256076736985659095t_unit(Uu,Uu) ).

% ATP.lambda_8
tff(fact_4297_ATP_Olambda__9,axiom,
    ! [Uu: ref_a] : aa_ref968167784541683456_ref_a(aTP_Lamm_fd,Uu) = produc1340070678782110931_ref_a(Uu,Uu) ).

% ATP.lambda_9
tff(fact_4298_ATP_Olambda__10,axiom,
    ! [Uu: $o] : aa_o_P8186867570850176203od_o_o(aTP_Lamm_eu,(Uu)) = aa_o_P8186867570850176203od_o_o(product_Pair_o_o((Uu)),(Uu)) ).

% ATP.lambda_10
tff(fact_4299_ATP_Olambda__11,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat] : aa_set1529286688856863004at_nat(aTP_Lamm_ew,Uu) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uu),Uu) ).

% ATP.lambda_11
tff(fact_4300_ATP_Olambda__12,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat] : aa_set8987774817048569474at_nat(aTP_Lamm_ex,Uu) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uu),Uu) ).

% ATP.lambda_12
tff(fact_4301_ATP_Olambda__13,axiom,
    ! [Uu: num] : aa_num3156681643174609981um_num(aTP_Lamm_ev,Uu) = product_Pair_num_num(Uu,Uu) ).

% ATP.lambda_13
tff(fact_4302_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_ey,Uu) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),Uu) ).

% ATP.lambda_14
tff(fact_4303_ATP_Olambda__15,axiom,
    ! [Uu: int] : aa_int7462565939943104719nt_int(aTP_Lamm_ez,Uu) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),Uu) ).

% ATP.lambda_15
tff(fact_4304_ATP_Olambda__16,axiom,
    ! [Uu: product_unit] : aa_Pro234974305641502940t_unit(aTP_Lamm_lv,Uu) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),bot_bo3957492148770167129t_unit) ).

% ATP.lambda_16
tff(fact_4305_ATP_Olambda__17,axiom,
    ! [Uu: produc3843707927480180839at_nat] : aa_Pro2325065730410872156at_nat(aTP_Lamm_lw,Uu) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Uu),bot_bo228742789529271731at_nat) ).

% ATP.lambda_17
tff(fact_4306_ATP_Olambda__18,axiom,
    ! [Uu: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aTP_Lamm_lx,Uu) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Uu),bot_bo2099793752762293965at_nat) ).

% ATP.lambda_18
tff(fact_4307_ATP_Olambda__19,axiom,
    ! [Uu: $o] : aa_o_set_o(aTP_Lamm_ly,(Uu)) = aa_set_o_set_o(insert_o((Uu)),bot_bot_set_o) ).

% ATP.lambda_19
tff(fact_4308_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_lz,Uu) = aa_set_nat_set_nat(insert_nat(Uu),bot_bot_set_nat) ).

% ATP.lambda_20
tff(fact_4309_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_aed,Uu) = aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_aec,Uu)) ).

% ATP.lambda_21
tff(fact_4310_ATP_Olambda__22,axiom,
    ! [Uu: fun_int_int] : aa_fun1083484306708752838nt_int(aTP_Lamm_abd,Uu) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aTP_Lamm_abc(Uu))) ).

% ATP.lambda_22
tff(fact_4311_ATP_Olambda__23,axiom,
    ! [Uu: fun_nat_nat] : aa_fun6680575331210251206at_nat(aTP_Lamm_abb,Uu) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aTP_Lamm_aba(Uu))) ).

% ATP.lambda_23
tff(fact_4312_ATP_Olambda__24,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aew,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_24
tff(fact_4313_ATP_Olambda__25,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_adt,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_25
tff(fact_4314_ATP_Olambda__26,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aef,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_26
tff(fact_4315_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_acz,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_27
tff(fact_4316_ATP_Olambda__28,axiom,
    ! [Uu: int] : aa_int_nat(aTP_Lamm_aee,Uu) = aa_int_nat(nat2,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_28
tff(fact_4317_ATP_Olambda__29,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_aez(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_aex(Uua),aTP_Lamm_aey(Uua),Uu) ).

% ATP.lambda_29
tff(fact_4318_ATP_Olambda__30,axiom,
    ! [Uu: pred_Product_unit,Uua: produc7822875418678951345atural] :
      aa_Pro6676232280479689400atural(aa_pre2304647123621358054atural(aTP_Lamm_aeu,Uu),Uua) = $ite(aa_Product_unit_o(eval_Product_unit(Uu),product_Unity),aa_Pro6676232280479689400atural(produc5069803637994805237atural(bot_bo2717538794563056311t_unit),Uua),aa_Pro6676232280479689400atural(produc5069803637994805237atural(single_Product_unit(product_Unity)),Uua)) ).

% ATP.lambda_30
tff(fact_4319_ATP_Olambda__31,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aeg,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_31
tff(fact_4320_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_afa,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_aez(Uua),Uu) ).

% ATP.lambda_32
tff(fact_4321_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_aex(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_aew,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_33
tff(fact_4322_ATP_Olambda__34,axiom,
    ! [Uu: fun_re5167877524888533480unit_o,Uua: produc7812382424243267045t_unit] :
      ( aa_Pro6713969180431817013unit_o(aTP_Lamm_gx(Uu),Uua)
    <=> aa_hea7994211679605476167unit_o(aa_ref4509144387093619241unit_o(Uu,aa_Pro7711270743135521984_ref_a(produc473321116828791851t_unit,Uua)),aa_Pro8575920914306358506t_unit(produc1126150082675451753t_unit,Uua)) ) ).

% ATP.lambda_34
tff(fact_4323_ATP_Olambda__35,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gy(Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ) ).

% ATP.lambda_35
tff(fact_4324_ATP_Olambda__36,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_aer,Uu),Uua) = $let(
        l: int,
        l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ).

% ATP.lambda_36
tff(fact_4325_ATP_Olambda__37,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set7135727717115484412at_nat(aTP_Lamm_mf(Uu),Uua) = insert291458823148189367at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uu),Uua),bot_bo4948859079157340979at_nat) ).

% ATP.lambda_37
tff(fact_4326_ATP_Olambda__38,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set1427696295806717282at_nat(aTP_Lamm_ml(Uu),Uua) = aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uu),Uua)),bot_bo228742789529271731at_nat) ).

% ATP.lambda_38
tff(fact_4327_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: num] : aa_num4662830302079825491at_num(aTP_Lamm_mh(Uu),Uua) = insert8920054152555992091at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Uu),Uua),bot_bo7038385379056416535at_num) ).

% ATP.lambda_39
tff(fact_4328_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_mn(Uu),Uua) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),Uua)),bot_bo2099793752762293965at_nat) ).

% ATP.lambda_40
tff(fact_4329_ATP_Olambda__41,axiom,
    ! [Uu: int,Uua: int] : aa_int8519581412571446831nt_int(aTP_Lamm_mj(Uu),Uua) = insert5033312907999012233nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),Uua),bot_bo1796632182523588997nt_int) ).

% ATP.lambda_41
tff(fact_4330_ATP_Olambda__42,axiom,
    ! [Uu: fun_num_fun_num_o,Uua: fun_in2248087180373909186um_num] :
      ( aa_fun7085617330537795730_num_o(aTP_Lamm_aj(Uu),Uua)
    <=> ord_le880128212290418581um_num(image_7722540232540426402um_num(Uua,top_top_set_int),collec2230928802738392704um_num(produc5703948589228662326_num_o(Uu))) ) ).

% ATP.lambda_42
tff(fact_4331_ATP_Olambda__43,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: fun_ch7922468903353698889at_nat] :
      ( aa_fun4726276236897182795_nat_o(aTP_Lamm_af(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_8382823969816519465at_nat(Uua,top_top_set_char)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Uu))) ) ).

% ATP.lambda_43
tff(fact_4332_ATP_Olambda__44,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: fun_li4460313466401262074at_nat] :
      ( aa_fun1862623989650871514_nat_o(aTP_Lamm_ao(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_2795870212791037530at_nat(Uua,top_top_set_list_nat)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Uu))) ) ).

% ATP.lambda_44
tff(fact_4333_ATP_Olambda__45,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: fun_ra4806459883200776834at_nat] :
      ( aa_fun6434711774563529938_nat_o(aTP_Lamm_ag(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_9175953278516507234at_nat(Uua,top_top_set_rat)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Uu))) ) ).

% ATP.lambda_45
tff(fact_4334_ATP_Olambda__46,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: fun_na1476630412504255114at_nat] :
      ( aa_fun4041494780444525258_nat_o(aTP_Lamm_ah(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_5846123807819985514at_nat(Uua,top_top_set_nat)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Uu))) ) ).

% ATP.lambda_46
tff(fact_4335_ATP_Olambda__47,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: fun_in4514472496968388910at_nat] :
      ( aa_fun6008925269331974950_nat_o(aTP_Lamm_ai(Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,image_8883965892284119310at_nat(Uua,top_top_set_int)),aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,Uu))) ) ).

% ATP.lambda_47
tff(fact_4336_ATP_Olambda__48,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: fun_ch7619307333114993921nt_int] :
      ( aa_fun7194017090715014035_int_o(aTP_Lamm_ak(Uu),Uua)
    <=> ord_le2843351958646193337nt_int(image_5204326662208377313nt_int(Uua,top_top_set_char),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Uu))) ) ).

% ATP.lambda_48
tff(fact_4337_ATP_Olambda__49,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: fun_ra4503298312962071866nt_int] :
      ( aa_fun8902452628381361178_int_o(aTP_Lamm_al(Uu),Uua)
    <=> ord_le2843351958646193337nt_int(image_5997455970908365082nt_int(Uua,top_top_set_rat),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Uu))) ) ).

% ATP.lambda_49
tff(fact_4338_ATP_Olambda__50,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: fun_na1173468842265550146nt_int] :
      ( aa_fun6509235634262356498_int_o(aTP_Lamm_am(Uu),Uua)
    <=> ord_le2843351958646193337nt_int(image_2667626500211843362nt_int(Uua,top_top_set_nat),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Uu))) ) ).

% ATP.lambda_50
tff(fact_4339_ATP_Olambda__51,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: fun_in4211310926729683942nt_int] :
      ( aa_fun8476666123149806190_int_o(aTP_Lamm_an(Uu),Uua)
    <=> ord_le2843351958646193337nt_int(image_5705468584675977158nt_int(Uua,top_top_set_int),aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,Uu))) ) ).

% ATP.lambda_51
tff(fact_4340_ATP_Olambda__52,axiom,
    ! [Uu: set_o,Uua: fun_nat_o] :
      ( aa_fun_nat_o_o(aTP_Lamm_fg(Uu),Uua)
    <=> aa_set_o_o(ord_less_eq_set_o(image_nat_o(Uua,top_top_set_nat)),Uu) ) ).

% ATP.lambda_52
tff(fact_4341_ATP_Olambda__53,axiom,
    ! [Uu: set_num,Uua: fun_int_num] :
      ( aa_fun_int_num_o(aTP_Lamm_fk(Uu),Uua)
    <=> ord_less_eq_set_num(image_int_num(Uua,top_top_set_int),Uu) ) ).

% ATP.lambda_53
tff(fact_4342_ATP_Olambda__54,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat] :
      ( aa_fun_nat_nat_o(aTP_Lamm_fh(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(aa_set_nat_set_nat(image_nat_nat(Uua),top_top_set_nat)),Uu) ) ).

% ATP.lambda_54
tff(fact_4343_ATP_Olambda__55,axiom,
    ! [Uu: set_nat,Uua: fun_int_nat] :
      ( aa_fun_int_nat_o(aTP_Lamm_fi(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(aa_set_int_set_nat(image_int_nat(Uua),top_top_set_int)),Uu) ) ).

% ATP.lambda_55
tff(fact_4344_ATP_Olambda__56,axiom,
    ! [Uu: set_int,Uua: fun_int_int] :
      ( aa_fun_int_int_o(aTP_Lamm_fj(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(aa_set_int_set_int(image_int_int(Uua),top_top_set_int)),Uu) ) ).

% ATP.lambda_56
tff(fact_4345_ATP_Olambda__57,axiom,
    ! [Uu: set_fu5818728815147633517nteger,Uua: fun_Co1981900195452921617nteger] :
      ( aa_fun7635108473159255689eger_o(aTP_Lamm_ff(Uu),Uua)
    <=> ord_le2952524580166011853nteger(image_2716445779319712667nteger(Uua,top_to4645266643341252675nteger),Uu) ) ).

% ATP.lambda_57
tff(fact_4346_ATP_Olambda__58,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_set_rat(aTP_Lamm_mx(Uu),Uua) = insert_rat(aa_nat_rat(Uu,Uua),bot_bot_set_rat) ).

% ATP.lambda_58
tff(fact_4347_ATP_Olambda__59,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_na(Uu),Uua) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(Uu,Uua)),bot_bot_set_nat) ).

% ATP.lambda_59
tff(fact_4348_ATP_Olambda__60,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_set_int(aTP_Lamm_my(Uu),Uua) = aa_set_int_set_int(insert_int(aa_nat_int(Uu,Uua)),bot_bot_set_int) ).

% ATP.lambda_60
tff(fact_4349_ATP_Olambda__61,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_set_nat(aTP_Lamm_nb(Uu),Uua) = aa_set_nat_set_nat(insert_nat(aa_int_nat(Uu,Uua)),bot_bot_set_nat) ).

% ATP.lambda_61
tff(fact_4350_ATP_Olambda__62,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_set_int(aTP_Lamm_mz(Uu),Uua) = aa_set_int_set_int(insert_int(aa_int_int(Uu,Uua)),bot_bot_set_int) ).

% ATP.lambda_62
tff(fact_4351_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aeh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uu) ).

% ATP.lambda_63
tff(fact_4352_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_aej,Uu),Uua) = aa_int_rat(fract(aa_nat_int(nat_int_decode,Uu)),aa_nat_int(nat_int_decode,Uua)) ).

% ATP.lambda_64
tff(fact_4353_ATP_Olambda__65,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_aeo,Uu),Uua) = $let(
        l: num,
        l:= aa_Code_integer_num(code_num_of_integer,Uu),
        $let(
          l2: num,
          l2:= aa_num_num(plus_plus_num(l),l),
          $ite(Uua = zero_z3403309356797280102nteger,l2,aa_num_num(plus_plus_num(l2),one)) ) ) ).

% ATP.lambda_65
tff(fact_4354_ATP_Olambda__66,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_aen,Uu),Uua) = $let(
        l: nat,
        l:= code_nat_of_integer(Uu),
        $let(
          l2: nat,
          l2:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l),l),
          $ite(Uua = zero_z3403309356797280102nteger,l2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),one_one_nat)) ) ) ).

% ATP.lambda_66
tff(fact_4355_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: product_unit] : aa_Pro6064029160526250199et_nat(aTP_Lamm_abg(Uu),Uua) = vimage4884490618288580032t_unit(Uu,aa_set8324764978855220220t_unit(insert_Product_unit(Uua),bot_bo3957492148770167129t_unit)) ).

% ATP.lambda_67
tff(fact_4356_ATP_Olambda__68,axiom,
    ! [Uu: fun_na367238361360896496at_nat,Uua: produc3843707927480180839at_nat] : aa_Pro1692590229789857841et_nat(aTP_Lamm_abh(Uu),Uua) = vimage6435164912253009178at_nat(Uu,aa_set4883462369968598652at_nat(insert9069300056098147895at_nat(Uua),bot_bo228742789529271731at_nat)) ).

% ATP.lambda_68
tff(fact_4357_ATP_Olambda__69,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aTP_Lamm_abi(Uu),Uua) = vimage8013328719654469172at_nat(Uu,aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Uua),bot_bo2099793752762293965at_nat)) ).

% ATP.lambda_69
tff(fact_4358_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_o,Uua: $o] : aa_o_set_nat(aTP_Lamm_abj(Uu),(Uua)) = aa_set_o_set_nat(vimage_nat_o(Uu),aa_set_o_set_o(insert_o((Uua)),bot_bot_set_o)) ).

% ATP.lambda_70
tff(fact_4359_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abk(Uu),Uua) = aa_set_nat_set_nat(vimage_nat_nat(Uu),aa_set_nat_set_nat(insert_nat(Uua),bot_bot_set_nat)) ).

% ATP.lambda_71
tff(fact_4360_ATP_Olambda__72,axiom,
    ! [Uu: fun_se6846035212209387651at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set1529286688856863004at_nat(aTP_Lamm_wg(Uu),Uua) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uua),aa_set4883462369968598652at_nat(Uu,Uua)) ).

% ATP.lambda_72
tff(fact_4361_ATP_Olambda__73,axiom,
    ! [Uu: fun_se320340636488258563at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set8987774817048569474at_nat(aTP_Lamm_wh(Uu),Uua) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uua),aa_set8868280134176360700at_nat(Uu,Uua)) ).

% ATP.lambda_73
tff(fact_4362_ATP_Olambda__74,axiom,
    ! [Uu: fun_nat_num,Uua: nat] : aa_nat8311465100035950781at_num(aTP_Lamm_wi(Uu),Uua) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Uua),aa_nat_num(Uu,Uua)) ).

% ATP.lambda_74
tff(fact_4363_ATP_Olambda__75,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_wj(Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_75
tff(fact_4364_ATP_Olambda__76,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int7462565939943104719nt_int(aTP_Lamm_wk(Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),aa_int_int(Uu,Uua)) ).

% ATP.lambda_76
tff(fact_4365_ATP_Olambda__77,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aev,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_77
tff(fact_4366_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_aei,Uu),Uua) = aa_list_nat_list_nat(cons_nat(Uu),aa_nat_list_nat(nat_list_decode,Uua)) ).

% ATP.lambda_78
tff(fact_4367_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acu,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_79
tff(fact_4368_ATP_Olambda__80,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_80
tff(fact_4369_ATP_Olambda__81,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_act,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_81
tff(fact_4370_ATP_Olambda__82,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(aTP_Lamm_qj,Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uua),Uu) ).

% ATP.lambda_82
tff(fact_4371_ATP_Olambda__83,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_qi(Uu),Uua) = aa_rat_rat(minus_minus_rat(Uua),Uu) ).

% ATP.lambda_83
tff(fact_4372_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_acx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_84
tff(fact_4373_ATP_Olambda__85,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_qh,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uua),Uu) ).

% ATP.lambda_85
tff(fact_4374_ATP_Olambda__86,axiom,
    ! [Uu: product_unit,Uua: product_unit] : aa_Pro8294289767544744956t_unit(aTP_Lamm_xg(Uu),Uua) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(Uua),Uu) ).

% ATP.lambda_86
tff(fact_4375_ATP_Olambda__87,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat] : aa_set5535132180132642428at_nat(aTP_Lamm_xd(Uu),Uua) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(Uua),Uu) ).

% ATP.lambda_87
tff(fact_4376_ATP_Olambda__88,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set4883462369968598652at_nat(aTP_Lamm_xe(Uu),Uua) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(Uua),Uu) ).

% ATP.lambda_88
tff(fact_4377_ATP_Olambda__89,axiom,
    ! [Uu: set_nat,Uua: set_nat] : aa_set_nat_set_nat(aTP_Lamm_xf(Uu),Uua) = aa_set_nat_set_nat(sup_sup_set_nat(Uua),Uu) ).

% ATP.lambda_89
tff(fact_4378_ATP_Olambda__90,axiom,
    ! [Uu: product_unit,Uua: product_unit] : aa_Pro8294289767544744956t_unit(aTP_Lamm_xc(Uu),Uua) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uua),Uu) ).

% ATP.lambda_90
tff(fact_4379_ATP_Olambda__91,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aTP_Lamm_xa(Uu),Uua) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uua),Uu) ).

% ATP.lambda_91
tff(fact_4380_ATP_Olambda__92,axiom,
    ! [Uu: set_nat,Uua: set_nat] : aa_set_nat_set_nat(aTP_Lamm_xb(Uu),Uua) = aa_set_nat_set_nat(inf_inf_set_nat(Uua),Uu) ).

% ATP.lambda_92
tff(fact_4381_ATP_Olambda__93,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ada,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_93
tff(fact_4382_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aec,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_94
tff(fact_4383_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat] : aa_set1405191528097542557at_nat(aTP_Lamm_me(Uu),Uua) = produc457027306803732586at_nat(Uua,Uu) ).

% ATP.lambda_95
tff(fact_4384_ATP_Olambda__96,axiom,
    ! [Uu: heap_e7401611519738050253t_unit,Uua: ref_a] : aa_ref2093636359556205760t_unit(aa_hea6306597601026987024t_unit(aTP_Lamm_fc,Uu),Uua) = aa_hea1409956391221964662t_unit(produc6544324848051098391t_unit(Uua),Uu) ).

% ATP.lambda_96
tff(fact_4385_ATP_Olambda__97,axiom,
    ! [Uu: ref_a,Uua: heap_e7401611519738050253t_unit] : aa_hea4620145581493207210_ref_a(aa_ref7656232950593978334_ref_a(aTP_Lamm_fb,Uu),Uua) = produc2564672488694626579_ref_a(Uua,Uu) ).

% ATP.lambda_97
tff(fact_4386_ATP_Olambda__98,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(aTP_Lamm_ec,Uu),Uua) = aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uua),Uu) ).

% ATP.lambda_98
tff(fact_4387_ATP_Olambda__99,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(aTP_Lamm_ed,Uu),Uua) = aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uua),Uu) ).

% ATP.lambda_99
tff(fact_4388_ATP_Olambda__100,axiom,
    ! [Uu: num,Uua: nat] : aa_nat8311465100035950781at_num(aa_num725332942028738172at_num(aTP_Lamm_ee,Uu),Uua) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Uua),Uu) ).

% ATP.lambda_100
tff(fact_4389_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ef,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uu) ).

% ATP.lambda_101
tff(fact_4390_ATP_Olambda__102,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_eg,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uu) ).

% ATP.lambda_102
tff(fact_4391_ATP_Olambda__103,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_set6353399069419443519_nat_o(aTP_Lamm_a,Uu),Uua)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uua),Uu) ) ).

% ATP.lambda_103
tff(fact_4392_ATP_Olambda__104,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_set1857767853180004543_nat_o(aTP_Lamm_ac,Uu),Uua)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uua),Uu) ) ).

% ATP.lambda_104
tff(fact_4393_ATP_Olambda__105,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Uua),Uu) ) ).

% ATP.lambda_105
tff(fact_4394_ATP_Olambda__106,axiom,
    ! [Uu: set_o,Uua: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_aa,Uu),(Uua))
    <=> aa_set_o_o(member_o((Uua)),Uu) ) ).

% ATP.lambda_106
tff(fact_4395_ATP_Olambda__107,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ad,Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_107
tff(fact_4396_ATP_Olambda__108,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ab,Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_108
tff(fact_4397_ATP_Olambda__109,axiom,
    ! [Uu: ref_a,Uua: heap_e7401611519738050253t_unit] : aa_hea5100639427662539565unit_a(aTP_Lamm_fa(Uu),Uua) = get_a(Uua,Uu) ).

% ATP.lambda_109
tff(fact_4398_ATP_Olambda__110,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_acv,Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_110
tff(fact_4399_ATP_Olambda__111,axiom,
    ! [Uu: set_Pr1133966238863718811t_unit,Uua: ref_a] : aa_ref8820618686110234386t_unit(aTP_Lamm_go(Uu),Uua) = image_17660834435998945t_unit(produc1126150082675451753t_unit,Uu) ).

% ATP.lambda_111
tff(fact_4400_ATP_Olambda__112,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_gn(Uu),Uua) = image_2486076414777270412at_nat(product_snd_nat_nat,Uu) ).

% ATP.lambda_112
tff(fact_4401_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_aey(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_113
tff(fact_4402_ATP_Olambda__114,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_ado,Uu),Uua) = aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_nat8314798964848609180_nat_o(aTP_Lamm_adn(Uu),Uua)) ).

% ATP.lambda_114
tff(fact_4403_ATP_Olambda__115,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_adm,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_adl(Uu),Uua)) ).

% ATP.lambda_115
tff(fact_4404_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_adk,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_adj(Uu),Uua)) ).

% ATP.lambda_116
tff(fact_4405_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_adi,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_adh(Uu),Uua)) ).

% ATP.lambda_117
tff(fact_4406_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_ade,Uu),Uua) = aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_nat8314798964848609180_nat_o(aTP_Lamm_add(Uu),Uua)) ).

% ATP.lambda_118
tff(fact_4407_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_adc,Uu),Uua) = aa_fun7564604662015400456_nat_o(produc6081775807080527818_nat_o,aa_nat8314798964848609180_nat_o(aTP_Lamm_adb(Uu),Uua)) ).

% ATP.lambda_119
tff(fact_4408_ATP_Olambda__120,axiom,
    ! [Uu: fun_na1234882596965831319t_unit,Uua: fun_rat_nat] : aa_fun130284467773092074t_unit(aTP_Lamm_pc(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_fun8822970404168604523t_unit(aTP_Lamm_pb(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_120
tff(fact_4409_ATP_Olambda__121,axiom,
    ! [Uu: fun_na1234882596965831319t_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_pv(Uu),Uua) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(aa_rat717851780304320512t_unit(aTP_Lamm_oz(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_121
tff(fact_4410_ATP_Olambda__122,axiom,
    ! [Uu: fun_na6561490796278613342t_unit,Uua: fun_char_nat] : aa_fun8036554417978873393t_unit(aTP_Lamm_oy(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_fun6946345169459377913t_unit(aTP_Lamm_ox(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_122
tff(fact_4411_ATP_Olambda__123,axiom,
    ! [Uu: fun_na6561490796278613342t_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_pt(Uu),Uua) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(aa_cha2763129282670189127t_unit(aTP_Lamm_ov(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_123
tff(fact_4412_ATP_Olambda__124,axiom,
    ! [Uu: fun_ra6895802966681190715t_unit,Uua: fun_int_rat] : aa_fun1394555006410168718t_unit(aTP_Lamm_ou(Uu),Uua) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(aa_fun1175253307200421051t_unit(aTP_Lamm_ot(Uu),Uua),top_top_set_int)) ).

% ATP.lambda_124
tff(fact_4413_ATP_Olambda__125,axiom,
    ! [Uu: fun_ra6895802966681190715t_unit,Uua: int] : aa_int_Product_unit(aTP_Lamm_pr(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_int8781220593113384100t_unit(aTP_Lamm_or(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_125
tff(fact_4414_ATP_Olambda__126,axiom,
    ! [Uu: fun_ra5341016771204120215t_unit,Uua: fun_nat_rat] : aa_fun6783687942380917994t_unit(aTP_Lamm_oq(Uu),Uua) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(aa_fun5470655599014116211t_unit(aTP_Lamm_op(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_126
tff(fact_4415_ATP_Olambda__127,axiom,
    ! [Uu: fun_ra5341016771204120215t_unit,Uua: nat] : aa_nat_Product_unit(aTP_Lamm_pp(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_nat1149729683777882112t_unit(aTP_Lamm_on(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_127
tff(fact_4416_ATP_Olambda__128,axiom,
    ! [Uu: fun_ra3623852323372631695t_unit,Uua: fun_rat_rat] : aa_fun7636492056796379362t_unit(aTP_Lamm_om(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_fun1131103075899691875t_unit(aTP_Lamm_ol(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_128
tff(fact_4417_ATP_Olambda__129,axiom,
    ! [Uu: fun_ra3623852323372631695t_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_pn(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_rat3538699410184682488t_unit(aTP_Lamm_oj(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_129
tff(fact_4418_ATP_Olambda__130,axiom,
    ! [Uu: fun_ra2707149721221939542t_unit,Uua: fun_char_rat] : aa_fun6319389970147384873t_unit(aTP_Lamm_oi(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_fun1052802603301123825t_unit(aTP_Lamm_oh(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_130
tff(fact_4419_ATP_Olambda__131,axiom,
    ! [Uu: fun_ra2707149721221939542t_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_pl(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_cha5583976912550551103t_unit(aTP_Lamm_of(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_131
tff(fact_4420_ATP_Olambda__132,axiom,
    ! [Uu: fun_ch1452000917118098306t_unit,Uua: fun_int_char] : aa_fun3113555303782821973t_unit(aTP_Lamm_oe(Uu),Uua) = comple4687483117567863418t_unit(image_5846526143423701914t_unit(aa_fun7378965330574502274t_unit(aTP_Lamm_od(Uu),Uua),top_top_set_int)) ).

% ATP.lambda_132
tff(fact_4421_ATP_Olambda__133,axiom,
    ! [Uu: fun_ch1452000917118098306t_unit,Uua: int] : aa_int_Product_unit(aTP_Lamm_pj(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_int444616083545322219t_unit(aTP_Lamm_ob(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_133
tff(fact_4422_ATP_Olambda__134,axiom,
    ! [Uu: fun_ch9120586758495803614t_unit,Uua: fun_nat_char] : aa_fun3304261219015086001t_unit(aTP_Lamm_oa(Uu),Uua) = comple4687483117567863418t_unit(image_8730104196221521654t_unit(aa_fun5279340021634321466t_unit(aTP_Lamm_nz(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_134
tff(fact_4423_ATP_Olambda__135,axiom,
    ! [Uu: fun_ch9120586758495803614t_unit,Uua: nat] : aa_nat_Product_unit(aTP_Lamm_ph(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_nat2699529241734896199t_unit(aTP_Lamm_nx(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_135
tff(fact_4424_ATP_Olambda__136,axiom,
    ! [Uu: fun_ch7403422310664315094t_unit,Uua: fun_rat_char] : aa_fun5644859648297204649t_unit(aTP_Lamm_nw(Uu),Uua) = comple4687483117567863418t_unit(image_2327579789247107822t_unit(aa_fun5856161259265371178t_unit(aTP_Lamm_nv(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_136
tff(fact_4425_ATP_Olambda__137,axiom,
    ! [Uu: fun_ch7403422310664315094t_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_pf(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_rat8068560203532998207t_unit(aTP_Lamm_nt(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_137
tff(fact_4426_ATP_Olambda__138,axiom,
    ! [Uu: fun_ch5514019212406764701t_unit,Uua: fun_char_char] : aa_fun3833878087877442032t_unit(aTP_Lamm_ns(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_fun2762813673030238648t_unit(aTP_Lamm_nr(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_138
tff(fact_4427_ATP_Olambda__139,axiom,
    ! [Uu: fun_ch5514019212406764701t_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_pd(Uu),Uua) = comple4687483117567863418t_unit(image_4337914935673063477t_unit(aa_cha4064768540349698438t_unit(aTP_Lamm_np(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_139
tff(fact_4428_ATP_Olambda__140,axiom,
    ! [Uu: fun_na1234882596965831319t_unit,Uua: fun_rat_nat] : aa_fun130284467773092074t_unit(aTP_Lamm_pw(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_fun8822970404168604523t_unit(aTP_Lamm_pb(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_140
tff(fact_4429_ATP_Olambda__141,axiom,
    ! [Uu: fun_na1234882596965831319t_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_pa(Uu),Uua) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_rat717851780304320512t_unit(aTP_Lamm_oz(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_141
tff(fact_4430_ATP_Olambda__142,axiom,
    ! [Uu: fun_na6561490796278613342t_unit,Uua: fun_char_nat] : aa_fun8036554417978873393t_unit(aTP_Lamm_pu(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_fun6946345169459377913t_unit(aTP_Lamm_ox(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_142
tff(fact_4431_ATP_Olambda__143,axiom,
    ! [Uu: fun_na6561490796278613342t_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_ow(Uu),Uua) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_cha2763129282670189127t_unit(aTP_Lamm_ov(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_143
tff(fact_4432_ATP_Olambda__144,axiom,
    ! [Uu: fun_ra6895802966681190715t_unit,Uua: fun_int_rat] : aa_fun1394555006410168718t_unit(aTP_Lamm_ps(Uu),Uua) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(aa_fun1175253307200421051t_unit(aTP_Lamm_ot(Uu),Uua),top_top_set_int)) ).

% ATP.lambda_144
tff(fact_4433_ATP_Olambda__145,axiom,
    ! [Uu: fun_ra6895802966681190715t_unit,Uua: int] : aa_int_Product_unit(aTP_Lamm_os(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_int8781220593113384100t_unit(aTP_Lamm_or(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_145
tff(fact_4434_ATP_Olambda__146,axiom,
    ! [Uu: fun_ra5341016771204120215t_unit,Uua: fun_nat_rat] : aa_fun6783687942380917994t_unit(aTP_Lamm_pq(Uu),Uua) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_fun5470655599014116211t_unit(aTP_Lamm_op(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_146
tff(fact_4435_ATP_Olambda__147,axiom,
    ! [Uu: fun_ra5341016771204120215t_unit,Uua: nat] : aa_nat_Product_unit(aTP_Lamm_oo(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_nat1149729683777882112t_unit(aTP_Lamm_on(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_147
tff(fact_4436_ATP_Olambda__148,axiom,
    ! [Uu: fun_ra3623852323372631695t_unit,Uua: fun_rat_rat] : aa_fun7636492056796379362t_unit(aTP_Lamm_po(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_fun1131103075899691875t_unit(aTP_Lamm_ol(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_148
tff(fact_4437_ATP_Olambda__149,axiom,
    ! [Uu: fun_ra3623852323372631695t_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_ok(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_rat3538699410184682488t_unit(aTP_Lamm_oj(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_149
tff(fact_4438_ATP_Olambda__150,axiom,
    ! [Uu: fun_ra2707149721221939542t_unit,Uua: fun_char_rat] : aa_fun6319389970147384873t_unit(aTP_Lamm_pm(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_fun1052802603301123825t_unit(aTP_Lamm_oh(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_150
tff(fact_4439_ATP_Olambda__151,axiom,
    ! [Uu: fun_ra2707149721221939542t_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_og(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_cha5583976912550551103t_unit(aTP_Lamm_of(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_151
tff(fact_4440_ATP_Olambda__152,axiom,
    ! [Uu: fun_ch1452000917118098306t_unit,Uua: fun_int_char] : aa_fun3113555303782821973t_unit(aTP_Lamm_pk(Uu),Uua) = comple2584293577114468500t_unit(image_5846526143423701914t_unit(aa_fun7378965330574502274t_unit(aTP_Lamm_od(Uu),Uua),top_top_set_int)) ).

% ATP.lambda_152
tff(fact_4441_ATP_Olambda__153,axiom,
    ! [Uu: fun_ch1452000917118098306t_unit,Uua: int] : aa_int_Product_unit(aTP_Lamm_oc(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_int444616083545322219t_unit(aTP_Lamm_ob(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_153
tff(fact_4442_ATP_Olambda__154,axiom,
    ! [Uu: fun_ch9120586758495803614t_unit,Uua: fun_nat_char] : aa_fun3304261219015086001t_unit(aTP_Lamm_pi(Uu),Uua) = comple2584293577114468500t_unit(image_8730104196221521654t_unit(aa_fun5279340021634321466t_unit(aTP_Lamm_nz(Uu),Uua),top_top_set_nat)) ).

% ATP.lambda_154
tff(fact_4443_ATP_Olambda__155,axiom,
    ! [Uu: fun_ch9120586758495803614t_unit,Uua: nat] : aa_nat_Product_unit(aTP_Lamm_ny(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_nat2699529241734896199t_unit(aTP_Lamm_nx(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_155
tff(fact_4444_ATP_Olambda__156,axiom,
    ! [Uu: fun_ch7403422310664315094t_unit,Uua: fun_rat_char] : aa_fun5644859648297204649t_unit(aTP_Lamm_pg(Uu),Uua) = comple2584293577114468500t_unit(image_2327579789247107822t_unit(aa_fun5856161259265371178t_unit(aTP_Lamm_nv(Uu),Uua),top_top_set_rat)) ).

% ATP.lambda_156
tff(fact_4445_ATP_Olambda__157,axiom,
    ! [Uu: fun_ch7403422310664315094t_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_nu(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_rat8068560203532998207t_unit(aTP_Lamm_nt(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_157
tff(fact_4446_ATP_Olambda__158,axiom,
    ! [Uu: fun_ch5514019212406764701t_unit,Uua: fun_char_char] : aa_fun3833878087877442032t_unit(aTP_Lamm_pe(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_fun2762813673030238648t_unit(aTP_Lamm_nr(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_158
tff(fact_4447_ATP_Olambda__159,axiom,
    ! [Uu: fun_ch5514019212406764701t_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_nq(Uu),Uua) = comple2584293577114468500t_unit(image_4337914935673063477t_unit(aa_cha4064768540349698438t_unit(aTP_Lamm_np(Uu),Uua),top_top_set_char)) ).

% ATP.lambda_159
tff(fact_4448_ATP_Olambda__160,axiom,
    ! [Uu: fun_nat_set_nat,Uua: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_mo(Uu),Uua) = comple5685304695842803022at_nat(image_7178329752028323786at_nat(aTP_Lamm_mn(Uua),aa_nat_set_nat(Uu,Uua))) ).

% ATP.lambda_160
tff(fact_4449_ATP_Olambda__161,axiom,
    ! [Uu: fun_se953073389798102371at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set1427696295806717282at_nat(aTP_Lamm_mm(Uu),Uua) = comple4364388969068068532at_nat(image_6898982319499645995at_nat(aTP_Lamm_ml(Uua),aa_set6242432332095302236at_nat(Uu,Uua))) ).

% ATP.lambda_161
tff(fact_4450_ATP_Olambda__162,axiom,
    ! [Uu: fun_int_set_int,Uua: int] : aa_int8519581412571446831nt_int(aTP_Lamm_mk(Uu),Uua) = comple5382143125604098054nt_int(image_8635204845542730022nt_int(aTP_Lamm_mj(Uua),aa_int_set_int(Uu,Uua))) ).

% ATP.lambda_162
tff(fact_4451_ATP_Olambda__163,axiom,
    ! [Uu: fun_nat_set_num,Uua: nat] : aa_nat2777925908496387357at_num(aTP_Lamm_mi(Uu),Uua) = comple1400524285282149784at_num(image_4778453735051108682at_num(aTP_Lamm_mh(Uua),aa_nat_set_num(Uu,Uua))) ).

% ATP.lambda_163
tff(fact_4452_ATP_Olambda__164,axiom,
    ! [Uu: fun_se3430100612660269923at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set7135727717115484412at_nat(aTP_Lamm_mg(Uu),Uua) = comple7342073961973170484at_nat(image_1124969593608450885at_nat(aTP_Lamm_mf(Uua),aa_set7518756442812098908at_nat(Uu,Uua))) ).

% ATP.lambda_164
tff(fact_4453_ATP_Olambda__165,axiom,
    ! [Uu: set_se5479201105039950375at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_qv(Uu),Uua)
    <=> aa_set_o_o(complete_Sup_Sup_o,image_3568060456209098518_nat_o(member8757157785044589968at_nat(Uua),Uu)) ) ).

% ATP.lambda_165
tff(fact_4454_ATP_Olambda__166,axiom,
    ! [Uu: set_se7855581050983116737at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_qy(Uu),Uua)
    <=> aa_set_o_o(complete_Sup_Sup_o,image_7457375456213313148_nat_o(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_166
tff(fact_4455_ATP_Olambda__167,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_qz(Uu),Uua)
    <=> aa_set_o_o(complete_Sup_Sup_o,image_701744273176368580_int_o(member5262025264175285858nt_int(Uua),Uu)) ) ).

% ATP.lambda_167
tff(fact_4456_ATP_Olambda__168,axiom,
    ! [Uu: set_set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_qw(Uu),(Uua))
    <=> aa_set_o_o(complete_Sup_Sup_o,image_set_o_o(member_o((Uua)),Uu)) ) ).

% ATP.lambda_168
tff(fact_4457_ATP_Olambda__169,axiom,
    ! [Uu: set_set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ra(Uu),Uua)
    <=> aa_set_o_o(complete_Sup_Sup_o,image_set_nat_o(member_nat(Uua),Uu)) ) ).

% ATP.lambda_169
tff(fact_4458_ATP_Olambda__170,axiom,
    ! [Uu: set_set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_qx(Uu),Uua)
    <=> aa_set_o_o(complete_Sup_Sup_o,image_set_int_o(member_int(Uua),Uu)) ) ).

% ATP.lambda_170
tff(fact_4459_ATP_Olambda__171,axiom,
    ! [Uu: set_se5479201105039950375at_nat,Uua: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aTP_Lamm_rb(Uu),Uua)
    <=> aa_set_o_o(complete_Inf_Inf_o,image_3568060456209098518_nat_o(member8757157785044589968at_nat(Uua),Uu)) ) ).

% ATP.lambda_171
tff(fact_4460_ATP_Olambda__172,axiom,
    ! [Uu: set_se7855581050983116737at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_re(Uu),Uua)
    <=> aa_set_o_o(complete_Inf_Inf_o,image_7457375456213313148_nat_o(member8440522571783428010at_nat(Uua),Uu)) ) ).

% ATP.lambda_172
tff(fact_4461_ATP_Olambda__173,axiom,
    ! [Uu: set_se6260736226359567993nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_rf(Uu),Uua)
    <=> aa_set_o_o(complete_Inf_Inf_o,image_701744273176368580_int_o(member5262025264175285858nt_int(Uua),Uu)) ) ).

% ATP.lambda_173
tff(fact_4462_ATP_Olambda__174,axiom,
    ! [Uu: set_set_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_rc(Uu),(Uua))
    <=> aa_set_o_o(complete_Inf_Inf_o,image_set_o_o(member_o((Uua)),Uu)) ) ).

% ATP.lambda_174
tff(fact_4463_ATP_Olambda__175,axiom,
    ! [Uu: set_set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_rg(Uu),Uua)
    <=> aa_set_o_o(complete_Inf_Inf_o,image_set_nat_o(member_nat(Uua),Uu)) ) ).

% ATP.lambda_175
tff(fact_4464_ATP_Olambda__176,axiom,
    ! [Uu: set_set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_rd(Uu),Uua)
    <=> aa_set_o_o(complete_Inf_Inf_o,image_set_int_o(member_int(Uua),Uu)) ) ).

% ATP.lambda_176
tff(fact_4465_ATP_Olambda__177,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat] : aa_set_nat_set_nat(aTP_Lamm_nm(Uu),Uua) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(Uu,Uua)) ).

% ATP.lambda_177
tff(fact_4466_ATP_Olambda__178,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat] : aa_set_nat_set_nat(aTP_Lamm_ms(Uu),Uua) = comple7806235888213564991et_nat(image_nat_set_nat(Uu,Uua)) ).

% ATP.lambda_178
tff(fact_4467_ATP_Olambda__179,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aTP_Lamm_ij(Uu),Uua) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_Pro989595756709598047_nat_o(Uu,Uua)) ).

% ATP.lambda_179
tff(fact_4468_ATP_Olambda__180,axiom,
    ! [Uu: fun_Pr4496145827618669174_int_o,Uua: product_prod_nat_nat] : aa_Pro2442717743127763220nt_int(aTP_Lamm_in(Uu),Uua) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_Pro8654398858934478503_int_o(Uu,Uua)) ).

% ATP.lambda_180
tff(fact_4469_ATP_Olambda__181,axiom,
    ! [Uu: fun_Pr6984107991879650211_nat_o,Uua: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aTP_Lamm_il(Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,aa_Pro4733806262894120092_nat_o(Uu,Uua)) ).

% ATP.lambda_181
tff(fact_4470_ATP_Olambda__182,axiom,
    ! [Uu: fun_Pr6550376307508196838_nat_o,Uua: product_prod_int_int] : aa_Pro6675321895958450596at_nat(aTP_Lamm_it(Uu),Uua) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_Pro5511567090416956951_nat_o(Uu,Uua)) ).

% ATP.lambda_182
tff(fact_4471_ATP_Olambda__183,axiom,
    ! [Uu: fun_Pr9018117161326028078_int_o,Uua: product_prod_int_int] : aa_Pro6372160325719745628nt_int(aTP_Lamm_ix(Uu),Uua) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_Pro3952998155787061599_int_o(Uu,Uua)) ).

% ATP.lambda_183
tff(fact_4472_ATP_Olambda__184,axiom,
    ! [Uu: fun_Pr8286563733112057579_nat_o,Uua: product_prod_int_int] : aa_Pro3599192995643711491et_nat(aTP_Lamm_iv(Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,aa_Pro6036262004126527460_nat_o(Uu,Uua)) ).

% ATP.lambda_184
tff(fact_4473_ATP_Olambda__185,axiom,
    ! [Uu: fun_na3970208099681795815_nat_o,Uua: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_ip(Uu),Uua) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,aa_nat7695548938202568488_nat_o(Uu,Uua)) ).

% ATP.lambda_185
tff(fact_4474_ATP_Olambda__186,axiom,
    ! [Uu: fun_na6437948953499627055_int_o,Uua: nat] : aa_nat6759544748818335627nt_int(aTP_Lamm_ir(Uu),Uua) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,aa_nat6136980003572673136_int_o(Uu,Uua)) ).

% ATP.lambda_186
tff(fact_4475_ATP_Olambda__187,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: nat] : aa_nat_set_nat(aTP_Lamm_iz(Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(Uu,Uua)) ).

% ATP.lambda_187
tff(fact_4476_ATP_Olambda__188,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: int] : aa_int_set_int(aTP_Lamm_jb(Uu),Uua) = aa_fun_int_o_set_int(collect_int,aa_int_fun_int_o(Uu,Uua)) ).

% ATP.lambda_188
tff(fact_4477_ATP_Olambda__189,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_ui,Uu),Uua)
    <=> ~ aa_Pro6150637604445964947_nat_o(Uu,Uua) ) ).

% ATP.lambda_189
tff(fact_4478_ATP_Olambda__190,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ug(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_190
tff(fact_4479_ATP_Olambda__191,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ub(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_191
tff(fact_4480_ATP_Olambda__192,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_ty(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_192
tff(fact_4481_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_acw,Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,aa_nat_fun_nat_o(aTP_Lamm_act,Uu)) ).

% ATP.lambda_193
tff(fact_4482_ATP_Olambda__194,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aTP_Lamm_ha(Uu),Uua) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,Uu) ).

% ATP.lambda_194
tff(fact_4483_ATP_Olambda__195,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: product_prod_int_int] : aa_Pro6675321895958450596at_nat(aTP_Lamm_hk(Uu),Uua) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,Uu) ).

% ATP.lambda_195
tff(fact_4484_ATP_Olambda__196,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: nat] : aa_nat7062706319057040595at_nat(aTP_Lamm_hg(Uu),Uua) = aa_fun7046812543386002057at_nat(collec3392354462482085612at_nat,Uu) ).

% ATP.lambda_196
tff(fact_4485_ATP_Olambda__197,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_nat_nat] : aa_Pro2442717743127763220nt_int(aTP_Lamm_he(Uu),Uua) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,Uu) ).

% ATP.lambda_197
tff(fact_4486_ATP_Olambda__198,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] : aa_Pro6372160325719745628nt_int(aTP_Lamm_ho(Uu),Uua) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,Uu) ).

% ATP.lambda_198
tff(fact_4487_ATP_Olambda__199,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: nat] : aa_nat6759544748818335627nt_int(aTP_Lamm_hi(Uu),Uua) = aa_fun4369364273680771065nt_int(collec213857154873943460nt_int,Uu) ).

% ATP.lambda_199
tff(fact_4488_ATP_Olambda__200,axiom,
    ! [Uu: fun_nat_o,Uua: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aTP_Lamm_hc(Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,Uu) ).

% ATP.lambda_200
tff(fact_4489_ATP_Olambda__201,axiom,
    ! [Uu: fun_nat_o,Uua: product_prod_int_int] : aa_Pro3599192995643711491et_nat(aTP_Lamm_hm(Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,Uu) ).

% ATP.lambda_201
tff(fact_4490_ATP_Olambda__202,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_set_nat(aTP_Lamm_hq(Uu),Uua) = aa_fun_nat_o_set_nat(collect_nat,Uu) ).

% ATP.lambda_202
tff(fact_4491_ATP_Olambda__203,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_set_int(aTP_Lamm_hs(Uu),Uua) = aa_fun_int_o_set_int(collect_int,Uu) ).

% ATP.lambda_203
tff(fact_4492_ATP_Olambda__204,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aem(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_204
tff(fact_4493_ATP_Olambda__205,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ael(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_205
tff(fact_4494_ATP_Olambda__206,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aek(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_206
tff(fact_4495_ATP_Olambda__207,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aeq(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_207
tff(fact_4496_ATP_Olambda__208,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aet(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).

% ATP.lambda_208
tff(fact_4497_ATP_Olambda__209,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_aep(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),Uub))) ).

% ATP.lambda_209
tff(fact_4498_ATP_Olambda__210,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: $o] :
      aa_o_s7415647395592896235at_nat(aa_set4424263627741053053at_nat(aTP_Lamm_wl(Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_210
tff(fact_4499_ATP_Olambda__211,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: $o] :
      aa_o_s4772616771316612843at_nat(aa_set7164036131237892221at_nat(aTP_Lamm_wm(Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_211
tff(fact_4500_ATP_Olambda__212,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: $o] :
      aa_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_wn(Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_212
tff(fact_4501_ATP_Olambda__213,axiom,
    ! [Uu: fun_ch5514019212406764701t_unit,Uua: fun_char_char,Uub: char] : aa_char_Product_unit(aa_fun2762813673030238648t_unit(aTP_Lamm_nr(Uu),Uua),Uub) = aa_char_Product_unit(aa_cha4064768540349698438t_unit(Uu,aa_char_char(Uua,Uub)),Uub) ).

% ATP.lambda_213
tff(fact_4502_ATP_Olambda__214,axiom,
    ! [Uu: fun_ch7403422310664315094t_unit,Uua: fun_rat_char,Uub: rat] : aa_rat_Product_unit(aa_fun5856161259265371178t_unit(aTP_Lamm_nv(Uu),Uua),Uub) = aa_rat_Product_unit(aa_cha5583976912550551103t_unit(Uu,aa_rat_char(Uua,Uub)),Uub) ).

% ATP.lambda_214
tff(fact_4503_ATP_Olambda__215,axiom,
    ! [Uu: fun_ch9120586758495803614t_unit,Uua: fun_nat_char,Uub: nat] : aa_nat_Product_unit(aa_fun5279340021634321466t_unit(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_Product_unit(aa_cha2763129282670189127t_unit(Uu,aa_nat_char(Uua,Uub)),Uub) ).

% ATP.lambda_215
tff(fact_4504_ATP_Olambda__216,axiom,
    ! [Uu: fun_ch1452000917118098306t_unit,Uua: fun_int_char,Uub: int] : aa_int_Product_unit(aa_fun7378965330574502274t_unit(aTP_Lamm_od(Uu),Uua),Uub) = aa_int_Product_unit(aa_cha9102923266727145195t_unit(Uu,aa_int_char(Uua,Uub)),Uub) ).

% ATP.lambda_216
tff(fact_4505_ATP_Olambda__217,axiom,
    ! [Uu: fun_ra2707149721221939542t_unit,Uua: fun_char_rat,Uub: char] : aa_char_Product_unit(aa_fun1052802603301123825t_unit(aTP_Lamm_oh(Uu),Uua),Uub) = aa_char_Product_unit(aa_rat8068560203532998207t_unit(Uu,aa_char_rat(Uua,Uub)),Uub) ).

% ATP.lambda_217
tff(fact_4506_ATP_Olambda__218,axiom,
    ! [Uu: fun_ra3623852323372631695t_unit,Uua: fun_rat_rat,Uub: rat] : aa_rat_Product_unit(aa_fun1131103075899691875t_unit(aTP_Lamm_ol(Uu),Uua),Uub) = aa_rat_Product_unit(aa_rat3538699410184682488t_unit(Uu,aa_rat_rat(Uua,Uub)),Uub) ).

% ATP.lambda_218
tff(fact_4507_ATP_Olambda__219,axiom,
    ! [Uu: fun_ra5341016771204120215t_unit,Uua: fun_nat_rat,Uub: nat] : aa_nat_Product_unit(aa_fun5470655599014116211t_unit(aTP_Lamm_op(Uu),Uua),Uub) = aa_nat_Product_unit(aa_rat717851780304320512t_unit(Uu,aa_nat_rat(Uua,Uub)),Uub) ).

% ATP.lambda_219
tff(fact_4508_ATP_Olambda__220,axiom,
    ! [Uu: fun_ra6895802966681190715t_unit,Uua: fun_int_rat,Uub: int] : aa_int_Product_unit(aa_fun1175253307200421051t_unit(aTP_Lamm_ot(Uu),Uua),Uub) = aa_int_Product_unit(aa_rat7057645764361276580t_unit(Uu,aa_int_rat(Uua,Uub)),Uub) ).

% ATP.lambda_220
tff(fact_4509_ATP_Olambda__221,axiom,
    ! [Uu: fun_na6561490796278613342t_unit,Uua: fun_char_nat,Uub: char] : aa_char_Product_unit(aa_fun6946345169459377913t_unit(aTP_Lamm_ox(Uu),Uua),Uub) = aa_char_Product_unit(aa_nat2699529241734896199t_unit(Uu,aa_char_nat(Uua,Uub)),Uub) ).

% ATP.lambda_221
tff(fact_4510_ATP_Olambda__222,axiom,
    ! [Uu: fun_na1234882596965831319t_unit,Uua: fun_rat_nat,Uub: rat] : aa_rat_Product_unit(aa_fun8822970404168604523t_unit(aTP_Lamm_pb(Uu),Uua),Uub) = aa_rat_Product_unit(aa_nat1149729683777882112t_unit(Uu,aa_rat_nat(Uua,Uub)),Uub) ).

% ATP.lambda_222
tff(fact_4511_ATP_Olambda__223,axiom,
    ! [Uu: fun_ch5514019212406764701t_unit,Uua: char,Uub: char] : aa_char_Product_unit(aa_cha4064768540349698438t_unit(aTP_Lamm_np(Uu),Uua),Uub) = aa_char_Product_unit(aa_cha4064768540349698438t_unit(Uu,Uub),Uua) ).

% ATP.lambda_223
tff(fact_4512_ATP_Olambda__224,axiom,
    ! [Uu: fun_ch7403422310664315094t_unit,Uua: rat,Uub: char] : aa_char_Product_unit(aa_rat8068560203532998207t_unit(aTP_Lamm_nt(Uu),Uua),Uub) = aa_rat_Product_unit(aa_cha5583976912550551103t_unit(Uu,Uub),Uua) ).

% ATP.lambda_224
tff(fact_4513_ATP_Olambda__225,axiom,
    ! [Uu: fun_ch9120586758495803614t_unit,Uua: nat,Uub: char] : aa_char_Product_unit(aa_nat2699529241734896199t_unit(aTP_Lamm_nx(Uu),Uua),Uub) = aa_nat_Product_unit(aa_cha2763129282670189127t_unit(Uu,Uub),Uua) ).

% ATP.lambda_225
tff(fact_4514_ATP_Olambda__226,axiom,
    ! [Uu: fun_ch1452000917118098306t_unit,Uua: int,Uub: char] : aa_char_Product_unit(aa_int444616083545322219t_unit(aTP_Lamm_ob(Uu),Uua),Uub) = aa_int_Product_unit(aa_cha9102923266727145195t_unit(Uu,Uub),Uua) ).

% ATP.lambda_226
tff(fact_4515_ATP_Olambda__227,axiom,
    ! [Uu: fun_ra2707149721221939542t_unit,Uua: char,Uub: rat] : aa_rat_Product_unit(aa_cha5583976912550551103t_unit(aTP_Lamm_of(Uu),Uua),Uub) = aa_char_Product_unit(aa_rat8068560203532998207t_unit(Uu,Uub),Uua) ).

% ATP.lambda_227
tff(fact_4516_ATP_Olambda__228,axiom,
    ! [Uu: fun_ra3623852323372631695t_unit,Uua: rat,Uub: rat] : aa_rat_Product_unit(aa_rat3538699410184682488t_unit(aTP_Lamm_oj(Uu),Uua),Uub) = aa_rat_Product_unit(aa_rat3538699410184682488t_unit(Uu,Uub),Uua) ).

% ATP.lambda_228
tff(fact_4517_ATP_Olambda__229,axiom,
    ! [Uu: fun_ra5341016771204120215t_unit,Uua: nat,Uub: rat] : aa_rat_Product_unit(aa_nat1149729683777882112t_unit(aTP_Lamm_on(Uu),Uua),Uub) = aa_nat_Product_unit(aa_rat717851780304320512t_unit(Uu,Uub),Uua) ).

% ATP.lambda_229
tff(fact_4518_ATP_Olambda__230,axiom,
    ! [Uu: fun_ra6895802966681190715t_unit,Uua: int,Uub: rat] : aa_rat_Product_unit(aa_int8781220593113384100t_unit(aTP_Lamm_or(Uu),Uua),Uub) = aa_int_Product_unit(aa_rat7057645764361276580t_unit(Uu,Uub),Uua) ).

% ATP.lambda_230
tff(fact_4519_ATP_Olambda__231,axiom,
    ! [Uu: fun_na6561490796278613342t_unit,Uua: char,Uub: nat] : aa_nat_Product_unit(aa_cha2763129282670189127t_unit(aTP_Lamm_ov(Uu),Uua),Uub) = aa_char_Product_unit(aa_nat2699529241734896199t_unit(Uu,Uub),Uua) ).

% ATP.lambda_231
tff(fact_4520_ATP_Olambda__232,axiom,
    ! [Uu: fun_na1234882596965831319t_unit,Uua: rat,Uub: nat] : aa_nat_Product_unit(aa_rat717851780304320512t_unit(aTP_Lamm_oz(Uu),Uua),Uub) = aa_rat_Product_unit(aa_nat1149729683777882112t_unit(Uu,Uub),Uua) ).

% ATP.lambda_232
tff(fact_4521_ATP_Olambda__233,axiom,
    ! [Uu: fun_na1668100129766719005at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_as(Uu),Uua),Uub) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uua) ).

% ATP.lambda_233
tff(fact_4522_ATP_Olambda__234,axiom,
    ! [Uu: fun_na5914473681949881652_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_at(Uu),Uua),Uub) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uua) ).

% ATP.lambda_234
tff(fact_4523_ATP_Olambda__235,axiom,
    ! [Uu: fun_in1696471567231710611nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_au(Uu),Uua),Uub) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uu,Uub),Uua) ).

% ATP.lambda_235
tff(fact_4524_ATP_Olambda__236,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_av(Uu),Uua),Uub)
    <=> aa_int_o(aa_int_fun_int_o(Uu,Uub),Uua) ) ).

% ATP.lambda_236
tff(fact_4525_ATP_Olambda__237,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_237
tff(fact_4526_ATP_Olambda__238,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_aes(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_238
tff(fact_4527_ATP_Olambda__239,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aeb(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_aea(Uua),Uub)),aa_rat7754553326175492643nt_int(quotient_of,Uu)) ).

% ATP.lambda_239
tff(fact_4528_ATP_Olambda__240,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_adz(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_ady(Uua),Uub)),aa_rat7754553326175492643nt_int(quotient_of,Uu)) ).

% ATP.lambda_240
tff(fact_4529_ATP_Olambda__241,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_adx(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_adw(Uua),Uub)),aa_rat7754553326175492643nt_int(quotient_of,Uu)) ).

% ATP.lambda_241
tff(fact_4530_ATP_Olambda__242,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_adv(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(aa_fun8026974734089330650nt_int(produc4245557441103728435nt_int,aa_int1966403025003302569nt_int(aTP_Lamm_adu(Uua),Uub)),aa_rat7754553326175492643nt_int(quotient_of,Uu)) ).

% ATP.lambda_242
tff(fact_4531_ATP_Olambda__243,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ads(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_adr(Uua),Uub)),aa_rat7754553326175492643nt_int(quotient_of,Uu)) ) ).

% ATP.lambda_243
tff(fact_4532_ATP_Olambda__244,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adq(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(aa_fun7329327620659528856_int_o(produc4947309494688390418_int_o,aa_int2464330468366073224_int_o(aTP_Lamm_adp(Uua),Uub)),aa_rat7754553326175492643nt_int(quotient_of,Uu)) ) ).

% ATP.lambda_244
tff(fact_4533_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_acs(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_245
tff(fact_4534_ATP_Olambda__246,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adg(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_246
tff(fact_4535_ATP_Olambda__247,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adf(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_247
tff(fact_4536_ATP_Olambda__248,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: fun_Pr3059359313727237556_nat_o,Uub: produc3843707927480180839at_nat] :
      ( aa_Pro5947980084102686253_nat_o(aa_fun2566793329686429606_nat_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(Uub),Uu)
        & aa_Pro5947980084102686253_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_248
tff(fact_4537_ATP_Olambda__249,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uub),Uu)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_249
tff(fact_4538_ATP_Olambda__250,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Uub),Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_250
tff(fact_4539_ATP_Olambda__251,axiom,
    ! [Uu: set_o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_cg(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_251
tff(fact_4540_ATP_Olambda__252,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_252
tff(fact_4541_ATP_Olambda__253,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_253
tff(fact_4542_ATP_Olambda__254,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_Pro6150637604445964947_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_254
tff(fact_4543_ATP_Olambda__255,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_Pro4592068669816069595_int_o(Uu,Uua) ) ) ).

% ATP.lambda_255
tff(fact_4544_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_256
tff(fact_4545_ATP_Olambda__257,axiom,
    ! [Uu: fun_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ce(Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa_int_o(Uu,Uua) ) ) ).

% ATP.lambda_257
tff(fact_4546_ATP_Olambda__258,axiom,
    ! [Uu: set_ref_a,Uua: fun_re9091073251222685225t_unit,Uub: ref_a] :
      ( aa_ref_a_o(aa_fun7429455494379205479ef_a_o(aTP_Lamm_kk(Uu),Uua),Uub)
    <=> ( member_ref_a(Uub,Uu)
        & ( aa_ref8820618686110234386t_unit(Uua,Uub) != bot_bo3686245532977126937t_unit ) ) ) ).

% ATP.lambda_258
tff(fact_4547_ATP_Olambda__259,axiom,
    ! [Uu: set_o,Uua: fun_o_1833192751255691672t_unit,Uub: $o] :
      ( aa_o_o(aa_fun67325214945128089un_o_o(aTP_Lamm_kb(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_s3492205133957624465t_unit(Uua,(Uub)) != bot_bo3957492148770167129t_unit ) ) ) ).

% ATP.lambda_259
tff(fact_4548_ATP_Olambda__260,axiom,
    ! [Uu: set_o,Uua: fun_o_set_o,Uub: $o] :
      ( aa_o_o(aa_fun4619127644298054564un_o_o(aTP_Lamm_ke(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_set_o(Uua,(Uub)) != bot_bot_set_o ) ) ) ).

% ATP.lambda_260
tff(fact_4549_ATP_Olambda__261,axiom,
    ! [Uu: set_o,Uua: fun_o_set_nat,Uub: $o] :
      ( aa_o_o(aa_fun8423158570077026744un_o_o(aTP_Lamm_kh(Uu),Uua),(Uub))
    <=> ( aa_set_o_o(member_o((Uub)),Uu)
        & ( aa_o_set_nat(Uua,(Uub)) != bot_bot_set_nat ) ) ) ).

% ATP.lambda_261
tff(fact_4550_ATP_Olambda__262,axiom,
    ! [Uu: set_nat,Uua: fun_na896991816813928310t_unit,Uub: nat] :
      ( aa_nat_o(aa_fun4664615919704916071_nat_o(aTP_Lamm_kd(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat2714212862977473631t_unit(Uua,Uub) != bot_bo3957492148770167129t_unit ) ) ) ).

% ATP.lambda_262
tff(fact_4551_ATP_Olambda__263,axiom,
    ! [Uu: set_nat,Uua: fun_nat_set_o,Uub: nat] :
      ( aa_nat_o(aa_fun4904187145036355100_nat_o(aTP_Lamm_kg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_set_o(Uua,Uub) != bot_bot_set_o ) ) ) ).

% ATP.lambda_263
tff(fact_4552_ATP_Olambda__264,axiom,
    ! [Uu: set_nat,Uua: fun_nat_set_nat,Uub: nat] :
      ( aa_nat_o(aa_fun2108954047564320208_nat_o(aTP_Lamm_kj(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_set_nat(Uua,Uub) != bot_bot_set_nat ) ) ) ).

% ATP.lambda_264
tff(fact_4553_ATP_Olambda__265,axiom,
    ! [Uu: set_int,Uua: fun_in4731230917697954842t_unit,Uub: int] :
      ( aa_int_o(aa_fun1317521766530534759_int_o(aTP_Lamm_kc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int6548451963861500163t_unit(Uua,Uub) != bot_bo3957492148770167129t_unit ) ) ) ).

% ATP.lambda_265
tff(fact_4554_ATP_Olambda__266,axiom,
    ! [Uu: set_int,Uua: fun_int_set_o,Uub: int] :
      ( aa_int_o(aa_fun4291261932532047132_int_o(aTP_Lamm_kf(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_set_o(Uua,Uub) != bot_bot_set_o ) ) ) ).

% ATP.lambda_266
tff(fact_4555_ATP_Olambda__267,axiom,
    ! [Uu: set_int,Uua: fun_int_set_nat,Uub: int] :
      ( aa_int_o(aa_fun1965424933913381656_int_o(aTP_Lamm_ki(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_set_nat(Uua,Uub) != bot_bot_set_nat ) ) ) ).

% ATP.lambda_267
tff(fact_4556_ATP_Olambda__268,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_aaz(Uu),Uua),Uub)
    <=> member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_268
tff(fact_4557_ATP_Olambda__269,axiom,
    ! [Uu: set_Pr7459493094073627847at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aa_set4399942904220837859_nat_o(aTP_Lamm_ep,Uu),Uua),Uub)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_269
tff(fact_4558_ATP_Olambda__270,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aa_set5605188905385065315_nat_o(aTP_Lamm_eq,Uu),Uua),Uub)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_270
tff(fact_4559_ATP_Olambda__271,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aa_set108017733112466993_num_o(aTP_Lamm_er,Uu),Uua),Uub)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Uua),Uub),Uu) ) ).

% ATP.lambda_271
tff(fact_4560_ATP_Olambda__272,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_set2634951030311766213_nat_o(aTP_Lamm_es,Uu),Uua),Uub)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_272
tff(fact_4561_ATP_Olambda__273,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_set742265667465749333_int_o(aTP_Lamm_et,Uu),Uua),Uub)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)),Uu) ) ).

% ATP.lambda_273
tff(fact_4562_ATP_Olambda__274,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_qu(Uu),Uua),Uub) = aa_set_nat_set_nat(minus_minus_set_nat(aa_nat_set_nat(Uu,Uub)),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_274
tff(fact_4563_ATP_Olambda__275,axiom,
    ! [Uu: fun_Pr2191293291710260995t_unit,Uua: fun_Pr2191293291710260995t_unit,Uub: product_unit] : aa_Pro8294289767544744956t_unit(aa_fun6153996563283759940t_unit(aTP_Lamm_wq(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_Pro8294289767544744956t_unit(Uu,Uub)),aa_Pro8294289767544744956t_unit(Uua,Uub)) ).

% ATP.lambda_275
tff(fact_4564_ATP_Olambda__276,axiom,
    ! [Uu: fun_Pr1628945989756059814et_nat,Uua: fun_Pr1628945989756059814et_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_wt(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro6064029160526250199et_nat(Uu,Uub)),aa_Pro6064029160526250199et_nat(Uua,Uub)) ).

% ATP.lambda_276
tff(fact_4565_ATP_Olambda__277,axiom,
    ! [Uu: fun_Product_unit_int,Uua: fun_Product_unit_int,Uub: product_unit] : aa_Product_unit_int(aa_fun3641207808737740476it_int(aTP_Lamm_wo(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_Product_unit_int(Uu,Uub)),aa_Product_unit_int(Uua,Uub)) ).

% ATP.lambda_277
tff(fact_4566_ATP_Olambda__278,axiom,
    ! [Uu: fun_Pr1469247711798767971at_nat,Uua: fun_Pr1469247711798767971at_nat,Uub: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aa_fun5344049784256257540at_nat(aTP_Lamm_wb(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(aa_Pro2745879313366468188at_nat(Uu,Uub)),aa_Pro2745879313366468188at_nat(Uua,Uub)) ).

% ATP.lambda_278
tff(fact_4567_ATP_Olambda__279,axiom,
    ! [Uu: fun_o_Product_unit,Uua: fun_o_Product_unit,Uub: $o] : aa_o_Product_unit(aa_fun4820891174006887598t_unit(aTP_Lamm_wr(Uu),Uua),(Uub)) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_o_Product_unit(Uu,(Uub))),aa_o_Product_unit(Uua,(Uub))) ).

% ATP.lambda_279
tff(fact_4568_ATP_Olambda__280,axiom,
    ! [Uu: fun_o_int,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_wp(Uu),Uua),(Uub)) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_o_int(Uu,(Uub))),aa_o_int(Uua,(Uub))) ).

% ATP.lambda_280
tff(fact_4569_ATP_Olambda__281,axiom,
    ! [Uu: fun_se953073389798102371at_nat,Uua: fun_se953073389798102371at_nat,Uub: set_Pr1261947904930325089at_nat] : aa_set6242432332095302236at_nat(aa_fun3786497492886271492at_nat(aTP_Lamm_wc(Uu),Uua),Uub) = sup_su3642409539654194069at_nat(aa_set6242432332095302236at_nat(Uu,Uub),aa_set6242432332095302236at_nat(Uua,Uub)) ).

% ATP.lambda_281
tff(fact_4570_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_Product_unit,Uub: nat] : aa_nat_Product_unit(aa_fun4858661924651670332t_unit(aTP_Lamm_ws(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(sup_sup_Product_unit(aa_nat_Product_unit(Uu,Uub)),aa_nat_Product_unit(Uua,Uub)) ).

% ATP.lambda_282
tff(fact_4571_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_wa(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(Uu,Uub)),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_283
tff(fact_4572_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_uq(Uu),Uua),Uub) = aa_rat_rat(sup_sup_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_284
tff(fact_4573_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ur(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_285
tff(fact_4574_ATP_Olambda__286,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_us(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_286
tff(fact_4575_ATP_Olambda__287,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_ut(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_287
tff(fact_4576_ATP_Olambda__288,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_uu(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_288
tff(fact_4577_ATP_Olambda__289,axiom,
    ! [Uu: fun_Pr2191293291710260995t_unit,Uua: fun_Pr2191293291710260995t_unit,Uub: product_unit] : aa_Pro8294289767544744956t_unit(aa_fun6153996563283759940t_unit(aTP_Lamm_ww(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro8294289767544744956t_unit(Uu,Uub)),aa_Pro8294289767544744956t_unit(Uua,Uub)) ).

% ATP.lambda_289
tff(fact_4578_ATP_Olambda__290,axiom,
    ! [Uu: fun_Pr1628945989756059814et_nat,Uua: fun_Pr1628945989756059814et_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_wz(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro6064029160526250199et_nat(Uu,Uub)),aa_Pro6064029160526250199et_nat(Uua,Uub)) ).

% ATP.lambda_290
tff(fact_4579_ATP_Olambda__291,axiom,
    ! [Uu: fun_Product_unit_int,Uua: fun_Product_unit_int,Uub: product_unit] : aa_Product_unit_int(aa_fun3641207808737740476it_int(aTP_Lamm_wu(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_Product_unit_int(Uu,Uub)),aa_Product_unit_int(Uua,Uub)) ).

% ATP.lambda_291
tff(fact_4580_ATP_Olambda__292,axiom,
    ! [Uu: fun_o_Product_unit,Uua: fun_o_Product_unit,Uub: $o] : aa_o_Product_unit(aa_fun4820891174006887598t_unit(aTP_Lamm_wx(Uu),Uua),(Uub)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_o_Product_unit(Uu,(Uub))),aa_o_Product_unit(Uua,(Uub))) ).

% ATP.lambda_292
tff(fact_4581_ATP_Olambda__293,axiom,
    ! [Uu: fun_o_int,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_wv(Uu),Uua),(Uub)) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_o_int(Uu,(Uub))),aa_o_int(Uua,(Uub))) ).

% ATP.lambda_293
tff(fact_4582_ATP_Olambda__294,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_Product_unit,Uub: nat] : aa_nat_Product_unit(aa_fun4858661924651670332t_unit(aTP_Lamm_wy(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_nat_Product_unit(Uu,Uub)),aa_nat_Product_unit(Uua,Uub)) ).

% ATP.lambda_294
tff(fact_4583_ATP_Olambda__295,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_wf(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(aa_nat_set_nat(Uu,Uub)),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_295
tff(fact_4584_ATP_Olambda__296,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ul(Uu),Uua),Uub) = aa_rat_rat(inf_inf_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_296
tff(fact_4585_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_um(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_297
tff(fact_4586_ATP_Olambda__298,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_un(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_298
tff(fact_4587_ATP_Olambda__299,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_uo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_299
tff(fact_4588_ATP_Olambda__300,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_up(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_300
tff(fact_4589_ATP_Olambda__301,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_rat,Uub: nat] : aa_nat4199610535024615586it_rat(aa_fun7557576995580850244it_rat(aTP_Lamm_zv(Uu),Uua),Uub) = produc4168158269727034510it_rat(aa_nat_Product_unit(Uu,Uub),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_301
tff(fact_4590_ATP_Olambda__302,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_nat,Uub: nat] : aa_nat7707839093418012314it_nat(aa_fun2380121109533724724it_nat(aTP_Lamm_zx(Uu),Uua),Uub) = produc4803288329813530246it_nat(aa_nat_Product_unit(Uu,Uub),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_302
tff(fact_4591_ATP_Olambda__303,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat3778437960758850435at_rat(aa_fun8989050709201395191at_rat(aTP_Lamm_zf(Uu),Uua),Uub) = product_Pair_rat_rat(aa_nat_rat(Uu,Uub),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_303
tff(fact_4592_ATP_Olambda__304,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_nat,Uub: nat] : aa_nat7286666519152247163at_nat(aa_fun3500786184992586743at_nat(aTP_Lamm_zh(Uu),Uua),Uub) = product_Pair_rat_nat(aa_nat_rat(Uu,Uub),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_304
tff(fact_4593_ATP_Olambda__305,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_int,Uub: nat] : aa_nat3108815499643050455at_int(aa_fun8155003486069834487at_int(aTP_Lamm_zj(Uu),Uua),Uub) = product_Pair_rat_int(aa_nat_rat(Uu,Uub),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_305
tff(fact_4594_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_rat,Uub: nat] : aa_nat5094346316594770903nt_rat(aa_fun6353284437548643403nt_rat(aTP_Lamm_zl(Uu),Uua),Uub) = product_Pair_int_rat(aa_nat_int(Uu,Uub),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_306
tff(fact_4595_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_nat,Uub: nat] : aa_nat8602574874988167631nt_nat(aa_fun865019913339834955nt_nat(aTP_Lamm_zn(Uu),Uua),Uub) = aa_nat8602574874988167631nt_nat(product_Pair_int_nat(aa_nat_int(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_307
tff(fact_4596_ATP_Olambda__308,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat4424723855478970923nt_int(aa_fun5519237214417082699nt_int(aTP_Lamm_zr(Uu),Uua),Uub) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_308
tff(fact_4597_ATP_Olambda__309,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_nat,Uub: int] : aa_int2417044922597525619nt_nat(aa_fun1008949613753415571nt_nat(aTP_Lamm_zp(Uu),Uua),Uub) = aa_nat8602574874988167631nt_nat(product_Pair_int_nat(aa_int_int(Uu,Uub)),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_309
tff(fact_4598_ATP_Olambda__310,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int7462565939943104719nt_int(aa_fun5663166914830663315nt_int(aTP_Lamm_zt(Uu),Uua),Uub) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_310
tff(fact_4599_ATP_Olambda__311,axiom,
    ! [Uu: fun_Pr533032343101338535t_char,Uua: set_char,Uub: product_unit] : aa_Pro9168384818938857560t_char(aa_set6291933337142777762t_char(aTP_Lamm_qk(Uu),Uua),Uub) = minus_minus_set_char(aa_Pro9168384818938857560t_char(Uu,Uub),Uua) ).

% ATP.lambda_311
tff(fact_4600_ATP_Olambda__312,axiom,
    ! [Uu: fun_Pr4472331689133158062et_rat,Uua: set_rat,Uub: product_unit] : aa_Pro2555800602132853471et_rat(aa_set3117609736885039074et_rat(aTP_Lamm_qn(Uu),Uua),Uub) = minus_minus_set_rat(aa_Pro2555800602132853471et_rat(Uu,Uub),Uua) ).

% ATP.lambda_312
tff(fact_4601_ATP_Olambda__313,axiom,
    ! [Uu: fun_Pr1628945989756059814et_nat,Uua: set_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_qt(Uu),Uua),Uub) = aa_set_nat_set_nat(minus_minus_set_nat(aa_Pro6064029160526250199et_nat(Uu,Uub)),Uua) ).

% ATP.lambda_313
tff(fact_4602_ATP_Olambda__314,axiom,
    ! [Uu: fun_Pr8362234881679108866et_int,Uua: set_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_set8624687020482735842et_int(aTP_Lamm_qq(Uu),Uua),Uub) = minus_minus_set_int(aa_Pro1886178141017053491et_int(Uu,Uub),Uua) ).

% ATP.lambda_314
tff(fact_4603_ATP_Olambda__315,axiom,
    ! [Uu: fun_o_set_char,Uua: set_char,Uub: $o] : aa_o_set_char(aa_set4325695674101962541t_char(aTP_Lamm_ql(Uu),Uua),(Uub)) = minus_minus_set_char(aa_o_set_char(Uu,(Uub)),Uua) ).

% ATP.lambda_315
tff(fact_4604_ATP_Olambda__316,axiom,
    ! [Uu: fun_o_set_rat,Uua: set_rat,Uub: $o] : aa_o_set_rat(aa_set6880101219114951661et_rat(aTP_Lamm_qo(Uu),Uua),(Uub)) = minus_minus_set_rat(aa_o_set_rat(Uu,(Uub)),Uua) ).

% ATP.lambda_316
tff(fact_4605_ATP_Olambda__317,axiom,
    ! [Uu: fun_o_set_int,Uua: set_int,Uub: $o] : aa_o_set_int(aa_set8238700067146519789et_int(aTP_Lamm_qr(Uu),Uua),(Uub)) = minus_minus_set_int(aa_o_set_int(Uu,(Uub)),Uua) ).

% ATP.lambda_317
tff(fact_4606_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_set_char,Uua: set_char,Uub: nat] : aa_nat_set_char(aa_set8633863032000686489t_char(aTP_Lamm_qm(Uu),Uua),Uub) = minus_minus_set_char(aa_nat_set_char(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_4607_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_set_rat,Uua: set_rat,Uub: nat] : aa_nat_set_rat(aa_set2783155014080441127et_rat(aTP_Lamm_qp(Uu),Uua),Uub) = minus_minus_set_rat(aa_nat_set_rat(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_4608_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_set_int,Uua: set_int,Uub: nat] : aa_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_qs(Uu),Uua),Uub) = minus_minus_set_int(aa_nat_set_int(Uu,Uub),Uua) ).

% ATP.lambda_320
tff(fact_4609_ATP_Olambda__321,axiom,
    ! [Uu: fun_Pr3013102738158121955t_unit,Uua: set_Product_unit,Uub: product_unit] : aa_Pro234974305641502940t_unit(aa_set7533529402238418324t_unit(aTP_Lamm_vf(Uu),Uua),Uub) = sup_su793286257634532545t_unit(aa_Pro234974305641502940t_unit(Uu,Uub),Uua) ).

% ATP.lambda_321
tff(fact_4610_ATP_Olambda__322,axiom,
    ! [Uu: fun_Pr53831763054929981at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: product_unit] : aa_Pro2353044888477550134at_nat(aa_set3561981400713889608at_nat(aTP_Lamm_ya(Uu),Uua),Uub) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro2353044888477550134at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_322
tff(fact_4611_ATP_Olambda__323,axiom,
    ! [Uu: fun_Pr1686037257031381053at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: product_unit] : aa_Pro2875264725485350454at_nat(aa_set6512053633918173000at_nat(aTP_Lamm_yd(Uu),Uua),Uub) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_Pro2875264725485350454at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_323
tff(fact_4612_ATP_Olambda__324,axiom,
    ! [Uu: fun_Pr3332065415989118679at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_unit] : aa_Pro5062749757247690704at_nat(aa_set9196459572129487356at_nat(aTP_Lamm_vo(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(aa_Pro5062749757247690704at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_324
tff(fact_4613_ATP_Olambda__325,axiom,
    ! [Uu: fun_Pr6739565440344758894_set_o,Uua: set_o,Uub: product_unit] : aa_Pro536174410435844839_set_o(aa_set5869638957405543594_set_o(aTP_Lamm_vg(Uu),Uua),Uub) = sup_sup_set_o(aa_Pro536174410435844839_set_o(Uu,Uub),Uua) ).

% ATP.lambda_325
tff(fact_4614_ATP_Olambda__326,axiom,
    ! [Uu: fun_Pr1628945989756059814et_nat,Uua: set_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_vl(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro6064029160526250199et_nat(Uu,Uub)),Uua) ).

% ATP.lambda_326
tff(fact_4615_ATP_Olambda__327,axiom,
    ! [Uu: fun_Pr6108405906935189705at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: product_prod_nat_nat] : aa_Pro1853492592809407170at_nat(aa_set2180152308823829844at_nat(aTP_Lamm_ye(Uu),Uua),Uub) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_Pro1853492592809407170at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_327
tff(fact_4616_ATP_Olambda__328,axiom,
    ! [Uu: fun_Pr9038118721296872730et_nat,Uua: set_nat,Uub: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aa_set3599674627694894166et_nat(aTP_Lamm_xz(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(aa_Pro9123573313595863371et_nat(Uu,Uub)),Uua) ).

% ATP.lambda_328
tff(fact_4617_ATP_Olambda__329,axiom,
    ! [Uu: fun_o_1833192751255691672t_unit,Uua: set_Product_unit,Uub: $o] : aa_o_s3492205133957624465t_unit(aa_set1706710359383640265t_unit(aTP_Lamm_vh(Uu),Uua),(Uub)) = sup_su793286257634532545t_unit(aa_o_s3492205133957624465t_unit(Uu,(Uub)),Uua) ).

% ATP.lambda_329
tff(fact_4618_ATP_Olambda__330,axiom,
    ! [Uu: fun_o_7500966680246433010at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: $o] : aa_o_s7415647395592896235at_nat(aa_set4424263627741053053at_nat(aTP_Lamm_yb(Uu),Uua),(Uub)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_o_s7415647395592896235at_nat(Uu,(Uub))),Uua) ).

% ATP.lambda_330
tff(fact_4619_ATP_Olambda__331,axiom,
    ! [Uu: fun_o_7855627885593154802at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: $o] : aa_o_s4772616771316612843at_nat(aa_set7164036131237892221at_nat(aTP_Lamm_yf(Uu),Uua),(Uub)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(aa_o_s4772616771316612843at_nat(Uu,(Uub))),Uua) ).

% ATP.lambda_331
tff(fact_4620_ATP_Olambda__332,axiom,
    ! [Uu: fun_o_set_o,Uua: set_o,Uub: $o] : aa_o_set_o(aa_set_o_fun_o_set_o(aTP_Lamm_vi(Uu),Uua),(Uub)) = sup_sup_set_o(aa_o_set_o(Uu,(Uub)),Uua) ).

% ATP.lambda_332
tff(fact_4621_ATP_Olambda__333,axiom,
    ! [Uu: fun_o_set_nat,Uua: set_nat,Uub: $o] : aa_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_vm(Uu),Uua),(Uub)) = aa_set_nat_set_nat(sup_sup_set_nat(aa_o_set_nat(Uu,(Uub))),Uua) ).

% ATP.lambda_333
tff(fact_4622_ATP_Olambda__334,axiom,
    ! [Uu: fun_na896991816813928310t_unit,Uua: set_Product_unit,Uub: nat] : aa_nat2714212862977473631t_unit(aa_set4829896985964406405t_unit(aTP_Lamm_vj(Uu),Uua),Uub) = sup_su793286257634532545t_unit(aa_nat2714212862977473631t_unit(Uu,Uub),Uua) ).

% ATP.lambda_334
tff(fact_4623_ATP_Olambda__335,axiom,
    ! [Uu: fun_na317059118361677008at_nat,Uua: set_Pr8693737435421807431at_nat,Uub: nat] : aa_nat1866645186517922489at_nat(aa_set911474328896584197at_nat(aTP_Lamm_yc(Uu),Uua),Uub) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(aa_nat1866645186517922489at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_335
tff(fact_4624_ATP_Olambda__336,axiom,
    ! [Uu: fun_nat_set_o,Uua: set_o,Uub: nat] : aa_nat_set_o(aa_set557987488673405253_set_o(aTP_Lamm_vk(Uu),Uua),Uub) = sup_sup_set_o(aa_nat_set_o(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_4625_ATP_Olambda__337,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_vn(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(aa_nat_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_337
tff(fact_4626_ATP_Olambda__338,axiom,
    ! [Uu: fun_Pr2191293291710260995t_unit,Uua: product_unit,Uub: product_unit] : aa_Pro8294289767544744956t_unit(aa_Pro1375803928533059540t_unit(aTP_Lamm_xh(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro8294289767544744956t_unit(Uu,Uub)),Uua) ).

% ATP.lambda_338
tff(fact_4627_ATP_Olambda__339,axiom,
    ! [Uu: fun_Pr3332065415989118679at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_unit] : aa_Pro5062749757247690704at_nat(aa_set9196459572129487356at_nat(aTP_Lamm_xn(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro5062749757247690704at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_339
tff(fact_4628_ATP_Olambda__340,axiom,
    ! [Uu: fun_Pr533032343101338535t_char,Uua: set_char,Uub: product_unit] : aa_Pro9168384818938857560t_char(aa_set6291933337142777762t_char(aTP_Lamm_sj(Uu),Uua),Uub) = inf_inf_set_char(aa_Pro9168384818938857560t_char(Uu,Uub),Uua) ).

% ATP.lambda_340
tff(fact_4629_ATP_Olambda__341,axiom,
    ! [Uu: fun_Pr4472331689133158062et_rat,Uua: set_rat,Uub: product_unit] : aa_Pro2555800602132853471et_rat(aa_set3117609736885039074et_rat(aTP_Lamm_sm(Uu),Uua),Uub) = inf_inf_set_rat(aa_Pro2555800602132853471et_rat(Uu,Uub),Uua) ).

% ATP.lambda_341
tff(fact_4630_ATP_Olambda__342,axiom,
    ! [Uu: fun_Pr1628945989756059814et_nat,Uua: set_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_ss(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro6064029160526250199et_nat(Uu,Uub)),Uua) ).

% ATP.lambda_342
tff(fact_4631_ATP_Olambda__343,axiom,
    ! [Uu: fun_Pr8362234881679108866et_int,Uua: set_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_set8624687020482735842et_int(aTP_Lamm_sp(Uu),Uua),Uub) = inf_inf_set_int(aa_Pro1886178141017053491et_int(Uu,Uub),Uua) ).

% ATP.lambda_343
tff(fact_4632_ATP_Olambda__344,axiom,
    ! [Uu: fun_Pr6212429452034653949at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: produc3843707927480180839at_nat] : aa_Pro441628658106787830at_nat(aa_set2412372282518133026at_nat(aTP_Lamm_yn(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro441628658106787830at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_344
tff(fact_4633_ATP_Olambda__345,axiom,
    ! [Uu: fun_Pr6764980218223570944et_nat,Uua: set_nat,Uub: produc3843707927480180839at_nat] : aa_Pro1692590229789857841et_nat(aa_set3215376554459393340et_nat(aTP_Lamm_yp(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro1692590229789857841et_nat(Uu,Uub)),Uua) ).

% ATP.lambda_345
tff(fact_4634_ATP_Olambda__346,axiom,
    ! [Uu: fun_Pr2481279687482350479t_unit,Uua: product_unit,Uub: product_prod_nat_nat] : aa_Pro676242055593665672t_unit(aa_Pro4251916131025800416t_unit(aTP_Lamm_xm(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_Pro676242055593665672t_unit(Uu,Uub)),Uua) ).

% ATP.lambda_346
tff(fact_4635_ATP_Olambda__347,axiom,
    ! [Uu: fun_Pr1469247711798767971at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aa_set6004322782020247304at_nat(aTP_Lamm_yo(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_Pro2745879313366468188at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_347
tff(fact_4636_ATP_Olambda__348,axiom,
    ! [Uu: fun_Pr9038118721296872730et_nat,Uua: set_nat,Uub: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aa_set3599674627694894166et_nat(aTP_Lamm_yq(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(aa_Pro9123573313595863371et_nat(Uu,Uub)),Uua) ).

% ATP.lambda_348
tff(fact_4637_ATP_Olambda__349,axiom,
    ! [Uu: fun_o_Product_unit,Uua: product_unit,Uub: $o] : aa_o_Product_unit(aa_Pro1198011010171541257t_unit(aTP_Lamm_xi(Uu),Uua),(Uub)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_o_Product_unit(Uu,(Uub))),Uua) ).

% ATP.lambda_349
tff(fact_4638_ATP_Olambda__350,axiom,
    ! [Uu: fun_o_4630291819367765772at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: $o] : aa_o_s3322794332960822277at_nat(aa_set7523300704893535665at_nat(aTP_Lamm_xo(Uu),Uua),(Uub)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_o_s3322794332960822277at_nat(Uu,(Uub))),Uua) ).

% ATP.lambda_350
tff(fact_4639_ATP_Olambda__351,axiom,
    ! [Uu: fun_o_set_char,Uua: set_char,Uub: $o] : aa_o_set_char(aa_set4325695674101962541t_char(aTP_Lamm_sk(Uu),Uua),(Uub)) = inf_inf_set_char(aa_o_set_char(Uu,(Uub)),Uua) ).

% ATP.lambda_351
tff(fact_4640_ATP_Olambda__352,axiom,
    ! [Uu: fun_o_set_rat,Uua: set_rat,Uub: $o] : aa_o_set_rat(aa_set6880101219114951661et_rat(aTP_Lamm_sn(Uu),Uua),(Uub)) = inf_inf_set_rat(aa_o_set_rat(Uu,(Uub)),Uua) ).

% ATP.lambda_352
tff(fact_4641_ATP_Olambda__353,axiom,
    ! [Uu: fun_o_set_nat,Uua: set_nat,Uub: $o] : aa_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_xk(Uu),Uua),(Uub)) = aa_set_nat_set_nat(inf_inf_set_nat(aa_o_set_nat(Uu,(Uub))),Uua) ).

% ATP.lambda_353
tff(fact_4642_ATP_Olambda__354,axiom,
    ! [Uu: fun_o_set_int,Uua: set_int,Uub: $o] : aa_o_set_int(aa_set8238700067146519789et_int(aTP_Lamm_sq(Uu),Uua),(Uub)) = inf_inf_set_int(aa_o_set_int(Uu,(Uub)),Uua) ).

% ATP.lambda_354
tff(fact_4643_ATP_Olambda__355,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: product_unit,Uub: nat] : aa_nat_Product_unit(aa_Pro6532773420561294981t_unit(aTP_Lamm_xj(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(aa_nat_Product_unit(Uu,Uub)),Uua) ).

% ATP.lambda_355
tff(fact_4644_ATP_Olambda__356,axiom,
    ! [Uu: fun_na7044673412300895210at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: nat] : aa_nat7062706319057040595at_nat(aa_set340465950006982789at_nat(aTP_Lamm_xp(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(aa_nat7062706319057040595at_nat(Uu,Uub)),Uua) ).

% ATP.lambda_356
tff(fact_4645_ATP_Olambda__357,axiom,
    ! [Uu: fun_nat_set_char,Uua: set_char,Uub: nat] : aa_nat_set_char(aa_set8633863032000686489t_char(aTP_Lamm_sl(Uu),Uua),Uub) = inf_inf_set_char(aa_nat_set_char(Uu,Uub),Uua) ).

% ATP.lambda_357
tff(fact_4646_ATP_Olambda__358,axiom,
    ! [Uu: fun_nat_set_rat,Uua: set_rat,Uub: nat] : aa_nat_set_rat(aa_set2783155014080441127et_rat(aTP_Lamm_so(Uu),Uua),Uub) = inf_inf_set_rat(aa_nat_set_rat(Uu,Uub),Uua) ).

% ATP.lambda_358
tff(fact_4647_ATP_Olambda__359,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_xl(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(aa_nat_set_nat(Uu,Uub)),Uua) ).

% ATP.lambda_359
tff(fact_4648_ATP_Olambda__360,axiom,
    ! [Uu: fun_nat_set_int,Uua: set_int,Uub: nat] : aa_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_sr(Uu),Uua),Uub) = inf_inf_set_int(aa_nat_set_int(Uu,Uub),Uua) ).

% ATP.lambda_360
tff(fact_4649_ATP_Olambda__361,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_abp(Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member8206827879206165904at_nat(aa_Pro314188276017538018at_nat(produc6161850002892822231at_nat(Uua),Uub),Uu) ) ) ).

% ATP.lambda_361
tff(fact_4650_ATP_Olambda__362,axiom,
    ! [Uu: set_Pr2560585780119916871nt_int,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_abq(Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member8566619992076573584nt_int(produc3646306378393792727nt_int(Uua,Uub),Uu) ) ) ).

% ATP.lambda_362
tff(fact_4651_ATP_Olambda__363,axiom,
    ! [Uu: set_Pr7459493094073627847at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aTP_Lamm_abr(Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uua),Uub),Uu) ) ) ).

% ATP.lambda_363
tff(fact_4652_ATP_Olambda__364,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_abs(Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uua),Uub)),Uu) ) ) ).

% ATP.lambda_364
tff(fact_4653_ATP_Olambda__365,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abt(Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)),Uu) ) ) ).

% ATP.lambda_365
tff(fact_4654_ATP_Olambda__366,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_abu(Uu),Uua),Uub)
    <=> ( ( Uub != Uua )
        & aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)),Uu) ) ) ).

% ATP.lambda_366
tff(fact_4655_ATP_Olambda__367,axiom,
    ! [Uu: set_char,Uua: fun_Pr533032343101338535t_char,Uub: product_unit] : aa_Pro9168384818938857560t_char(aa_fun3708311092942545532t_char(aTP_Lamm_px(Uu),Uua),Uub) = minus_minus_set_char(Uu,aa_Pro9168384818938857560t_char(Uua,Uub)) ).

% ATP.lambda_367
tff(fact_4656_ATP_Olambda__368,axiom,
    ! [Uu: set_char,Uua: fun_o_set_char,Uub: $o] : aa_o_set_char(aa_fun8606171068324639164t_char(aTP_Lamm_py(Uu),Uua),(Uub)) = minus_minus_set_char(Uu,aa_o_set_char(Uua,(Uub))) ).

% ATP.lambda_368
tff(fact_4657_ATP_Olambda__369,axiom,
    ! [Uu: set_char,Uua: fun_nat_set_char,Uub: nat] : aa_nat_set_char(aa_fun6435767835650977254t_char(aTP_Lamm_pz(Uu),Uua),Uub) = minus_minus_set_char(Uu,aa_nat_set_char(Uua,Uub)) ).

% ATP.lambda_369
tff(fact_4658_ATP_Olambda__370,axiom,
    ! [Uu: set_rat,Uua: fun_Pr4472331689133158062et_rat,Uub: product_unit] : aa_Pro2555800602132853471et_rat(aa_fun1353959048015916348et_rat(aTP_Lamm_qa(Uu),Uua),Uub) = minus_minus_set_rat(Uu,aa_Pro2555800602132853471et_rat(Uua,Uub)) ).

% ATP.lambda_370
tff(fact_4659_ATP_Olambda__371,axiom,
    ! [Uu: set_rat,Uua: fun_o_set_rat,Uub: $o] : aa_o_set_rat(aa_fun5436817229860788988et_rat(aTP_Lamm_qb(Uu),Uua),(Uub)) = minus_minus_set_rat(Uu,aa_o_set_rat(Uua,(Uub))) ).

% ATP.lambda_371
tff(fact_4660_ATP_Olambda__372,axiom,
    ! [Uu: set_rat,Uua: fun_nat_set_rat,Uub: nat] : aa_nat_set_rat(aa_fun57241613143730932et_rat(aTP_Lamm_qc(Uu),Uua),Uub) = minus_minus_set_rat(Uu,aa_nat_set_rat(Uua,Uub)) ).

% ATP.lambda_372
tff(fact_4661_ATP_Olambda__373,axiom,
    ! [Uu: set_nat,Uua: fun_Pr1628945989756059814et_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_qg(Uu),Uua),Uub) = aa_set_nat_set_nat(minus_minus_set_nat(Uu),aa_Pro6064029160526250199et_nat(Uua,Uub)) ).

% ATP.lambda_373
tff(fact_4662_ATP_Olambda__374,axiom,
    ! [Uu: set_nat,Uua: fun_Pr6764980218223570944et_nat,Uub: produc3843707927480180839at_nat] : aa_Pro1692590229789857841et_nat(aa_fun2592980488468313788et_nat(aTP_Lamm_rh(Uu),Uua),Uub) = aa_set_nat_set_nat(minus_minus_set_nat(Uu),aa_Pro1692590229789857841et_nat(Uua,Uub)) ).

% ATP.lambda_374
tff(fact_4663_ATP_Olambda__375,axiom,
    ! [Uu: set_nat,Uua: fun_Pr9038118721296872730et_nat,Uub: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aa_fun994922058233850812et_nat(aTP_Lamm_ri(Uu),Uua),Uub) = aa_set_nat_set_nat(minus_minus_set_nat(Uu),aa_Pro9123573313595863371et_nat(Uua,Uub)) ).

% ATP.lambda_375
tff(fact_4664_ATP_Olambda__376,axiom,
    ! [Uu: set_nat,Uua: fun_o_set_nat,Uub: $o] : aa_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_rj(Uu),Uua),(Uub)) = aa_set_nat_set_nat(minus_minus_set_nat(Uu),aa_o_set_nat(Uua,(Uub))) ).

% ATP.lambda_376
tff(fact_4665_ATP_Olambda__377,axiom,
    ! [Uu: set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_rk(Uu),Uua),Uub) = aa_set_nat_set_nat(minus_minus_set_nat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_377
tff(fact_4666_ATP_Olambda__378,axiom,
    ! [Uu: set_int,Uua: fun_Pr8362234881679108866et_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_fun3247719957694404156et_int(aTP_Lamm_qd(Uu),Uua),Uub) = minus_minus_set_int(Uu,aa_Pro1886178141017053491et_int(Uua,Uub)) ).

% ATP.lambda_378
tff(fact_4667_ATP_Olambda__379,axiom,
    ! [Uu: set_int,Uua: fun_o_set_int,Uub: $o] : aa_o_set_int(aa_fun1981618354277381628et_int(aTP_Lamm_qe(Uu),Uua),(Uub)) = minus_minus_set_int(Uu,aa_o_set_int(Uua,(Uub))) ).

% ATP.lambda_379
tff(fact_4668_ATP_Olambda__380,axiom,
    ! [Uu: set_int,Uua: fun_nat_set_int,Uub: nat] : aa_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_qf(Uu),Uua),Uub) = minus_minus_set_int(Uu,aa_nat_set_int(Uua,Uub)) ).

% ATP.lambda_380
tff(fact_4669_ATP_Olambda__381,axiom,
    ! [Uu: set_Product_unit,Uua: fun_Pr3013102738158121955t_unit,Uub: product_unit] : aa_Pro234974305641502940t_unit(aa_fun8117516366632191236t_unit(aTP_Lamm_uv(Uu),Uua),Uub) = sup_su793286257634532545t_unit(Uu,aa_Pro234974305641502940t_unit(Uua,Uub)) ).

% ATP.lambda_381
tff(fact_4670_ATP_Olambda__382,axiom,
    ! [Uu: set_Product_unit,Uua: fun_o_1833192751255691672t_unit,Uub: $o] : aa_o_s3492205133957624465t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ux(Uu),Uua),(Uub)) = sup_su793286257634532545t_unit(Uu,aa_o_s3492205133957624465t_unit(Uua,(Uub))) ).

% ATP.lambda_382
tff(fact_4671_ATP_Olambda__383,axiom,
    ! [Uu: set_Product_unit,Uua: fun_na896991816813928310t_unit,Uub: nat] : aa_nat2714212862977473631t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_uz(Uu),Uua),Uub) = sup_su793286257634532545t_unit(Uu,aa_nat2714212862977473631t_unit(Uua,Uub)) ).

% ATP.lambda_383
tff(fact_4672_ATP_Olambda__384,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: fun_Pr53831763054929981at_nat,Uub: product_unit] : aa_Pro2353044888477550134at_nat(aa_fun3999407453037076664at_nat(aTP_Lamm_yh(Uu),Uua),Uub) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(Uu),aa_Pro2353044888477550134at_nat(Uua,Uub)) ).

% ATP.lambda_384
tff(fact_4673_ATP_Olambda__385,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: fun_Pr6108405906935189705at_nat,Uub: product_prod_nat_nat] : aa_Pro1853492592809407170at_nat(aa_fun393720513254445520at_nat(aTP_Lamm_yl(Uu),Uua),Uub) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(Uu),aa_Pro1853492592809407170at_nat(Uua,Uub)) ).

% ATP.lambda_385
tff(fact_4674_ATP_Olambda__386,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: fun_o_7500966680246433010at_nat,Uub: $o] : aa_o_s7415647395592896235at_nat(aa_fun1001176853729534498at_nat(aTP_Lamm_yi(Uu),Uua),(Uub)) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(Uu),aa_o_s7415647395592896235at_nat(Uua,(Uub))) ).

% ATP.lambda_386
tff(fact_4675_ATP_Olambda__387,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: fun_na317059118361677008at_nat,Uub: nat] : aa_nat1866645186517922489at_nat(aa_fun379109724618789564at_nat(aTP_Lamm_yj(Uu),Uua),Uub) = aa_set5535132180132642428at_nat(sup_su718114333110466843at_nat(Uu),aa_nat1866645186517922489at_nat(Uua,Uub)) ).

% ATP.lambda_387
tff(fact_4676_ATP_Olambda__388,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: fun_Pr1686037257031381053at_nat,Uub: product_unit] : aa_Pro2875264725485350454at_nat(aa_fun3072769712689967288at_nat(aTP_Lamm_yk(Uu),Uua),Uub) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(Uu),aa_Pro2875264725485350454at_nat(Uua,Uub)) ).

% ATP.lambda_388
tff(fact_4677_ATP_Olambda__389,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: fun_o_7855627885593154802at_nat,Uub: $o] : aa_o_s4772616771316612843at_nat(aa_fun4159517961099164194at_nat(aTP_Lamm_ym(Uu),Uua),(Uub)) = aa_set4883462369968598652at_nat(sup_su5525570899277871387at_nat(Uu),aa_o_s4772616771316612843at_nat(Uua,(Uub))) ).

% ATP.lambda_389
tff(fact_4678_ATP_Olambda__390,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr3332065415989118679at_nat,Uub: product_unit] : aa_Pro5062749757247690704at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_ve(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(sup_su6327502436637775413at_nat(Uu),aa_Pro5062749757247690704at_nat(Uua,Uub)) ).

% ATP.lambda_390
tff(fact_4679_ATP_Olambda__391,axiom,
    ! [Uu: set_o,Uua: fun_Pr6739565440344758894_set_o,Uub: product_unit] : aa_Pro536174410435844839_set_o(aa_fun5942798580971356698_set_o(aTP_Lamm_uw(Uu),Uua),Uub) = sup_sup_set_o(Uu,aa_Pro536174410435844839_set_o(Uua,Uub)) ).

% ATP.lambda_391
tff(fact_4680_ATP_Olambda__392,axiom,
    ! [Uu: set_o,Uua: fun_o_set_o,Uub: $o] : aa_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_uy(Uu),Uua),(Uub)) = sup_sup_set_o(Uu,aa_o_set_o(Uua,(Uub))) ).

% ATP.lambda_392
tff(fact_4681_ATP_Olambda__393,axiom,
    ! [Uu: set_o,Uua: fun_nat_set_o,Uub: nat] : aa_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_va(Uu),Uua),Uub) = sup_sup_set_o(Uu,aa_nat_set_o(Uua,Uub)) ).

% ATP.lambda_393
tff(fact_4682_ATP_Olambda__394,axiom,
    ! [Uu: set_nat,Uua: fun_Pr1628945989756059814et_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_vb(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(Uu),aa_Pro6064029160526250199et_nat(Uua,Uub)) ).

% ATP.lambda_394
tff(fact_4683_ATP_Olambda__395,axiom,
    ! [Uu: set_nat,Uua: fun_Pr9038118721296872730et_nat,Uub: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aa_fun994922058233850812et_nat(aTP_Lamm_yg(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(Uu),aa_Pro9123573313595863371et_nat(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_4684_ATP_Olambda__396,axiom,
    ! [Uu: set_nat,Uua: fun_o_set_nat,Uub: $o] : aa_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_vc(Uu),Uua),(Uub)) = aa_set_nat_set_nat(sup_sup_set_nat(Uu),aa_o_set_nat(Uua,(Uub))) ).

% ATP.lambda_396
tff(fact_4685_ATP_Olambda__397,axiom,
    ! [Uu: set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_vd(Uu),Uua),Uub) = aa_set_nat_set_nat(sup_sup_set_nat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_4686_ATP_Olambda__398,axiom,
    ! [Uu: product_unit,Uua: fun_Pr2191293291710260995t_unit,Uub: product_unit] : aa_Pro8294289767544744956t_unit(aa_fun6153996563283759940t_unit(aTP_Lamm_xq(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uu),aa_Pro8294289767544744956t_unit(Uua,Uub)) ).

% ATP.lambda_398
tff(fact_4687_ATP_Olambda__399,axiom,
    ! [Uu: product_unit,Uua: fun_Pr2481279687482350479t_unit,Uub: product_prod_nat_nat] : aa_Pro676242055593665672t_unit(aa_fun7695145498385251164t_unit(aTP_Lamm_xv(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uu),aa_Pro676242055593665672t_unit(Uua,Uub)) ).

% ATP.lambda_399
tff(fact_4688_ATP_Olambda__400,axiom,
    ! [Uu: product_unit,Uua: fun_o_Product_unit,Uub: $o] : aa_o_Product_unit(aa_fun4820891174006887598t_unit(aTP_Lamm_xr(Uu),Uua),(Uub)) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uu),aa_o_Product_unit(Uua,(Uub))) ).

% ATP.lambda_400
tff(fact_4689_ATP_Olambda__401,axiom,
    ! [Uu: product_unit,Uua: fun_nat_Product_unit,Uub: nat] : aa_nat_Product_unit(aa_fun4858661924651670332t_unit(aTP_Lamm_xs(Uu),Uua),Uub) = aa_Pro8294289767544744956t_unit(inf_inf_Product_unit(Uu),aa_nat_Product_unit(Uua,Uub)) ).

% ATP.lambda_401
tff(fact_4690_ATP_Olambda__402,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr3332065415989118679at_nat,Uub: product_unit] : aa_Pro5062749757247690704at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_xw(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uu),aa_Pro5062749757247690704at_nat(Uua,Uub)) ).

% ATP.lambda_402
tff(fact_4691_ATP_Olambda__403,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr6212429452034653949at_nat,Uub: produc3843707927480180839at_nat] : aa_Pro441628658106787830at_nat(aa_fun3339055349710477880at_nat(aTP_Lamm_yr(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uu),aa_Pro441628658106787830at_nat(Uua,Uub)) ).

% ATP.lambda_403
tff(fact_4692_ATP_Olambda__404,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr1469247711798767971at_nat,Uub: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aa_fun5344049784256257540at_nat(aTP_Lamm_ys(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uu),aa_Pro2745879313366468188at_nat(Uua,Uub)) ).

% ATP.lambda_404
tff(fact_4693_ATP_Olambda__405,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_o_4630291819367765772at_nat,Uub: $o] : aa_o_s3322794332960822277at_nat(aa_fun213375806635181270at_nat(aTP_Lamm_xx(Uu),Uua),(Uub)) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uu),aa_o_s3322794332960822277at_nat(Uua,(Uub))) ).

% ATP.lambda_405
tff(fact_4694_ATP_Olambda__406,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_na7044673412300895210at_nat,Uub: nat] : aa_nat7062706319057040595at_nat(aa_fun5076269500744484796at_nat(aTP_Lamm_xy(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uu),aa_nat7062706319057040595at_nat(Uua,Uub)) ).

% ATP.lambda_406
tff(fact_4695_ATP_Olambda__407,axiom,
    ! [Uu: set_char,Uua: fun_Pr533032343101338535t_char,Uub: product_unit] : aa_Pro9168384818938857560t_char(aa_fun3708311092942545532t_char(aTP_Lamm_rz(Uu),Uua),Uub) = inf_inf_set_char(Uu,aa_Pro9168384818938857560t_char(Uua,Uub)) ).

% ATP.lambda_407
tff(fact_4696_ATP_Olambda__408,axiom,
    ! [Uu: set_char,Uua: fun_o_set_char,Uub: $o] : aa_o_set_char(aa_fun8606171068324639164t_char(aTP_Lamm_sa(Uu),Uua),(Uub)) = inf_inf_set_char(Uu,aa_o_set_char(Uua,(Uub))) ).

% ATP.lambda_408
tff(fact_4697_ATP_Olambda__409,axiom,
    ! [Uu: set_char,Uua: fun_nat_set_char,Uub: nat] : aa_nat_set_char(aa_fun6435767835650977254t_char(aTP_Lamm_sb(Uu),Uua),Uub) = inf_inf_set_char(Uu,aa_nat_set_char(Uua,Uub)) ).

% ATP.lambda_409
tff(fact_4698_ATP_Olambda__410,axiom,
    ! [Uu: set_rat,Uua: fun_Pr4472331689133158062et_rat,Uub: product_unit] : aa_Pro2555800602132853471et_rat(aa_fun1353959048015916348et_rat(aTP_Lamm_sc(Uu),Uua),Uub) = inf_inf_set_rat(Uu,aa_Pro2555800602132853471et_rat(Uua,Uub)) ).

% ATP.lambda_410
tff(fact_4699_ATP_Olambda__411,axiom,
    ! [Uu: set_rat,Uua: fun_o_set_rat,Uub: $o] : aa_o_set_rat(aa_fun5436817229860788988et_rat(aTP_Lamm_sd(Uu),Uua),(Uub)) = inf_inf_set_rat(Uu,aa_o_set_rat(Uua,(Uub))) ).

% ATP.lambda_411
tff(fact_4700_ATP_Olambda__412,axiom,
    ! [Uu: set_rat,Uua: fun_nat_set_rat,Uub: nat] : aa_nat_set_rat(aa_fun57241613143730932et_rat(aTP_Lamm_se(Uu),Uua),Uub) = inf_inf_set_rat(Uu,aa_nat_set_rat(Uua,Uub)) ).

% ATP.lambda_412
tff(fact_4701_ATP_Olambda__413,axiom,
    ! [Uu: set_nat,Uua: fun_Pr1628945989756059814et_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_si(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(Uu),aa_Pro6064029160526250199et_nat(Uua,Uub)) ).

% ATP.lambda_413
tff(fact_4702_ATP_Olambda__414,axiom,
    ! [Uu: set_nat,Uua: fun_Pr6764980218223570944et_nat,Uub: produc3843707927480180839at_nat] : aa_Pro1692590229789857841et_nat(aa_fun2592980488468313788et_nat(aTP_Lamm_yt(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(Uu),aa_Pro1692590229789857841et_nat(Uua,Uub)) ).

% ATP.lambda_414
tff(fact_4703_ATP_Olambda__415,axiom,
    ! [Uu: set_nat,Uua: fun_Pr9038118721296872730et_nat,Uub: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aa_fun994922058233850812et_nat(aTP_Lamm_yu(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(Uu),aa_Pro9123573313595863371et_nat(Uua,Uub)) ).

% ATP.lambda_415
tff(fact_4704_ATP_Olambda__416,axiom,
    ! [Uu: set_nat,Uua: fun_o_set_nat,Uub: $o] : aa_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_xt(Uu),Uua),(Uub)) = aa_set_nat_set_nat(inf_inf_set_nat(Uu),aa_o_set_nat(Uua,(Uub))) ).

% ATP.lambda_416
tff(fact_4705_ATP_Olambda__417,axiom,
    ! [Uu: set_nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_xu(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_417
tff(fact_4706_ATP_Olambda__418,axiom,
    ! [Uu: set_int,Uua: fun_Pr8362234881679108866et_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_fun3247719957694404156et_int(aTP_Lamm_sf(Uu),Uua),Uub) = inf_inf_set_int(Uu,aa_Pro1886178141017053491et_int(Uua,Uub)) ).

% ATP.lambda_418
tff(fact_4707_ATP_Olambda__419,axiom,
    ! [Uu: set_int,Uua: fun_o_set_int,Uub: $o] : aa_o_set_int(aa_fun1981618354277381628et_int(aTP_Lamm_sg(Uu),Uua),(Uub)) = inf_inf_set_int(Uu,aa_o_set_int(Uua,(Uub))) ).

% ATP.lambda_419
tff(fact_4708_ATP_Olambda__420,axiom,
    ! [Uu: set_int,Uua: fun_nat_set_int,Uub: nat] : aa_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_sh(Uu),Uua),Uub) = inf_inf_set_int(Uu,aa_nat_set_int(Uua,Uub)) ).

% ATP.lambda_420
tff(fact_4709_ATP_Olambda__421,axiom,
    ! [Uu: product_unit,Uua: fun_Pr3013102738158121955t_unit,Uub: product_unit] : aa_Pro234974305641502940t_unit(aa_fun8117516366632191236t_unit(aTP_Lamm_ll(Uu),Uua),Uub) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),aa_Pro234974305641502940t_unit(Uua,Uub)) ).

% ATP.lambda_421
tff(fact_4710_ATP_Olambda__422,axiom,
    ! [Uu: product_unit,Uua: fun_Pr2387170064179968111t_unit,Uub: product_prod_nat_nat] : aa_Pro1567413966374740328t_unit(aa_fun7169552063946493212t_unit(aTP_Lamm_md(Uu),Uua),Uub) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),aa_Pro1567413966374740328t_unit(Uua,Uub)) ).

% ATP.lambda_422
tff(fact_4711_ATP_Olambda__423,axiom,
    ! [Uu: product_unit,Uua: fun_o_1833192751255691672t_unit,Uub: $o] : aa_o_s3492205133957624465t_unit(aa_fun2742321542510090862t_unit(aTP_Lamm_ln(Uu),Uua),(Uub)) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),aa_o_s3492205133957624465t_unit(Uua,(Uub))) ).

% ATP.lambda_423
tff(fact_4712_ATP_Olambda__424,axiom,
    ! [Uu: product_unit,Uua: fun_na896991816813928310t_unit,Uub: nat] : aa_nat2714212862977473631t_unit(aa_fun4128477375200405308t_unit(aTP_Lamm_lp(Uu),Uua),Uub) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),aa_nat2714212862977473631t_unit(Uua,Uub)) ).

% ATP.lambda_424
tff(fact_4713_ATP_Olambda__425,axiom,
    ! [Uu: product_unit,Uua: fun_in4731230917697954842t_unit,Uub: int] : aa_int6548451963861500163t_unit(aa_fun7057315224923419708t_unit(aTP_Lamm_ma(Uu),Uua),Uub) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),aa_int6548451963861500163t_unit(Uua,Uub)) ).

% ATP.lambda_425
tff(fact_4714_ATP_Olambda__426,axiom,
    ! [Uu: product_prod_nat_nat,Uua: fun_Pr3332065415989118679at_nat,Uub: product_unit] : aa_Pro5062749757247690704at_nat(aa_fun2878973870771325420at_nat(aTP_Lamm_lu(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(insert8211810215607154385at_nat(Uu),aa_Pro5062749757247690704at_nat(Uua,Uub)) ).

% ATP.lambda_426
tff(fact_4715_ATP_Olambda__427,axiom,
    ! [Uu: $o,Uua: fun_Pr6739565440344758894_set_o,Uub: product_unit] : aa_Pro536174410435844839_set_o(aa_fun5942798580971356698_set_o(aTP_Lamm_lm((Uu)),Uua),Uub) = aa_set_o_set_o(insert_o((Uu)),aa_Pro536174410435844839_set_o(Uua,Uub)) ).

% ATP.lambda_427
tff(fact_4716_ATP_Olambda__428,axiom,
    ! [Uu: $o,Uua: fun_o_set_o,Uub: $o] : aa_o_set_o(aa_fun8872376298180992644_set_o(aTP_Lamm_lo((Uu)),Uua),(Uub)) = aa_set_o_set_o(insert_o((Uu)),aa_o_set_o(Uua,(Uub))) ).

% ATP.lambda_428
tff(fact_4717_ATP_Olambda__429,axiom,
    ! [Uu: $o,Uua: fun_nat_set_o,Uub: nat] : aa_nat_set_o(aa_fun1660089829521737980_set_o(aTP_Lamm_lq((Uu)),Uua),Uub) = aa_set_o_set_o(insert_o((Uu)),aa_nat_set_o(Uua,Uub)) ).

% ATP.lambda_429
tff(fact_4718_ATP_Olambda__430,axiom,
    ! [Uu: $o,Uua: fun_int_set_o,Uub: int] : aa_int_set_o(aa_fun9190404755809275900_set_o(aTP_Lamm_mb((Uu)),Uua),Uub) = aa_set_o_set_o(insert_o((Uu)),aa_int_set_o(Uua,Uub)) ).

% ATP.lambda_430
tff(fact_4719_ATP_Olambda__431,axiom,
    ! [Uu: nat,Uua: fun_Pr1628945989756059814et_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_fun754599740888130364et_nat(aTP_Lamm_lr(Uu),Uua),Uub) = aa_set_nat_set_nat(insert_nat(Uu),aa_Pro6064029160526250199et_nat(Uua,Uub)) ).

% ATP.lambda_431
tff(fact_4720_ATP_Olambda__432,axiom,
    ! [Uu: nat,Uua: fun_o_set_nat,Uub: $o] : aa_o_set_nat(aa_fun4234124357558533372et_nat(aTP_Lamm_ls(Uu),Uua),(Uub)) = aa_set_nat_set_nat(insert_nat(Uu),aa_o_set_nat(Uua,(Uub))) ).

% ATP.lambda_432
tff(fact_4721_ATP_Olambda__433,axiom,
    ! [Uu: nat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_nat(aa_fun850356524293783268et_nat(aTP_Lamm_lt(Uu),Uua),Uub) = aa_set_nat_set_nat(insert_nat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_433
tff(fact_4722_ATP_Olambda__434,axiom,
    ! [Uu: nat,Uua: fun_int_set_nat,Uub: int] : aa_int_set_nat(aa_fun5565777959088800668et_nat(aTP_Lamm_mc(Uu),Uua),Uub) = aa_set_nat_set_nat(insert_nat(Uu),aa_int_set_nat(Uua,Uub)) ).

% ATP.lambda_434
tff(fact_4723_ATP_Olambda__435,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_o_set_nat,Uub: $o] : aa_o_set_rat(aa_fun7077510056935631620et_rat(aTP_Lamm_zb(Uu),Uua),(Uub)) = aa_set_nat_set_rat(image_nat_rat(Uu),aa_o_set_nat(Uua,(Uub))) ).

% ATP.lambda_435
tff(fact_4724_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_rat(aa_fun3693742223670881516et_rat(aTP_Lamm_zd(Uu),Uua),Uub) = aa_set_nat_set_rat(image_nat_rat(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_436
tff(fact_4725_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_int,Uua: fun_o_set_nat,Uub: $o] : aa_o_set_int(aa_fun1744041212626806616et_int(aTP_Lamm_zc(Uu),Uua),(Uub)) = aa_set_nat_set_int(image_nat_int(Uu),aa_o_set_nat(Uua,(Uub))) ).

% ATP.lambda_437
tff(fact_4726_ATP_Olambda__438,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_set_nat,Uub: nat] : aa_nat_set_int(aa_fun7583645416216832320et_int(aTP_Lamm_ze(Uu),Uua),Uub) = aa_set_nat_set_int(image_nat_int(Uu),aa_nat_set_nat(Uua,Uub)) ).

% ATP.lambda_438
tff(fact_4727_ATP_Olambda__439,axiom,
    ! [Uu: fun_int_nat,Uua: fun_o_set_int,Uub: $o] : aa_o_set_nat(aa_fun4471701499209108384et_nat(aTP_Lamm_yy(Uu),Uua),(Uub)) = aa_set_int_set_nat(image_int_nat(Uu),aa_o_set_int(Uua,(Uub))) ).

% ATP.lambda_439
tff(fact_4728_ATP_Olambda__440,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_set_int,Uub: nat] : aa_nat_set_nat(aa_fun77027002720559424et_nat(aTP_Lamm_yz(Uu),Uua),Uub) = aa_set_int_set_nat(image_int_nat(Uu),aa_nat_set_int(Uua,Uub)) ).

% ATP.lambda_440
tff(fact_4729_ATP_Olambda__441,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_set_int,Uub: int] : aa_int_set_nat(aa_fun4792448437515576824et_nat(aTP_Lamm_za(Uu),Uua),Uub) = aa_set_int_set_nat(image_int_nat(Uu),aa_int_set_int(Uua,Uub)) ).

% ATP.lambda_441
tff(fact_4730_ATP_Olambda__442,axiom,
    ! [Uu: fun_int_int,Uua: fun_o_set_int,Uub: $o] : aa_o_set_int(aa_fun1981618354277381628et_int(aTP_Lamm_yv(Uu),Uua),(Uub)) = aa_set_int_set_int(image_int_int(Uu),aa_o_set_int(Uua,(Uub))) ).

% ATP.lambda_442
tff(fact_4731_ATP_Olambda__443,axiom,
    ! [Uu: fun_int_int,Uua: fun_nat_set_int,Uub: nat] : aa_nat_set_int(aa_fun6810315894643608476et_int(aTP_Lamm_yw(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uu),aa_nat_set_int(Uua,Uub)) ).

% ATP.lambda_443
tff(fact_4732_ATP_Olambda__444,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_set_int,Uub: int] : aa_int_set_int(aa_fun2302365292583850068et_int(aTP_Lamm_yx(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uu),aa_int_set_int(Uua,Uub)) ).

% ATP.lambda_444
tff(fact_4733_ATP_Olambda__445,axiom,
    ! [Uu: fun_Pr3013102738158121955t_unit,Uua: fun_Product_unit_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_fun2049105527398774130et_int(aTP_Lamm_tb(Uu),Uua),Uub) = image_873079544045703924it_int(Uua,aa_Pro234974305641502940t_unit(Uu,Uub)) ).

% ATP.lambda_445
tff(fact_4734_ATP_Olambda__446,axiom,
    ! [Uu: fun_Pr6739565440344758894_set_o,Uua: fun_o_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_fun7391356117566996647et_int(aTP_Lamm_td(Uu),Uua),Uub) = image_o_int(Uua,aa_Pro536174410435844839_set_o(Uu,Uub)) ).

% ATP.lambda_446
tff(fact_4735_ATP_Olambda__447,axiom,
    ! [Uu: fun_Pr8362234881679108866et_int,Uua: fun_int_int,Uub: product_unit] : aa_Pro1886178141017053491et_int(aa_fun339368410111030063et_int(aTP_Lamm_sz(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uua),aa_Pro1886178141017053491et_int(Uu,Uub)) ).

% ATP.lambda_447
tff(fact_4736_ATP_Olambda__448,axiom,
    ! [Uu: fun_o_1833192751255691672t_unit,Uua: fun_Product_unit_int,Uub: $o] : aa_o_set_int(aa_fun5570681940130100093et_int(aTP_Lamm_th(Uu),Uua),(Uub)) = image_873079544045703924it_int(Uua,aa_o_s3492205133957624465t_unit(Uu,(Uub))) ).

% ATP.lambda_448
tff(fact_4737_ATP_Olambda__449,axiom,
    ! [Uu: fun_o_set_o,Uua: fun_o_int,Uub: $o] : aa_o_set_int(aa_fun4317264022763090482et_int(aTP_Lamm_tj(Uu),Uua),(Uub)) = image_o_int(Uua,aa_o_set_o(Uu,(Uub))) ).

% ATP.lambda_449
tff(fact_4738_ATP_Olambda__450,axiom,
    ! [Uu: fun_o_set_int,Uua: fun_int_int,Uub: $o] : aa_o_set_int(aa_fun1632018885420323002et_int(aTP_Lamm_tf(Uu),Uua),(Uub)) = aa_set_int_set_int(image_int_int(Uua),aa_o_set_int(Uu,(Uub))) ).

% ATP.lambda_450
tff(fact_4739_ATP_Olambda__451,axiom,
    ! [Uu: fun_nat_set_int,Uua: fun_int_int,Uub: nat] : aa_nat_set_int(aa_fun8970103232618661698et_int(aTP_Lamm_tl(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uua),aa_nat_set_int(Uu,Uub)) ).

% ATP.lambda_451
tff(fact_4740_ATP_Olambda__452,axiom,
    ! [Uu: fun_in4731230917697954842t_unit,Uua: fun_Product_unit_int,Uub: int] : aa_int_set_int(aa_fun7610309373818225051et_int(aTP_Lamm_sv(Uu),Uua),Uub) = image_873079544045703924it_int(Uua,aa_int6548451963861500163t_unit(Uu,Uub)) ).

% ATP.lambda_452
tff(fact_4741_ATP_Olambda__453,axiom,
    ! [Uu: fun_int_set_o,Uua: fun_o_int,Uub: int] : aa_int_set_int(aa_fun4704607026696346150et_int(aTP_Lamm_sx(Uu),Uua),Uub) = image_o_int(Uua,aa_int_set_o(Uu,Uub)) ).

% ATP.lambda_453
tff(fact_4742_ATP_Olambda__454,axiom,
    ! [Uu: fun_int_set_int,Uua: fun_int_int,Uub: int] : aa_int_set_int(aa_fun118384756000730526et_int(aTP_Lamm_st(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uua),aa_int_set_int(Uu,Uub)) ).

% ATP.lambda_454
tff(fact_4743_ATP_Olambda__455,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aba(Uu),Uua),Uub)
    <=> ( Uub = aa_nat_nat(Uu,Uua) ) ) ).

% ATP.lambda_455
tff(fact_4744_ATP_Olambda__456,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_abc(Uu),Uua),Uub)
    <=> ( Uub = aa_int_int(Uu,Uua) ) ) ).

% ATP.lambda_456
tff(fact_4745_ATP_Olambda__457,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aa_set6004322782020247304at_nat(aTP_Lamm_wd(Uu),Uua),Uub) = aa_set8868280134176360700at_nat(inf_in2572325071724192079at_nat(Uu),Uua) ).

% ATP.lambda_457
tff(fact_4746_ATP_Olambda__458,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_we(Uu),Uua),Uub) = aa_set_nat_set_nat(inf_inf_set_nat(Uu),Uua) ).

% ATP.lambda_458
tff(fact_4747_ATP_Olambda__459,axiom,
    ! [Uu: product_unit,Uua: set_Product_unit,Uub: product_unit] : aa_Pro234974305641502940t_unit(aa_set7533529402238418324t_unit(aTP_Lamm_rs(Uu),Uua),Uub) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),Uua) ).

% ATP.lambda_459
tff(fact_4748_ATP_Olambda__460,axiom,
    ! [Uu: product_unit,Uua: set_Product_unit,Uub: $o] : aa_o_s3492205133957624465t_unit(aa_set1706710359383640265t_unit(aTP_Lamm_rv(Uu),Uua),(Uub)) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),Uua) ).

% ATP.lambda_460
tff(fact_4749_ATP_Olambda__461,axiom,
    ! [Uu: product_unit,Uua: set_Product_unit,Uub: nat] : aa_nat2714212862977473631t_unit(aa_set4829896985964406405t_unit(aTP_Lamm_rp(Uu),Uua),Uub) = aa_set8324764978855220220t_unit(insert_Product_unit(Uu),Uua) ).

% ATP.lambda_461
tff(fact_4750_ATP_Olambda__462,axiom,
    ! [Uu: $o,Uua: set_o,Uub: product_unit] : aa_Pro536174410435844839_set_o(aa_set5869638957405543594_set_o(aTP_Lamm_rt((Uu)),Uua),Uub) = aa_set_o_set_o(insert_o((Uu)),Uua) ).

% ATP.lambda_462
tff(fact_4751_ATP_Olambda__463,axiom,
    ! [Uu: $o,Uua: set_o,Uub: $o] : aa_o_set_o(aa_set_o_fun_o_set_o(aTP_Lamm_rw((Uu)),Uua),(Uub)) = aa_set_o_set_o(insert_o((Uu)),Uua) ).

% ATP.lambda_463
tff(fact_4752_ATP_Olambda__464,axiom,
    ! [Uu: $o,Uua: set_o,Uub: nat] : aa_nat_set_o(aa_set557987488673405253_set_o(aTP_Lamm_rq((Uu)),Uua),Uub) = aa_set_o_set_o(insert_o((Uu)),Uua) ).

% ATP.lambda_464
tff(fact_4753_ATP_Olambda__465,axiom,
    ! [Uu: num,Uua: set_num,Uub: nat] : aa_nat_set_num(aa_set6533290090869974187et_num(aTP_Lamm_rx(Uu),Uua),Uub) = insert_num(Uu,Uua) ).

% ATP.lambda_465
tff(fact_4754_ATP_Olambda__466,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: product_unit] : aa_Pro6064029160526250199et_nat(aa_set7097633925648574946et_nat(aTP_Lamm_rr(Uu),Uua),Uub) = aa_set_nat_set_nat(insert_nat(Uu),Uua) ).

% ATP.lambda_466
tff(fact_4755_ATP_Olambda__467,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: $o] : aa_o_set_nat(aa_set1696117090455734765et_nat(aTP_Lamm_ru(Uu),Uua),(Uub)) = aa_set_nat_set_nat(insert_nat(Uu),Uua) ).

% ATP.lambda_467
tff(fact_4756_ATP_Olambda__468,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] : aa_int_set_int(aa_set2836339071814736939et_int(aTP_Lamm_ry(Uu),Uua),Uub) = aa_set_int_set_int(insert_int(Uu),Uua) ).

% ATP.lambda_468
tff(fact_4757_ATP_Olambda__469,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat] : aa_nat_set_rat(aa_set3405691356855822623et_rat(aTP_Lamm_gb(Uu),Uua),Uub) = aa_set_nat_set_rat(image_nat_rat(Uu),Uua) ).

% ATP.lambda_469
tff(fact_4758_ATP_Olambda__470,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: int] : aa_int_set_rat(aa_set3777344917092667259et_rat(aTP_Lamm_gh(Uu),Uua),Uub) = aa_set_nat_set_rat(image_nat_rat(Uu),Uua) ).

% ATP.lambda_470
tff(fact_4759_ATP_Olambda__471,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: int] : aa_int_set_nat(aa_set933959217715569011et_nat(aTP_Lamm_gj(Uu),Uua),Uub) = aa_set_nat_set_nat(image_nat_nat(Uu),Uua) ).

% ATP.lambda_471
tff(fact_4760_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: nat] : aa_nat_set_int(aa_set7295594549401773427et_int(aTP_Lamm_gd(Uu),Uua),Uub) = aa_set_nat_set_int(image_nat_int(Uu),Uua) ).

% ATP.lambda_472
tff(fact_4761_ATP_Olambda__473,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: rat] : aa_rat_set_nat(aa_set5726841466071138683et_nat(aTP_Lamm_fs(Uu),Uua),Uub) = aa_set_int_set_nat(image_int_nat(Uu),Uua) ).

% ATP.lambda_473
tff(fact_4762_ATP_Olambda__474,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int] : aa_int_set_nat(aa_set5326422216746463695et_nat(aTP_Lamm_fy(Uu),Uua),Uub) = aa_set_int_set_nat(image_int_nat(Uu),Uua) ).

% ATP.lambda_474
tff(fact_4763_ATP_Olambda__475,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: rat] : aa_rat_set_int(aa_set3236758321139411927et_int(aTP_Lamm_fu(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uu),Uua) ).

% ATP.lambda_475
tff(fact_4764_ATP_Olambda__476,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: nat] : aa_nat_set_int(aa_set2464685511577892303et_int(aTP_Lamm_fw(Uu),Uua),Uub) = aa_set_int_set_int(image_int_int(Uu),Uua) ).

% ATP.lambda_476
tff(fact_4765_ATP_Olambda__477,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_dn(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_477
tff(fact_4766_ATP_Olambda__478,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_do(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_478
tff(fact_4767_ATP_Olambda__479,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dp(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_479
tff(fact_4768_ATP_Olambda__480,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_480
tff(fact_4769_ATP_Olambda__481,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dr(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_481
tff(fact_4770_ATP_Olambda__482,axiom,
    ! [Uu: fun_Product_unit_nat,Uua: fun_Pr2191293291710260995t_unit,Uub: product_unit] : aa_Product_unit_nat(aa_fun7205615774095635567it_nat(aTP_Lamm_abm(Uu),Uua),Uub) = aa_Product_unit_nat(Uu,aa_Pro8294289767544744956t_unit(Uua,Uub)) ).

% ATP.lambda_482
tff(fact_4771_ATP_Olambda__483,axiom,
    ! [Uu: fun_Product_unit_nat,Uua: fun_o_Product_unit,Uub: $o] : aa_o_nat(aa_fun6071669926149765765_o_nat(aTP_Lamm_abn(Uu),Uua),(Uub)) = aa_Product_unit_nat(Uu,aa_o_Product_unit(Uua,(Uub))) ).

% ATP.lambda_483
tff(fact_4772_ATP_Olambda__484,axiom,
    ! [Uu: fun_Product_unit_nat,Uua: fun_nat_Product_unit,Uub: nat] : aa_nat_nat(aa_fun2105170996421215863at_nat(aTP_Lamm_abo(Uu),Uua),Uub) = aa_Product_unit_nat(Uu,aa_nat_Product_unit(Uua,Uub)) ).

% ATP.lambda_484
tff(fact_4773_ATP_Olambda__485,axiom,
    ! [Uu: fun_Product_unit_nat,Uua: fun_int_Product_unit,Uub: int] : aa_int_nat(aa_fun1692700469306394999nt_nat(aTP_Lamm_abl(Uu),Uua),Uub) = aa_Product_unit_nat(Uu,aa_int_Product_unit(Uua,Uub)) ).

% ATP.lambda_485
tff(fact_4774_ATP_Olambda__486,axiom,
    ! [Uu: fun_rat_set_nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_set_nat(aa_fun7287459611079639094et_nat(aTP_Lamm_nc(Uu),Uua),Uub) = aa_rat_set_nat(Uu,aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_486
tff(fact_4775_ATP_Olambda__487,axiom,
    ! [Uu: fun_rat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ax(Uu),Uua),Uub) = aa_rat_rat(Uu,aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_487
tff(fact_4776_ATP_Olambda__488,axiom,
    ! [Uu: fun_rat_nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_nat(aa_fun4273987348596450176at_nat(aTP_Lamm_ay(Uu),Uua),Uub) = aa_rat_nat(Uu,aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_488
tff(fact_4777_ATP_Olambda__489,axiom,
    ! [Uu: fun_rat_int,Uua: fun_nat_rat,Uub: nat] : aa_nat_int(aa_fun96136329087253468at_int(aTP_Lamm_az(Uu),Uua),Uub) = aa_rat_int(Uu,aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_489
tff(fact_4778_ATP_Olambda__490,axiom,
    ! [Uu: fun_num_fun_num_o,Uua: fun_num_num,Uub: num] : aa_num_fun_num_o(aa_fun7420891108566521959_num_o(aTP_Lamm_dk(Uu),Uua),Uub) = aa_num_fun_num_o(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_490
tff(fact_4779_ATP_Olambda__491,axiom,
    ! [Uu: fun_num_fun_num_o,Uua: fun_nat_num,Uub: nat] : aa_nat_fun_num_o(aa_fun7501294707364756987_num_o(aTP_Lamm_dj(Uu),Uua),Uub) = aa_num_fun_num_o(Uu,aa_nat_num(Uua,Uub)) ).

% ATP.lambda_491
tff(fact_4780_ATP_Olambda__492,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_nat,Uub: nat] : aa_nat_Product_unit(aa_fun3939150890428253051t_unit(aTP_Lamm_aau(Uu),Uua),Uub) = aa_nat_Product_unit(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_492
tff(fact_4781_ATP_Olambda__493,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_int_nat,Uub: int] : aa_int_Product_unit(aa_fun8867120635469333699t_unit(aTP_Lamm_aat(Uu),Uua),Uub) = aa_nat_Product_unit(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_493
tff(fact_4782_ATP_Olambda__494,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_set_nat(aa_fun7909995953855020590et_nat(aTP_Lamm_nd(Uu),Uua),Uub) = aa_nat_set_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_494
tff(fact_4783_ATP_Olambda__495,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_int_nat,Uub: int] : aa_int_set_nat(aa_fun7439376938756338406et_nat(aTP_Lamm_nf(Uu),Uua),Uub) = aa_nat_set_nat(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_495
tff(fact_4784_ATP_Olambda__496,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_char_nat,Uub: char] : aa_char_rat(aa_fun7940347660735252098ar_rat(aTP_Lamm_cz(Uu),Uua),Uub) = aa_nat_rat(Uu,aa_char_nat(Uua,Uub)) ).

% ATP.lambda_496
tff(fact_4785_ATP_Olambda__497,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_rat_nat,Uub: rat] : aa_rat_rat(aa_fun6175819412704661392at_rat(aTP_Lamm_de(Uu),Uua),Uub) = aa_nat_rat(Uu,aa_rat_nat(Uua,Uub)) ).

% ATP.lambda_497
tff(fact_4786_ATP_Olambda__498,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_nat,Uub: nat] : aa_nat_rat(aa_fun2283220954820195200at_rat(aTP_Lamm_bb(Uu),Uua),Uub) = aa_nat_rat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_498
tff(fact_4787_ATP_Olambda__499,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_int_nat,Uub: int] : aa_int_rat(aa_fun2998523592351139896nt_rat(aTP_Lamm_bc(Uu),Uua),Uub) = aa_nat_rat(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_499
tff(fact_4788_ATP_Olambda__500,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_char_nat,Uub: char] : aa_char_nat(aa_fun2225204182273873018ar_nat(aTP_Lamm_da(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_char_nat(Uua,Uub)) ).

% ATP.lambda_500
tff(fact_4789_ATP_Olambda__501,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_rat_nat,Uub: rat] : aa_rat_nat(aa_fun460675934243282312at_nat(aTP_Lamm_df(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_rat_nat(Uua,Uub)) ).

% ATP.lambda_501
tff(fact_4790_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_bd(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_502
tff(fact_4791_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_be(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_503
tff(fact_4792_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_int,Uua: fun_char_nat,Uub: char] : aa_char_int(aa_fun7270725199619452118ar_int(aTP_Lamm_db(Uu),Uua),Uub) = aa_nat_int(Uu,aa_char_nat(Uua,Uub)) ).

% ATP.lambda_504
tff(fact_4793_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_int,Uua: fun_rat_nat,Uub: rat] : aa_rat_int(aa_fun5506196951588861412at_int(aTP_Lamm_dg(Uu),Uua),Uub) = aa_nat_int(Uu,aa_rat_nat(Uua,Uub)) ).

% ATP.lambda_505
tff(fact_4794_ATP_Olambda__506,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_bf(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_506
tff(fact_4795_ATP_Olambda__507,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_bg(Uu),Uua),Uub) = aa_nat_int(Uu,aa_int_nat(Uua,Uub)) ).

% ATP.lambda_507
tff(fact_4796_ATP_Olambda__508,axiom,
    ! [Uu: fun_int_set_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_set_nat(aa_fun3079086916031139466et_nat(aTP_Lamm_ne(Uu),Uua),Uub) = aa_int_set_nat(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_508
tff(fact_4797_ATP_Olambda__509,axiom,
    ! [Uu: fun_int_set_nat,Uua: fun_int_int,Uub: int] : aa_int_set_nat(aa_fun2608467900932457282et_nat(aTP_Lamm_ng(Uu),Uua),Uub) = aa_int_set_nat(Uu,aa_int_int(Uua,Uub)) ).

% ATP.lambda_509
tff(fact_4798_ATP_Olambda__510,axiom,
    ! [Uu: fun_int_rat,Uua: fun_nat_int,Uub: nat] : aa_nat_rat(aa_fun6234665583235904476at_rat(aTP_Lamm_ba(Uu),Uua),Uub) = aa_int_rat(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_510
tff(fact_4799_ATP_Olambda__511,axiom,
    ! [Uu: fun_int_nat,Uua: fun_Product_unit_int,Uub: product_unit] : aa_Product_unit_nat(aa_fun7819058828246937184it_nat(aTP_Lamm_aaw(Uu),Uua),Uub) = aa_int_nat(Uu,aa_Product_unit_int(Uua,Uub)) ).

% ATP.lambda_511
tff(fact_4800_ATP_Olambda__512,axiom,
    ! [Uu: fun_int_nat,Uua: fun_char_int,Uub: char] : aa_char_nat(aa_fun3120538043794061526ar_nat(aTP_Lamm_dc(Uu),Uua),Uub) = aa_int_nat(Uu,aa_char_int(Uua,Uub)) ).

% ATP.lambda_512
tff(fact_4801_ATP_Olambda__513,axiom,
    ! [Uu: fun_int_nat,Uua: fun_o_int,Uub: $o] : aa_o_nat(aa_fun8905617046332959520_o_nat(aTP_Lamm_aax(Uu),Uua),(Uub)) = aa_int_nat(Uu,aa_o_int(Uua,(Uub))) ).

% ATP.lambda_513
tff(fact_4802_ATP_Olambda__514,axiom,
    ! [Uu: fun_int_nat,Uua: fun_rat_int,Uub: rat] : aa_rat_nat(aa_fun4412120562658991588at_nat(aTP_Lamm_dh(Uu),Uua),Uub) = aa_int_nat(Uu,aa_rat_int(Uua,Uub)) ).

% ATP.lambda_514
tff(fact_4803_ATP_Olambda__515,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_nat(aa_fun519522104774525396at_nat(aTP_Lamm_aay(Uu),Uua),Uub) = aa_int_nat(Uu,aa_nat_int(Uua,Uub)) ).

% ATP.lambda_515
tff(fact_4804_ATP_Olambda__516,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_int,Uub: int] : aa_int_nat(aa_fun1234824742305470092nt_nat(aTP_Lamm_aav(Uu),Uua),Uub) = aa_int_nat(Uu,aa_int_int(Uua,Uub)) ).

% ATP.lambda_516
tff(fact_4805_ATP_Olambda__517,axiom,
    ! [Uu: fun_int_int,Uua: fun_char_int,Uub: char] : aa_char_int(aa_fun8166059061139640626ar_int(aTP_Lamm_dd(Uu),Uua),Uub) = aa_int_int(Uu,aa_char_int(Uua,Uub)) ).

% ATP.lambda_517
tff(fact_4806_ATP_Olambda__518,axiom,
    ! [Uu: fun_int_int,Uua: fun_rat_int,Uub: rat] : aa_rat_int(aa_fun234269543149794880at_int(aTP_Lamm_di(Uu),Uua),Uub) = aa_int_int(Uu,aa_rat_int(Uua,Uub)) ).

% ATP.lambda_518
tff(fact_4807_ATP_Olambda__519,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun3836547582011492095_nat_o(aTP_Lamm_eo(Uu),Uua),Uub)
    <=> aa_nat_o(Uua,aa_Pro4243173770044397973at_nat(Uu,Uub)) ) ).

% ATP.lambda_519
tff(fact_4808_ATP_Olambda__520,axiom,
    ! [Uu: fun_Pr7647638413817135296at_int,Uua: fun_int_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun3324632943032829859_nat_o(aTP_Lamm_em(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_Pro4240683299535347697at_int(Uu,Uub)) ) ).

% ATP.lambda_520
tff(fact_4809_ATP_Olambda__521,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun1766064008402934507_int_o(aTP_Lamm_en(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_Pro6799258434465214121nt_int(Uu,Uub)) ) ).

% ATP.lambda_521
tff(fact_4810_ATP_Olambda__522,axiom,
    ! [Uu: fun_char_nat,Uua: fun_nat_nat,Uub: char] : aa_char_nat(aa_fun8194089762075812217ar_nat(aTP_Lamm_acf(Uu),Uua),Uub) = aa_nat_nat(Uua,aa_char_nat(Uu,Uub)) ).

% ATP.lambda_522
tff(fact_4811_ATP_Olambda__523,axiom,
    ! [Uu: fun_rat_nat,Uua: fun_nat_nat,Uub: rat] : aa_rat_nat(aa_fun5474894869278726016at_nat(aTP_Lamm_acg(Uu),Uua),Uub) = aa_nat_nat(Uua,aa_rat_nat(Uu,Uub)) ).

% ATP.lambda_523
tff(fact_4812_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_o,Uua: fun_o_o,Uub: nat] :
      ( aa_nat_o(aa_fun_o_o_fun_nat_o(aTP_Lamm_ej(Uu),Uua),Uub)
    <=> aa_o_o(Uua,aa_nat_o(Uu,Uub)) ) ).

% ATP.lambda_524
tff(fact_4813_ATP_Olambda__525,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_rat_rat,Uub: nat] : aa_nat_rat(aa_fun4974911892022385552at_rat(aTP_Lamm_ach(Uu),Uua),Uub) = aa_rat_rat(Uua,aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_525
tff(fact_4814_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_rat_nat,Uub: nat] : aa_nat_nat(aa_fun777230578178148224at_nat(aTP_Lamm_acl(Uu),Uua),Uub) = aa_rat_nat(Uua,aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_526
tff(fact_4815_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_rat_int,Uub: nat] : aa_nat_int(aa_fun550824187084660792at_int(aTP_Lamm_aci(Uu),Uua),Uub) = aa_rat_int(Uua,aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_527
tff(fact_4816_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_eh(Uu),Uua),Uub)
    <=> aa_nat_o(Uua,aa_nat_nat(Uu,Uub)) ) ).

% ATP.lambda_528
tff(fact_4817_ATP_Olambda__529,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_acn(Uu),Uua),Uub) = aa_nat_rat(Uua,aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_529
tff(fact_4818_ATP_Olambda__530,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_aco(Uu),Uua),Uub) = aa_nat_int(Uua,aa_nat_nat(Uu,Uub)) ).

% ATP.lambda_530
tff(fact_4819_ATP_Olambda__531,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_o,Uub: nat] :
      ( aa_nat_o(aa_fun8064560132910005592_nat_o(aTP_Lamm_el(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_nat_int(Uu,Uub)) ) ).

% ATP.lambda_531
tff(fact_4820_ATP_Olambda__532,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_rat,Uub: nat] : aa_nat_rat(aa_fun481707715832943588at_rat(aTP_Lamm_acj(Uu),Uua),Uub) = aa_int_rat(Uua,aa_nat_int(Uu,Uub)) ).

% ATP.lambda_532
tff(fact_4821_ATP_Olambda__533,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_nat,Uub: nat] : aa_nat_nat(aa_fun5507398438843482068at_nat(aTP_Lamm_acm(Uu),Uua),Uub) = aa_int_nat(Uua,aa_nat_int(Uu,Uub)) ).

% ATP.lambda_533
tff(fact_4822_ATP_Olambda__534,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_int,Uub: nat] : aa_nat_int(aa_fun5280992047749994636at_int(aTP_Lamm_ack(Uu),Uua),Uub) = aa_int_int(Uua,aa_nat_int(Uu,Uub)) ).

% ATP.lambda_534
tff(fact_4823_ATP_Olambda__535,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_o,Uub: int] :
      ( aa_int_o(aa_fun4116151057475255200_int_o(aTP_Lamm_ek(Uu),Uua),Uub)
    <=> aa_nat_o(Uua,aa_int_nat(Uu,Uub)) ) ).

% ATP.lambda_535
tff(fact_4824_ATP_Olambda__536,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ei(Uu),Uua),Uub)
    <=> aa_int_o(Uua,aa_int_int(Uu,Uub)) ) ).

% ATP.lambda_536
tff(fact_4825_ATP_Olambda__537,axiom,
    ! [Uu: fun_nat_set_nat,Uua: nat,Uub: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(aTP_Lamm_mp(Uu),Uua),Uub) = aa_nat_set_nat(Uu,Uua) ).

% ATP.lambda_537
tff(fact_4826_ATP_Olambda__538,axiom,
    ! [Uu: fun_Pr3013102738158121955t_unit,Uua: fun_Product_unit_int,Uub: product_unit] : aa_Product_unit_int(aa_fun3641207808737740476it_int(aTP_Lamm_tc(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(Uua,aa_Pro234974305641502940t_unit(Uu,Uub))) ).

% ATP.lambda_538
tff(fact_4827_ATP_Olambda__539,axiom,
    ! [Uu: fun_Pr6739565440344758894_set_o,Uua: fun_o_int,Uub: product_unit] : aa_Product_unit_int(aa_fun3625413194728170993it_int(aTP_Lamm_te(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(Uua,aa_Pro536174410435844839_set_o(Uu,Uub))) ).

% ATP.lambda_539
tff(fact_4828_ATP_Olambda__540,axiom,
    ! [Uu: fun_Pr8362234881679108866et_int,Uua: fun_int_int,Uub: product_unit] : aa_Product_unit_int(aa_fun2487178249965271161it_int(aTP_Lamm_ta(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(Uua),aa_Pro1886178141017053491et_int(Uu,Uub))) ).

% ATP.lambda_540
tff(fact_4829_ATP_Olambda__541,axiom,
    ! [Uu: fun_o_1833192751255691672t_unit,Uua: fun_Product_unit_int,Uub: $o] : aa_o_int(aa_fun4152134955913539783_o_int(aTP_Lamm_ti(Uu),Uua),(Uub)) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(Uua,aa_o_s3492205133957624465t_unit(Uu,(Uub)))) ).

% ATP.lambda_541
tff(fact_4830_ATP_Olambda__542,axiom,
    ! [Uu: fun_o_set_o,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_tk(Uu),Uua),(Uub)) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(Uua,aa_o_set_o(Uu,(Uub)))) ).

% ATP.lambda_542
tff(fact_4831_ATP_Olambda__543,axiom,
    ! [Uu: fun_o_set_int,Uua: fun_int_int,Uub: $o] : aa_o_int(aa_fun2290039529095201284_o_int(aTP_Lamm_tg(Uu),Uua),(Uub)) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(Uua),aa_o_set_int(Uu,(Uub)))) ).

% ATP.lambda_543
tff(fact_4832_ATP_Olambda__544,axiom,
    ! [Uu: fun_nat_set_int,Uua: fun_int_int,Uub: nat] : aa_nat_int(aa_fun5280992047749994636at_int(aTP_Lamm_tm(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(Uua),aa_nat_set_int(Uu,Uub))) ).

% ATP.lambda_544
tff(fact_4833_ATP_Olambda__545,axiom,
    ! [Uu: fun_in4731230917697954842t_unit,Uua: fun_Product_unit_int,Uub: int] : aa_int_int(aa_fun5430718389669344997nt_int(aTP_Lamm_sw(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,image_873079544045703924it_int(Uua,aa_int6548451963861500163t_unit(Uu,Uub))) ).

% ATP.lambda_545
tff(fact_4834_ATP_Olambda__546,axiom,
    ! [Uu: fun_int_set_o,Uua: fun_o_int,Uub: int] : aa_int_int(aa_fun1998554717657234288nt_int(aTP_Lamm_sy(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,image_o_int(Uua,aa_int_set_o(Uu,Uub))) ).

% ATP.lambda_546
tff(fact_4835_ATP_Olambda__547,axiom,
    ! [Uu: fun_int_set_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_su(Uu),Uua),Uub) = aa_set_int_int(complete_Sup_Sup_int,aa_set_int_set_int(image_int_int(Uua),aa_int_set_int(Uu,Uub))) ).

% ATP.lambda_547
tff(fact_4836_ATP_Olambda__548,axiom,
    ! [Uu: fun_Pr3013102738158121955t_unit,Uua: fun_Product_unit_int,Uub: product_unit] : aa_Product_unit_int(aa_fun3641207808737740476it_int(aTP_Lamm_tr(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(Uua,aa_Pro234974305641502940t_unit(Uu,Uub))) ).

% ATP.lambda_548
tff(fact_4837_ATP_Olambda__549,axiom,
    ! [Uu: fun_Pr6739565440344758894_set_o,Uua: fun_o_int,Uub: product_unit] : aa_Product_unit_int(aa_fun3625413194728170993it_int(aTP_Lamm_ts(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(Uua,aa_Pro536174410435844839_set_o(Uu,Uub))) ).

% ATP.lambda_549
tff(fact_4838_ATP_Olambda__550,axiom,
    ! [Uu: fun_Pr8362234881679108866et_int,Uua: fun_int_int,Uub: product_unit] : aa_Product_unit_int(aa_fun2487178249965271161it_int(aTP_Lamm_tq(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(Uua),aa_Pro1886178141017053491et_int(Uu,Uub))) ).

% ATP.lambda_550
tff(fact_4839_ATP_Olambda__551,axiom,
    ! [Uu: fun_o_1833192751255691672t_unit,Uua: fun_Product_unit_int,Uub: $o] : aa_o_int(aa_fun4152134955913539783_o_int(aTP_Lamm_tu(Uu),Uua),(Uub)) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(Uua,aa_o_s3492205133957624465t_unit(Uu,(Uub)))) ).

% ATP.lambda_551
tff(fact_4840_ATP_Olambda__552,axiom,
    ! [Uu: fun_o_set_o,Uua: fun_o_int,Uub: $o] : aa_o_int(aa_fun4727766026823762812_o_int(aTP_Lamm_tv(Uu),Uua),(Uub)) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(Uua,aa_o_set_o(Uu,(Uub)))) ).

% ATP.lambda_552
tff(fact_4841_ATP_Olambda__553,axiom,
    ! [Uu: fun_o_set_int,Uua: fun_int_int,Uub: $o] : aa_o_int(aa_fun2290039529095201284_o_int(aTP_Lamm_tt(Uu),Uua),(Uub)) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(Uua),aa_o_set_int(Uu,(Uub)))) ).

% ATP.lambda_553
tff(fact_4842_ATP_Olambda__554,axiom,
    ! [Uu: fun_nat_set_int,Uua: fun_int_int,Uub: nat] : aa_nat_int(aa_fun5280992047749994636at_int(aTP_Lamm_tw(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(Uua),aa_nat_set_int(Uu,Uub))) ).

% ATP.lambda_554
tff(fact_4843_ATP_Olambda__555,axiom,
    ! [Uu: fun_in4731230917697954842t_unit,Uua: fun_Product_unit_int,Uub: int] : aa_int_int(aa_fun5430718389669344997nt_int(aTP_Lamm_to(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,image_873079544045703924it_int(Uua,aa_int6548451963861500163t_unit(Uu,Uub))) ).

% ATP.lambda_555
tff(fact_4844_ATP_Olambda__556,axiom,
    ! [Uu: fun_int_set_o,Uua: fun_o_int,Uub: int] : aa_int_int(aa_fun1998554717657234288nt_int(aTP_Lamm_tp(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,image_o_int(Uua,aa_int_set_o(Uu,Uub))) ).

% ATP.lambda_556
tff(fact_4845_ATP_Olambda__557,axiom,
    ! [Uu: fun_int_set_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_tn(Uu),Uua),Uub) = aa_set_int_int(complete_Inf_Inf_int,aa_set_int_set_int(image_int_int(Uua),aa_int_set_int(Uu,Uub))) ).

% ATP.lambda_557
tff(fact_4846_ATP_Olambda__558,axiom,
    ! [Uu: fun_nat_set_nat,Uua: set_nat,Uub: nat] : aa_nat_set_nat(aa_set562305657478724375et_nat(aTP_Lamm_mr(Uu),Uua),Uub) = aa_set6364576160234933248et_nat(comple7399068483239264473et_nat,image_nat_set_nat(Uu,Uua)) ).

% ATP.lambda_558
tff(fact_4847_ATP_Olambda__559,axiom,
    ! [Uu: fun_na1668100129766719005at_nat,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_cm(Uu),Uua),Uub),Uuc) = aa_Pro262675715343573244at_nat(aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uuc),Uua) ).

% ATP.lambda_559
tff(fact_4848_ATP_Olambda__560,axiom,
    ! [Uu: fun_na5914473681949881652_nat_o,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_cn(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_560
tff(fact_4849_ATP_Olambda__561,axiom,
    ! [Uu: fun_Pr4311113161502339697t_char,Uua: set_Product_unit,Uub: product_unit,Uuc: char] :
      aa_char_Product_unit(aa_Pro30065888502685636t_unit(aa_set5681920794977454460t_unit(aTP_Lamm_aam(Uu),Uua),Uub),Uuc) = $ite(member_char(Uuc,image_7248553556944771225t_char(Uu,Uua)),aa_char_Product_unit(the_in3194898024646563179t_char(Uua,Uu),Uuc),Uub) ).

% ATP.lambda_561
tff(fact_4850_ATP_Olambda__562,axiom,
    ! [Uu: fun_Product_unit_o,Uua: set_Product_unit,Uub: product_unit,Uuc: $o] :
      aa_o_Product_unit(aa_Pro1198011010171541257t_unit(aa_set4357366655728970727t_unit(aTP_Lamm_aaj(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),image_Product_unit_o(Uu,Uua)),aa_o_Product_unit(the_in43325605794470270unit_o(Uua,Uu),(Uuc)),Uub) ).

% ATP.lambda_562
tff(fact_4851_ATP_Olambda__563,axiom,
    ! [Uu: fun_Product_unit_rat,Uua: set_Product_unit,Uub: product_unit,Uuc: rat] :
      aa_rat_Product_unit(aa_Pro130249013586881149t_unit(aa_set475620726854874293t_unit(aTP_Lamm_aap(Uu),Uua),Uub),Uuc) = $ite(aa_set_rat_o(member_rat(Uuc),image_240439954468258464it_rat(Uu,Uua)),aa_rat_Product_unit(the_in3915408195775746418it_rat(Uua,Uu),Uuc),Uub) ).

% ATP.lambda_563
tff(fact_4852_ATP_Olambda__564,axiom,
    ! [Uu: fun_o_char,Uua: set_o,Uub: $o,Uuc: char] :
      ( aa_char_o(aa_o_fun_char_o(aa_set1568417695978065863char_o(aTP_Lamm_aan(Uu),Uua),(Uub)),Uuc)
    <=> $ite(member_char(Uuc,image_o_char(Uu,Uua)),aa_char_o(the_inv_into_o_char(Uua,Uu),Uuc),(Uub)) ) ).

% ATP.lambda_564
tff(fact_4853_ATP_Olambda__565,axiom,
    ! [Uu: fun_o_o,Uua: set_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_set4838170466384450098un_o_o(aTP_Lamm_aak(Uu),Uua),(Uub)),(Uuc))
    <=> $ite(aa_set_o_o(member_o((Uuc)),aa_set_o_set_o(image_o_o(Uu),Uua)),aa_o_o(the_inv_into_o_o(Uua,Uu),(Uuc)),(Uub)) ) ).

% ATP.lambda_565
tff(fact_4854_ATP_Olambda__566,axiom,
    ! [Uu: fun_o_rat,Uua: set_o,Uub: $o,Uuc: rat] :
      ( aa_rat_o(aa_o_fun_rat_o(aa_set4692966247657309312_rat_o(aTP_Lamm_aaq(Uu),Uua),(Uub)),Uuc)
    <=> $ite(aa_set_rat_o(member_rat(Uuc),image_o_rat(Uu,Uua)),aa_rat_o(the_inv_into_o_rat(Uua,Uu),Uuc),(Uub)) ) ).

% ATP.lambda_566
tff(fact_4855_ATP_Olambda__567,axiom,
    ! [Uu: fun_nat_char,Uua: set_nat,Uub: nat,Uuc: char] :
      aa_char_nat(aa_nat_fun_char_nat(aa_set484384187446105903ar_nat(aTP_Lamm_aao(Uu),Uua),Uub),Uuc) = $ite(member_char(Uuc,image_nat_char(Uu,Uua)),aa_char_nat(the_in9023905271258939222t_char(Uua,Uu),Uuc),Uub) ).

% ATP.lambda_567
tff(fact_4856_ATP_Olambda__568,axiom,
    ! [Uu: fun_nat_o,Uua: set_nat,Uub: nat,Uuc: $o] :
      aa_o_nat(aa_nat_fun_o_nat(aa_set7102390759555960768_o_nat(aTP_Lamm_aal(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),image_nat_o(Uu,Uua)),aa_o_nat(the_inv_into_nat_o(Uua,Uu),(Uuc)),Uub) ).

% ATP.lambda_568
tff(fact_4857_ATP_Olambda__569,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: nat,Uuc: rat] :
      aa_rat_nat(aa_nat_fun_rat_nat(aa_set58180079379113142at_nat(aTP_Lamm_aar(Uu),Uua),Uub),Uuc) = $ite(aa_set_rat_o(member_rat(Uuc),aa_set_nat_set_rat(image_nat_rat(Uu),Uua)),aa_rat_nat(the_inv_into_nat_rat(Uua,Uu),Uuc),Uub) ).

% ATP.lambda_569
tff(fact_4858_ATP_Olambda__570,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_set123921758692524994at_int(aTP_Lamm_aas(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),aa_set_int_set_nat(image_int_nat(Uu),Uua)),aa_nat_int(the_inv_into_int_nat(Uua,Uu),Uuc),Uub) ).

% ATP.lambda_570
tff(fact_4859_ATP_Olambda__571,axiom,
    ! [Uu: set_o,Uua: fun_o_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun7163513353746205997un_o_o(aTP_Lamm_aac(Uu),Uua),(Uub)),(Uuc))
    <=> $ite(aa_set_o_o(member_o((Uuc)),aa_set_o_set_o(image_o_o(Uua),Uu)),aa_o_o(the_inv_into_o_o(Uu,Uua),(Uuc)),(Uub)) ) ).

% ATP.lambda_571
tff(fact_4860_ATP_Olambda__572,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o,Uuc: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aa_fun336398880946066315_nat_o(aTP_Lamm_aad(Uu),Uua),(Uub)),Uuc)
    <=> $ite(aa_set_nat_o(member_nat(Uuc),image_o_nat(Uua,Uu)),aa_nat_o(the_inv_into_o_nat(Uu,Uua),Uuc),(Uub)) ) ).

% ATP.lambda_572
tff(fact_4861_ATP_Olambda__573,axiom,
    ! [Uu: set_o,Uua: fun_o_int,Uub: $o,Uuc: int] :
      ( aa_int_o(aa_o_fun_int_o(aa_fun5004808018037445259_int_o(aTP_Lamm_aae(Uu),Uua),(Uub)),Uuc)
    <=> $ite(aa_set_int_o(member_int(Uuc),image_o_int(Uua,Uu)),aa_int_o(the_inv_into_o_int(Uu,Uua),Uuc),(Uub)) ) ).

% ATP.lambda_573
tff(fact_4862_ATP_Olambda__574,axiom,
    ! [Uu: set_num,Uua: fun_num_o,Uub: num,Uuc: $o] :
      aa_o_num(aa_num_fun_o_num(aa_fun4126176751244095145_o_num(aTP_Lamm_zz(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),image_num_o(Uua,Uu)),aa_o_num(the_inv_into_num_o(Uu,Uua),(Uuc)),Uub) ).

% ATP.lambda_574
tff(fact_4863_ATP_Olambda__575,axiom,
    ! [Uu: set_num,Uua: fun_num_nat,Uub: num,Uuc: nat] :
      aa_nat_num(aa_num_fun_nat_num(aa_fun5532158766709364899at_num(aTP_Lamm_aaa(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),image_num_nat(Uua,Uu)),aa_nat_num(the_inv_into_num_nat(Uu,Uua),Uuc),Uub) ).

% ATP.lambda_575
tff(fact_4864_ATP_Olambda__576,axiom,
    ! [Uu: set_num,Uua: fun_num_int,Uub: num,Uuc: int] :
      aa_int_num(aa_num_fun_int_num(aa_fun463138793435060571nt_num(aTP_Lamm_aab(Uu),Uua),Uub),Uuc) = $ite(aa_set_int_o(member_int(Uuc),image_num_int(Uua,Uu)),aa_int_num(the_inv_into_num_int(Uu,Uua),Uuc),Uub) ).

% ATP.lambda_576
tff(fact_4865_ATP_Olambda__577,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat,Uuc: $o] :
      aa_o_nat(aa_nat_fun_o_nat(aa_fun3953650679475262431_o_nat(aTP_Lamm_aag(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),image_nat_o(Uua,Uu)),aa_o_nat(the_inv_into_nat_o(Uu,Uua),(Uuc)),Uub) ).

% ATP.lambda_577
tff(fact_4866_ATP_Olambda__578,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat,Uuc: rat] :
      aa_rat_nat(aa_nat_fun_rat_nat(aa_fun4643191702187387861at_nat(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc) = $ite(aa_set_rat_o(member_rat(Uuc),aa_set_nat_set_rat(image_nat_rat(Uua),Uu)),aa_rat_nat(the_inv_into_nat_rat(Uu,Uua),Uuc),Uub) ).

% ATP.lambda_578
tff(fact_4867_ATP_Olambda__579,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aah(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),aa_set_nat_set_nat(image_nat_nat(Uua),Uu)),aa_nat_nat(the_inv_into_nat_nat(Uu,Uua),Uuc),Uub) ).

% ATP.lambda_579
tff(fact_4868_ATP_Olambda__580,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat,Uuc: int] :
      aa_int_nat(aa_nat_fun_int_nat(aa_fun1581638925303081085nt_nat(aTP_Lamm_aai(Uu),Uua),Uub),Uuc) = $ite(aa_set_int_o(member_int(Uuc),aa_set_nat_set_int(image_nat_int(Uua),Uu)),aa_int_nat(the_inv_into_nat_int(Uu,Uua),Uuc),Uub) ).

% ATP.lambda_580
tff(fact_4869_ATP_Olambda__581,axiom,
    ! [Uu: fun_o_Product_unit,Uua: set_o,Uub: fun_o_Product_unit,Uuc: $o] :
      aa_o_Product_unit(aa_fun4820891174006887598t_unit(aa_set5856121796315333033t_unit(aTP_Lamm_vt(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),Uua),aa_o_Product_unit(Uu,(Uuc)),aa_o_Product_unit(Uub,(Uuc))) ).

% ATP.lambda_581
tff(fact_4870_ATP_Olambda__582,axiom,
    ! [Uu: fun_o_o,Uua: set_o,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_vw(Uu),Uua),Uub),(Uuc))
    <=> $ite(aa_set_o_o(member_o((Uuc)),Uua),aa_o_o(Uu,(Uuc)),aa_o_o(Uub,(Uuc))) ) ).

% ATP.lambda_582
tff(fact_4871_ATP_Olambda__583,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: fun_o_nat,Uuc: $o] :
      aa_o_nat(aa_fun6176012476734143100_o_nat(aa_set4039056531748102335_o_nat(aTP_Lamm_vz(Uu),Uua),Uub),(Uuc)) = $ite(aa_set_o_o(member_o((Uuc)),Uua),aa_o_nat(Uu,(Uuc)),aa_o_nat(Uub,(Uuc))) ).

% ATP.lambda_583
tff(fact_4872_ATP_Olambda__584,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: set_nat,Uub: fun_nat_Product_unit,Uuc: nat] :
      aa_nat_Product_unit(aa_fun4858661924651670332t_unit(aa_set7546214604051456903t_unit(aTP_Lamm_vu(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_Product_unit(Uu,Uuc),aa_nat_Product_unit(Uub,Uuc)) ).

% ATP.lambda_584
tff(fact_4873_ATP_Olambda__585,axiom,
    ! [Uu: fun_nat_o,Uua: set_nat,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_vx(Uu),Uua),Uub),Uuc)
    <=> $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_o(Uu,Uuc),aa_nat_o(Uub,Uuc)) ) ).

% ATP.lambda_585
tff(fact_4874_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_set8926561665770780043at_rat(aTP_Lamm_vq(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_rat(Uu,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_586
tff(fact_4875_ATP_Olambda__587,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_set9061999614494806067at_int(aTP_Lamm_vr(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_int(Uu,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_587
tff(fact_4876_ATP_Olambda__588,axiom,
    ! [Uu: fun_int_Product_unit,Uua: set_int,Uub: fun_int_Product_unit,Uuc: int] :
      aa_int_Product_unit(aa_fun3184267698828548156t_unit(aa_set2257548544956115171t_unit(aTP_Lamm_vv(Uu),Uua),Uub),Uuc) = $ite(aa_set_int_o(member_int(Uuc),Uua),aa_int_Product_unit(Uu,Uuc),aa_int_Product_unit(Uub,Uuc)) ).

% ATP.lambda_588
tff(fact_4877_ATP_Olambda__589,axiom,
    ! [Uu: fun_int_o,Uua: set_int,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_vy(Uu),Uua),Uub),Uuc)
    <=> $ite(aa_set_int_o(member_int(Uuc),Uua),aa_int_o(Uu,Uuc),aa_int_o(Uub,Uuc)) ) ).

% ATP.lambda_589
tff(fact_4878_ATP_Olambda__590,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: fun_int_int,Uuc: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aa_set7789521219411994439nt_int(aTP_Lamm_vs(Uu),Uua),Uub),Uuc) = $ite(aa_set_int_o(member_int(Uuc),Uua),aa_int_int(Uu,Uuc),aa_int_int(Uub,Uuc)) ).

% ATP.lambda_590
tff(fact_4879_ATP_Olambda__591,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr5303981572562559465at_nat,Uub: fun_Pr5303981572562559465at_nat,Uuc: product_prod_nat_nat] :
      aa_Pro314188276017538018at_nat(aa_fun5472858705586414608at_nat(aa_fun8506152981838162806at_nat(aTP_Lamm_uh(Uu),Uua),Uub),Uuc) = $ite(aa_Pro6150637604445964947_nat_o(Uu,Uuc),aa_Pro314188276017538018at_nat(Uua,Uuc),aa_Pro314188276017538018at_nat(Uub,Uuc)) ).

% ATP.lambda_591
tff(fact_4880_ATP_Olambda__592,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3818778794767778097at_nat,Uub: fun_Pr3818778794767778097at_nat,Uuc: product_prod_int_int] :
      aa_Pro8052357535077532458at_nat(aa_fun2261957746281396896at_nat(aa_fun1759909663267528014at_nat(aTP_Lamm_uf(Uu),Uua),Uub),Uuc) = $ite(aa_Pro4592068669816069595_int_o(Uu,Uuc),aa_Pro8052357535077532458at_nat(Uua,Uuc),aa_Pro8052357535077532458at_nat(Uub,Uuc)) ).

% ATP.lambda_592
tff(fact_4881_ATP_Olambda__593,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr6044903459216281905at_nat,Uub: fun_Pr6044903459216281905at_nat,Uuc: product_prod_int_int] :
      aa_Pro790483803792192298at_nat(aa_fun5130711716664441504at_nat(aa_fun4358810537669686606at_nat(aTP_Lamm_uk(Uu),Uua),Uub),Uuc) = $ite(aa_Pro4592068669816069595_int_o(Uu,Uuc),aa_Pro790483803792192298at_nat(Uua,Uuc),aa_Pro790483803792192298at_nat(Uub,Uuc)) ).

% ATP.lambda_593
tff(fact_4882_ATP_Olambda__594,axiom,
    ! [Uu: fun_nat_o,Uua: fun_na3202795955365919216at_nat,Uub: fun_na3202795955365919216at_nat,Uuc: nat] :
      aa_nat4793067608094048217at_nat(aa_fun2085067056331341500at_nat(aa_fun2877972393441065591at_nat(aTP_Lamm_ue(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat4793067608094048217at_nat(Uua,Uuc),aa_nat4793067608094048217at_nat(Uub,Uuc)) ).

% ATP.lambda_594
tff(fact_4883_ATP_Olambda__595,axiom,
    ! [Uu: fun_nat_o,Uua: fun_na367238361360896496at_nat,Uub: fun_na367238361360896496at_nat,Uuc: nat] :
      aa_nat5613602927182297049at_nat(aa_fun2206166324314207932at_nat(aa_fun4824349113117479543at_nat(aTP_Lamm_uj(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat5613602927182297049at_nat(Uua,Uuc),aa_nat5613602927182297049at_nat(Uub,Uuc)) ).

% ATP.lambda_595
tff(fact_4884_ATP_Olambda__596,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ua(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_596
tff(fact_4885_ATP_Olambda__597,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uc(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_597
tff(fact_4886_ATP_Olambda__598,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ud(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(Uu,Uuc),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_598
tff(fact_4887_ATP_Olambda__599,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_nat,Uub: fun_int_nat,Uuc: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aa_fun5407591398502139302nt_nat(aTP_Lamm_tx(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_nat(Uua,Uuc),aa_int_nat(Uub,Uuc)) ).

% ATP.lambda_599
tff(fact_4888_ATP_Olambda__600,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_int,Uub: fun_int_int,Uuc: int] :
      aa_int_int(aa_fun6280345759651049192nt_int(aa_fun3693603986580398266nt_int(aTP_Lamm_tz(Uu),Uua),Uub),Uuc) = $ite(aa_int_o(Uu,Uuc),aa_int_int(Uua,Uuc),aa_int_int(Uub,Uuc)) ).

% ATP.lambda_600
tff(fact_4889_ATP_Olambda__601,axiom,
    ! [Uu: fun_num_fun_num_o,Uua: fun_num_num,Uub: num,Uuc: num] :
      ( aa_num_o(aa_num_fun_num_o(aa_fun7420891108566521959_num_o(aTP_Lamm_dm(Uu),Uua),Uub),Uuc)
    <=> aa_num_o(aa_num_fun_num_o(Uu,Uub),aa_num_num(Uua,Uuc)) ) ).

% ATP.lambda_601
tff(fact_4890_ATP_Olambda__602,axiom,
    ! [Uu: fun_num_fun_num_o,Uua: fun_nat_num,Uub: num,Uuc: nat] :
      ( aa_nat_o(aa_num_fun_nat_o(aa_fun8523389415946384635_nat_o(aTP_Lamm_dl(Uu),Uua),Uub),Uuc)
    <=> aa_num_o(aa_num_fun_num_o(Uu,Uub),aa_nat_num(Uua,Uuc)) ) ).

% ATP.lambda_602
tff(fact_4891_ATP_Olambda__603,axiom,
    ! [Uu: fun_se5639961202347202350_nat_o,Uua: set_se7855581050983116737at_nat,Uub: set_Pr1261947904930325089at_nat,Uuc: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aa_set5424408346876907241_nat_o(aTP_Lamm_rm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,Uub),Uuc)),produc7726050920695093128at_nat(Uua,aTP_Lamm_jj(Uua)))
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(Uu,Uub),Uuc) ) ) ).

% ATP.lambda_603
tff(fact_4892_ATP_Olambda__604,axiom,
    ! [Uu: fun_se5672428230568430382_nat_o,Uua: set_se5479201105039950375at_nat,Uub: set_Pr4329608150637261639at_nat,Uuc: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aa_set5310703806256024963_nat_o(aTP_Lamm_rl(Uu),Uua),Uub),Uuc)
    <=> ( member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,Uub),Uuc),produc6258361011560675592at_nat(Uua,aTP_Lamm_ji(Uua)))
        & aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(Uu,Uub),Uuc) ) ) ).

% ATP.lambda_604
tff(fact_4893_ATP_Olambda__605,axiom,
    ! [Uu: fun_nat_fun_nat_o,Uua: set_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_set8149993881734264422_nat_o(aTP_Lamm_rn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uuc)),produc457027306803732586at_nat(Uua,aTP_Lamm_gm(Uua)))
        & aa_nat_o(aa_nat_fun_nat_o(Uu,Uub),Uuc) ) ) ).

% ATP.lambda_605
tff(fact_4894_ATP_Olambda__606,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: set_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_set5204721951351315026_int_o(aTP_Lamm_ro(Uu),Uua),Uub),Uuc)
    <=> ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc)),produc1453890548195736866nt_int(Uua,aTP_Lamm_gf(Uua)))
        & aa_int_o(aa_int_fun_int_o(Uu,Uub),Uuc) ) ) ).

% ATP.lambda_606
tff(fact_4895_ATP_Olambda__607,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_adl(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_607
tff(fact_4896_ATP_Olambda__608,axiom,
    ! [Uu: set_Pr7459493094073627847at_nat,Uua: fun_na5670658592214072528at_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun1198561302286688022_nat_o(aTP_Lamm_gp(Uu),Uua),Uub),Uuc)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_nat8843416860741599929at_nat(Uua,Uub)),aa_nat8843416860741599929at_nat(Uua,Uuc)),Uu) ) ).

% ATP.lambda_608
tff(fact_4897_ATP_Olambda__609,axiom,
    ! [Uu: set_Pr7459493094073627847at_nat,Uua: fun_in2433280179646926196at_nat,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6626869587057797306_int_o(aTP_Lamm_gt(Uu),Uua),Uub),Uuc)
    <=> member1466754251312161552at_nat(aa_set1529286688856863004at_nat(aa_set8477542300186285614at_nat(produc9060074326276436823at_nat,aa_int5606038448174453597at_nat(Uua,Uub)),aa_int5606038448174453597at_nat(Uua,Uuc)),Uu) ) ).

% ATP.lambda_609
tff(fact_4898_ATP_Olambda__610,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: fun_na7044673412300895210at_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun2628974918839883004_nat_o(aTP_Lamm_gq(Uu),Uua),Uub),Uuc)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_nat7062706319057040595at_nat(Uua,Uub)),aa_nat7062706319057040595at_nat(Uua,Uuc))),Uu) ) ).

% ATP.lambda_610
tff(fact_4899_ATP_Olambda__611,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: fun_in8804710076054006414at_nat,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun7652824614168344736_int_o(aTP_Lamm_gu(Uu),Uua),Uub),Uuc)
    <=> aa_set5844364881195198285_nat_o(member8757157785044589968at_nat(aa_set8987774817048569474at_nat(aa_set914776049116413614at_nat(produc2922128104949294807at_nat,aa_int8822742982810151799at_nat(Uua,Uub)),aa_int8822742982810151799at_nat(Uua,Uuc))),Uu) ) ).

% ATP.lambda_611
tff(fact_4900_ATP_Olambda__612,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_gr(Uu),Uua),Uub),Uuc)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(Uua,Uub)),aa_nat_nat(Uua,Uuc))),Uu) ) ).

% ATP.lambda_612
tff(fact_4901_ATP_Olambda__613,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_int_nat,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun927631379751134075_int_o(aTP_Lamm_gv(Uu),Uua),Uub),Uuc)
    <=> aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_int_nat(Uua,Uub)),aa_int_nat(Uua,Uuc))),Uu) ) ).

% ATP.lambda_613
tff(fact_4902_ATP_Olambda__614,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_gs(Uu),Uua),Uub),Uuc)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uua,Uuc))),Uu) ) ).

% ATP.lambda_614
tff(fact_4903_ATP_Olambda__615,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_gw(Uu),Uua),Uub),Uuc)
    <=> aa_set6170482117830564987_int_o(member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(Uua,Uub)),aa_int_int(Uua,Uuc))),Uu) ) ).

% ATP.lambda_615
tff(fact_4904_ATP_Olambda__616,axiom,
    ! [Uu: fun_o_o,Uua: set_o,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_bh(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),aa_set_o_set_o(image_o_o(Uu),Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_616
tff(fact_4905_ATP_Olambda__617,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1339306056044713023_nat_o(aTP_Lamm_bv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),image_o_nat(Uu,Uua))
        & aa_nat_o(Uub,Uuc) ) ) ).

% ATP.lambda_617
tff(fact_4906_ATP_Olambda__618,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set1221527818376329535_int_o(aTP_Lamm_bl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),image_o_int(Uu,Uua))
        & aa_int_o(Uub,Uuc) ) ) ).

% ATP.lambda_618
tff(fact_4907_ATP_Olambda__619,axiom,
    ! [Uu: fun_nat_o,Uua: set_nat,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set1656415104011699143un_o_o(aTP_Lamm_br(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),image_nat_o(Uu,Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_619
tff(fact_4908_ATP_Olambda__620,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: fun_rat_o,Uuc: rat] :
      ( aa_rat_o(aa_fun2875410490525819644_rat_o(aa_set1331966092573250503_rat_o(aTP_Lamm_bp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_rat_o(member_rat(Uuc),aa_set_nat_set_rat(image_nat_rat(Uu),Uua))
        & aa_rat_o(Uub,Uuc) ) ) ).

% ATP.lambda_620
tff(fact_4909_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_bz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),aa_set_nat_set_nat(image_nat_nat(Uu),Uua))
        & aa_nat_o(Uub,Uuc) ) ) ).

% ATP.lambda_621
tff(fact_4910_ATP_Olambda__622,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set1288072335105258183_int_o(aTP_Lamm_bt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),aa_set_nat_set_int(image_nat_int(Uu),Uua))
        & aa_int_o(Uub,Uuc) ) ) ).

% ATP.lambda_622
tff(fact_4911_ATP_Olambda__623,axiom,
    ! [Uu: fun_int_o,Uua: set_int,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set2864532236940079139un_o_o(aTP_Lamm_bj(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),image_int_o(Uu,Uua))
        & aa_o_o(Uub,(Uuc)) ) ) ).

% ATP.lambda_623
tff(fact_4912_ATP_Olambda__624,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set563577997438114851_nat_o(aTP_Lamm_bx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),aa_set_int_set_nat(image_int_nat(Uu),Uua))
        & aa_nat_o(Uub,Uuc) ) ) ).

% ATP.lambda_624
tff(fact_4913_ATP_Olambda__625,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_bn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),aa_set_int_set_int(image_int_int(Uu),Uua))
        & aa_int_o(Uub,Uuc) ) ) ).

% ATP.lambda_625
tff(fact_4914_ATP_Olambda__626,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun2769474628653446662_nat_o(aTP_Lamm_ace(Uu),Uua),Uub),Uuc)
    <=> ( aa_set3702741264012733747_nat_o(member8440522571783428010at_nat(Uuc),Uu)
        & ( aa_Pro4243173770044397973at_nat(Uua,Uuc) = aa_Pro4243173770044397973at_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_626
tff(fact_4915_ATP_Olambda__627,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aa_fun4260993738209906510_int_o(aTP_Lamm_acd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set6170482117830564987_int_o(member5262025264175285858nt_int(Uuc),Uu)
        & ( aa_Pro6801748904974264397nt_nat(Uua,Uuc) = aa_Pro6801748904974264397nt_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_627
tff(fact_4916_ATP_Olambda__628,axiom,
    ! [Uu: set_o,Uua: fun_o_67843798866261420at_nat,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun4676553822554939524un_o_o(aTP_Lamm_acb(Uu),Uua),(Uub)),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_o_P6312935399938478757at_nat(Uua,(Uuc)) = aa_o_P6312935399938478757at_nat(Uua,(Uub)) ) ) ) ).

% ATP.lambda_628
tff(fact_4917_ATP_Olambda__629,axiom,
    ! [Uu: set_o,Uua: fun_o_nat,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun8960405049393213551un_o_o(aTP_Lamm_abw(Uu),Uua),(Uub)),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uu)
        & ( aa_o_nat(Uua,(Uuc)) = aa_o_nat(Uua,(Uub)) ) ) ) ).

% ATP.lambda_629
tff(fact_4918_ATP_Olambda__630,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun2969884873976723847_nat_o(aTP_Lamm_aby(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_rat(Uua,Uuc) = aa_nat_rat(Uua,Uub) ) ) ) ).

% ATP.lambda_630
tff(fact_4919_ATP_Olambda__631,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6824225949033397647_nat_o(aTP_Lamm_aca(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_nat(Uua,Uuc) = aa_nat_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_631
tff(fact_4920_ATP_Olambda__632,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun4569312790843823667_nat_o(aTP_Lamm_abz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_int(Uua,Uuc) = aa_nat_int(Uua,Uub) ) ) ) ).

% ATP.lambda_632
tff(fact_4921_ATP_Olambda__633,axiom,
    ! [Uu: set_int,Uua: fun_in4514472496968388910at_nat,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun1661645502662368768_int_o(aTP_Lamm_acc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int1417691210696471063at_nat(Uua,Uuc) = aa_int1417691210696471063at_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_633
tff(fact_4922_ATP_Olambda__634,axiom,
    ! [Uu: set_int,Uua: fun_int_nat,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun927631379751134075_int_o(aTP_Lamm_abx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_nat(Uua,Uuc) = aa_int_nat(Uua,Uub) ) ) ) ).

% ATP.lambda_634
tff(fact_4923_ATP_Olambda__635,axiom,
    ! [Uu: set_int,Uua: fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aTP_Lamm_abv(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_int_int(Uua,Uuc) = aa_int_int(Uua,Uub) ) ) ) ).

% ATP.lambda_635
tff(fact_4924_ATP_Olambda__636,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_adr(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_636
tff(fact_4925_ATP_Olambda__637,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_adb(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_637
tff(fact_4926_ATP_Olambda__638,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_adp(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_638
tff(fact_4927_ATP_Olambda__639,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_add(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_639
tff(fact_4928_ATP_Olambda__640,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_adj(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_640
tff(fact_4929_ATP_Olambda__641,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_adh(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_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_641
tff(fact_4930_ATP_Olambda__642,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: set_Pr4329608150637261639at_nat,Uub: set_Pr4329608150637261639at_nat,Uuc: set_Pr4329608150637261639at_nat] :
      ( aa_set5844364881195198285_nat_o(aa_set7096950279330728543_nat_o(aa_set2950039546221647971_nat_o(aTP_Lamm_id(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_642
tff(fact_4931_ATP_Olambda__643,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat,Uuc: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aa_set8959908230745118793_nat_o(aTP_Lamm_ie(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_643
tff(fact_4932_ATP_Olambda__644,axiom,
    ! [Uu: nat,Uua: num,Uub: nat,Uuc: num] :
      ( aa_num_o(aa_nat_fun_num_o(aa_num3446304307616776968_num_o(aTP_Lamm_if(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_644
tff(fact_4933_ATP_Olambda__645,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ig(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_645
tff(fact_4934_ATP_Olambda__646,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ih(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_646
tff(fact_4935_ATP_Olambda__647,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_adn(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_647
tff(fact_4936_ATP_Olambda__648,axiom,
    ! [Uu: fun_o_o,Uua: set_o,Uub: fun_o_o,Uuc: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aa_set8294393945046410815un_o_o(aTP_Lamm_bi(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uua)
        & aa_o_o(Uub,aa_o_o(Uu,(Uuc))) ) ) ).

% ATP.lambda_648
tff(fact_4937_ATP_Olambda__649,axiom,
    ! [Uu: fun_o_nat,Uua: set_o,Uub: fun_nat_o,Uuc: $o] :
      ( aa_o_o(aa_fun_nat_o_fun_o_o(aa_set26024703028173499un_o_o(aTP_Lamm_bw(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uua)
        & aa_nat_o(Uub,aa_o_nat(Uu,(Uuc))) ) ) ).

% ATP.lambda_649
tff(fact_4938_ATP_Olambda__650,axiom,
    ! [Uu: fun_o_int,Uua: set_o,Uub: fun_int_o,Uuc: $o] :
      ( aa_o_o(aa_fun_int_o_fun_o_o(aa_set6365818687085129567un_o_o(aTP_Lamm_bm(Uu),Uua),Uub),(Uuc))
    <=> ( aa_set_o_o(member_o((Uuc)),Uua)
        & aa_int_o(Uub,aa_o_int(Uu,(Uuc))) ) ) ).

% ATP.lambda_650
tff(fact_4939_ATP_Olambda__651,axiom,
    ! [Uu: fun_nat_o,Uua: set_nat,Uub: fun_o_o,Uuc: nat] :
      ( aa_nat_o(aa_fun_o_o_fun_nat_o(aa_set2987053916987590195_nat_o(aTP_Lamm_bs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_o_o(Uub,aa_nat_o(Uu,Uuc)) ) ) ).

% ATP.lambda_651
tff(fact_4940_ATP_Olambda__652,axiom,
    ! [Uu: fun_nat_rat,Uua: set_nat,Uub: fun_rat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun6612743417842387716_nat_o(aa_set2081400587708086223_nat_o(aTP_Lamm_bq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_rat_o(Uub,aa_nat_rat(Uu,Uuc)) ) ) ).

% ATP.lambda_652
tff(fact_4941_ATP_Olambda__653,axiom,
    ! [Uu: fun_nat_nat,Uua: set_nat,Uub: fun_nat_o,Uuc: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aa_set1405850572773641671_nat_o(aTP_Lamm_ca(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_nat_o(Uub,aa_nat_nat(Uu,Uuc)) ) ) ).

% ATP.lambda_653
tff(fact_4942_ATP_Olambda__654,axiom,
    ! [Uu: fun_nat_int,Uua: set_nat,Uub: fun_int_o,Uuc: nat] :
      ( aa_nat_o(aa_fun8064560132910005592_nat_o(aa_set3533217302775704099_nat_o(aTP_Lamm_bu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uua)
        & aa_int_o(Uub,aa_nat_int(Uu,Uuc)) ) ) ).

% ATP.lambda_654
tff(fact_4943_ATP_Olambda__655,axiom,
    ! [Uu: fun_int_o,Uua: set_int,Uub: fun_o_o,Uuc: int] :
      ( aa_int_o(aa_fun_o_o_fun_int_o(aa_set2419422181098893691_int_o(aTP_Lamm_bk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_o_o(Uub,aa_int_o(Uu,Uuc)) ) ) ).

% ATP.lambda_655
tff(fact_4944_ATP_Olambda__656,axiom,
    ! [Uu: fun_int_nat,Uua: set_int,Uub: fun_nat_o,Uuc: int] :
      ( aa_int_o(aa_fun4116151057475255200_int_o(aa_set7541805066622444743_int_o(aTP_Lamm_by(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_nat_o(Uub,aa_int_nat(Uu,Uuc)) ) ) ).

% ATP.lambda_656
tff(fact_4945_ATP_Olambda__657,axiom,
    ! [Uu: fun_int_int,Uua: set_int,Uub: fun_int_o,Uuc: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aa_set445799759769731363_int_o(aTP_Lamm_bo(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uua)
        & aa_int_o(Uub,aa_int_int(Uu,Uuc)) ) ) ).

% ATP.lambda_657
tff(fact_4946_ATP_Olambda__658,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr2028404973800837934_nat_o,Uub: product_prod_nat_nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun443525351149406012_nat_o(aTP_Lamm_ii(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_658
tff(fact_4947_ATP_Olambda__659,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4496145827618669174_int_o,Uub: product_prod_nat_nat,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro8654398858934478503_int_o(aa_fun5031764673572090172_int_o(aTP_Lamm_im(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(aa_Pro8654398858934478503_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_659
tff(fact_4948_ATP_Olambda__660,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr6984107991879650211_nat_o,Uub: product_prod_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_Pro4733806262894120092_nat_o(aa_fun5503210535175347332_nat_o(aTP_Lamm_ik(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_nat_o(aa_Pro4733806262894120092_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_660
tff(fact_4949_ATP_Olambda__661,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr6550376307508196838_nat_o,Uub: product_prod_int_int,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro5511567090416956951_nat_o(aa_fun4736202344062180668_nat_o(aTP_Lamm_is(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(aa_Pro5511567090416956951_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_661
tff(fact_4950_ATP_Olambda__662,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr9018117161326028078_int_o,Uub: product_prod_int_int,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aa_fun101069629630089020_int_o(aTP_Lamm_iw(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_662
tff(fact_4951_ATP_Olambda__663,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr8286563733112057579_nat_o,Uub: product_prod_int_int,Uuc: nat] :
      ( aa_nat_o(aa_Pro6036262004126527460_nat_o(aa_fun8858419945204301588_nat_o(aTP_Lamm_iu(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_nat_o(aa_Pro6036262004126527460_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_663
tff(fact_4952_ATP_Olambda__664,axiom,
    ! [Uu: fun_nat_o,Uua: fun_na3970208099681795815_nat_o,Uub: nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_fun2376078316387139852_nat_o(aTP_Lamm_io(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_664
tff(fact_4953_ATP_Olambda__665,axiom,
    ! [Uu: fun_nat_o,Uua: fun_na6437948953499627055_int_o,Uub: nat,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_nat6136980003572673136_int_o(aa_fun6971596671186205596_int_o(aTP_Lamm_iq(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(aa_nat6136980003572673136_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_665
tff(fact_4954_ATP_Olambda__666,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_iy(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_666
tff(fact_4955_ATP_Olambda__667,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ja(Uu),Uua),Uub),Uuc)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_667
tff(fact_4956_ATP_Olambda__668,axiom,
    ! [Uu: fun_nat_set_nat,Uua: fun_nat_set_nat,Uub: nat,Uuc: nat] : aa_nat7062706319057040595at_nat(aa_nat7495903657192774236at_nat(aa_fun2373699841287868962at_nat(aTP_Lamm_mq(Uu),Uua),Uub),Uuc) = produc457027306803732586at_nat(aa_nat_set_nat(Uu,Uub),aa_nat1818925681583257441et_nat(aTP_Lamm_mp(Uua),Uuc)) ).

% ATP.lambda_668
tff(fact_4957_ATP_Olambda__669,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_int_nat,Uub: nat,Uuc: int] : aa_int1101136566761605151at_nat(aa_nat7440946206073622952at_nat(aa_fun8497543784391190088at_nat(aTP_Lamm_fq(Uu),Uua),Uub),Uuc) = product_Pair_rat_nat(aa_nat_rat(Uu,Uub),aa_int_nat(Uua,Uuc)) ).

% ATP.lambda_669
tff(fact_4958_ATP_Olambda__670,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_int_int,Uub: nat,Uuc: int] : aa_int6146657584107184251at_int(aa_nat4950863061141896196at_int(aa_fun6462367883577042760at_int(aTP_Lamm_ft(Uu),Uua),Uub),Uuc) = product_Pair_rat_int(aa_nat_rat(Uu,Uub),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_670
tff(fact_4959_ATP_Olambda__671,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_int_int,Uub: nat,Uuc: int] : aa_int6463212228042050163at_int(aa_nat128175214796122620at_int(aa_fun566144318431074624at_int(aTP_Lamm_fv(Uu),Uua),Uub),Uuc) = product_Pair_nat_int(aa_nat_nat(Uu,Uub),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_671
tff(fact_4960_ATP_Olambda__672,axiom,
    ! [Uu: fun_nat_int,Uua: fun_int_nat,Uub: nat,Uuc: int] : aa_int2417044922597525619nt_nat(aa_nat4805179934420871164nt_nat(aa_fun5820023813409591964nt_nat(aTP_Lamm_fx(Uu),Uua),Uub),Uuc) = aa_nat8602574874988167631nt_nat(product_Pair_int_nat(aa_nat_int(Uu,Uub)),aa_int_nat(Uua,Uuc)) ).

% ATP.lambda_672
tff(fact_4961_ATP_Olambda__673,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_rat,Uub: int,Uuc: nat] : aa_nat4094992604693716347at_rat(aa_int2674150736754365864at_rat(aa_fun1750146759014209948at_rat(aTP_Lamm_fz(Uu),Uua),Uub),Uuc) = product_Pair_nat_rat(aa_int_nat(Uu,Uub),aa_nat_rat(Uua,Uuc)) ).

% ATP.lambda_673
tff(fact_4962_ATP_Olambda__674,axiom,
    ! [Uu: fun_int_nat,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat3425370143577916367at_int(aa_int6564053929300316668at_int(aa_fun2564171078181162140at_int(aTP_Lamm_gc(Uu),Uua),Uub),Uuc) = product_Pair_nat_int(aa_int_nat(Uu,Uub),aa_nat_int(Uua,Uuc)) ).

% ATP.lambda_674
tff(fact_4963_ATP_Olambda__675,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int6463212228042050163at_int(aa_int378523976909674656at_int(aa_fun816493080544626660at_int(aTP_Lamm_ge(Uu),Uua),Uub),Uuc) = product_Pair_nat_int(aa_int_nat(Uu,Uub),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_675
tff(fact_4964_ATP_Olambda__676,axiom,
    ! [Uu: fun_int_int,Uua: fun_nat_rat,Uub: int,Uuc: nat] : aa_nat5094346316594770903nt_rat(aa_int4861072311447387652nt_rat(aa_fun4968850353178579960nt_rat(aTP_Lamm_gg(Uu),Uua),Uub),Uuc) = product_Pair_int_rat(aa_int_int(Uu,Uub),aa_nat_rat(Uua,Uuc)) ).

% ATP.lambda_676
tff(fact_4965_ATP_Olambda__677,axiom,
    ! [Uu: fun_int_int,Uua: fun_nat_nat,Uub: int,Uuc: nat] : aa_nat8602574874988167631nt_nat(aa_int2017686612070289404nt_nat(aa_fun7818050573159679480nt_nat(aTP_Lamm_gi(Uu),Uua),Uub),Uuc) = aa_nat8602574874988167631nt_nat(product_Pair_int_nat(aa_int_int(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_677
tff(fact_4966_ATP_Olambda__678,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_nat,Uub: int,Uuc: int] : aa_int2417044922597525619nt_nat(aa_int5055528696534423200nt_nat(aa_fun6070372575523144000nt_nat(aTP_Lamm_gk(Uu),Uua),Uub),Uuc) = aa_nat8602574874988167631nt_nat(product_Pair_int_nat(aa_int_int(Uu,Uub)),aa_int_nat(Uua,Uuc)) ).

% ATP.lambda_678
tff(fact_4967_ATP_Olambda__679,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4035196674708996672nt_int(aTP_Lamm_dw(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_679
tff(fact_4968_ATP_Olambda__680,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aa_fun140556562124455766_nat_o(aTP_Lamm_gz(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uuc) ) ) ).

% ATP.lambda_680
tff(fact_4969_ATP_Olambda__681,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_nat_nat,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro8654398858934478503_int_o(aa_fun483471867002406230_int_o(aTP_Lamm_hd(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uuc) ) ) ).

% ATP.lambda_681
tff(fact_4970_ATP_Olambda__682,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_nat_o,Uub: product_prod_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_Pro4733806262894120092_nat_o(aa_fun1047528069869899400_nat_o(aTP_Lamm_hb(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro6150637604445964947_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uuc) ) ) ).

% ATP.lambda_682
tff(fact_4971_ATP_Olambda__683,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_int_int,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro5511567090416956951_nat_o(aa_fun4662527895831814670_nat_o(aTP_Lamm_hj(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uuc) ) ) ).

% ATP.lambda_683
tff(fact_4972_ATP_Olambda__684,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aa_fun5005443200709765134_int_o(aTP_Lamm_hn(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uuc) ) ) ).

% ATP.lambda_684
tff(fact_4973_ATP_Olambda__685,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_nat_o,Uub: product_prod_int_int,Uuc: nat] :
      ( aa_nat_o(aa_Pro6036262004126527460_nat_o(aa_fun2349983811102306768_nat_o(aTP_Lamm_hl(Uu),Uua),Uub),Uuc)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_nat_o(Uua,Uuc) ) ) ).

% ATP.lambda_685
tff(fact_4974_ATP_Olambda__686,axiom,
    ! [Uu: fun_nat_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: nat,Uuc: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_fun6835597611960591551_nat_o(aTP_Lamm_hf(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_Pro6150637604445964947_nat_o(Uua,Uuc) ) ) ).

% ATP.lambda_686
tff(fact_4975_ATP_Olambda__687,axiom,
    ! [Uu: fun_nat_o,Uua: fun_Pr3238374608701176162_int_o,Uub: nat,Uuc: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_nat6136980003572673136_int_o(aa_fun5166854214830555983_int_o(aTP_Lamm_hh(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uuc) ) ) ).

% ATP.lambda_687
tff(fact_4976_ATP_Olambda__688,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun5001253801653566085_nat_o(aTP_Lamm_hp(Uu),Uua),Uub),Uuc)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uuc) ) ) ).

% ATP.lambda_688
tff(fact_4977_ATP_Olambda__689,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6736451341614545449_int_o(aTP_Lamm_hr(Uu),Uua),Uub),Uuc)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uuc) ) ) ).

% ATP.lambda_689
tff(fact_4978_ATP_Olambda__690,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_ar((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_690
tff(fact_4979_ATP_Olambda__691,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_rat,Uub: set_nat,Uuc: product_unit] : aa_Pro2555800602132853471et_rat(aa_set717647588170897386et_rat(aa_fun1928299002622888286et_rat(aTP_Lamm_zw(Uu),Uua),Uub),Uuc) = aa_set_nat_set_rat(image_nat_rat(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage4884490618288580032t_unit(Uu,aa_set8324764978855220220t_unit(insert_Product_unit(Uuc),bot_bo3957492148770167129t_unit))),Uub)) ).

% ATP.lambda_691
tff(fact_4980_ATP_Olambda__692,axiom,
    ! [Uu: fun_nat_Product_unit,Uua: fun_nat_nat,Uub: set_nat,Uuc: product_unit] : aa_Pro6064029160526250199et_nat(aa_set7097633925648574946et_nat(aa_fun5826417969294899038et_nat(aTP_Lamm_zy(Uu),Uua),Uub),Uuc) = aa_set_nat_set_nat(image_nat_nat(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage4884490618288580032t_unit(Uu,aa_set8324764978855220220t_unit(insert_Product_unit(Uuc),bot_bo3957492148770167129t_unit))),Uub)) ).

% ATP.lambda_692
tff(fact_4981_ATP_Olambda__693,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: set_nat,Uuc: rat] : aa_rat_set_rat(aa_set4177764166417342247et_rat(aa_fun4037799688509761353et_rat(aTP_Lamm_zg(Uu),Uua),Uub),Uuc) = aa_set_nat_set_rat(image_nat_rat(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_rat(Uu,insert_rat(Uuc,bot_bot_set_rat))),Uub)) ).

% ATP.lambda_693
tff(fact_4982_ATP_Olambda__694,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_nat,Uub: set_nat,Uuc: rat] : aa_rat_set_nat(aa_set1334378467040243999et_nat(aa_fun2282505651424660281et_nat(aTP_Lamm_zi(Uu),Uua),Uub),Uuc) = aa_set_nat_set_nat(image_nat_nat(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_rat(Uu,insert_rat(Uuc,bot_bot_set_rat))),Uub)) ).

% ATP.lambda_694
tff(fact_4983_ATP_Olambda__695,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_int,Uub: set_nat,Uuc: rat] : aa_rat_set_int(aa_set8067667358963293051et_int(aa_fun8515194321480695793et_int(aTP_Lamm_zk(Uu),Uua),Uub),Uuc) = aa_set_nat_set_int(image_nat_int(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_rat(Uu,insert_rat(Uuc,bot_bot_set_rat))),Uub)) ).

% ATP.lambda_695
tff(fact_4984_ATP_Olambda__696,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_rat,Uub: set_nat,Uuc: int] : aa_int_set_rat(aa_set3777344917092667259et_rat(aa_fun1292496876412202397et_rat(aTP_Lamm_zm(Uu),Uua),Uub),Uuc) = aa_set_nat_set_rat(image_nat_rat(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_int(Uu,aa_set_int_set_int(insert_int(Uuc),bot_bot_set_int))),Uub)) ).

% ATP.lambda_696
tff(fact_4985_ATP_Olambda__697,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_nat,Uub: set_nat,Uuc: int] : aa_int_set_nat(aa_set933959217715569011et_nat(aa_fun8760574876181877133et_nat(aTP_Lamm_zo(Uu),Uua),Uub),Uuc) = aa_set_nat_set_nat(image_nat_nat(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_int(Uu,aa_set_int_set_int(insert_int(Uuc),bot_bot_set_int))),Uub)) ).

% ATP.lambda_697
tff(fact_4986_ATP_Olambda__698,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: set_nat,Uuc: int] : aa_int_set_int(aa_set7667248109638618063et_int(aa_fun5769891509383136837et_int(aTP_Lamm_zs(Uu),Uua),Uub),Uuc) = aa_set_nat_set_int(image_nat_int(Uua),aa_set_nat_set_nat(inf_inf_set_nat(vimage_nat_int(Uu,aa_set_int_set_int(insert_int(Uuc),bot_bot_set_int))),Uub)) ).

% ATP.lambda_698
tff(fact_4987_ATP_Olambda__699,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_nat,Uub: set_int,Uuc: int] : aa_int_set_nat(aa_set5326422216746463695et_nat(aa_fun7229874773247449157et_nat(aTP_Lamm_zq(Uu),Uua),Uub),Uuc) = aa_set_int_set_nat(image_int_nat(Uua),inf_inf_set_int(aa_set_int_set_int(vimage_int_int(Uu),aa_set_int_set_int(insert_int(Uuc),bot_bot_set_int)),Uub)) ).

% ATP.lambda_699
tff(fact_4988_ATP_Olambda__700,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_set_int(aa_set2836339071814736939et_int(aa_fun4239191406448708861et_int(aTP_Lamm_zu(Uu),Uua),Uub),Uuc) = aa_set_int_set_int(image_int_int(Uua),inf_inf_set_int(aa_set_int_set_int(vimage_int_int(Uu),aa_set_int_set_int(insert_int(Uuc),bot_bot_set_int)),Uub)) ).

% ATP.lambda_700
tff(fact_4989_ATP_Olambda__701,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_cw(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_701
tff(fact_4990_ATP_Olambda__702,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_cs(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_702
tff(fact_4991_ATP_Olambda__703,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_ct(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_703
tff(fact_4992_ATP_Olambda__704,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_cu(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_704
tff(fact_4993_ATP_Olambda__705,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_co(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_705
tff(fact_4994_ATP_Olambda__706,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_cp(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_706
tff(fact_4995_ATP_Olambda__707,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_cv(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_707
tff(fact_4996_ATP_Olambda__708,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_cq(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_708
tff(fact_4997_ATP_Olambda__709,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_cr(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_709
tff(fact_4998_ATP_Olambda__710,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_cx(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_710
tff(fact_4999_ATP_Olambda__711,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_adu(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_711
tff(fact_5000_ATP_Olambda__712,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_adw(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_712
tff(fact_5001_ATP_Olambda__713,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aea(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_713
tff(fact_5002_ATP_Olambda__714,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ady(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_714
tff(fact_5003_ATP_Olambda__715,axiom,
    ! [Uu: fun_na1668100129766719005at_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat,Uud: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aa_fun5672751633241614456at_nat(aa_fun6003775592413651218at_nat(aTP_Lamm_dx(Uu),Uua),Uub),Uuc),Uud) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uud)) ).

% ATP.lambda_715
tff(fact_5004_ATP_Olambda__716,axiom,
    ! [Uu: fun_na5914473681949881652_nat_o,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat,Uud: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun5827507165208764047_nat_o(aa_fun83116112763158185_nat_o(aTP_Lamm_dy(Uu),Uua),Uub),Uuc),Uud) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uud)) ).

% ATP.lambda_716
tff(fact_5005_ATP_Olambda__717,axiom,
    ! [Uu: fun_in1696471567231710611nt_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int,Uud: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4035196674708996672nt_int(aa_fun2536542299119233284nt_int(aTP_Lamm_dz(Uu),Uua),Uub),Uuc),Uud) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uu,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uud)) ).

% ATP.lambda_717
tff(fact_5006_ATP_Olambda__718,axiom,
    ! [Uu: fun_int_fun_int_o,Uua: fun_int_int,Uub: fun_int_int,Uuc: int,Uud: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun7896090258416335903_int_o(aa_fun8092296977475930211_int_o(aTP_Lamm_ea(Uu),Uua),Uub),Uuc),Uud)
    <=> aa_int_o(aa_int_fun_int_o(Uu,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uud)) ) ).

% ATP.lambda_718
tff(fact_5007_ATP_Olambda__719,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: fun_int_int,Uub: fun_int_int,Uuc: int,Uud: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aa_fun8505558176664197859nt_int(aTP_Lamm_eb(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(Uu,aa_int_int(Uua,Uuc)),aa_int_int(Uub,Uud)) ).

% ATP.lambda_719
tff(fact_5008_ATP_Olambda__720,axiom,
    ! [Uu: $o,Uua: set_Pr1261947904930325089at_nat,Uub: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(aTP_Lamm_jy((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_720
tff(fact_5009_ATP_Olambda__721,axiom,
    ! [Uu: $o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_jz((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_721
tff(fact_5010_ATP_Olambda__722,axiom,
    ! [Uu: $o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ka((Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_722
tff(fact_5011_ATP_Olambda__723,axiom,
    ! [Uu: product_unit,Uua: product_unit] : aa_Pro8294289767544744956t_unit(aTP_Lamm_lh(Uu),Uua) = Uu ).

% ATP.lambda_723
tff(fact_5012_ATP_Olambda__724,axiom,
    ! [Uu: product_unit,Uua: produc3843707927480180839at_nat] : aa_Pro1279885795167260962t_unit(aTP_Lamm_li(Uu),Uua) = Uu ).

% ATP.lambda_724
tff(fact_5013_ATP_Olambda__725,axiom,
    ! [Uu: product_unit,Uua: product_prod_nat_nat] : aa_Pro676242055593665672t_unit(aTP_Lamm_lj(Uu),Uua) = Uu ).

% ATP.lambda_725
tff(fact_5014_ATP_Olambda__726,axiom,
    ! [Uu: product_unit,Uua: char] : aa_char_Product_unit(aTP_Lamm_kl(Uu),Uua) = Uu ).

% ATP.lambda_726
tff(fact_5015_ATP_Olambda__727,axiom,
    ! [Uu: product_unit,Uua: $o] : aa_o_Product_unit(aTP_Lamm_lk(Uu),(Uua)) = Uu ).

% ATP.lambda_727
tff(fact_5016_ATP_Olambda__728,axiom,
    ! [Uu: product_unit,Uua: rat] : aa_rat_Product_unit(aTP_Lamm_ko(Uu),Uua) = Uu ).

% ATP.lambda_728
tff(fact_5017_ATP_Olambda__729,axiom,
    ! [Uu: product_unit,Uua: nat] : aa_nat_Product_unit(aTP_Lamm_kt(Uu),Uua) = Uu ).

% ATP.lambda_729
tff(fact_5018_ATP_Olambda__730,axiom,
    ! [Uu: product_unit,Uua: int] : aa_int_Product_unit(aTP_Lamm_nk(Uu),Uua) = Uu ).

% ATP.lambda_730
tff(fact_5019_ATP_Olambda__731,axiom,
    ! [Uu: ref_a,Uua: heap_e7401611519738050253t_unit] : aa_hea4634405385475018798_ref_a(aa_ref7899946105209721698_ref_a(aTP_Lamm_ds,Uu),Uua) = Uu ).

% ATP.lambda_731
tff(fact_5020_ATP_Olambda__732,axiom,
    ! [Uu: $o,Uua: product_unit] :
      ( aa_Product_unit_o(aTP_Lamm_nh((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_732
tff(fact_5021_ATP_Olambda__733,axiom,
    ! [Uu: $o,Uua: char] :
      ( aa_char_o(aTP_Lamm_km((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_733
tff(fact_5022_ATP_Olambda__734,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_fl,(Uu)),(Uua))
    <=> (Uu) ) ).

% ATP.lambda_734
tff(fact_5023_ATP_Olambda__735,axiom,
    ! [Uu: $o,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_kp((Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_735
tff(fact_5024_ATP_Olambda__736,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_ku,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_736
tff(fact_5025_ATP_Olambda__737,axiom,
    ! [Uu: $o,Uua: int] :
      ( aa_int_o(aa_o_fun_int_o(aTP_Lamm_nl,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_737
tff(fact_5026_ATP_Olambda__738,axiom,
    ! [Uu: set_Code_integer,Uua: code_integer] : aa_Cod2813182003028390446nteger(aTP_Lamm_vp(Uu),Uua) = Uu ).

% ATP.lambda_738
tff(fact_5027_ATP_Olambda__739,axiom,
    ! [Uu: set_he8644174446132254637t_unit,Uua: ref_a] : aa_ref8820618686110234386t_unit(aTP_Lamm_ju(Uu),Uua) = Uu ).

% ATP.lambda_739
tff(fact_5028_ATP_Olambda__740,axiom,
    ! [Uu: set_Product_unit,Uua: product_unit] : aa_Pro234974305641502940t_unit(aTP_Lamm_jp(Uu),Uua) = Uu ).

% ATP.lambda_740
tff(fact_5029_ATP_Olambda__741,axiom,
    ! [Uu: set_Product_unit,Uua: $o] : aa_o_s3492205133957624465t_unit(aTP_Lamm_js(Uu),(Uua)) = Uu ).

% ATP.lambda_741
tff(fact_5030_ATP_Olambda__742,axiom,
    ! [Uu: set_Product_unit,Uua: nat] : aa_nat2714212862977473631t_unit(aTP_Lamm_jt(Uu),Uua) = Uu ).

% ATP.lambda_742
tff(fact_5031_ATP_Olambda__743,axiom,
    ! [Uu: set_Pr4329608150637261639at_nat,Uua: produc3843707927480180839at_nat] : aa_Pro2325065730410872156at_nat(aTP_Lamm_lb(Uu),Uua) = Uu ).

% ATP.lambda_743
tff(fact_5032_ATP_Olambda__744,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_unit] : aa_Pro5062749757247690704at_nat(aTP_Lamm_jx(Uu),Uua) = Uu ).

% ATP.lambda_744
tff(fact_5033_ATP_Olambda__745,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] : aa_Pro2745879313366468188at_nat(aTP_Lamm_lc(Uu),Uua) = Uu ).

% ATP.lambda_745
tff(fact_5034_ATP_Olambda__746,axiom,
    ! [Uu: set_char,Uua: product_unit] : aa_Pro9168384818938857560t_char(aTP_Lamm_kv(Uu),Uua) = Uu ).

% ATP.lambda_746
tff(fact_5035_ATP_Olambda__747,axiom,
    ! [Uu: set_char,Uua: $o] : aa_o_set_char(aTP_Lamm_kw(Uu),(Uua)) = Uu ).

% ATP.lambda_747
tff(fact_5036_ATP_Olambda__748,axiom,
    ! [Uu: set_char,Uua: nat] : aa_nat_set_char(aTP_Lamm_kx(Uu),Uua) = Uu ).

% ATP.lambda_748
tff(fact_5037_ATP_Olambda__749,axiom,
    ! [Uu: set_o,Uua: product_unit] : aa_Pro536174410435844839_set_o(aTP_Lamm_jq(Uu),Uua) = Uu ).

% ATP.lambda_749
tff(fact_5038_ATP_Olambda__750,axiom,
    ! [Uu: set_o,Uua: product_prod_nat_nat] : aa_Pro5970036531051701619_set_o(aTP_Lamm_jh(Uu),Uua) = Uu ).

% ATP.lambda_750
tff(fact_5039_ATP_Olambda__751,axiom,
    ! [Uu: set_o,Uua: $o] : aa_o_set_o(aTP_Lamm_jc(Uu),(Uua)) = Uu ).

% ATP.lambda_751
tff(fact_5040_ATP_Olambda__752,axiom,
    ! [Uu: set_o,Uua: nat] : aa_nat_set_o(aTP_Lamm_jf(Uu),Uua) = Uu ).

% ATP.lambda_752
tff(fact_5041_ATP_Olambda__753,axiom,
    ! [Uu: set_o,Uua: int] : aa_int_set_o(aTP_Lamm_jg(Uu),Uua) = Uu ).

% ATP.lambda_753
tff(fact_5042_ATP_Olambda__754,axiom,
    ! [Uu: set_se5479201105039950375at_nat,Uua: set_Pr4329608150637261639at_nat] : aa_set7518756442812098908at_nat(aTP_Lamm_ji(Uu),Uua) = Uu ).

% ATP.lambda_754
tff(fact_5043_ATP_Olambda__755,axiom,
    ! [Uu: set_se7855581050983116737at_nat,Uua: set_Pr1261947904930325089at_nat] : aa_set6242432332095302236at_nat(aTP_Lamm_jj(Uu),Uua) = Uu ).

% ATP.lambda_755
tff(fact_5044_ATP_Olambda__756,axiom,
    ! [Uu: set_rat,Uua: product_unit] : aa_Pro2555800602132853471et_rat(aTP_Lamm_ky(Uu),Uua) = Uu ).

% ATP.lambda_756
tff(fact_5045_ATP_Olambda__757,axiom,
    ! [Uu: set_rat,Uua: $o] : aa_o_set_rat(aTP_Lamm_kz(Uu),(Uua)) = Uu ).

% ATP.lambda_757
tff(fact_5046_ATP_Olambda__758,axiom,
    ! [Uu: set_rat,Uua: nat] : aa_nat_set_rat(aTP_Lamm_gl(Uu),Uua) = Uu ).

% ATP.lambda_758
tff(fact_5047_ATP_Olambda__759,axiom,
    ! [Uu: set_rat,Uua: int] : aa_int_set_rat(aTP_Lamm_jo(Uu),Uua) = Uu ).

% ATP.lambda_759
tff(fact_5048_ATP_Olambda__760,axiom,
    ! [Uu: set_num,Uua: nat] : aa_nat_set_num(aTP_Lamm_jl(Uu),Uua) = Uu ).

% ATP.lambda_760
tff(fact_5049_ATP_Olambda__761,axiom,
    ! [Uu: set_nat,Uua: product_unit] : aa_Pro6064029160526250199et_nat(aTP_Lamm_jr(Uu),Uua) = Uu ).

% ATP.lambda_761
tff(fact_5050_ATP_Olambda__762,axiom,
    ! [Uu: set_nat,Uua: produc3843707927480180839at_nat] : aa_Pro1692590229789857841et_nat(aTP_Lamm_lf(Uu),Uua) = Uu ).

% ATP.lambda_762
tff(fact_5051_ATP_Olambda__763,axiom,
    ! [Uu: set_nat,Uua: product_prod_nat_nat] : aa_Pro9123573313595863371et_nat(aTP_Lamm_lg(Uu),Uua) = Uu ).

% ATP.lambda_763
tff(fact_5052_ATP_Olambda__764,axiom,
    ! [Uu: set_nat,Uua: $o] : aa_o_set_nat(aTP_Lamm_jd(Uu),(Uua)) = Uu ).

% ATP.lambda_764
tff(fact_5053_ATP_Olambda__765,axiom,
    ! [Uu: set_nat,Uua: rat] : aa_rat_set_nat(aTP_Lamm_jm(Uu),Uua) = Uu ).

% ATP.lambda_765
tff(fact_5054_ATP_Olambda__766,axiom,
    ! [Uu: set_nat,Uua: num] : aa_num_set_nat(aTP_Lamm_jk(Uu),Uua) = Uu ).

% ATP.lambda_766
tff(fact_5055_ATP_Olambda__767,axiom,
    ! [Uu: set_nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_gm(Uu),Uua) = Uu ).

% ATP.lambda_767
tff(fact_5056_ATP_Olambda__768,axiom,
    ! [Uu: set_nat,Uua: int] : aa_int_set_nat(aTP_Lamm_ga(Uu),Uua) = Uu ).

% ATP.lambda_768
tff(fact_5057_ATP_Olambda__769,axiom,
    ! [Uu: set_int,Uua: product_unit] : aa_Pro1886178141017053491et_int(aTP_Lamm_la(Uu),Uua) = Uu ).

% ATP.lambda_769
tff(fact_5058_ATP_Olambda__770,axiom,
    ! [Uu: set_int,Uua: $o] : aa_o_set_int(aTP_Lamm_je(Uu),(Uua)) = Uu ).

% ATP.lambda_770
tff(fact_5059_ATP_Olambda__771,axiom,
    ! [Uu: set_int,Uua: rat] : aa_rat_set_int(aTP_Lamm_jn(Uu),Uua) = Uu ).

% ATP.lambda_771
tff(fact_5060_ATP_Olambda__772,axiom,
    ! [Uu: set_int,Uua: nat] : aa_nat_set_int(aTP_Lamm_fr(Uu),Uua) = Uu ).

% ATP.lambda_772
tff(fact_5061_ATP_Olambda__773,axiom,
    ! [Uu: set_int,Uua: int] : aa_int_set_int(aTP_Lamm_gf(Uu),Uua) = Uu ).

% ATP.lambda_773
tff(fact_5062_ATP_Olambda__774,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kr(Uu),Uua) = Uu ).

% ATP.lambda_774
tff(fact_5063_ATP_Olambda__775,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_num(aTP_Lamm_fm(Uu),Uua) = Uu ).

% ATP.lambda_775
tff(fact_5064_ATP_Olambda__776,axiom,
    ! [Uu: nat,Uua: product_unit] : aa_Product_unit_nat(aTP_Lamm_ni(Uu),Uua) = Uu ).

% ATP.lambda_776
tff(fact_5065_ATP_Olambda__777,axiom,
    ! [Uu: nat,Uua: char] : aa_char_nat(aTP_Lamm_kn(Uu),Uua) = Uu ).

% ATP.lambda_777
tff(fact_5066_ATP_Olambda__778,axiom,
    ! [Uu: nat,Uua: $o] : aa_o_nat(aTP_Lamm_nj(Uu),(Uua)) = Uu ).

% ATP.lambda_778
tff(fact_5067_ATP_Olambda__779,axiom,
    ! [Uu: nat,Uua: rat] : aa_rat_nat(aTP_Lamm_kq(Uu),Uua) = Uu ).

% ATP.lambda_779
tff(fact_5068_ATP_Olambda__780,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fn,Uu),Uua) = Uu ).

% ATP.lambda_780
tff(fact_5069_ATP_Olambda__781,axiom,
    ! [Uu: nat,Uua: int] : aa_int_nat(aTP_Lamm_fo(Uu),Uua) = Uu ).

% ATP.lambda_781
tff(fact_5070_ATP_Olambda__782,axiom,
    ! [Uu: int,Uua: product_unit] : aa_Product_unit_int(aTP_Lamm_nn(Uu),Uua) = Uu ).

% ATP.lambda_782
tff(fact_5071_ATP_Olambda__783,axiom,
    ! [Uu: int,Uua: $o] : aa_o_int(aTP_Lamm_no(Uu),(Uua)) = Uu ).

% ATP.lambda_783
tff(fact_5072_ATP_Olambda__784,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ks(Uu),Uua) = Uu ).

% ATP.lambda_784
tff(fact_5073_ATP_Olambda__785,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dt,Uu),Uua) = Uu ).

% ATP.lambda_785
tff(fact_5074_ATP_Olambda__786,axiom,
    ! [Uu: fun_Pr5876161306168176259nteger,Uua: fun_Co8499238673658658103nteger] : aa_fun2509385170507305080nteger(aTP_Lamm_fp(Uu),Uua) = Uu ).

% ATP.lambda_786
tff(fact_5075_ATP_Olambda__787,axiom,
    ! [Uu: ref_a,Uua: heap_e7401611519738050253t_unit] : aa_hea2136485429539994620t_unit(aa_ref6778525820557750750t_unit(aTP_Lamm_du,Uu),Uua) = Uua ).

% ATP.lambda_787
tff(fact_5076_ATP_Olambda__788,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dv,Uu),Uua) = Uua ).

% ATP.lambda_788
tff(fact_5077_ATP_Olambda__789,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cy,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_789
tff(fact_5078_ATP_Olambda__790,axiom,
    ! [Uu: product_unit] : aa_Pro8294289767544744956t_unit(aTP_Lamm_le,Uu) = Uu ).

% ATP.lambda_790
tff(fact_5079_ATP_Olambda__791,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_cl,(Uu))
    <=> (Uu) ) ).

% ATP.lambda_791
tff(fact_5080_ATP_Olambda__792,axiom,
    ! [Uu: set_nat] : aa_set_nat_set_nat(aTP_Lamm_ld,Uu) = Uu ).

% ATP.lambda_792
tff(fact_5081_ATP_Olambda__793,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_793
tff(fact_5082_ATP_Olambda__794,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aq,Uu) = Uu ).

% ATP.lambda_794
tff(fact_5083_ATP_Olambda__795,axiom,
    ! [Uu: char] : aa_char_set_char(aTP_Lamm_ht,Uu) = top_top_set_char ).

% ATP.lambda_795
tff(fact_5084_ATP_Olambda__796,axiom,
    ! [Uu: char] : aa_char_set_rat(aTP_Lamm_hu,Uu) = top_top_set_rat ).

% ATP.lambda_796
tff(fact_5085_ATP_Olambda__797,axiom,
    ! [Uu: char] : aa_char_set_nat(aTP_Lamm_hv,Uu) = top_top_set_nat ).

% ATP.lambda_797
tff(fact_5086_ATP_Olambda__798,axiom,
    ! [Uu: char] : aa_char_set_int(aTP_Lamm_hw,Uu) = top_top_set_int ).

% ATP.lambda_798
tff(fact_5087_ATP_Olambda__799,axiom,
    ! [Uu: ref_a] : aa_ref8820618686110234386t_unit(aTP_Lamm_abe,Uu) = top_to3956321331435058429t_unit ).

% ATP.lambda_799
tff(fact_5088_ATP_Olambda__800,axiom,
    ! [Uu: rat] : aa_rat_set_char(aTP_Lamm_hx,Uu) = top_top_set_char ).

% ATP.lambda_800
tff(fact_5089_ATP_Olambda__801,axiom,
    ! [Uu: rat] : aa_rat_set_rat(aTP_Lamm_hy,Uu) = top_top_set_rat ).

% ATP.lambda_801
tff(fact_5090_ATP_Olambda__802,axiom,
    ! [Uu: rat] : aa_rat_set_nat(aTP_Lamm_hz,Uu) = top_top_set_nat ).

% ATP.lambda_802
tff(fact_5091_ATP_Olambda__803,axiom,
    ! [Uu: rat] : aa_rat_set_int(aTP_Lamm_ia,Uu) = top_top_set_int ).

% ATP.lambda_803
tff(fact_5092_ATP_Olambda__804,axiom,
    ! [Uu: nat] : aa_nat_set_char(aTP_Lamm_ib,Uu) = top_top_set_char ).

% ATP.lambda_804
tff(fact_5093_ATP_Olambda__805,axiom,
    ! [Uu: nat] : aa_nat_set_rat(aTP_Lamm_ic,Uu) = top_top_set_rat ).

% ATP.lambda_805
tff(fact_5094_ATP_Olambda__806,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_abf,Uu) = top_top_set_nat ).

% ATP.lambda_806
tff(fact_5095_ATP_Olambda__807,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat] : aa_set6242432332095302236at_nat(aTP_Lamm_jv,Uu) = bot_bo3083307316010499117at_nat ).

% ATP.lambda_807
tff(fact_5096_ATP_Olambda__808,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_jw,Uu) = bot_bot_set_nat ).

% ATP.lambda_808

% Helper facts (30)
tff(help_fNot_2_1_U,axiom,
    ! [P2: $o] :
      ( (P2)
      | aa_o_o(fNot,(P2)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P2: $o] :
      ( ~ aa_o_o(fNot,(P2))
      | ~ (P2) ) ).

tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X7: $o,Y2: $o] :
      ( ( (X7) != (Y2) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X7)),(Y2)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X7: $o,Y2: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X7)),(Y2))
      | ( (X7) = (Y2) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X7: int,Y2: int] :
      ( ( X7 != Y2 )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X7: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X7: nat,Y2: nat] :
      ( ( X7 != Y2 )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X7: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X7: num,Y2: num] :
      ( ( X7 != Y2 )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X7: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X7: rat,Y2: rat] :
      ( ( X7 != Y2 )
      | aa_rat_o(aa_rat_fun_rat_o(fequal_rat,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Rat__Orat_T,axiom,
    ! [X7: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(fequal_rat,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__String__Ochar_T,axiom,
    ! [X7: char,Y2: char] :
      ( ( X7 != Y2 )
      | aa_char_o(aa_char_fun_char_o(fequal_char,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__String__Ochar_T,axiom,
    ! [X7: char,Y2: char] :
      ( ~ aa_char_o(aa_char_fun_char_o(fequal_char,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_I_Eo_M_Eo_J_T,axiom,
    ! [X7: fun_o_o,Y2: fun_o_o] :
      ( ( X7 != Y2 )
      | aa_fun_o_o_o(aa_fun1576665407785211693_o_o_o(fequal_fun_o_o,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__fun_I_Eo_M_Eo_J_T,axiom,
    ! [X7: fun_o_o,Y2: fun_o_o] :
      ( ~ aa_fun_o_o_o(aa_fun1576665407785211693_o_o_o(fequal_fun_o_o,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X7: code_integer,Y2: code_integer] :
      ( ( X7 != Y2 )
      | aa_Code_integer_o(aa_Cod326357343816142457eger_o(fequal_Code_integer,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X7: code_integer,Y2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(fequal_Code_integer,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X7: list_nat,Y2: list_nat] :
      ( ( X7 != Y2 )
      | aa_list_nat_o(aa_lis557685189327716851_nat_o(fequal_list_nat,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X7: list_nat,Y2: list_nat] :
      ( ~ aa_list_nat_o(aa_lis557685189327716851_nat_o(fequal_list_nat,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_T,axiom,
    ! [X7: fun_int_nat,Y2: fun_int_nat] :
      ( ( X7 != Y2 )
      | aa_fun_int_nat_o(aa_fun7045208173470077249_nat_o(fequal_fun_int_nat,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Int__Oint_Mt__Nat__Onat_J_T,axiom,
    ! [X7: fun_int_nat,Y2: fun_int_nat] :
      ( ~ aa_fun_int_nat_o(aa_fun7045208173470077249_nat_o(fequal_fun_int_nat,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X7: fun_nat_nat,Y2: fun_nat_nat] :
      ( ( X7 != Y2 )
      | aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(fequal_fun_nat_nat,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X7: fun_nat_nat,Y2: fun_nat_nat] :
      ( ~ aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(fequal_fun_nat_nat,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Nat__Onat_Mt__Num__Onum_J_T,axiom,
    ! [X7: fun_nat_num,Y2: fun_nat_num] :
      ( ( X7 != Y2 )
      | aa_fun_nat_num_o(aa_fun3018564658994998373_num_o(fequal_fun_nat_num,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Nat__Onat_Mt__Num__Onum_J_T,axiom,
    ! [X7: fun_nat_num,Y2: fun_nat_num] :
      ( ~ aa_fun_nat_num_o(aa_fun3018564658994998373_num_o(fequal_fun_nat_num,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Num__Onum_Mt__Num__Onum_J_T,axiom,
    ! [X7: fun_num_num,Y2: fun_num_num] :
      ( ( X7 != Y2 )
      | aa_fun_num_num_o(aa_fun1934075535957799633_num_o(fequal_fun_num_num,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Num__Onum_Mt__Num__Onum_J_T,axiom,
    ! [X7: fun_num_num,Y2: fun_num_num] :
      ( ~ aa_fun_num_num_o(aa_fun1934075535957799633_num_o(fequal_fun_num_num,X7),Y2)
      | ( X7 = Y2 ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X7: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( X7 != Y2 )
      | aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X7),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X7: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(fequal9188845342289841878at_nat,X7),Y2)
      | ( X7 = Y2 ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    $true ).

tff(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------