%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Combinable_Wands/0001_Mask/prob_00256_007596__7133920_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2005 ( 603 unt; 716 typ;   0 def)
%            Number of atoms       : 4371 (1437 equ;   0 cnn)
%            Maximal formula atoms :   13 (   3 avg)
%            Number of connectives : 15416 ( 323   ~;  49   |; 390   &;13015   @)
%                                         (   0 <=>;1639  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   7 avg)
%            Number of types       :   65 (  64 usr)
%            Number of type conns  : 4253 (4253   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  655 ( 652 usr;  40 con; 0-4 aty)
%            Number of variables   : 5031 (1614   ^;3262   !; 155   ?;5031   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:05:22.273
%------------------------------------------------------------------------------
% Could-be-implicit typings (64)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    produc5803078220529002682t_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J_J,type,
    set_Pr1685704261205283783r_prat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    produc3336119888008106215r_prat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    produc2489117125269924006t_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    set_Pr3996677194857090567t_char: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    produc6071853487833700903t_char: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr8693737435421807431at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc859450856879609959at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    produc3933091914578511633r_prat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    set_Pr3433046674199179443r_prat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    set_Pr4480099295215233283t_char: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    set_na6246771325765140825t_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr2539167527615954998at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr7717912310451564380at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    set_se253819897320413138t_char: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_M_Eo_J_J,type,
    set_Pr2094409768014413473char_o: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    produc6859378861088059299t_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8373899037510109440at_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    set_Pr2507339680178222706t_char: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_J,type,
    set_Pr8803067691650843250ar_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr3667225130493526178at_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_na8845969913329135560at_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr3601174868274201367at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr400265656397884439et_nat: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J_J,type,
    set_nat_list_char_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
    set_Pr5582243495563764594_nat_o: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__PosRat__Oprat_Mt__PosRat__Oprat_J_J,type,
    set_Pr3298805672376727137t_prat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    produc1457211279475724562t_char: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J,type,
    produc7606469099237612818ar_nat: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc7491599851749785783at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc2400336064389900727et_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J_J,type,
    set_set_filter_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_set_nat_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    set_nat_nat_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Filter__Ofilter_It__Nat__Onat_J_M_Eo_J_J,type,
    set_filter_nat_o: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    set_set_nat_o: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_J,type,
    set_nat_prat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
    set_filter_nat: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_nat_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Set__Oset_It__PosRat__Oprat_J_J,type,
    set_set_prat: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    set_nat_o: $tType ).

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

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

thf(ty_n_t__Set__Oset_It__PosRat__Oprat_J,type,
    set_prat: $tType ).

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

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

thf(ty_n_t__PosRat__Oprat,type,
    prat: $tType ).

thf(ty_n_t__Mask__Oval,type,
    val: $tType ).

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

thf(ty_n_tf__b,type,
    b: $tType ).

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

% Explicit typings (652)
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr_nat_list_char: set_nat > ( nat > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr_nat_nat: set_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Def_OGr_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_Gr_nat_set_nat: set_nat > ( nat > set_nat ) > set_Pr400265656397884439et_nat ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_I_Eo_M_Eo_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_J,type,
    bNF_re3400349493146995663_val_o: ( ( $o > $o ) > ( $o > $o ) > $o ) > ( ( ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_I_Eo_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_J,type,
    bNF_re8530354864449075465al_nat: ( ( $o > $o ) > ( $o > nat ) > $o ) > ( ( ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_M_Eo_J_001_062_I_Eo_Mt__PosRat__Oprat_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_J_001_062_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_M_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_J,type,
    bNF_re1381473823880018185l_prat: ( ( $o > $o ) > ( $o > prat ) > $o ) > ( ( ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ) > ( ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_Mt__Nat__Onat_J_001_062_I_Eo_M_Eo_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_J,type,
    bNF_re2233505157435102499_val_o: ( ( $o > nat ) > ( $o > $o ) > $o ) > ( ( ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ) > ( ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_Mt__Nat__Onat_J_001_062_I_Eo_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_J,type,
    bNF_re5390656012764358813al_nat: ( ( $o > nat ) > ( $o > nat ) > $o ) > ( ( ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ) > ( ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_Mt__Nat__Onat_J_001_062_I_Eo_Mt__PosRat__Oprat_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_J_001_062_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_M_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_J,type,
    bNF_re5807499856486624669l_prat: ( ( $o > nat ) > ( $o > prat ) > $o ) > ( ( ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ) > ( ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_Mt__PosRat__Oprat_J_001_062_I_Eo_M_Eo_J_001_062_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_M_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_J,type,
    bNF_re1988916615843939875_val_o: ( ( $o > prat ) > ( $o > $o ) > $o ) > ( ( ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ) > ( ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_Mt__PosRat__Oprat_J_001_062_I_Eo_Mt__Nat__Onat_J_001_062_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_M_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_J_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_J,type,
    bNF_re3676624960930530973al_nat: ( ( $o > prat ) > ( $o > nat ) > $o ) > ( ( ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ) > ( ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_Eo_Mt__PosRat__Oprat_J_001_062_I_Eo_Mt__PosRat__Oprat_J_001_062_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_M_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_J_001_062_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_M_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_J,type,
    bNF_re8674362571971103645l_prat: ( ( $o > prat ) > ( $o > prat ) > $o ) > ( ( ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ) > ( ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_001_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J,type,
    bNF_re4064708811108098077_val_o: ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( ( prat > $o ) > val > $o ) > ( ( prat > $o ) > val > $o ) > $o ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_001_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J,type,
    bNF_re9138709833252949743al_nat: ( ( nat > $o ) > ( nat > nat ) > $o ) > ( ( ( prat > $o ) > val > $o ) > ( ( prat > nat ) > val > nat ) > $o ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J_001_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J_001_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J,type,
    bNF_re8023842771505706095l_prat: ( ( nat > $o ) > ( nat > prat ) > $o ) > ( ( ( prat > $o ) > val > $o ) > ( ( prat > prat ) > val > prat ) > $o ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_001_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J,type,
    bNF_re7947119355560860307_val_o: ( ( nat > nat ) > ( nat > $o ) > $o ) > ( ( ( prat > nat ) > val > nat ) > ( ( prat > $o ) > val > $o ) > $o ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_001_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J,type,
    bNF_re3142472647042036189al_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( ( ( prat > nat ) > val > nat ) > ( ( prat > nat ) > val > nat ) > $o ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J_001_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J_001_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J,type,
    bNF_re3017882675219251037l_prat: ( ( nat > nat ) > ( nat > prat ) > $o ) > ( ( ( prat > nat ) > val > nat ) > ( ( prat > prat ) > val > prat ) > $o ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_001_062_I_062_It__PosRat__Oprat_M_Eo_J_M_062_It__Mask__Oval_M_Eo_J_J,type,
    bNF_re5697875234658812691_val_o: ( ( nat > prat ) > ( nat > $o ) > $o ) > ( ( ( prat > prat ) > val > prat ) > ( ( prat > $o ) > val > $o ) > $o ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( nat > $o ) > ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_001_062_I_062_It__PosRat__Oprat_Mt__Nat__Onat_J_M_062_It__Mask__Oval_Mt__Nat__Onat_J_J,type,
    bNF_re3377346888436407133al_nat: ( ( nat > prat ) > ( nat > nat ) > $o ) > ( ( ( prat > prat ) > val > prat ) > ( ( prat > nat ) > val > nat ) > $o ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( nat > nat ) > ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J_001_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J_001_062_I_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_M_062_It__Mask__Oval_Mt__PosRat__Oprat_J_J,type,
    bNF_re6019836462720788701l_prat: ( ( nat > prat ) > ( nat > prat ) > $o ) > ( ( ( prat > prat ) > val > prat ) > ( ( prat > prat ) > val > prat ) > $o ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > ( ( nat > prat ) > ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_M_Eo_J_001_062_It__PosRat__Oprat_M_Eo_J_001_062_It__Mask__Oval_M_Eo_J_001_062_It__Mask__Oval_M_Eo_J,type,
    bNF_re6376993983822893079_val_o: ( ( prat > $o ) > ( prat > $o ) > $o ) > ( ( val > $o ) > ( val > $o ) > $o ) > ( ( prat > $o ) > val > $o ) > ( ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_M_Eo_J_001_062_It__PosRat__Oprat_Mt__Nat__Onat_J_001_062_It__Mask__Oval_M_Eo_J_001_062_It__Mask__Oval_Mt__Nat__Onat_J,type,
    bNF_re4353138970906064163al_nat: ( ( prat > $o ) > ( prat > nat ) > $o ) > ( ( val > $o ) > ( val > nat ) > $o ) > ( ( prat > $o ) > val > $o ) > ( ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_M_Eo_J_001_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_001_062_It__Mask__Oval_M_Eo_J_001_062_It__Mask__Oval_Mt__PosRat__Oprat_J,type,
    bNF_re8035667688264635171l_prat: ( ( prat > $o ) > ( prat > prat ) > $o ) > ( ( val > $o ) > ( val > prat ) > $o ) > ( ( prat > $o ) > val > $o ) > ( ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_Mt__Nat__Onat_J_001_062_It__PosRat__Oprat_M_Eo_J_001_062_It__Mask__Oval_Mt__Nat__Onat_J_001_062_It__Mask__Oval_M_Eo_J,type,
    bNF_re7761813834571909981_val_o: ( ( prat > nat ) > ( prat > $o ) > $o ) > ( ( val > nat ) > ( val > $o ) > $o ) > ( ( prat > nat ) > val > nat ) > ( ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_Mt__Nat__Onat_J_001_062_It__PosRat__Oprat_Mt__Nat__Onat_J_001_062_It__Mask__Oval_Mt__Nat__Onat_J_001_062_It__Mask__Oval_Mt__Nat__Onat_J,type,
    bNF_re6117652819335925469al_nat: ( ( prat > nat ) > ( prat > nat ) > $o ) > ( ( val > nat ) > ( val > nat ) > $o ) > ( ( prat > nat ) > val > nat ) > ( ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_Mt__Nat__Onat_J_001_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_001_062_It__Mask__Oval_Mt__Nat__Onat_J_001_062_It__Mask__Oval_Mt__PosRat__Oprat_J,type,
    bNF_re5989471590236891101l_prat: ( ( prat > nat ) > ( prat > prat ) > $o ) > ( ( val > nat ) > ( val > prat ) > $o ) > ( ( prat > nat ) > val > nat ) > ( ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_001_062_It__PosRat__Oprat_M_Eo_J_001_062_It__Mask__Oval_Mt__PosRat__Oprat_J_001_062_It__Mask__Oval_M_Eo_J,type,
    bNF_re1271012312234361181_val_o: ( ( prat > prat ) > ( prat > $o ) > $o ) > ( ( val > prat ) > ( val > $o ) > $o ) > ( ( prat > prat ) > val > prat ) > ( ( prat > $o ) > val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_001_062_It__PosRat__Oprat_Mt__Nat__Onat_J_001_062_It__Mask__Oval_Mt__PosRat__Oprat_J_001_062_It__Mask__Oval_Mt__Nat__Onat_J,type,
    bNF_re4810291329183323613al_nat: ( ( prat > prat ) > ( prat > nat ) > $o ) > ( ( val > prat ) > ( val > nat ) > $o ) > ( ( prat > prat ) > val > prat ) > ( ( prat > nat ) > val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_001_062_It__PosRat__Oprat_Mt__PosRat__Oprat_J_001_062_It__Mask__Oval_Mt__PosRat__Oprat_J_001_062_It__Mask__Oval_Mt__PosRat__Oprat_J,type,
    bNF_re5071352437643341533l_prat: ( ( prat > prat ) > ( prat > prat ) > $o ) > ( ( val > prat ) > ( val > prat ) > $o ) > ( ( prat > prat ) > val > prat ) > ( ( prat > prat ) > val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001_Eo,type,
    bNF_rel_fun_o_o_o_o: ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o ) > ( $o > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001t__Nat__Onat,type,
    bNF_re9072777675879608139_o_nat: ( $o > $o > $o ) > ( $o > nat > $o ) > ( $o > $o ) > ( $o > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001_Eo_001t__PosRat__Oprat,type,
    bNF_re7136572702137213643o_prat: ( $o > $o > $o ) > ( $o > prat > $o ) > ( $o > $o ) > ( $o > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001_Eo,type,
    bNF_re2952077000133017641_nat_o: ( $o > $o > $o ) > ( nat > $o > $o ) > ( $o > nat ) > ( $o > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re552426215088010623at_nat: ( $o > $o > $o ) > ( nat > nat > $o ) > ( $o > nat ) > ( $o > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__PosRat__Oprat,type,
    bNF_re3012592894602769663t_prat: ( $o > $o > $o ) > ( nat > prat > $o ) > ( $o > nat ) > ( $o > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__PosRat__Oprat_001_Eo,type,
    bNF_re3555829217104990121prat_o: ( $o > $o > $o ) > ( prat > $o > $o ) > ( $o > prat ) > ( $o > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__PosRat__Oprat_001t__Nat__Onat,type,
    bNF_re4232132340546668031at_nat: ( $o > $o > $o ) > ( prat > nat > $o ) > ( $o > prat ) > ( $o > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    bNF_re2544178717384787327t_prat: ( $o > $o > $o ) > ( prat > prat > $o ) > ( $o > prat ) > ( $o > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001_Eo_001_Eo,type,
    bNF_re3282371258598777373al_o_o: ( val > val > $o ) > ( $o > $o > $o ) > ( val > $o ) > ( val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001_Eo_001t__Nat__Onat,type,
    bNF_re8984071810037758091_o_nat: ( val > val > $o ) > ( $o > nat > $o ) > ( val > $o ) > ( val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001_Eo_001t__PosRat__Oprat,type,
    bNF_re1819705386720226315o_prat: ( val > val > $o ) > ( $o > prat > $o ) > ( val > $o ) > ( val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001t__Nat__Onat_001_Eo,type,
    bNF_re2863371134291167593_nat_o: ( val > val > $o ) > ( nat > $o > $o ) > ( val > nat ) > ( val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re478083327121898559at_nat: ( val > val > $o ) > ( nat > nat > $o ) > ( val > nat ) > ( val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001t__Nat__Onat_001t__PosRat__Oprat,type,
    bNF_re462759099702918079t_prat: ( val > val > $o ) > ( nat > prat > $o ) > ( val > nat ) > ( val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001t__PosRat__Oprat_001_Eo,type,
    bNF_re7462333938542778601prat_o: ( val > val > $o ) > ( prat > $o > $o ) > ( val > prat ) > ( val > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001t__PosRat__Oprat_001t__Nat__Onat,type,
    bNF_re1682298545646816447at_nat: ( val > val > $o ) > ( prat > nat > $o ) > ( val > prat ) > ( val > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Mask__Oval_001t__Mask__Oval_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    bNF_re8661857836907285567t_prat: ( val > val > $o ) > ( prat > prat > $o ) > ( val > prat ) > ( val > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    bNF_re2942033592892840063char_o: ( nat > list_char > $o ) > ( ( nat > $o ) > ( list_char > $o ) > $o ) > ( nat > nat > $o ) > ( list_char > list_char > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re1113502957280211792_nat_o: ( nat > list_char > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( list_char > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001_Eo_001_Eo,type,
    bNF_re1135979186550299152ar_o_o: ( nat > list_char > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( list_char > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    bNF_re4495586573714376046char_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( list_char > $o ) > $o ) > ( nat > nat > $o ) > ( nat > list_char > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
    bNF_re7523573913982466217_o_nat: ( nat > nat > $o ) > ( $o > nat > $o ) > ( nat > $o ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001t__PosRat__Oprat,type,
    bNF_re8284746473853242921o_prat: ( nat > nat > $o ) > ( $o > prat > $o ) > ( nat > $o ) > ( nat > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    bNF_re1402873238235875719_nat_o: ( nat > nat > $o ) > ( nat > $o > $o ) > ( nat > nat ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_re2076907181175516082t_char: ( nat > nat > $o ) > ( nat > list_char > $o ) > ( nat > nat ) > ( nat > list_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__PosRat__Oprat,type,
    bNF_re4147878452613618977t_prat: ( nat > nat > $o ) > ( nat > prat > $o ) > ( nat > nat ) > ( nat > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__PosRat__Oprat_001_Eo,type,
    bNF_re4704002988821019399prat_o: ( nat > nat > $o ) > ( prat > $o > $o ) > ( nat > prat ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__PosRat__Oprat_001t__Nat__Onat,type,
    bNF_re5367417898557517345at_nat: ( nat > nat > $o ) > ( prat > nat > $o ) > ( nat > prat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    bNF_re2294495614902143905t_prat: ( nat > nat > $o ) > ( prat > prat > $o ) > ( nat > prat ) > ( nat > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001_Eo_001_Eo,type,
    bNF_re2178160303036436095at_o_o: ( prat > prat > $o ) > ( $o > $o > $o ) > ( prat > $o ) > ( prat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001_Eo_001t__Nat__Onat,type,
    bNF_re2521994545097566377_o_nat: ( prat > prat > $o ) > ( $o > nat > $o ) > ( prat > $o ) > ( prat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001_Eo_001t__PosRat__Oprat,type,
    bNF_re2805847409588092457o_prat: ( prat > prat > $o ) > ( $o > prat > $o ) > ( prat > $o ) > ( prat > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__Nat__Onat_001_Eo,type,
    bNF_re5624665906205751687_nat_o: ( prat > prat > $o ) > ( nat > $o > $o ) > ( prat > nat ) > ( prat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re1361929969394864033at_nat: ( prat > prat > $o ) > ( nat > nat > $o ) > ( prat > nat ) > ( prat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__Nat__Onat_001t__PosRat__Oprat,type,
    bNF_re2796068022358551841t_prat: ( prat > prat > $o ) > ( nat > prat > $o ) > ( prat > nat ) > ( prat > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__PosRat__Oprat_001_Eo,type,
    bNF_re8448475961410644743prat_o: ( prat > prat > $o ) > ( prat > $o > $o ) > ( prat > prat ) > ( prat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__Nat__Onat,type,
    bNF_re4015607468302450209at_nat: ( prat > prat > $o ) > ( prat > nat > $o ) > ( prat > prat ) > ( prat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    bNF_re2220458561309875105t_prat: ( prat > prat > $o ) > ( prat > prat > $o ) > ( prat > prat ) > ( prat > prat ) > $o ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr7397543130437154909t_char: set_nat > ( nat > nat ) > ( nat > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr7318156773085685964at_nat: set_nat > ( nat > nat ) > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr3738443398878377306t_char: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr8954129895687480649at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr3178947973406375814at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_Gr553682781689516860et_nat: set_Pr1261947904930325089at_nat > ( nat > set_nat ) > set_Pr5488025237498180813et_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr8020361688147858275at_nat: set_nat > set_Pr1261947904930325089at_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    bNF_Gr7866296307673166617et_nat: set_nat > set_Pr5488025237498180813et_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_Gr4292214848008494032at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Nat__Onat_001_Eo,type,
    bNF_We709690948610045618_nat_o: set_nat > set_o > set_nat_o ).

thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We7540516734559056694at_nat: set_nat > set_nat > set_nat_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Nat__Onat_001t__PosRat__Oprat,type,
    bNF_We4121571284332944310t_prat: set_nat > set_prat > set_nat_prat ).

thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_We3695879855305361469at_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr3667225130493526178at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OFunc_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_We5625579243924517612at_nat: set_set_nat > set_nat > set_set_nat_nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    comple7993840426618028700char_o: set_nat_list_char_o > nat > list_char > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    comple1675732732025788397_nat_o: set_nat_nat_o > nat > nat > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Nat__Onat_M_Eo_J,type,
    comple6214475593288795910_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    comple6265855634338933877_nat_o: set_Pr5582243495563764594_nat_o > product_prod_nat_nat > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001_Eo,type,
    complete_Inf_Inf_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    comple6612786467464916249er_nat: set_filter_nat > filter_nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    comple4824088763117934073t_char: set_se253819897320413138t_char > set_Pr2507339680178222706t_char ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple1930127330929219688at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    comple1065008630642458357et_nat: set_set_set_nat > set_set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    comple724453453808405122char_o: set_nat_list_char_o > nat > list_char > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    comple3592611370556534995_nat_o: set_nat_nat_o > nat > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
    comple8317665133742190828_nat_o: set_nat_o > nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    comple1901448594430626575_nat_o: set_Pr5582243495563764594_nat_o > product_prod_nat_nat > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    comple2454491259644266675er_nat: set_filter_nat > filter_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    comple9178978156834306527t_char: set_se253819897320413138t_char > set_Pr2507339680178222706t_char ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple5685304695842803022at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).

thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    comple787645200385910945er_nat: ( filter_nat > filter_nat ) > filter_nat > $o ).

thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp_001t__Set__Oset_It__Nat__Onat_J,type,
    comple8270374135192184263et_nat: ( set_nat > set_nat ) > set_nat > $o ).

thf(sy_c_Complete__Partial__Order_Ochain_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    comple7182774465050373296er_nat: ( filter_nat > filter_nat > $o ) > set_filter_nat > $o ).

thf(sy_c_Complete__Partial__Order_Ochain_001t__Nat__Onat,type,
    comple7016393980872852640in_nat: ( nat > nat > $o ) > set_nat > $o ).

thf(sy_c_Complete__Partial__Order_Ochain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comple3588573705683770417at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Complete__Partial__Order_Ochain_001t__Set__Oset_It__Nat__Onat_J,type,
    comple675287286122868822et_nat: ( set_nat > set_nat > $o ) > set_set_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    condit4029664723990530684er_nat: set_filter_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
    condit5477540289124974626et_nat: set_set_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    condit3133331395780078992er_nat: set_filter_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Nat__Onat,type,
    condit1738341127787009408ow_nat: set_nat > $o ).

thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below_001t__Set__Oset_It__Nat__Onat_J,type,
    condit68592940725977398et_nat: set_set_nat > $o ).

thf(sy_c_Equiv__Relations_Ocongruent2_001t__Nat__Onat_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_4377714518549004389et_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > ( nat > nat > set_nat ) > $o ).

thf(sy_c_Equiv__Relations_Ocongruent2_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_2960386318186113850et_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > ( nat > product_prod_nat_nat > set_nat ) > $o ).

thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_6930685768344799580et_nat: set_Pr8693737435421807431at_nat > set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat > set_nat ) > $o ).

thf(sy_c_Equiv__Relations_Ocongruent2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_4637926981076332931et_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > product_prod_nat_nat > set_nat ) > $o ).

thf(sy_c_Equiv__Relations_Ocongruent_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_4704879099140956910et_nat: set_Pr1261947904930325089at_nat > ( nat > set_nat ) > $o ).

thf(sy_c_Equiv__Relations_Ocongruent_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    equiv_7243916823904213361et_nat: set_Pr8693737435421807431at_nat > ( product_prod_nat_nat > set_nat ) > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
    equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__PosRat__Oprat,type,
    equiv_equiv_prat: set_prat > set_Pr3298805672376727137t_prat > $o ).

thf(sy_c_Equiv__Relations_Oequiv_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    equiv_5407799809235660215at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Equiv__Relations_Oproj_001t__Nat__Onat_001t__Nat__Onat,type,
    equiv_proj_nat_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Equiv__Relations_Oproj_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    equiv_proj_prat_prat: set_Pr3298805672376727137t_prat > prat > set_prat ).

thf(sy_c_Equiv__Relations_Oproj_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    equiv_7265450603036132554at_nat: set_Pr8693737435421807431at_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Equiv__Relations_Oquotient_001t__Nat__Onat,type,
    equiv_quotient_nat: set_nat > set_Pr1261947904930325089at_nat > set_set_nat ).

thf(sy_c_Equiv__Relations_Oquotient_001t__PosRat__Oprat,type,
    equiv_quotient_prat: set_prat > set_Pr3298805672376727137t_prat > set_set_prat ).

thf(sy_c_Equiv__Relations_Oquotient_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    equiv_3811336339175339080at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > set_se7855581050983116737at_nat ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Set__Oset_It__Nat__Onat_J,type,
    eventually_set_nat: ( set_nat > $o ) > filter_set_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    filter3212408913953519116et_nat: ( nat > set_nat ) > filter_set_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    monoto7809364549836873959t_char: set_se253819897320413138t_char > ( set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > $o ) > ( set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > $o ) > ( set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ) > $o ).

thf(sy_c_Fun_Omonotone__on_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    monoto3897784414697140647at_nat: set_se7855581050983116737at_nat > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > $o ).

thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
    the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_in1052143192868928060at_nat: set_nat > ( nat > product_prod_nat_nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    the_in5057678521256355851et_nat: set_nat > ( nat > set_nat ) > set_nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    minus_4207664762107033000at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
    uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    uminus8676089048583255045_nat_o: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Nat__Onat_001t__Nat__Onat,type,
    groups1986416967739987077at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    groups3963206862555359278at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
    the_nat: ( nat > $o ) > nat ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    the_Pr202245514940600089t_char: ( produc1457211279475724562t_char > $o ) > produc1457211279475724562t_char ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr7557018466319803784at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
    uniq_nat: ( nat > $o ) > $o ).

thf(sy_c_HOL_OUniq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    uniq_P1720416203664110854at_nat: ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_HOL_Oundefined_001_Eo,type,
    undefined_o: $o ).

thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
    undefined_nat: nat ).

thf(sy_c_HOL_Oundefined_001t__PosRat__Oprat,type,
    undefined_prat: prat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__PosRat__Oprat,type,
    if_prat: $o > prat > prat > prat ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    if_set1248378584462554936t_char: $o > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).

thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    if_set4744415642439091239at_nat: $o > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    comple3856628597079623266er_nat: ( filter_nat > filter_nat ) > filter_nat ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Nat__Onat_J,type,
    comple1596078789208929544et_nat: ( set_nat > set_nat ) > set_nat ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple8166066674555273823at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001_062_It__Filter__Ofilter_It__Nat__Onat_J_M_Eo_J,type,
    comple6377367993240666408_nat_o: ( ( filter_nat > $o ) > filter_nat > $o ) > filter_nat > $o ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    comple7308811355306212098_nat_o: ( ( set_nat > $o ) > set_nat > $o ) > set_nat > $o ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    comple7665627409031812199er_nat: ( filter_nat > filter_nat ) > filter_nat ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7975543026063415949et_nat: ( set_nat > set_nat ) > set_nat ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    comple3227850406620335403t_char: ( set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    comple1738246912304005786at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    inf_in8805929356528176899char_o: ( nat > list_char > $o ) > ( nat > list_char > $o ) > nat > list_char > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    inf_inf_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
    inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    inf_in5163264567034779214_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    inf_inf_filter_nat: filter_nat > filter_nat > filter_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_in7187749990401937269at_nat: filter1242075044329608583at_nat > filter1242075044329608583at_nat > filter1242075044329608583at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Filter__Ofilter_It__Set__Oset_It__Nat__Onat_J_J,type,
    inf_in335088806127669288et_nat: filter_set_nat > filter_set_nat > filter_set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    inf_in3544766959038609504t_char: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    sup_su1536542383718553321char_o: ( nat > list_char > $o ) > ( nat > list_char > $o ) > nat > list_char > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
    sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    sup_su798857527126471912_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    sup_su7899656352754981958t_char: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).

thf(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 > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(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 ).

thf(sy_c_List_OBleast_001t__Nat__Onat,type,
    bleast_nat: set_nat > ( nat > $o ) > nat ).

thf(sy_c_List_Oabort__Bleast_001t__Nat__Onat,type,
    abort_Bleast_nat: set_nat > ( nat > $o ) > nat ).

thf(sy_c_Mask_Oadd__acc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    add_ac6043276672051819257t_char: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat > produc1457211279475724562t_char > prat ).

thf(sy_c_Mask_Oadd__masks_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    add_ma6586698345353345811t_char: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ).

thf(sy_c_Mask_Oadd__masks__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    add_ma2412268806269088278t_char: produc5803078220529002682t_char > produc5803078220529002682t_char > $o ).

thf(sy_c_Mask_Oempty__bmask_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    empty_7634413185502380953r_prat: ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Mask_Oempty__bmask__rel_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    empty_2051610590842377682r_prat: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Mask_Oempty__mask_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    empty_6616431780912394577r_prat: ( produc1457211279475724562t_char > prat ) > prat ).

thf(sy_c_Mask_Oempty__mask_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    empty_3446695950879338768t_char: produc1457211279475724562t_char > prat ).

thf(sy_c_Mask_Oempty__mask__rel_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    empty_3412554051312619034r_prat: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Mask_Oempty__mask__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    empty_8141283858255108185t_char: produc1457211279475724562t_char > produc1457211279475724562t_char > $o ).

thf(sy_c_Mask_Oequal__on__mask_001tf__a_001tf__b,type,
    equal_on_mask_a_b: ( a > prat ) > ( a > b ) > ( a > b ) > $o ).

thf(sy_c_Mask_Ofull__mask,type,
    full_mask: produc1457211279475724562t_char > prat ).

thf(sy_c_Mask_Ogreater__mask_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    greate7680731506090375811t_char: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Mask_Omultiply__mask,type,
    multiply_mask: prat > ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ).

thf(sy_c_Mask_Onull_001t__Nat__Onat,type,
    null_nat: nat ).

thf(sy_c_Mask_Opre__get__m_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    pre_ge2740421702909910421t_char: produc1457211279475724562t_char > nat ).

thf(sy_c_Mask_Opre__get__m__rel_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    pre_ge8748858970926381898t_char: produc1457211279475724562t_char > produc1457211279475724562t_char > $o ).

thf(sy_c_Mask_Ouni__mask_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    uni_ma6578582744724537101t_char: produc1457211279475724562t_char > prat > produc1457211279475724562t_char > prat ).

thf(sy_c_Mask_Ouni__mask__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    uni_ma1788645471123007132t_char: produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o ).

thf(sy_c_Mask_Oupper__bounded,type,
    upper_bounded: ( produc1457211279475724562t_char > prat ) > prat > $o ).

thf(sy_c_Mask_Oval_OAddress,type,
    address: nat > val ).

thf(sy_c_Mask_Oval_OBool,type,
    bool: $o > val ).

thf(sy_c_Mask_Oval_ORat,type,
    rat: prat > val ).

thf(sy_c_Mask_Oval_Ocase__val_001_Eo,type,
    case_val_o: ( $o > $o ) > ( nat > $o ) > ( prat > $o ) > val > $o ).

thf(sy_c_Mask_Oval_Ocase__val_001t__Nat__Onat,type,
    case_val_nat: ( $o > nat ) > ( nat > nat ) > ( prat > nat ) > val > nat ).

thf(sy_c_Mask_Oval_Ocase__val_001t__PosRat__Oprat,type,
    case_val_prat: ( $o > prat ) > ( nat > prat ) > ( prat > prat ) > val > prat ).

thf(sy_c_Mask_Oval_Ois__Address,type,
    is_Address: val > $o ).

thf(sy_c_Mask_Oval_Ois__Bool,type,
    is_Bool: val > $o ).

thf(sy_c_Mask_Oval_Ois__Rat,type,
    is_Rat: val > $o ).

thf(sy_c_Mask_Oval_Osize__val,type,
    size_val: val > nat ).

thf(sy_c_Mask_Oval_Othe__address,type,
    the_address: val > nat ).

thf(sy_c_Mask_Oval_Othe__bool,type,
    the_bool: val > $o ).

thf(sy_c_Mask_Oval_Othe__rat,type,
    the_rat: val > prat ).

thf(sy_c_Mask_Ovalid__mask,type,
    valid_mask: ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Mask_Ovalid__mask__rel,type,
    valid_mask_rel: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Mask_Ovalid__null,type,
    valid_null: ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Ocompow_001_062_It__Filter__Ofilter_It__Nat__Onat_J_Mt__Filter__Ofilter_It__Nat__Onat_J_J,type,
    compow8527595912636063980er_nat: nat > ( filter_nat > filter_nat ) > filter_nat > filter_nat ).

thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    compow_nat_nat_o: nat > ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Ocompow_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    compow6694791279094763314at_nat: nat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Nat_Ocompow_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    compow8708494347934031032et_nat: nat > ( set_nat > set_nat ) > set_nat > set_nat ).

thf(sy_c_Nat_Ocompow_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    compow1820118003163928468at_nat: nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Mask__Oval,type,
    size_size_val: val > nat ).

thf(sy_c_Order__Relation_OaboveS_001t__Nat__Onat,type,
    order_aboveS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Order__Relation_OaboveS_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    order_4940263126897443436at_nat: set_Pr8693737435421807431at_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Order__Relation_Oabove_001t__Nat__Onat,type,
    order_above_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Order__Relation_Oabove_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    order_793026671032835073at_nat: set_Pr8693737435421807431at_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Order__Relation_Orelation__of_001t__Nat__Onat,type,
    order_1962693441708344834of_nat: ( nat > nat > $o ) > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    bot_bo8065403328713304961char_o: nat > list_char > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bot_nat_nat_o: nat > nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bot_set_nat_o: set_nat_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bot_set_nat_nat: set_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__PosRat__Oprat_J_J,type,
    bot_bot_set_nat_prat: set_nat_prat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
    bot_bo498966703094740906er_nat: set_filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
    bot_bo8503570744482963627t_char: set_list_char ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__PosRat__Oprat_J,type,
    bot_bot_set_prat: set_prat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    bot_bo1414094081888755934t_char: set_Pr2507339680178222706t_char ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_Least_filter_nat: ( filter_nat > $o ) > filter_nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_Least_set_nat: ( set_nat > $o ) > set_nat ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_less_filter_nat: filter_nat > filter_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J_J,type,
    ord_le6723339807950303054prat_o: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J,type,
    ord_le2830794348966088778prat_o: ( ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Filter__Ofilter_It__Nat__Onat_J_M_Eo_J,type,
    ord_le2350800233650606735_nat_o: ( filter_nat > $o ) > ( filter_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    ord_le2712619360209801397char_o: ( nat > list_char > $o ) > ( nat > list_char > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le5085569704386948391at_nat: filter1242075044329608583at_nat > filter1242075044329608583at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_J,type,
    ord_le3387867619833322002ar_nat: set_Pr8803067691650843250ar_nat > set_Pr8803067691650843250ar_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    ord_le6315511645215477266t_char: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    ord_le3845944159117341623et_nat: set_Pr400265656397884439et_nat > set_Pr400265656397884439et_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_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,
    ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    ord_le4402255868550298733et_nat: set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    orderi2807725203697798511er_nat: ( filter_nat > filter_nat > $o ) > ( filter_nat > filter_nat > $o ) > filter_nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__Nat__Onat_J,type,
    ordering_top_set_nat: ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > set_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    top_to1537031869075220069char_o: nat > list_char > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    top_top_nat_nat_o: nat > nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    top_top_filter_nat: filter_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Filter__Ofilter_It__Nat__Onat_J_M_Eo_J_J,type,
    top_to8962845839458735263_nat_o: set_filter_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_top_set_nat_nat: set_nat_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
    top_to28523156410905465_nat_o: set_set_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
    top_to4409321577746088390er_nat: set_filter_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
    top_to3190553582279535303t_char: set_list_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    top_to1250346684925019074t_char: set_Pr2507339680178222706t_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_top_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    top_to6005184720552305954t_char: set_se253819897320413138t_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    top_to7629004291339433233at_nat: set_se7855581050983116737at_nat ).

thf(sy_c_Partial__Function_Oflat__lub_001t__Nat__Onat,type,
    partial_flat_lub_nat: nat > set_nat > nat ).

thf(sy_c_Partial__Function_Oflat__lub_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    partia4823134289123629261at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > product_prod_nat_nat ).

thf(sy_c_PosRat_Ocomp__one,type,
    comp_one: prat > prat ).

thf(sy_c_PosRat_Ohalf,type,
    half: prat ).

thf(sy_c_PosRat_Opadd,type,
    padd: prat > prat > prat ).

thf(sy_c_PosRat_Opdiv,type,
    pdiv: prat > prat > prat ).

thf(sy_c_PosRat_Opgt,type,
    pgt: prat > prat > $o ).

thf(sy_c_PosRat_Opgte,type,
    pgte: prat > prat > $o ).

thf(sy_c_PosRat_Opinv,type,
    pinv: prat > prat ).

thf(sy_c_PosRat_Opmax,type,
    pmax: prat > prat > prat ).

thf(sy_c_PosRat_Opmin,type,
    pmin: prat > prat > prat ).

thf(sy_c_PosRat_Opmult,type,
    pmult: prat > prat > prat ).

thf(sy_c_PosRat_Opnone,type,
    pnone: prat ).

thf(sy_c_PosRat_Oppos,type,
    ppos: prat > $o ).

thf(sy_c_PosRat_Opwrite,type,
    pwrite: prat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    produc1164665353665699927r_prat: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > produc3336119888008106215r_prat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    produc2957999048406202538t_char: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > produc5803078220529002682t_char ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    produc2591414526589101846t_char: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > produc2489117125269924006t_char ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    produc1606739095562851931t_char: list_char > list_char > produc6859378861088059299t_char ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    produc8702345860417625034ar_nat: list_char > nat > produc7606469099237612818ar_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    produc120197956887798346t_char: nat > list_char > produc1457211279475724562t_char ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    image_8573581984095792621er_nat: ( set_nat > filter_nat ) > set_set_nat > set_filter_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    image_7123945608888165160char_o: ( set_Pr2507339680178222706t_char > nat > list_char > $o ) > set_se253819897320413138t_char > set_nat_list_char_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    image_6923511907955291850_nat_o: ( set_Pr1261947904930325089at_nat > nat > nat > $o ) > set_se7855581050983116737at_nat > set_nat_nat_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    image_5253903537071781336_nat_o: ( set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o ) > set_se7855581050983116737at_nat > set_Pr5582243495563764594_nat_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
    image_7457375456213313148_nat_o: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_3778355557410424805at_nat: ( set_Pr1261947904930325089at_nat > product_prod_nat_nat ) > set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_5853224409075402146et_nat: ( set_Pr1261947904930325089at_nat > set_nat ) > set_se7855581050983116737at_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_M_Eo_J,type,
    insert_nat_o: ( nat > $o ) > set_nat_o > set_nat_o ).

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__PosRat__Oprat_J,type,
    insert_nat_prat: ( nat > prat ) > set_nat_prat > set_nat_prat ).

thf(sy_c_Set_Oinsert_001t__List__Olist_It__String__Ochar_J,type,
    insert_list_char: list_char > set_list_char > set_list_char ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    insert3372640069787401954t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vimage8013328719654469172at_nat: ( nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    vimage_nat_set_nat: ( nat > set_nat ) > set_set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat,type,
    vimage2352271050577936999ar_nat: ( produc1457211279475724562t_char > nat ) > set_nat > set_Pr2507339680178222706t_char ).

thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    vimage4653281326611754070at_nat: ( product_prod_nat_nat > nat ) > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Ovimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vimage2449269961533847803at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    transi8609417484261754244at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Oacc_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    acc_Pr1479980713244937752r_prat: set_Pr1685704261205283783r_prat > set_Pr3433046674199179443r_prat ).

thf(sy_c_Wellfounded_Oacc_001t__Nat__Onat,type,
    acc_nat: set_Pr1261947904930325089at_nat > set_nat ).

thf(sy_c_Wellfounded_Oacc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    acc_Pr2806007504492836550at_nat: set_Pr8693737435421807431at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Oaccp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    accp_P4765339447491148764r_prat: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
    accp_P224314924008452035t_char: ( produc5803078220529002682t_char > produc5803078220529002682t_char > $o ) > produc5803078220529002682t_char > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    accp_P8892882183480857371t_char: ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > produc1457211279475724562t_char > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    accp_P8956218777217113800r_prat: ( produc3933091914578511633r_prat > produc3933091914578511633r_prat > $o ) > produc3933091914578511633r_prat > $o ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    lex_pr1681056561245511266t_char: set_Pr1261947904930325089at_nat > set_Pr4480099295215233283t_char > set_Pr3996677194857090567t_char ).

thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
    lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Wellfounded_Omax__ext_001t__Nat__Onat,type,
    max_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Omax__extp_001t__Nat__Onat,type,
    max_extp_nat: ( nat > nat > $o ) > set_nat > set_nat > $o ).

thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Omlex__prod_001t__List__Olist_It__String__Ochar_J,type,
    mlex_prod_list_char: ( list_char > nat ) > set_Pr4480099295215233283t_char > set_Pr4480099295215233283t_char ).

thf(sy_c_Wellfounded_Omlex__prod_001t__Nat__Onat,type,
    mlex_prod_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Nat__Onat,type,
    same_fst_nat_nat: ( nat > $o ) > ( nat > set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_fChoice_001t__Nat__Onat,type,
    fChoice_nat: ( nat > $o ) > nat ).

thf(sy_c_fChoice_001t__PosRat__Oprat,type,
    fChoice_prat: ( prat > $o ) > prat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    fChoic2476838209665704038t_char: ( produc1457211279475724562t_char > $o ) > produc1457211279475724562t_char ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fChoic6978938873391328853at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    member3158901901048444578t_char: ( nat > produc1457211279475724562t_char ) > set_na6246771325765140825t_char > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member6743353555990375057at_nat: ( nat > product_prod_nat_nat ) > set_na8845969913329135560at_nat > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    member2336342582916103932r_prat: ( produc1457211279475724562t_char > prat ) > set_Pr3433046674199179443r_prat > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member7868840539957676139at_nat: ( product_prod_nat_nat > nat ) > set_Pr3667225130493526178at_nat > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member_set_nat_nat: ( set_nat > nat ) > set_set_nat_nat > $o ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    member_filter_nat: filter_nat > set_filter_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
    member_list_char: list_char > set_list_char > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__PosRat__Oprat,type,
    member_prat: prat > set_prat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    member1381939458143960592r_prat: produc3336119888008106215r_prat > set_Pr1685704261205283783r_prat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    member6023471406134104908t_char: produc6859378861088059299t_char > set_Pr4480099295215233283t_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J,type,
    member6633493567087309371ar_nat: produc7606469099237612818ar_nat > set_Pr8803067691650843250ar_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    member484235747325421115t_char: produc1457211279475724562t_char > set_Pr2507339680178222706t_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2223272150424702269at_nat: produc7248412053542808358at_nat > set_Pr7717912310451564380at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    member151853725258021968t_char: produc6071853487833700903t_char > set_Pr3996677194857090567t_char > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member3348759134392003351at_nat: produc8373899037510109440at_nat > set_Pr2539167527615954998at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Filter__Ofilter_It__Nat__Onat_J_J,type,
    member8572727008022913325er_nat: set_filter_nat > set_set_filter_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__PosRat__Oprat_J,type,
    member_set_prat: set_prat > set_set_prat > $o ).

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

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

thf(sy_v__092_060pi_062,type,
    pi: a > prat ).

thf(sy_v_a,type,
    a2: a > b ).

thf(sy_v_b,type,
    b2: a > b ).

% Relevant facts (1271)
thf(fact_0__092_060open_062_092_060And_062b_Aa_O_Aequal__on__mask_A_092_060pi_062_Aa_Ab_A_092_060Longrightarrow_062_Aequal__on__mask_A_092_060pi_062_Ab_Aa_092_060close_062,axiom,
    ! [A: a > b,B: a > b] :
      ( ( equal_on_mask_a_b @ pi @ A @ B )
     => ( equal_on_mask_a_b @ pi @ B @ A ) ) ).

% \<open>\<And>b a. equal_on_mask \<pi> a b \<Longrightarrow> equal_on_mask \<pi> b a\<close>
thf(fact_1_equal__on__mask__def,axiom,
    ( equal_on_mask_a_b
    = ( ^ [Pi: a > prat,H: a > b,H2: a > b] :
        ! [Hl: a] :
          ( ( ppos @ ( Pi @ Hl ) )
         => ( ( H @ Hl )
            = ( H2 @ Hl ) ) ) ) ) ).

% equal_on_mask_def
thf(fact_2_val_Oinject_I3_J,axiom,
    ! [X3: prat,Y3: prat] :
      ( ( ( rat @ X3 )
        = ( rat @ Y3 ) )
      = ( X3 = Y3 ) ) ).

% val.inject(3)
thf(fact_3_val_Oinject_I2_J,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( address @ X2 )
        = ( address @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% val.inject(2)
thf(fact_4_val_Oinject_I1_J,axiom,
    ! [X1: $o,Y1: $o] :
      ( ( ( bool @ X1 )
        = ( bool @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% val.inject(1)
thf(fact_5_val_Odistinct_I5_J,axiom,
    ! [X2: nat,X3: prat] :
      ( ( address @ X2 )
     != ( rat @ X3 ) ) ).

% val.distinct(5)
thf(fact_6_val_Odistinct_I3_J,axiom,
    ! [X1: $o,X3: prat] :
      ( ( bool @ X1 )
     != ( rat @ X3 ) ) ).

% val.distinct(3)
thf(fact_7_val_Odistinct_I1_J,axiom,
    ! [X1: $o,X2: nat] :
      ( ( bool @ X1 )
     != ( address @ X2 ) ) ).

% val.distinct(1)
thf(fact_8_val_Oexhaust,axiom,
    ! [Y: val] :
      ( ! [X12: $o] :
          ( Y
         != ( bool @ X12 ) )
     => ( ! [X22: nat] :
            ( Y
           != ( address @ X22 ) )
       => ~ ! [X32: prat] :
              ( Y
             != ( rat @ X32 ) ) ) ) ).

% val.exhaust
thf(fact_9_val_Osel_I3_J,axiom,
    ! [X3: prat] :
      ( ( the_rat @ ( rat @ X3 ) )
      = X3 ) ).

% val.sel(3)
thf(fact_10_val_Osel_I2_J,axiom,
    ! [X2: nat] :
      ( ( the_address @ ( address @ X2 ) )
      = X2 ) ).

% val.sel(2)
thf(fact_11_val_Osel_I1_J,axiom,
    ! [X1: $o] :
      ( ( the_bool @ ( bool @ X1 ) )
      = X1 ) ).

% val.sel(1)
thf(fact_12_val_Osimps_I12_J,axiom,
    ! [F1: $o > $o,F2: nat > $o,F3: prat > $o,X3: prat] :
      ( ( case_val_o @ F1 @ F2 @ F3 @ ( rat @ X3 ) )
      = ( F3 @ X3 ) ) ).

% val.simps(12)
thf(fact_13_val_Osimps_I12_J,axiom,
    ! [F1: $o > nat,F2: nat > nat,F3: prat > nat,X3: prat] :
      ( ( case_val_nat @ F1 @ F2 @ F3 @ ( rat @ X3 ) )
      = ( F3 @ X3 ) ) ).

% val.simps(12)
thf(fact_14_val_Osimps_I12_J,axiom,
    ! [F1: $o > prat,F2: nat > prat,F3: prat > prat,X3: prat] :
      ( ( case_val_prat @ F1 @ F2 @ F3 @ ( rat @ X3 ) )
      = ( F3 @ X3 ) ) ).

% val.simps(12)
thf(fact_15_val_Osimps_I11_J,axiom,
    ! [F1: $o > $o,F2: nat > $o,F3: prat > $o,X2: nat] :
      ( ( case_val_o @ F1 @ F2 @ F3 @ ( address @ X2 ) )
      = ( F2 @ X2 ) ) ).

% val.simps(11)
thf(fact_16_val_Osimps_I11_J,axiom,
    ! [F1: $o > nat,F2: nat > nat,F3: prat > nat,X2: nat] :
      ( ( case_val_nat @ F1 @ F2 @ F3 @ ( address @ X2 ) )
      = ( F2 @ X2 ) ) ).

% val.simps(11)
thf(fact_17_val_Osimps_I11_J,axiom,
    ! [F1: $o > prat,F2: nat > prat,F3: prat > prat,X2: nat] :
      ( ( case_val_prat @ F1 @ F2 @ F3 @ ( address @ X2 ) )
      = ( F2 @ X2 ) ) ).

% val.simps(11)
thf(fact_18_val_Osimps_I10_J,axiom,
    ! [F1: $o > $o,F2: nat > $o,F3: prat > $o,X1: $o] :
      ( ( case_val_o @ F1 @ F2 @ F3 @ ( bool @ X1 ) )
      = ( F1 @ X1 ) ) ).

% val.simps(10)
thf(fact_19_val_Osimps_I10_J,axiom,
    ! [F1: $o > nat,F2: nat > nat,F3: prat > nat,X1: $o] :
      ( ( case_val_nat @ F1 @ F2 @ F3 @ ( bool @ X1 ) )
      = ( F1 @ X1 ) ) ).

% val.simps(10)
thf(fact_20_val_Osimps_I10_J,axiom,
    ! [F1: $o > prat,F2: nat > prat,F3: prat > prat,X1: $o] :
      ( ( case_val_prat @ F1 @ F2 @ F3 @ ( bool @ X1 ) )
      = ( F1 @ X1 ) ) ).

% val.simps(10)
thf(fact_21_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B2 @ A2 ) ) ) ).

% add_masks_comm
thf(fact_22_add__masks__asso,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B ) @ C )
      = ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) ) ) ).

% add_masks_asso
thf(fact_23_val_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F1: $o > $o,F2: nat > $o,F3: prat > $o,Val: val] :
      ( ( P @ ( case_val_o @ F1 @ F2 @ F3 @ Val ) )
      = ( ~ ( ( ( Val
                = ( bool @ ( the_bool @ Val ) ) )
              & ~ ( P @ ( F1 @ ( the_bool @ Val ) ) ) )
            | ( ( Val
                = ( address @ ( the_address @ Val ) ) )
              & ~ ( P @ ( F2 @ ( the_address @ Val ) ) ) )
            | ( ( Val
                = ( rat @ ( the_rat @ Val ) ) )
              & ~ ( P @ ( F3 @ ( the_rat @ Val ) ) ) ) ) ) ) ).

% val.split_sel_asm
thf(fact_24_val_Osplit__sel__asm,axiom,
    ! [P: nat > $o,F1: $o > nat,F2: nat > nat,F3: prat > nat,Val: val] :
      ( ( P @ ( case_val_nat @ F1 @ F2 @ F3 @ Val ) )
      = ( ~ ( ( ( Val
                = ( bool @ ( the_bool @ Val ) ) )
              & ~ ( P @ ( F1 @ ( the_bool @ Val ) ) ) )
            | ( ( Val
                = ( address @ ( the_address @ Val ) ) )
              & ~ ( P @ ( F2 @ ( the_address @ Val ) ) ) )
            | ( ( Val
                = ( rat @ ( the_rat @ Val ) ) )
              & ~ ( P @ ( F3 @ ( the_rat @ Val ) ) ) ) ) ) ) ).

% val.split_sel_asm
thf(fact_25_val_Osplit__sel__asm,axiom,
    ! [P: prat > $o,F1: $o > prat,F2: nat > prat,F3: prat > prat,Val: val] :
      ( ( P @ ( case_val_prat @ F1 @ F2 @ F3 @ Val ) )
      = ( ~ ( ( ( Val
                = ( bool @ ( the_bool @ Val ) ) )
              & ~ ( P @ ( F1 @ ( the_bool @ Val ) ) ) )
            | ( ( Val
                = ( address @ ( the_address @ Val ) ) )
              & ~ ( P @ ( F2 @ ( the_address @ Val ) ) ) )
            | ( ( Val
                = ( rat @ ( the_rat @ Val ) ) )
              & ~ ( P @ ( F3 @ ( the_rat @ Val ) ) ) ) ) ) ) ).

% val.split_sel_asm
thf(fact_26_val_Osplit__sel,axiom,
    ! [P: $o > $o,F1: $o > $o,F2: nat > $o,F3: prat > $o,Val: val] :
      ( ( P @ ( case_val_o @ F1 @ F2 @ F3 @ Val ) )
      = ( ( ( Val
            = ( bool @ ( the_bool @ Val ) ) )
         => ( P @ ( F1 @ ( the_bool @ Val ) ) ) )
        & ( ( Val
            = ( address @ ( the_address @ Val ) ) )
         => ( P @ ( F2 @ ( the_address @ Val ) ) ) )
        & ( ( Val
            = ( rat @ ( the_rat @ Val ) ) )
         => ( P @ ( F3 @ ( the_rat @ Val ) ) ) ) ) ) ).

% val.split_sel
thf(fact_27_val_Osplit__sel,axiom,
    ! [P: nat > $o,F1: $o > nat,F2: nat > nat,F3: prat > nat,Val: val] :
      ( ( P @ ( case_val_nat @ F1 @ F2 @ F3 @ Val ) )
      = ( ( ( Val
            = ( bool @ ( the_bool @ Val ) ) )
         => ( P @ ( F1 @ ( the_bool @ Val ) ) ) )
        & ( ( Val
            = ( address @ ( the_address @ Val ) ) )
         => ( P @ ( F2 @ ( the_address @ Val ) ) ) )
        & ( ( Val
            = ( rat @ ( the_rat @ Val ) ) )
         => ( P @ ( F3 @ ( the_rat @ Val ) ) ) ) ) ) ).

% val.split_sel
thf(fact_28_val_Osplit__sel,axiom,
    ! [P: prat > $o,F1: $o > prat,F2: nat > prat,F3: prat > prat,Val: val] :
      ( ( P @ ( case_val_prat @ F1 @ F2 @ F3 @ Val ) )
      = ( ( ( Val
            = ( bool @ ( the_bool @ Val ) ) )
         => ( P @ ( F1 @ ( the_bool @ Val ) ) ) )
        & ( ( Val
            = ( address @ ( the_address @ Val ) ) )
         => ( P @ ( F2 @ ( the_address @ Val ) ) ) )
        & ( ( Val
            = ( rat @ ( the_rat @ Val ) ) )
         => ( P @ ( F3 @ ( the_rat @ Val ) ) ) ) ) ) ).

% val.split_sel
thf(fact_29_val_Oexhaust__sel,axiom,
    ! [Val: val] :
      ( ( Val
       != ( bool @ ( the_bool @ Val ) ) )
     => ( ( Val
         != ( address @ ( the_address @ Val ) ) )
       => ( Val
          = ( rat @ ( the_rat @ Val ) ) ) ) ) ).

% val.exhaust_sel
thf(fact_30_val_Ocollapse_I3_J,axiom,
    ! [Val: val] :
      ( ( is_Rat @ Val )
     => ( ( rat @ ( the_rat @ Val ) )
        = Val ) ) ).

% val.collapse(3)
thf(fact_31_val_Ocollapse_I1_J,axiom,
    ! [Val: val] :
      ( ( is_Bool @ Val )
     => ( ( bool @ ( the_bool @ Val ) )
        = Val ) ) ).

% val.collapse(1)
thf(fact_32_val_Ocollapse_I2_J,axiom,
    ! [Val: val] :
      ( ( is_Address @ Val )
     => ( ( address @ ( the_address @ Val ) )
        = Val ) ) ).

% val.collapse(2)
thf(fact_33_val_Ocase__eq__if,axiom,
    ( case_val_o
    = ( ^ [F12: $o > $o,F22: nat > $o,F32: prat > $o,Val2: val] :
          ( ( ( is_Bool @ Val2 )
           => ( F12 @ ( the_bool @ Val2 ) ) )
          & ( ~ ( is_Bool @ Val2 )
           => ( ( ( is_Address @ Val2 )
               => ( F22 @ ( the_address @ Val2 ) ) )
              & ( ~ ( is_Address @ Val2 )
               => ( F32 @ ( the_rat @ Val2 ) ) ) ) ) ) ) ) ).

% val.case_eq_if
thf(fact_34_val_Ocase__eq__if,axiom,
    ( case_val_nat
    = ( ^ [F12: $o > nat,F22: nat > nat,F32: prat > nat,Val2: val] : ( if_nat @ ( is_Bool @ Val2 ) @ ( F12 @ ( the_bool @ Val2 ) ) @ ( if_nat @ ( is_Address @ Val2 ) @ ( F22 @ ( the_address @ Val2 ) ) @ ( F32 @ ( the_rat @ Val2 ) ) ) ) ) ) ).

% val.case_eq_if
thf(fact_35_val_Ocase__eq__if,axiom,
    ( case_val_prat
    = ( ^ [F12: $o > prat,F22: nat > prat,F32: prat > prat,Val2: val] : ( if_prat @ ( is_Bool @ Val2 ) @ ( F12 @ ( the_bool @ Val2 ) ) @ ( if_prat @ ( is_Address @ Val2 ) @ ( F22 @ ( the_address @ Val2 ) ) @ ( F32 @ ( the_rat @ Val2 ) ) ) ) ) ) ).

% val.case_eq_if
thf(fact_36_minus__empty,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] :
      ( Pi2
      = ( add_ma6586698345353345811t_char @ Pi2 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_37_is__Rat__def,axiom,
    ( is_Rat
    = ( ^ [Val2: val] :
        ? [X33: prat] :
          ( Val2
          = ( rat @ X33 ) ) ) ) ).

% is_Rat_def
thf(fact_38_val_OdiscI_I3_J,axiom,
    ! [Val: val,X3: prat] :
      ( ( Val
        = ( rat @ X3 ) )
     => ( is_Rat @ Val ) ) ).

% val.discI(3)
thf(fact_39_val_Odisc_I7_J,axiom,
    ! [X1: $o] :
      ~ ( is_Rat @ ( bool @ X1 ) ) ).

% val.disc(7)
thf(fact_40_val_Odisc_I8_J,axiom,
    ! [X2: nat] :
      ~ ( is_Rat @ ( address @ X2 ) ) ).

% val.disc(8)
thf(fact_41_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_46_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_47_val_Odisc_I9_J,axiom,
    ! [X3: prat] : ( is_Rat @ ( rat @ X3 ) ) ).

% val.disc(9)
thf(fact_48_val_Odistinct__disc_I6_J,axiom,
    ! [Val: val] :
      ( ( is_Rat @ Val )
     => ~ ( is_Address @ Val ) ) ).

% val.distinct_disc(6)
thf(fact_49_val_Odistinct__disc_I5_J,axiom,
    ! [Val: val] :
      ( ( is_Address @ Val )
     => ~ ( is_Rat @ Val ) ) ).

% val.distinct_disc(5)
thf(fact_50_val_Odistinct__disc_I4_J,axiom,
    ! [Val: val] :
      ( ( is_Rat @ Val )
     => ~ ( is_Bool @ Val ) ) ).

% val.distinct_disc(4)
thf(fact_51_val_Odistinct__disc_I3_J,axiom,
    ! [Val: val] :
      ( ( is_Bool @ Val )
     => ~ ( is_Rat @ Val ) ) ).

% val.distinct_disc(3)
thf(fact_52_val_Odistinct__disc_I2_J,axiom,
    ! [Val: val] :
      ( ( is_Address @ Val )
     => ~ ( is_Bool @ Val ) ) ).

% val.distinct_disc(2)
thf(fact_53_val_Odistinct__disc_I1_J,axiom,
    ! [Val: val] :
      ( ( is_Bool @ Val )
     => ~ ( is_Address @ Val ) ) ).

% val.distinct_disc(1)
thf(fact_54_val_Oexhaust__disc,axiom,
    ! [Val: val] :
      ( ~ ( is_Bool @ Val )
     => ( ~ ( is_Address @ Val )
       => ( is_Rat @ Val ) ) ) ).

% val.exhaust_disc
thf(fact_55_val_Oexpand,axiom,
    ! [Val: val,Val3: val] :
      ( ( ( is_Bool @ Val )
        = ( is_Bool @ Val3 ) )
     => ( ( ( is_Bool @ Val )
         => ( ( is_Bool @ Val3 )
           => ( ( the_bool @ Val )
              = ( the_bool @ Val3 ) ) ) )
       => ( ( ( is_Address @ Val )
            = ( is_Address @ Val3 ) )
         => ( ( ( is_Address @ Val )
             => ( ( is_Address @ Val3 )
               => ( ( the_address @ Val )
                  = ( the_address @ Val3 ) ) ) )
           => ( ( ( is_Rat @ Val )
               => ( ( is_Rat @ Val3 )
                 => ( ( the_rat @ Val )
                    = ( the_rat @ Val3 ) ) ) )
             => ( Val = Val3 ) ) ) ) ) ) ).

% val.expand
thf(fact_56_val_Odisc_I6_J,axiom,
    ! [X3: prat] :
      ~ ( is_Address @ ( rat @ X3 ) ) ).

% val.disc(6)
thf(fact_57_val_Odisc_I5_J,axiom,
    ! [X2: nat] : ( is_Address @ ( address @ X2 ) ) ).

% val.disc(5)
thf(fact_58_val_Odisc_I4_J,axiom,
    ! [X1: $o] :
      ~ ( is_Address @ ( bool @ X1 ) ) ).

% val.disc(4)
thf(fact_59_val_Odisc_I3_J,axiom,
    ! [X3: prat] :
      ~ ( is_Bool @ ( rat @ X3 ) ) ).

% val.disc(3)
thf(fact_60_val_Odisc_I2_J,axiom,
    ! [X2: nat] :
      ~ ( is_Bool @ ( address @ X2 ) ) ).

% val.disc(2)
thf(fact_61_val_Odisc_I1_J,axiom,
    ! [X1: $o] : ( is_Bool @ ( bool @ X1 ) ) ).

% val.disc(1)
thf(fact_62_val_OdiscI_I2_J,axiom,
    ! [Val: val,X2: nat] :
      ( ( Val
        = ( address @ X2 ) )
     => ( is_Address @ Val ) ) ).

% val.discI(2)
thf(fact_63_val_OdiscI_I1_J,axiom,
    ! [Val: val,X1: $o] :
      ( ( Val
        = ( bool @ X1 ) )
     => ( is_Bool @ Val ) ) ).

% val.discI(1)
thf(fact_64_is__Bool__def,axiom,
    ( is_Bool
    = ( ^ [Val2: val] :
        ? [X13: $o] :
          ( Val2
          = ( bool @ X13 ) ) ) ) ).

% is_Bool_def
thf(fact_65_is__Address__def,axiom,
    ( is_Address
    = ( ^ [Val2: val] :
        ? [X23: nat] :
          ( Val2
          = ( address @ X23 ) ) ) ) ).

% is_Address_def
thf(fact_66_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
        ? [R: produc1457211279475724562t_char > prat] :
          ( Pi3
          = ( add_ma6586698345353345811t_char @ Pi @ R ) ) ) ) ).

% greater_mask_def
thf(fact_67_greater__mask__decomp,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( greate7680731506090375811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
     => ? [A1: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char > prat] :
          ( ( A
            = ( add_ma6586698345353345811t_char @ A1 @ A22 ) )
          & ( greate7680731506090375811t_char @ A1 @ B )
          & ( greate7680731506090375811t_char @ A22 @ C ) ) ) ).

% greater_mask_decomp
thf(fact_68_empty__mask_Oelims,axiom,
    ! [X5: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X5 )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_69_empty__mask_Osimps,axiom,
    ( empty_3446695950879338768t_char
    = ( ^ [Hl: produc1457211279475724562t_char] : pnone ) ) ).

% empty_mask.simps
thf(fact_70_add__acc__uni__mask,axiom,
    ( add_ac6043276672051819257t_char
    = ( ^ [Pi: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char,P2: prat] : ( add_ma6586698345353345811t_char @ Pi @ ( uni_ma6578582744724537101t_char @ Hl @ P2 ) ) ) ) ).

% add_acc_uni_mask
thf(fact_71_add__masks_Oelims,axiom,
    ! [X5: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X5 @ Xa @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X5 @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_72_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( padd @ ( Pi3 @ Hl ) @ ( Pi @ Hl ) ) ) ) ).

% add_masks.simps
thf(fact_73_ppos__inv,axiom,
    ! [P3: prat] :
      ( ( ppos @ P3 )
     => ( ppos @ ( pinv @ P3 ) ) ) ).

% ppos_inv
thf(fact_74_empty__bmask_Opelims_I2_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat] :
      ( ( empty_7634413185502380953r_prat @ X5 )
     => ~ ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X5 ) ) ).

% empty_bmask.pelims(2)
thf(fact_75_padd__cancellative,axiom,
    ! [A: prat,X5: prat,B: prat,Y: prat] :
      ( ( A
        = ( padd @ X5 @ B ) )
     => ( ( A
          = ( padd @ Y @ B ) )
       => ( X5 = Y ) ) ) ).

% padd_cancellative
thf(fact_76_p__greater__exists,axiom,
    ( pgte
    = ( ^ [A2: prat,B2: prat] :
        ? [R: prat] :
          ( A2
          = ( padd @ B2 @ R ) ) ) ) ).

% p_greater_exists
thf(fact_77_greater__sum__both,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( pgte @ A @ ( padd @ B @ C ) )
     => ? [A1: prat,A22: prat] :
          ( ( A
            = ( padd @ A1 @ A22 ) )
          & ( pgte @ A1 @ B )
          & ( pgte @ A22 @ C ) ) ) ).

% greater_sum_both
thf(fact_78_pgte__antisym,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pgte @ B @ A )
       => ( A = B ) ) ) ).

% pgte_antisym
thf(fact_79_padd__zero,axiom,
    ! [A: prat,B: prat] :
      ( ( pnone
        = ( padd @ A @ B ) )
      = ( ( A = pnone )
        & ( B = pnone ) ) ) ).

% padd_zero
thf(fact_80_padd__comm,axiom,
    ( padd
    = ( ^ [A2: prat,B2: prat] : ( padd @ B2 @ A2 ) ) ) ).

% padd_comm
thf(fact_81_padd__asso,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( padd @ ( padd @ A @ B ) @ C )
      = ( padd @ A @ ( padd @ B @ C ) ) ) ).

% padd_asso
thf(fact_82_pinv__inverts,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ B )
       => ( pgte @ ( pinv @ B ) @ ( pinv @ A ) ) ) ) ).

% pinv_inverts
thf(fact_83_val_Odisc__eq__case_I2_J,axiom,
    ( is_Address
    = ( case_val_o
      @ ^ [Uu: $o] : $false
      @ ^ [Uu: nat] : $true
      @ ^ [Uu: prat] : $false ) ) ).

% val.disc_eq_case(2)
thf(fact_84_val_Odisc__eq__case_I1_J,axiom,
    ( is_Bool
    = ( case_val_o
      @ ^ [Uu: $o] : $true
      @ ^ [Uu: nat] : $false
      @ ^ [Uu: prat] : $false ) ) ).

% val.disc_eq_case(1)
thf(fact_85_val_Ocase__distrib,axiom,
    ! [H3: $o > $o,F1: $o > $o,F2: nat > $o,F3: prat > $o,Val: val] :
      ( ( H3 @ ( case_val_o @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_o
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_86_val_Ocase__distrib,axiom,
    ! [H3: $o > nat,F1: $o > $o,F2: nat > $o,F3: prat > $o,Val: val] :
      ( ( H3 @ ( case_val_o @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_nat
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_87_val_Ocase__distrib,axiom,
    ! [H3: $o > prat,F1: $o > $o,F2: nat > $o,F3: prat > $o,Val: val] :
      ( ( H3 @ ( case_val_o @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_prat
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_88_val_Ocase__distrib,axiom,
    ! [H3: nat > $o,F1: $o > nat,F2: nat > nat,F3: prat > nat,Val: val] :
      ( ( H3 @ ( case_val_nat @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_o
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_89_val_Ocase__distrib,axiom,
    ! [H3: nat > nat,F1: $o > nat,F2: nat > nat,F3: prat > nat,Val: val] :
      ( ( H3 @ ( case_val_nat @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_nat
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_90_val_Ocase__distrib,axiom,
    ! [H3: nat > prat,F1: $o > nat,F2: nat > nat,F3: prat > nat,Val: val] :
      ( ( H3 @ ( case_val_nat @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_prat
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_91_val_Ocase__distrib,axiom,
    ! [H3: prat > $o,F1: $o > prat,F2: nat > prat,F3: prat > prat,Val: val] :
      ( ( H3 @ ( case_val_prat @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_o
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_92_val_Ocase__distrib,axiom,
    ! [H3: prat > nat,F1: $o > prat,F2: nat > prat,F3: prat > prat,Val: val] :
      ( ( H3 @ ( case_val_prat @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_nat
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_93_val_Ocase__distrib,axiom,
    ! [H3: prat > prat,F1: $o > prat,F2: nat > prat,F3: prat > prat,Val: val] :
      ( ( H3 @ ( case_val_prat @ F1 @ F2 @ F3 @ Val ) )
      = ( case_val_prat
        @ ^ [X: $o] : ( H3 @ ( F1 @ X ) )
        @ ^ [X: nat] : ( H3 @ ( F2 @ X ) )
        @ ^ [X: prat] : ( H3 @ ( F3 @ X ) )
        @ Val ) ) ).

% val.case_distrib
thf(fact_94_val_Odisc__eq__case_I3_J,axiom,
    ( is_Rat
    = ( case_val_o
      @ ^ [Uu: $o] : $false
      @ ^ [Uu: nat] : $false
      @ ^ [Uu: prat] : $true ) ) ).

% val.disc_eq_case(3)
thf(fact_95_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_96_ppos__eq__pnone,axiom,
    ( ppos
    = ( ^ [P2: prat] : ( P2 != pnone ) ) ) ).

% ppos_eq_pnone
thf(fact_97_empty__bmask_Opelims_I1_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( empty_7634413185502380953r_prat @ X5 )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X5 )
       => ~ ( ~ Y
           => ~ ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X5 ) ) ) ) ).

% empty_bmask.pelims(1)
thf(fact_98_empty__mask_Opelims,axiom,
    ! [X5: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X5 )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X5 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X5 ) ) ) ) ).

% empty_mask.pelims
thf(fact_99_empty__mask_Opelims,axiom,
    ! [X5: produc1457211279475724562t_char > prat,Y: prat] :
      ( ( ( empty_6616431780912394577r_prat @ X5 )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X5 )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X5 ) ) ) ) ).

% empty_mask.pelims
thf(fact_100_upper__bounded__def,axiom,
    ( upper_bounded
    = ( ^ [Pi: produc1457211279475724562t_char > prat,P2: prat] :
        ! [Hl: produc1457211279475724562t_char] : ( pgte @ P2 @ ( Pi @ Hl ) ) ) ) ).

% upper_bounded_def
thf(fact_101_pgte__pgt,axiom,
    ! [A: prat,B: prat,C: prat,D: prat] :
      ( ( pgt @ A @ B )
     => ( ( pgte @ C @ D )
       => ( pgt @ ( padd @ A @ C ) @ ( padd @ B @ D ) ) ) ) ).

% pgte_pgt
thf(fact_102_accp__induct__rule,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X4: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X4 )
           => ( ! [Y4: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y4 @ X4 )
                 => ( P @ Y4 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_103_not__accp__down,axiom,
    ! [R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X5: produc1457211279475724562t_char > prat] :
      ( ~ ( accp_P4765339447491148764r_prat @ R3 @ X5 )
     => ~ ! [Z: produc1457211279475724562t_char > prat] :
            ( ( R3 @ Z @ X5 )
           => ( accp_P4765339447491148764r_prat @ R3 @ Z ) ) ) ).

% not_accp_down
thf(fact_104_accp__downward,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B: produc1457211279475724562t_char > prat,A: produc1457211279475724562t_char > prat] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P4765339447491148764r_prat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_105_accp__induct,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X4: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X4 )
           => ( ! [Y4: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y4 @ X4 )
                 => ( P @ Y4 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_106_accpI,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X5: produc1457211279475724562t_char > prat] :
      ( ! [Y5: produc1457211279475724562t_char > prat] :
          ( ( R2 @ Y5 @ X5 )
         => ( accp_P4765339447491148764r_prat @ R2 @ Y5 ) )
     => ( accp_P4765339447491148764r_prat @ R2 @ X5 ) ) ).

% accpI
thf(fact_107_accp_Osimps,axiom,
    ( accp_P4765339447491148764r_prat
    = ( ^ [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A2: produc1457211279475724562t_char > prat] :
        ? [X: produc1457211279475724562t_char > prat] :
          ( ( A2 = X )
          & ! [Y6: produc1457211279475724562t_char > prat] :
              ( ( R @ Y6 @ X )
             => ( accp_P4765339447491148764r_prat @ R @ Y6 ) ) ) ) ) ).

% accp.simps
thf(fact_108_not__pgte__charact,axiom,
    ! [A: prat,B: prat] :
      ( ( ~ ( pgte @ A @ B ) )
      = ( pgt @ B @ A ) ) ).

% not_pgte_charact
thf(fact_109_pgt__implies__pgte,axiom,
    ! [A: prat,B: prat] :
      ( ( pgt @ A @ B )
     => ( pgte @ A @ B ) ) ).

% pgt_implies_pgte
thf(fact_110_accp_Ocases,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ! [Y4: produc1457211279475724562t_char > prat] :
          ( ( R2 @ Y4 @ A )
         => ( accp_P4765339447491148764r_prat @ R2 @ Y4 ) ) ) ).

% accp.cases
thf(fact_111_the__bool__def,axiom,
    ( the_bool
    = ( case_val_o
      @ ^ [X13: $o] : X13
      @ ^ [Uu2: nat] : undefined_o
      @ ^ [Uu2: prat] : undefined_o ) ) ).

% the_bool_def
thf(fact_112_uni__mask_Oelims,axiom,
    ! [X5: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( uni_ma6578582744724537101t_char @ X5 @ Xa )
        = Y )
     => ( Y
        = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X5 @ Xa ) ) ) ).

% uni_mask.elims
thf(fact_113_uni__mask_Osimps,axiom,
    ( uni_ma6578582744724537101t_char
    = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char ) ) ).

% uni_mask.simps
thf(fact_114_pinv__double__half,axiom,
    ! [P3: prat] :
      ( ( ppos @ P3 )
     => ( ( pmult @ half @ ( pinv @ P3 ) )
        = ( pinv @ ( padd @ P3 @ P3 ) ) ) ) ).

% pinv_double_half
thf(fact_115_pmin__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmin @ A @ B )
        = B ) ) ).

% pmin_is
thf(fact_116_pmin__greater,axiom,
    ! [A: prat,B: prat] : ( pgte @ A @ ( pmin @ A @ B ) ) ).

% pmin_greater
thf(fact_117_pmin__comm,axiom,
    ( pmin
    = ( ^ [A2: prat,B2: prat] : ( pmin @ B2 @ A2 ) ) ) ).

% pmin_comm
thf(fact_118_pmult__comm,axiom,
    ( pmult
    = ( ^ [A2: prat,B2: prat] : ( pmult @ B2 @ A2 ) ) ) ).

% pmult_comm
thf(fact_119_pmult__order,axiom,
    ! [A: prat,B: prat,P3: prat] :
      ( ( pgte @ A @ B )
     => ( pgte @ ( pmult @ P3 @ A ) @ ( pmult @ B @ P3 ) ) ) ).

% pmult_order
thf(fact_120_pmult__distr,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( pmult @ A @ ( padd @ B @ C ) )
      = ( padd @ ( pmult @ A @ B ) @ ( pmult @ A @ C ) ) ) ).

% pmult_distr
thf(fact_121_pmult__padd,axiom,
    ! [A: prat,B: prat,X5: prat,C: prat,Y: prat] :
      ( ( pmult @ A @ ( padd @ ( pmult @ B @ X5 ) @ ( pmult @ C @ Y ) ) )
      = ( padd @ ( pmult @ ( pmult @ A @ B ) @ X5 ) @ ( pmult @ ( pmult @ A @ C ) @ Y ) ) ) ).

% pmult_padd
thf(fact_122_pmult__special_I2_J,axiom,
    ! [X5: prat] :
      ( ( pmult @ pnone @ X5 )
      = pnone ) ).

% pmult_special(2)
thf(fact_123_pmult__ppos,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ppos @ ( pmult @ A @ B ) ) ) ) ).

% pmult_ppos
thf(fact_124_multiply__mask__def,axiom,
    ( multiply_mask
    = ( ^ [P2: prat,Pi: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( pmult @ P2 @ ( Pi @ Hl ) ) ) ) ).

% multiply_mask_def
thf(fact_125_half__between__0__1,axiom,
    ( ( ppos @ half )
    & ( pgt @ pwrite @ half ) ) ).

% half_between_0_1
thf(fact_126_pinv__pmult__ok,axiom,
    ! [P3: prat] :
      ( ( ppos @ P3 )
     => ( ( pmult @ P3 @ ( pinv @ P3 ) )
        = pwrite ) ) ).

% pinv_pmult_ok
thf(fact_127_pmult__pdiv__cancel,axiom,
    ! [A: prat,X5: prat] :
      ( ( ppos @ A )
     => ( ( pmult @ A @ ( pdiv @ X5 @ A ) )
        = X5 ) ) ).

% pmult_pdiv_cancel
thf(fact_128_padd__one__ineq__sum,axiom,
    ! [A: prat,B: prat,X5: prat,Aa: prat,Bb: prat] :
      ( ( ( padd @ A @ B )
        = pwrite )
     => ( ( pgte @ X5 @ Aa )
       => ( ( pgte @ X5 @ Bb )
         => ( pgte @ X5 @ ( padd @ ( pmult @ A @ Aa ) @ ( pmult @ B @ Bb ) ) ) ) ) ) ).

% padd_one_ineq_sum
thf(fact_129_pinv__def,axiom,
    ( pinv
    = ( pdiv @ pwrite ) ) ).

% pinv_def
thf(fact_130_pmult__special_I1_J,axiom,
    ! [X5: prat] :
      ( ( pmult @ pwrite @ X5 )
      = X5 ) ).

% pmult_special(1)
thf(fact_131_pdiv__smaller,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ A )
       => ( pgte @ pwrite @ ( pdiv @ B @ A ) ) ) ) ).

% pdiv_smaller
thf(fact_132_sum__coeff,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ( padd @ ( pdiv @ A @ ( padd @ A @ B ) ) @ ( pdiv @ B @ ( padd @ A @ B ) ) )
          = pwrite ) ) ) ).

% sum_coeff
thf(fact_133_pinv__pwrite,axiom,
    ( ( pinv @ pwrite )
    = pwrite ) ).

% pinv_pwrite
thf(fact_134_multiply__smaller__pwrite,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ pwrite @ A )
     => ( ( pgte @ pwrite @ B )
       => ( pgte @ pwrite @ ( pmult @ A @ B ) ) ) ) ).

% multiply_smaller_pwrite
thf(fact_135_half__plus__half,axiom,
    ( ( padd @ half @ half )
    = pwrite ) ).

% half_plus_half
thf(fact_136_padd__comp__one,axiom,
    ! [X5: prat] :
      ( ( pgte @ pwrite @ X5 )
     => ( ( padd @ X5 @ ( comp_one @ X5 ) )
        = pwrite ) ) ).

% padd_comp_one
thf(fact_137_pmin__pmax,axiom,
    ! [X5: prat,A: prat,B: prat] :
      ( ( pgte @ X5 @ ( pmin @ A @ B ) )
     => ( X5
        = ( pmin @ ( pmax @ X5 @ A ) @ ( pmax @ X5 @ B ) ) ) ) ).

% pmin_pmax
thf(fact_138_uni__mask_Opelims,axiom,
    ! [X5: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( uni_ma6578582744724537101t_char @ X5 @ Xa )
        = Y )
     => ( ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X5 @ Xa ) )
       => ~ ( ( Y
              = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X5 @ Xa ) )
           => ~ ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X5 @ Xa ) ) ) ) ) ).

% uni_mask.pelims
thf(fact_139_pmax__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmax @ A @ B )
        = A ) ) ).

% pmax_is
thf(fact_140_pmax__smaller,axiom,
    ! [A: prat,B: prat] : ( pgte @ ( pmax @ A @ B ) @ A ) ).

% pmax_smaller
thf(fact_141_pmax__is__smaller,axiom,
    ! [X5: prat,A: prat,B: prat] :
      ( ( pgte @ X5 @ A )
     => ( ( pgte @ X5 @ B )
       => ( pgte @ X5 @ ( pmax @ A @ B ) ) ) ) ).

% pmax_is_smaller
thf(fact_142_comp__one__def,axiom,
    ( comp_one
    = ( ^ [P2: prat] :
          ( fChoice_prat
          @ ^ [R: prat] :
              ( ( padd @ P2 @ R )
              = pwrite ) ) ) ) ).

% comp_one_def
thf(fact_143_the__address__def,axiom,
    ( the_address
    = ( case_val_nat
      @ ^ [Uu2: $o] : undefined_nat
      @ ^ [X23: nat] : X23
      @ ^ [Uu2: prat] : undefined_nat ) ) ).

% the_address_def
thf(fact_144_the__rat__def,axiom,
    ( the_rat
    = ( case_val_prat
      @ ^ [Uu2: $o] : undefined_prat
      @ ^ [Uu2: nat] : undefined_prat
      @ ^ [X33: prat] : X33 ) ) ).

% the_rat_def
thf(fact_145_pmax__comm,axiom,
    ( pmax
    = ( ^ [A2: prat,B2: prat] : ( pmax @ B2 @ A2 ) ) ) ).

% pmax_comm
thf(fact_146_some__equality,axiom,
    ! [P: prat > $o,A: prat] :
      ( ( P @ A )
     => ( ! [X4: prat] :
            ( ( P @ X4 )
           => ( X4 = A ) )
       => ( ( fChoice_prat @ P )
          = A ) ) ) ).

% some_equality
thf(fact_147_some__eq__trivial,axiom,
    ! [X5: prat] :
      ( ( fChoice_prat
        @ ^ [Y6: prat] : ( Y6 = X5 ) )
      = X5 ) ).

% some_eq_trivial
thf(fact_148_some__sym__eq__trivial,axiom,
    ! [X5: prat] :
      ( ( fChoice_prat
        @ ( ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
          @ X5 ) )
      = X5 ) ).

% some_sym_eq_trivial
thf(fact_149_verit__sko__ex_H,axiom,
    ! [P: prat > $o,A3: $o] :
      ( ( ( P @ ( fChoice_prat @ P ) )
        = A3 )
     => ( ( ? [X6: prat] : ( P @ X6 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_150_verit__sko__forall,axiom,
    ( ( ^ [P4: prat > $o] :
        ! [X7: prat] : ( P4 @ X7 ) )
    = ( ^ [P5: prat > $o] :
          ( P5
          @ ( fChoice_prat
            @ ^ [X: prat] :
                ~ ( P5 @ X ) ) ) ) ) ).

% verit_sko_forall
thf(fact_151_someI2,axiom,
    ! [P: prat > $o,A: prat,Q: prat > $o] :
      ( ( P @ A )
     => ( ! [X4: prat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( fChoice_prat @ P ) ) ) ) ).

% someI2
thf(fact_152_verit__sko__forall_H,axiom,
    ! [P: prat > $o,A3: $o] :
      ( ( ( P
          @ ( fChoice_prat
            @ ^ [X: prat] :
                ~ ( P @ X ) ) )
        = A3 )
     => ( ( ! [X6: prat] : ( P @ X6 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_153_verit__sko__forall_H_H,axiom,
    ! [B3: prat,A3: prat,P: prat > $o] :
      ( ( B3 = A3 )
     => ( ( ( fChoice_prat @ P )
          = A3 )
        = ( ( fChoice_prat @ P )
          = B3 ) ) ) ).

% verit_sko_forall''
thf(fact_154_someI__ex,axiom,
    ! [P: prat > $o] :
      ( ? [X_1: prat] : ( P @ X_1 )
     => ( P @ ( fChoice_prat @ P ) ) ) ).

% someI_ex
thf(fact_155_verit__sko__forall__indirect2,axiom,
    ! [X5: prat,P: prat > $o,P6: prat > $o] :
      ( ( X5
        = ( fChoice_prat
          @ ^ [X: prat] :
              ~ ( P @ X ) ) )
     => ( ! [X4: prat] :
            ( ( P @ X4 )
            = ( P6 @ X4 ) )
       => ( ( ! [X6: prat] : ( P6 @ X6 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_156_verit__sko__forall__indirect,axiom,
    ! [X5: prat,P: prat > $o] :
      ( ( X5
        = ( fChoice_prat
          @ ^ [X: prat] :
              ~ ( P @ X ) ) )
     => ( ( ! [X6: prat] : ( P @ X6 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_157_some1__equality,axiom,
    ! [P: prat > $o,A: prat] :
      ( ? [X8: prat] :
          ( ( P @ X8 )
          & ! [Y5: prat] :
              ( ( P @ Y5 )
             => ( Y5 = X8 ) ) )
     => ( ( P @ A )
       => ( ( fChoice_prat @ P )
          = A ) ) ) ).

% some1_equality
thf(fact_158_verit__sko__ex__indirect2,axiom,
    ! [X5: prat,P: prat > $o,P6: prat > $o] :
      ( ( X5
        = ( fChoice_prat @ P ) )
     => ( ! [X4: prat] :
            ( ( P @ X4 )
            = ( P6 @ X4 ) )
       => ( ( ? [X6: prat] : ( P6 @ X6 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_159_verit__sko__ex__indirect,axiom,
    ! [X5: prat,P: prat > $o] :
      ( ( X5
        = ( fChoice_prat @ P ) )
     => ( ( ? [X6: prat] : ( P @ X6 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_160_some__eq__ex,axiom,
    ! [P: prat > $o] :
      ( ( P @ ( fChoice_prat @ P ) )
      = ( ? [X6: prat] : ( P @ X6 ) ) ) ).

% some_eq_ex
thf(fact_161_someI2__bex,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ? [X8: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X8 @ A3 )
          & ( P @ X8 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_162_someI2__bex,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ? [X8: nat] :
          ( ( member_nat @ X8 @ A3 )
          & ( P @ X8 ) )
     => ( ! [X4: nat] :
            ( ( ( member_nat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_163_someI2__bex,axiom,
    ! [A3: set_prat,P: prat > $o,Q: prat > $o] :
      ( ? [X8: prat] :
          ( ( member_prat @ X8 @ A3 )
          & ( P @ X8 ) )
     => ( ! [X4: prat] :
            ( ( ( member_prat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoice_prat
            @ ^ [X: prat] :
                ( ( member_prat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_164_someI2__ex,axiom,
    ! [P: prat > $o,Q: prat > $o] :
      ( ? [X_1: prat] : ( P @ X_1 )
     => ( ! [X4: prat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( fChoice_prat @ P ) ) ) ) ).

% someI2_ex
thf(fact_165_pred__equals__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R3 ) )
        = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_166_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 ) )
        = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_167_accp__acc__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( accp_P4275260045618599050at_nat
        @ ^ [X: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y6 ) @ R2 ) )
      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_168_accp__acc__eq,axiom,
    ! [R2: set_Pr1685704261205283783r_prat] :
      ( ( accp_P4765339447491148764r_prat
        @ ^ [X: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X @ Y6 ) @ R2 ) )
      = ( ^ [X: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_169_accp__acc__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( accp_nat
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R2 ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( acc_nat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_170_acc__def,axiom,
    ( acc_Pr2806007504492836550at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat] :
          ( collec3392354462482085612at_nat
          @ ( accp_P4275260045618599050at_nat
            @ ^ [X: product_prod_nat_nat,Y6: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y6 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_171_acc__def,axiom,
    ( acc_Pr1479980713244937752r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat] :
          ( collec7093475260167521854r_prat
          @ ( accp_P4765339447491148764r_prat
            @ ^ [X: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X @ Y6 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_172_acc__def,axiom,
    ( acc_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ( accp_nat
            @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_173_in__lex__prod,axiom,
    ! [A: nat,B: list_char,A4: nat,B4: list_char,R2: set_Pr1261947904930325089at_nat,S2: set_Pr4480099295215233283t_char] :
      ( ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc120197956887798346t_char @ A4 @ B4 ) ) @ ( lex_pr1681056561245511266t_char @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_174_in__lex__prod,axiom,
    ! [A: nat,B: nat,A4: nat,B4: nat,R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A4 @ B4 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_175_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] :
              ( ( B2 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_176_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B2: nat] :
              ( ( B2 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_177_acc_Ocases,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ! [Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ A ) @ R2 )
         => ( member8440522571783428010at_nat @ Y4 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_178_acc_Ocases,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ! [Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ A ) @ R2 )
         => ( member_nat @ Y4 @ ( acc_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_179_acc_Osimps,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
      = ( ? [X: product_prod_nat_nat] :
            ( ( A = X )
            & ! [Y6: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y6 @ X ) @ R2 )
               => ( member8440522571783428010at_nat @ Y6 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_180_acc_Osimps,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
      = ( ? [X: nat] :
            ( ( A = X )
            & ! [Y6: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ X ) @ R2 )
               => ( member_nat @ Y6 @ ( acc_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_181_acc_Ointros,axiom,
    ! [X5: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [Y5: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X5 ) @ R2 )
         => ( member8440522571783428010at_nat @ Y5 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) )
     => ( member8440522571783428010at_nat @ X5 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_182_acc_Ointros,axiom,
    ! [X5: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [Y5: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X5 ) @ R2 )
         => ( member_nat @ Y5 @ ( acc_nat @ R2 ) ) )
     => ( member_nat @ X5 @ ( acc_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_183_acc__induct,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
           => ( ! [Y4: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X4 ) @ R2 )
                 => ( P @ Y4 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% acc_induct
thf(fact_184_acc__induct,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( acc_nat @ R2 ) )
           => ( ! [Y4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X4 ) @ R2 )
                 => ( P @ Y4 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% acc_induct
thf(fact_185_acc__downward,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_186_acc__downward,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( member_nat @ B @ ( acc_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( member_nat @ A @ ( acc_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_187_not__acc__down,axiom,
    ! [X5: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X5 @ ( acc_Pr2806007504492836550at_nat @ R3 ) )
     => ~ ! [Z: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z @ X5 ) @ R3 )
           => ( member8440522571783428010at_nat @ Z @ ( acc_Pr2806007504492836550at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_188_not__acc__down,axiom,
    ! [X5: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member_nat @ X5 @ ( acc_nat @ R3 ) )
     => ~ ! [Z: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ X5 ) @ R3 )
           => ( member_nat @ Z @ ( acc_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_189_acc__induct__rule,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
           => ( ! [Y4: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X4 ) @ R2 )
                 => ( P @ Y4 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% acc_induct_rule
thf(fact_190_acc__induct__rule,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( acc_nat @ R2 ) )
           => ( ! [Y4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X4 ) @ R2 )
                 => ( P @ Y4 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% acc_induct_rule
thf(fact_191_add__masks_Opelims,axiom,
    ! [X5: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X5 @ Xa @ Xb )
        = Y )
     => ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X5 @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X5 @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X5 @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_192_above__def,axiom,
    ( order_793026671032835073at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_193_above__def,axiom,
    ( order_above_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_194_valid__maskI,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi2 @ Hl2 ) )
     => ( ! [F: list_char] :
            ( ( Pi2 @ ( produc120197956887798346t_char @ null_nat @ F ) )
            = pnone )
       => ( valid_mask @ Pi2 ) ) ) ).

% valid_maskI
thf(fact_195_valid__mask_Osimps,axiom,
    ( valid_mask
    = ( ^ [Pi: produc1457211279475724562t_char > prat] :
          ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi @ Hl ) )
          & ! [F4: list_char] :
              ( ( Pi @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
              = pnone ) ) ) ) ).

% valid_mask.simps
thf(fact_196_valid__mask_Oelims_I1_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X5 )
        = Y )
     => ( Y
        = ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X5 @ Hl ) )
          & ! [F4: list_char] :
              ( ( X5 @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
              = pnone ) ) ) ) ).

% valid_mask.elims(1)
thf(fact_197_valid__mask_Oelims_I2_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X5 )
     => ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X5 @ Hl3 ) )
        & ! [F5: list_char] :
            ( ( X5 @ ( produc120197956887798346t_char @ null_nat @ F5 ) )
            = pnone ) ) ) ).

% valid_mask.elims(2)
thf(fact_198_valid__mask_Oelims_I3_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X5 )
     => ~ ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X5 @ Hl2 ) )
          & ! [F: list_char] :
              ( ( X5 @ ( produc120197956887798346t_char @ null_nat @ F ) )
              = pnone ) ) ) ).

% valid_mask.elims(3)
thf(fact_199_upper__valid__aux,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( valid_mask @ B ) ) ) ).

% upper_valid_aux
thf(fact_200_upper__valid,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( ( valid_mask @ B )
          & ( valid_mask @ C ) ) ) ) ).

% upper_valid
thf(fact_201_valid__empty,axiom,
    valid_mask @ empty_3446695950879338768t_char ).

% valid_empty
thf(fact_202_valid__mask_Opelims_I3_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X5 )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X5 )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X5 )
           => ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X5 @ Hl2 ) )
              & ! [F: list_char] :
                  ( ( X5 @ ( produc120197956887798346t_char @ null_nat @ F ) )
                  = pnone ) ) ) ) ) ).

% valid_mask.pelims(3)
thf(fact_203_valid__mask_Opelims_I2_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X5 )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X5 )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X5 )
           => ~ ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X5 @ Hl3 ) )
                & ! [F5: list_char] :
                    ( ( X5 @ ( produc120197956887798346t_char @ null_nat @ F5 ) )
                    = pnone ) ) ) ) ) ).

% valid_mask.pelims(2)
thf(fact_204_valid__mask_Opelims_I1_J,axiom,
    ! [X5: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X5 )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X5 )
       => ~ ( ( Y
              = ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X5 @ Hl ) )
                & ! [F4: list_char] :
                    ( ( X5 @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
                    = pnone ) ) )
           => ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X5 ) ) ) ) ).

% valid_mask.pelims(1)
thf(fact_205_valid__null__def,axiom,
    ( valid_null
    = ( ^ [Pi: produc1457211279475724562t_char > prat] :
        ! [F4: list_char] :
          ( ( Pi @ ( produc120197956887798346t_char @ null_nat @ F4 ) )
          = pnone ) ) ) ).

% valid_null_def
thf(fact_206_full__mask__def,axiom,
    ( full_mask
    = ( ^ [Hl: produc1457211279475724562t_char] :
          ( if_prat
          @ ( ( produc6204653154488278070t_char @ Hl )
            = null_nat )
          @ pnone
          @ pwrite ) ) ) ).

% full_mask_def
thf(fact_207_accp__eq__acc,axiom,
    ( accp_P4275260045618599050at_nat
    = ( ^ [R: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( acc_Pr2806007504492836550at_nat @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_208_accp__eq__acc,axiom,
    ( accp_P4765339447491148764r_prat
    = ( ^ [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X @ ( acc_Pr1479980713244937752r_prat @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_209_accp__eq__acc,axiom,
    ( accp_nat
    = ( ^ [R: nat > nat > $o,X: nat] : ( member_nat @ X @ ( acc_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_210_acc__subset__induct,axiom,
    ! [D2: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,X5: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ D2 @ ( acc_Pr2806007504492836550at_nat @ R3 ) )
     => ( ! [X4: product_prod_nat_nat,Z: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ D2 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z @ X4 ) @ R3 )
             => ( member8440522571783428010at_nat @ Z @ D2 ) ) )
       => ( ( member8440522571783428010at_nat @ X5 @ D2 )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ D2 )
               => ( ! [Z3: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z3 @ X4 ) @ R3 )
                     => ( P @ Z3 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X5 ) ) ) ) ) ).

% acc_subset_induct
thf(fact_211_acc__subset__induct,axiom,
    ! [D2: set_nat,R3: set_Pr1261947904930325089at_nat,X5: nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ D2 @ ( acc_nat @ R3 ) )
     => ( ! [X4: nat,Z: nat] :
            ( ( member_nat @ X4 @ D2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ X4 ) @ R3 )
             => ( member_nat @ Z @ D2 ) ) )
       => ( ( member_nat @ X5 @ D2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ D2 )
               => ( ! [Z3: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ X4 ) @ R3 )
                     => ( P @ Z3 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X5 ) ) ) ) ) ).

% acc_subset_induct
thf(fact_212_Eps__case__prod__eq,axiom,
    ! [X5: nat,Y: list_char] :
      ( ( fChoic2476838209665704038t_char
        @ ( produc8526122873445342073char_o
          @ ^ [X9: nat,Y8: list_char] :
              ( ( X5 = X9 )
              & ( Y = Y8 ) ) ) )
      = ( produc120197956887798346t_char @ X5 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_213_Eps__case__prod__eq,axiom,
    ! [X5: nat,Y: nat] :
      ( ( fChoic6978938873391328853at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X9: nat,Y8: nat] :
              ( ( X5 = X9 )
              & ( Y = Y8 ) ) ) )
      = ( product_Pair_nat_nat @ X5 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_214_pre__get__m_Osimps,axiom,
    pre_ge2740421702909910421t_char = produc6204653154488278070t_char ).

% pre_get_m.simps
thf(fact_215_pre__get__m_Oelims,axiom,
    ! [X5: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X5 )
        = Y )
     => ( Y
        = ( produc6204653154488278070t_char @ X5 ) ) ) ).

% pre_get_m.elims
thf(fact_216_subset__Collect__iff,axiom,
    ! [B3: set_nat,A3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( ord_less_eq_set_nat @ B3
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_217_subset__Collect__iff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ B3
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_218_subset__CollectI,axiom,
    ! [B3: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B3 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_219_subset__CollectI,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ B3 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_220_add__masks__equiv__valid__null,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( valid_null @ ( add_ma6586698345353345811t_char @ A @ B ) )
      = ( ( valid_null @ A )
        & ( valid_null @ B ) ) ) ).

% add_masks_equiv_valid_null
thf(fact_221_ssubsetI,axiom,
    ! [A3: set_Pr2507339680178222706t_char,B3: set_Pr2507339680178222706t_char] :
      ( ! [Pi4: nat,H4: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi4 @ H4 ) @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi4 @ H4 ) @ B3 ) )
     => ( ord_le6315511645215477266t_char @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_222_ssubsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [Pi4: nat,H4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi4 @ H4 ) @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi4 @ H4 ) @ B3 ) )
     => ( ord_le3146513528884898305at_nat @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_223_split__paired__Eps,axiom,
    ( fChoic2476838209665704038t_char
    = ( ^ [P5: produc1457211279475724562t_char > $o] :
          ( fChoic2476838209665704038t_char
          @ ( produc8526122873445342073char_o
            @ ^ [A2: nat,B2: list_char] : ( P5 @ ( produc120197956887798346t_char @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_224_split__paired__Eps,axiom,
    ( fChoic6978938873391328853at_nat
    = ( ^ [P5: product_prod_nat_nat > $o] :
          ( fChoic6978938873391328853at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] : ( P5 @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_225_case__prodI,axiom,
    ! [F6: nat > list_char > $o,A: nat,B: list_char] :
      ( ( F6 @ A @ B )
     => ( produc8526122873445342073char_o @ F6 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_226_case__prodI,axiom,
    ! [F6: nat > nat > $o,A: nat,B: nat] :
      ( ( F6 @ A @ B )
     => ( produc6081775807080527818_nat_o @ F6 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_227_case__prodI2,axiom,
    ! [P3: produc1457211279475724562t_char,C: nat > list_char > $o] :
      ( ! [A5: nat,B5: list_char] :
          ( ( P3
            = ( produc120197956887798346t_char @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc8526122873445342073char_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_228_case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A5: nat,B5: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc6081775807080527818_nat_o @ C @ P3 ) ) ).

% case_prodI2
thf(fact_229_case__prod__conv,axiom,
    ! [F6: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F6 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F6 @ A @ B ) ) ).

% case_prod_conv
thf(fact_230_pre__get__m_Opelims,axiom,
    ! [X5: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X5 )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X5 )
       => ~ ( ( Y
              = ( produc6204653154488278070t_char @ X5 ) )
           => ~ ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X5 ) ) ) ) ).

% pre_get_m.pelims
thf(fact_231_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A2: nat,B2: nat] :
            ( P
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_232_fst__def,axiom,
    ( produc6204653154488278070t_char
    = ( produc9163108120003956271ar_nat
      @ ^ [X13: nat,X23: list_char] : X13 ) ) ).

% fst_def
thf(fact_233_case__prodD,axiom,
    ! [F6: nat > list_char > $o,A: nat,B: list_char] :
      ( ( produc8526122873445342073char_o @ F6 @ ( produc120197956887798346t_char @ A @ B ) )
     => ( F6 @ A @ B ) ) ).

% case_prodD
thf(fact_234_case__prodD,axiom,
    ! [F6: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F6 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F6 @ A @ B ) ) ).

% case_prodD
thf(fact_235_case__prodE,axiom,
    ! [C: nat > list_char > $o,P3: produc1457211279475724562t_char] :
      ( ( produc8526122873445342073char_o @ C @ P3 )
     => ~ ! [X4: nat,Y5: list_char] :
            ( ( P3
              = ( produc120197956887798346t_char @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_236_case__prodE,axiom,
    ! [C: nat > nat > $o,P3: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P3 )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_237_mem__case__prodI2,axiom,
    ! [P3: produc1457211279475724562t_char,Z4: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B5: list_char] :
          ( ( P3
            = ( produc120197956887798346t_char @ A5 @ B5 ) )
         => ( member8440522571783428010at_nat @ Z4 @ ( C @ A5 @ B5 ) ) )
     => ( member8440522571783428010at_nat @ Z4 @ ( produc3665563422017311042at_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_238_mem__case__prodI2,axiom,
    ! [P3: produc1457211279475724562t_char,Z4: nat,C: nat > list_char > set_nat] :
      ( ! [A5: nat,B5: list_char] :
          ( ( P3
            = ( produc120197956887798346t_char @ A5 @ B5 ) )
         => ( member_nat @ Z4 @ ( C @ A5 @ B5 ) ) )
     => ( member_nat @ Z4 @ ( produc2160937464213208549et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_239_mem__case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,Z4: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A5 @ B5 ) )
         => ( member8440522571783428010at_nat @ Z4 @ ( C @ A5 @ B5 ) ) )
     => ( member8440522571783428010at_nat @ Z4 @ ( produc8197505143624133779at_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_240_mem__case__prodI2,axiom,
    ! [P3: product_prod_nat_nat,Z4: nat,C: nat > nat > set_nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( P3
            = ( product_Pair_nat_nat @ A5 @ B5 ) )
         => ( member_nat @ Z4 @ ( C @ A5 @ B5 ) ) )
     => ( member_nat @ Z4 @ ( produc6189476227299908564et_nat @ C @ P3 ) ) ) ).

% mem_case_prodI2
thf(fact_241_mem__case__prodI,axiom,
    ! [Z4: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat,A: nat,B: list_char] :
      ( ( member8440522571783428010at_nat @ Z4 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z4 @ ( produc3665563422017311042at_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_242_mem__case__prodI,axiom,
    ! [Z4: nat,C: nat > list_char > set_nat,A: nat,B: list_char] :
      ( ( member_nat @ Z4 @ ( C @ A @ B ) )
     => ( member_nat @ Z4 @ ( produc2160937464213208549et_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_243_mem__case__prodI,axiom,
    ! [Z4: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z4 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z4 @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_244_mem__case__prodI,axiom,
    ! [Z4: nat,C: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z4 @ ( C @ A @ B ) )
     => ( member_nat @ Z4 @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_245_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A2: nat,B2: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A6: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_246_prod_Ocase__distrib,axiom,
    ! [H3: $o > $o,F6: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc6081775807080527818_nat_o @ F6 @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X13: nat,X23: nat] : ( H3 @ ( F6 @ X13 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_247_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu: nat,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_248_cond__case__prod__eta,axiom,
    ! [F6: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F6 @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F6 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_249_case__prod__eta,axiom,
    ! [F6: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y6: nat] : ( F6 @ ( product_Pair_nat_nat @ X @ Y6 ) ) )
      = F6 ) ).

% case_prod_eta
thf(fact_250_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z4: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z4 ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z4
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_251_Collect__restrict,axiom,
    ! [X10: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X10 )
            & ( P @ X ) ) )
      @ X10 ) ).

% Collect_restrict
thf(fact_252_Collect__restrict,axiom,
    ! [X10: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ X10 )
            & ( P @ X ) ) )
      @ X10 ) ).

% Collect_restrict
thf(fact_253_prop__restrict,axiom,
    ! [X5: nat,Z5: set_nat,X10: set_nat,P: nat > $o] :
      ( ( member_nat @ X5 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X10 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_254_prop__restrict,axiom,
    ! [X5: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X10: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ Z5 )
     => ( ( ord_le3146513528884898305at_nat @ Z5
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ X10 )
                & ( P @ X ) ) ) )
       => ( P @ X5 ) ) ) ).

% prop_restrict
thf(fact_255_conj__subset__def,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_256_conj__subset__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) )
        & ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_257_Collect__subset,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_258_Collect__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_259_case__prod__Pair__iden,axiom,
    ! [P3: produc1457211279475724562t_char] :
      ( ( produc4071494739252518515t_char @ produc120197956887798346t_char @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_260_case__prod__Pair__iden,axiom,
    ! [P3: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P3 )
      = P3 ) ).

% case_prod_Pair_iden
thf(fact_261_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,F4: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y6 ) ) @ R ) ) ) ) ) ).

% inv_image_def
thf(fact_262_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P5: nat > $o,R4: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X9: nat,Y8: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X: nat,Y6: nat] :
                      ( ( X9 = X )
                      & ( P5 @ X )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y8 @ Y6 ) @ ( R4 @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_263_Eps__case__prod,axiom,
    ! [P: nat > list_char > $o] :
      ( ( fChoic2476838209665704038t_char @ ( produc8526122873445342073char_o @ P ) )
      = ( fChoic2476838209665704038t_char
        @ ^ [Xy: produc1457211279475724562t_char] : ( P @ ( produc6204653154488278070t_char @ Xy ) @ ( produc1900778367047502200t_char @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_264_Eps__case__prod,axiom,
    ! [P: nat > nat > $o] :
      ( ( fChoic6978938873391328853at_nat @ ( produc6081775807080527818_nat_o @ P ) )
      = ( fChoic6978938873391328853at_nat
        @ ^ [Xy: product_prod_nat_nat] : ( P @ ( product_fst_nat_nat @ Xy ) @ ( product_snd_nat_nat @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_265_The__split__eq,axiom,
    ! [X5: nat,Y: list_char] :
      ( ( the_Pr202245514940600089t_char
        @ ( produc8526122873445342073char_o
          @ ^ [X9: nat,Y8: list_char] :
              ( ( X5 = X9 )
              & ( Y = Y8 ) ) ) )
      = ( produc120197956887798346t_char @ X5 @ Y ) ) ).

% The_split_eq
thf(fact_266_The__split__eq,axiom,
    ! [X5: nat,Y: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X9: nat,Y8: nat] :
              ( ( X5 = X9 )
              & ( Y = Y8 ) ) ) )
      = ( product_Pair_nat_nat @ X5 @ Y ) ) ).

% The_split_eq
thf(fact_267_case__swap,axiom,
    ! [F6: nat > nat > $o,P3: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y6: nat,X: nat] : ( F6 @ X @ Y6 )
        @ ( product_swap_nat_nat @ P3 ) )
      = ( produc6081775807080527818_nat_o @ F6 @ P3 ) ) ).

% case_swap
thf(fact_268_pair__imageI,axiom,
    ! [A: nat,B: list_char,A3: set_Pr2507339680178222706t_char,F6: nat > list_char > product_prod_nat_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F6 @ A @ B ) @ ( image_2110972657701675764at_nat @ ( produc6775168648774419426at_nat @ F6 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_269_pair__imageI,axiom,
    ! [A: nat,B: list_char,A3: set_Pr2507339680178222706t_char,F6: nat > list_char > nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A3 )
     => ( member_nat @ ( F6 @ A @ B ) @ ( image_3548405822317538717ar_nat @ ( produc9163108120003956271ar_nat @ F6 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_270_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F6: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F6 @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F6 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_271_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F6: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_nat @ ( F6 @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F6 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_272_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F6: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_o @ ( F6 @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F6 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_273_image__ident,axiom,
    ! [Y9: set_nat] :
      ( ( image_nat_nat2
        @ ^ [X: nat] : X
        @ Y9 )
      = Y9 ) ).

% image_ident
thf(fact_274_imageE,axiom,
    ! [B: set_nat,F6: nat > set_nat,A3: set_nat] :
      ( ( member_set_nat @ B @ ( image_nat_set_nat @ F6 @ A3 ) )
     => ~ ! [X4: nat] :
            ( ( B
              = ( F6 @ X4 ) )
           => ~ ( member_nat @ X4 @ A3 ) ) ) ).

% imageE
thf(fact_275_imageE,axiom,
    ! [B: product_prod_nat_nat,F6: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_5168914502847457605at_nat @ F6 @ A3 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( B
              = ( F6 @ X4 ) )
           => ~ ( member8440522571783428010at_nat @ X4 @ A3 ) ) ) ).

% imageE
thf(fact_276_imageE,axiom,
    ! [B: product_prod_nat_nat,F6: nat > product_prod_nat_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F6 @ A3 ) )
     => ~ ! [X4: nat] :
            ( ( B
              = ( F6 @ X4 ) )
           => ~ ( member_nat @ X4 @ A3 ) ) ) ).

% imageE
thf(fact_277_imageE,axiom,
    ! [B: nat,F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ B @ ( image_2486076414777270412at_nat @ F6 @ A3 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( B
              = ( F6 @ X4 ) )
           => ~ ( member8440522571783428010at_nat @ X4 @ A3 ) ) ) ).

% imageE
thf(fact_278_imageE,axiom,
    ! [B: nat,F6: nat > nat,A3: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat2 @ F6 @ A3 ) )
     => ~ ! [X4: nat] :
            ( ( B
              = ( F6 @ X4 ) )
           => ~ ( member_nat @ X4 @ A3 ) ) ) ).

% imageE
thf(fact_279_image__image,axiom,
    ! [F6: set_nat > nat,G: nat > set_nat,A3: set_nat] :
      ( ( image_set_nat_nat @ F6 @ ( image_nat_set_nat @ G @ A3 ) )
      = ( image_nat_nat2
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_280_image__image,axiom,
    ! [F6: set_nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( image_7916887816326733075et_nat @ F6 @ ( image_nat_set_nat @ G @ A3 ) )
      = ( image_nat_set_nat
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_281_image__image,axiom,
    ! [F6: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( image_nat_nat2 @ F6 @ ( image_nat_nat2 @ G @ A3 ) )
      = ( image_nat_nat2
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_282_image__image,axiom,
    ! [F6: nat > set_nat,G: nat > nat,A3: set_nat] :
      ( ( image_nat_set_nat @ F6 @ ( image_nat_nat2 @ G @ A3 ) )
      = ( image_nat_set_nat
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ A3 ) ) ).

% image_image
thf(fact_283_Compr__image__eq,axiom,
    ! [F6: nat > set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ ( image_nat_set_nat @ F6 @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_set_nat @ F6
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F6 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_284_Compr__image__eq,axiom,
    ! [F6: nat > nat,A3: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat2 @ F6 @ A3 ) )
            & ( P @ X ) ) )
      = ( image_nat_nat2 @ F6
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F6 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_285_Compr__image__eq,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ F6 @ A3 ) )
            & ( P @ X ) ) )
      = ( image_2486076414777270412at_nat @ F6
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( P @ ( F6 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_286_Compr__image__eq,axiom,
    ! [F6: nat > product_prod_nat_nat,A3: set_nat,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( image_5846123807819985514at_nat @ F6 @ A3 ) )
            & ( P @ X ) ) )
      = ( image_5846123807819985514at_nat @ F6
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ ( F6 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_287_Compr__image__eq,axiom,
    ! [F6: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( image_5168914502847457605at_nat @ F6 @ A3 ) )
            & ( P @ X ) ) )
      = ( image_5168914502847457605at_nat @ F6
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( P @ ( F6 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_288_The__case__prod,axiom,
    ! [P: nat > list_char > $o] :
      ( ( the_Pr202245514940600089t_char @ ( produc8526122873445342073char_o @ P ) )
      = ( the_Pr202245514940600089t_char
        @ ^ [Xy: produc1457211279475724562t_char] : ( P @ ( produc6204653154488278070t_char @ Xy ) @ ( produc1900778367047502200t_char @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_289_The__case__prod,axiom,
    ! [P: nat > nat > $o] :
      ( ( the_Pr7557018466319803784at_nat @ ( produc6081775807080527818_nat_o @ P ) )
      = ( the_Pr7557018466319803784at_nat
        @ ^ [Xy: product_prod_nat_nat] : ( P @ ( product_fst_nat_nat @ Xy ) @ ( product_snd_nat_nat @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_290_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F6: nat > set_nat,B3: set_set_nat] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( member_set_nat @ ( F6 @ X4 ) @ B3 ) )
     => ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F6 @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_291_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F6: nat > product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( member8440522571783428010at_nat @ ( F6 @ X4 ) @ B3 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F6 @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_292_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F6: nat > nat,B3: set_nat] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( member_nat @ ( F6 @ X4 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F6 @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_293_image__Collect__subsetI,axiom,
    ! [P: product_prod_nat_nat > $o,F6: product_prod_nat_nat > product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( P @ X4 )
         => ( member8440522571783428010at_nat @ ( F6 @ X4 ) @ B3 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F6 @ ( collec3392354462482085612at_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_294_image__Collect__subsetI,axiom,
    ! [P: product_prod_nat_nat > $o,F6: product_prod_nat_nat > nat,B3: set_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( P @ X4 )
         => ( member_nat @ ( F6 @ X4 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F6 @ ( collec3392354462482085612at_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_295_case__prod__unfold,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [C2: nat > nat > $o,P2: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P2 ) @ ( product_snd_nat_nat @ P2 ) ) ) ) ).

% case_prod_unfold
thf(fact_296_case__prod__beta_H,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F4: nat > nat > $o,X: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_297_exE__realizer,axiom,
    ! [P: nat > nat > $o,P3: product_prod_nat_nat,Q: $o > $o,F6: nat > nat > $o] :
      ( ( P @ ( product_snd_nat_nat @ P3 ) @ ( product_fst_nat_nat @ P3 ) )
     => ( ! [X4: nat,Y5: nat] :
            ( ( P @ Y5 @ X4 )
           => ( Q @ ( F6 @ X4 @ Y5 ) ) )
       => ( Q @ ( produc6081775807080527818_nat_o @ F6 @ P3 ) ) ) ) ).

% exE_realizer
thf(fact_298_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_nat > nat,A3: set_nat] :
      ( ( Inf
        @ ( image_nat_nat2
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( Inf @ A3 ) ) ).

% Inf.INF_identity_eq
thf(fact_299_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_nat > nat,A3: set_nat] :
      ( ( Sup
        @ ( image_nat_nat2
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( Sup @ A3 ) ) ).

% Sup.SUP_identity_eq
thf(fact_300_subset__fst__snd,axiom,
    ! [A3: set_Pr2507339680178222706t_char] :
      ( ord_le6315511645215477266t_char @ A3
      @ ( produc6030673636645808891t_char @ ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ A3 )
        @ ^ [Uu: nat] : ( image_3034139768025083822t_char @ produc1900778367047502200t_char @ A3 ) ) ) ).

% subset_fst_snd
thf(fact_301_fst__snd__flip,axiom,
    ( produc5563429021163328950ar_nat
    = ( comp_P26140635296239040ar_nat @ produc1900778367047502200t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X: list_char,Y6: nat] : ( produc120197956887798346t_char @ Y6 @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_302_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_303_fst__snd__flip,axiom,
    ( produc6204653154488278070t_char
    = ( comp_P680174135349460753t_char @ produc1259554233722553080ar_nat
      @ ( produc997380522159630963ar_nat
        @ ^ [X: nat,Y6: list_char] : ( produc8702345860417625034ar_nat @ Y6 @ X ) ) ) ) ).

% fst_snd_flip
thf(fact_304_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_305_snd__fst__flip,axiom,
    ( produc1259554233722553080ar_nat
    = ( comp_P3187634502937433873ar_nat @ produc6204653154488278070t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X: list_char,Y6: nat] : ( produc120197956887798346t_char @ Y6 @ X ) ) ) ) ).

% snd_fst_flip
thf(fact_306_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [A2: nat,B2: product_prod_nat_nat] :
              ( ( P @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_307_Collect__case__prod,axiom,
    ! [P: product_prod_nat_nat > $o,Q: nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [A2: product_prod_nat_nat,B2: nat] :
              ( ( P @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_308_Collect__case__prod,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( P @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [Uu: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_309_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A2: nat,B2: nat] :
              ( ( P @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_310_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [X: nat,Y6: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X @ Y6 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P )
        @ ^ [X: nat] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_311_Collect__case__prod__Sigma,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [X: product_prod_nat_nat,Y6: nat] :
              ( ( P @ X )
              & ( Q @ X @ Y6 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [X: product_prod_nat_nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_312_Collect__case__prod__Sigma,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X: product_prod_nat_nat,Y6: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X @ Y6 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [X: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_313_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y6: nat] :
              ( ( P @ X )
              & ( Q @ X @ Y6 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [X: nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_314_mem__Times__iff,axiom,
    ! [X5: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X5
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 ) )
      = ( ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ X5 ) @ A3 )
        & ( member8440522571783428010at_nat @ ( produc6408287024330202629at_nat @ X5 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_315_mem__Times__iff,axiom,
    ! [X5: produc8373899037510109440at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_nat] :
      ( ( member3348759134392003351at_nat @ X5
        @ ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 ) )
      = ( ( member8440522571783428010at_nat @ ( produc4622195368937614990at_nat @ X5 ) @ A3 )
        & ( member_nat @ ( produc3066998791170923468at_nat @ X5 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_316_mem__Times__iff,axiom,
    ! [X5: produc7248412053542808358at_nat,A3: set_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member2223272150424702269at_nat @ X5
        @ ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ( member_nat @ ( produc7982242761980330092at_nat @ X5 ) @ A3 )
        & ( member8440522571783428010at_nat @ ( produc6427046184213638570at_nat @ X5 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_317_mem__Times__iff,axiom,
    ! [X5: product_prod_nat_nat,A3: set_nat,B3: set_nat] :
      ( ( member8440522571783428010at_nat @ X5
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ( member_nat @ ( product_fst_nat_nat @ X5 ) @ A3 )
        & ( member_nat @ ( product_snd_nat_nat @ X5 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_318_mem__Times__iff,axiom,
    ! [X5: produc1457211279475724562t_char,A3: set_nat,B3: set_list_char] :
      ( ( member484235747325421115t_char @ X5
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ( member_nat @ ( produc6204653154488278070t_char @ X5 ) @ A3 )
        & ( member_list_char @ ( produc1900778367047502200t_char @ X5 ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_319_swap__product,axiom,
    ! [A3: set_list_char,B3: set_nat] :
      ( ( image_688197443715608197t_char
        @ ( produc6166083679720228083t_char
          @ ^ [I: list_char,J: nat] : ( produc120197956887798346t_char @ J @ I ) )
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 ) )
      = ( produc6030673636645808891t_char @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% swap_product
thf(fact_320_swap__product,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% swap_product
thf(fact_321_image__paired__Times,axiom,
    ! [F6: nat > nat,G: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( image_6562270973368889851et_nat
        @ ( produc112521266228933609et_nat
          @ ^ [X: nat,Y6: nat] : ( produc4207506657711014383et_nat @ ( F6 @ X ) @ ( G @ Y6 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc8883945523214541856et_nat @ ( image_nat_nat2 @ F6 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_set_nat @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_322_image__paired__Times,axiom,
    ! [F6: nat > set_nat,G: nat > nat,A3: set_nat,B3: set_nat] :
      ( ( image_2430162723873999099at_nat
        @ ( produc5203785053588818665at_nat
          @ ^ [X: nat,Y6: nat] : ( produc641871753055645167at_nat @ ( F6 @ X ) @ ( G @ Y6 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc5318310618559172640at_nat @ ( image_nat_set_nat @ F6 @ A3 )
        @ ^ [Uu: set_nat] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_323_image__paired__Times,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( image_4566292076127975857et_nat
        @ ( produc8159404288152011679et_nat
          @ ^ [X: nat,Y6: nat] : ( produc4532415448927165861et_nat @ ( F6 @ X ) @ ( G @ Y6 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F6 @ A3 )
        @ ^ [Uu: set_nat] : ( image_nat_set_nat @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_324_image__paired__Times,axiom,
    ! [F6: nat > nat,G: nat > nat,A3: set_nat,B3: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y6: nat] : ( product_Pair_nat_nat @ ( F6 @ X ) @ ( G @ Y6 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F6 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_325_Gr__incl,axiom,
    ! [A3: set_nat,F6: nat > nat,B3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( bNF_Gr_nat_nat @ A3 @ F6 )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F6 @ A3 ) @ B3 ) ) ).

% Gr_incl
thf(fact_326_Gr__incl,axiom,
    ! [A3: set_nat,F6: nat > set_nat,B3: set_set_nat] :
      ( ( ord_le3845944159117341623et_nat @ ( bNF_Gr_nat_set_nat @ A3 @ F6 )
        @ ( produc8883945523214541856et_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F6 @ A3 ) @ B3 ) ) ).

% Gr_incl
thf(fact_327_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P5: nat > nat > $o,A7: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 )
                  @ ( produc457027306803732586at_nat @ A7
                    @ ^ [Uu: nat] : A7 ) )
                & ( P5 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_328_member__product,axiom,
    ! [X5: product_prod_nat_nat,A3: set_nat,B3: set_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ ( produc929334515565554804at_nat @ A3 @ B3 ) )
      = ( member8440522571783428010at_nat @ X5
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) ) ) ).

% member_product
thf(fact_329_snd__image__times,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = B3 ) ) ) ).

% snd_image_times
thf(fact_330_fst__image__times,axiom,
    ! [B3: set_list_char,A3: set_nat] :
      ( ( ( B3 = bot_bo8503570744482963627t_char )
       => ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char
            @ ( produc6030673636645808891t_char @ A3
              @ ^ [Uu: nat] : B3 ) )
          = bot_bot_set_nat ) )
      & ( ( B3 != bot_bo8503570744482963627t_char )
       => ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char
            @ ( produc6030673636645808891t_char @ A3
              @ ^ [Uu: nat] : B3 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_331_fst__image__times,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( ( B3 = bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = bot_bot_set_nat ) )
      & ( ( B3 != bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_332_Times__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ( B3 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_333_empty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $false ) ) ).

% empty_def
thf(fact_334_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_335_times__eq__iff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat,D2: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 )
        = ( produc457027306803732586at_nat @ C3
          @ ^ [Uu: nat] : D2 ) )
      = ( ( ( A3 = C3 )
          & ( B3 = D2 ) )
        | ( ( ( A3 = bot_bot_set_nat )
            | ( B3 = bot_bot_set_nat ) )
          & ( ( C3 = bot_bot_set_nat )
            | ( D2 = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_336_some__in__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat
        @ ( fChoic6978938873391328853at_nat
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo2099793752762293965at_nat ) ) ).

% some_in_eq
thf(fact_337_some__in__eq,axiom,
    ! [A3: set_prat] :
      ( ( member_prat
        @ ( fChoice_prat
          @ ^ [X: prat] : ( member_prat @ X @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_prat ) ) ).

% some_in_eq
thf(fact_338_some__in__eq,axiom,
    ! [A3: set_nat] :
      ( ( member_nat
        @ ( fChoice_nat
          @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_nat ) ) ).

% some_in_eq
thf(fact_339_times__subset__iff,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat,D2: set_nat] :
      ( ( ord_le3146513528884898305at_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : C3 )
        @ ( produc457027306803732586at_nat @ B3
          @ ^ [Uu: nat] : D2 ) )
      = ( ( A3 = bot_bot_set_nat )
        | ( C3 = bot_bot_set_nat )
        | ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_340_fst__image__Sigma,axiom,
    ! [A3: set_nat,B3: nat > set_list_char] :
      ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
      = ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bo8503570744482963627t_char ) ) ) ) ).

% fst_image_Sigma
thf(fact_341_fst__image__Sigma,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) )
      = ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bot_set_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_342_fst__image__Sigma,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( image_8369444115148354106at_nat @ produc4622195368937614990at_nat @ ( produc7672662199629908489at_nat @ A3 @ B3 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A3 )
            & ( ( B3 @ X )
             != bot_bot_set_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_343_flip__pred,axiom,
    ! [A3: set_Pr8803067691650843250ar_nat,R3: nat > list_char > $o] :
      ( ( ord_le3387867619833322002ar_nat @ A3 @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ ( conver4209408378539545768t_char @ R3 ) ) ) )
     => ( ord_le6315511645215477266t_char
        @ ( image_688197443715608197t_char
          @ ( produc6166083679720228083t_char
            @ ^ [X: list_char,Y6: nat] : ( produc120197956887798346t_char @ Y6 @ X ) )
          @ A3 )
        @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_344_flip__pred,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: nat > nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( conversep_nat_nat @ R3 ) ) ) )
     => ( ord_le3146513528884898305at_nat
        @ ( image_5168914502847457605at_nat
          @ ( produc2626176000494625587at_nat
            @ ^ [X: nat,Y6: nat] : ( product_Pair_nat_nat @ Y6 @ X ) )
          @ A3 )
        @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_345_snd__diag__fst,axiom,
    ( ( comp_P2122579405136797538t_char @ product_snd_nat_nat
      @ ( comp_n1645060710405774596t_char
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ produc6204653154488278070t_char ) )
    = produc6204653154488278070t_char ) ).

% snd_diag_fst
thf(fact_346_fst__diag__fst,axiom,
    ( ( comp_P2122579405136797538t_char @ product_fst_nat_nat
      @ ( comp_n1645060710405774596t_char
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ produc6204653154488278070t_char ) )
    = produc6204653154488278070t_char ) ).

% fst_diag_fst
thf(fact_347_refl__on__def,axiom,
    ( refl_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ R
            @ ( produc457027306803732586at_nat @ A7
              @ ^ [Uu: nat] : A7 ) )
          & ! [X: nat] :
              ( ( member_nat @ X @ A7 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_348_refl__onI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R2
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : A3 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R2 ) )
       => ( refl_o7122697186039111552at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_349_refl__onI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R2
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : A3 ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R2 ) )
       => ( refl_on_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_350_map__prod__surj__on,axiom,
    ! [F6: nat > nat,A3: set_nat,A8: set_nat,G: nat > nat,B3: set_nat,B7: set_nat] :
      ( ( ( image_nat_nat2 @ F6 @ A3 )
        = A8 )
     => ( ( ( image_nat_nat2 @ G @ B3 )
          = B7 )
       => ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F6 @ G )
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = ( produc457027306803732586at_nat @ A8
            @ ^ [Uu: nat] : B7 ) ) ) ) ).

% map_prod_surj_on
thf(fact_351_map__prod__surj__on,axiom,
    ! [F6: nat > nat,A3: set_nat,A8: set_nat,G: nat > set_nat,B3: set_nat,B7: set_set_nat] :
      ( ( ( image_nat_nat2 @ F6 @ A3 )
        = A8 )
     => ( ( ( image_nat_set_nat @ G @ B3 )
          = B7 )
       => ( ( image_6562270973368889851et_nat @ ( produc5294471305854144752et_nat @ F6 @ G )
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = ( produc8883945523214541856et_nat @ A8
            @ ^ [Uu: nat] : B7 ) ) ) ) ).

% map_prod_surj_on
thf(fact_352_map__prod__surj__on,axiom,
    ! [F6: nat > set_nat,A3: set_nat,A8: set_set_nat,G: nat > nat,B3: set_nat,B7: set_nat] :
      ( ( ( image_nat_set_nat @ F6 @ A3 )
        = A8 )
     => ( ( ( image_nat_nat2 @ G @ B3 )
          = B7 )
       => ( ( image_2430162723873999099at_nat @ ( produc9133550253898737904at_nat @ F6 @ G )
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = ( produc5318310618559172640at_nat @ A8
            @ ^ [Uu: set_nat] : B7 ) ) ) ) ).

% map_prod_surj_on
thf(fact_353_map__prod__surj__on,axiom,
    ! [F6: nat > set_nat,A3: set_nat,A8: set_set_nat,G: nat > set_nat,B3: set_nat,B7: set_set_nat] :
      ( ( ( image_nat_set_nat @ F6 @ A3 )
        = A8 )
     => ( ( ( image_nat_set_nat @ G @ B3 )
          = B7 )
       => ( ( image_4566292076127975857et_nat @ ( produc4836607750981702822et_nat @ F6 @ G )
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) )
          = ( produc105401717395143638et_nat @ A8
            @ ^ [Uu: set_nat] : B7 ) ) ) ) ).

% map_prod_surj_on
thf(fact_354_snd__image__Sigma,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ).

% snd_image_Sigma
thf(fact_355_SUP__identity__eq,axiom,
    ! [A3: set_nat] :
      ( ( complete_Sup_Sup_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( complete_Sup_Sup_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_356_SUP__identity__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [X: set_nat] : X
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_357_UN__I,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B @ ( B3 @ A ) )
       => ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_358_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: product_prod_nat_nat,B3: nat > set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B @ ( B3 @ A ) )
       => ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_359_UN__I,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: nat,B3: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member_nat @ B @ ( B3 @ A ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_360_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: nat,B3: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_nat @ B @ ( B3 @ A ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_361_UN__iff,axiom,
    ! [B: nat,B3: nat > set_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( member_nat @ B @ ( B3 @ X ) ) ) ) ) ).

% UN_iff
thf(fact_362_SUP__bot,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : bot_bot_set_nat
          @ A3 ) )
      = bot_bot_set_nat ) ).

% SUP_bot
thf(fact_363_SUP__bot__conv_I1_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_364_SUP__bot__conv_I2_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_365_SUP__const,axiom,
    ! [A3: set_nat,F6: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [I: nat] : F6
            @ A3 ) )
        = F6 ) ) ).

% SUP_const
thf(fact_366_UN__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_367_UN__extend__simps_I9_J,axiom,
    ! [C3: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( B3 @ X ) ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_368_UN__extend__simps_I8_J,axiom,
    ! [B3: nat > set_nat,A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [Y6: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ Y6 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_369_UN__E,axiom,
    ! [B: product_prod_nat_nat,B3: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B3 @ A3 ) ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ~ ( member8440522571783428010at_nat @ B @ ( B3 @ X4 ) ) ) ) ).

% UN_E
thf(fact_370_UN__E,axiom,
    ! [B: product_prod_nat_nat,B3: nat > set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B3 @ A3 ) ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ~ ( member8440522571783428010at_nat @ B @ ( B3 @ X4 ) ) ) ) ).

% UN_E
thf(fact_371_UN__E,axiom,
    ! [B: nat,B3: product_prod_nat_nat > set_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ~ ( member_nat @ B @ ( B3 @ X4 ) ) ) ) ).

% UN_E
thf(fact_372_UN__E,axiom,
    ! [B: nat,B3: nat > set_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ~ ( member_nat @ B @ ( B3 @ X4 ) ) ) ) ).

% UN_E
thf(fact_373_SUP__UNION,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y6: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( G @ Y6 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_374_UN__UN__flatten,axiom,
    ! [C3: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y6: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C3 @ ( B3 @ Y6 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_375_UN__Times__distrib,axiom,
    ! [E: nat > set_nat,F7: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( comple5685304695842803022at_nat
        @ ( image_5423882813909314213at_nat
          @ ( produc8197505143624133779at_nat
            @ ^ [A2: nat,B2: nat] :
                ( produc457027306803732586at_nat @ ( E @ A2 )
                @ ^ [Uu: nat] : ( F7 @ B2 ) ) )
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B3 ) ) )
      = ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ E @ A3 ) )
        @ ^ [Uu: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F7 @ B3 ) ) ) ) ).

% UN_Times_distrib
thf(fact_376_SUP__commute,axiom,
    ! [F6: nat > nat > set_nat,B3: set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F6 @ I ) @ B3 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [J: nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [I: nat] : ( F6 @ I @ J )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_377_image__Union,axiom,
    ! [F6: nat > set_nat,S: set_set_nat] :
      ( ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ S ) )
      = ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F6 ) @ S ) ) ) ).

% image_Union
thf(fact_378_image__Union,axiom,
    ! [F6: nat > nat,S: set_set_nat] :
      ( ( image_nat_nat2 @ F6 @ ( comple7399068483239264473et_nat @ S ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F6 ) @ S ) ) ) ).

% image_Union
thf(fact_379_image__UN,axiom,
    ! [F6: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple548664676211718543et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X: nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_380_image__UN,axiom,
    ! [F6: nat > nat,B3: nat > set_nat,A3: set_nat] :
      ( ( image_nat_nat2 @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_381_UN__extend__simps_I10_J,axiom,
    ! [B3: set_nat > set_nat,F6: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B3 @ ( F6 @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B3 @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_382_UN__extend__simps_I10_J,axiom,
    ! [B3: nat > set_nat,F6: nat > nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B3 @ ( F6 @ A2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( image_nat_nat2 @ F6 @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_383_Union__natural,axiom,
    ! [F6: nat > set_nat] :
      ( ( comp_s5357857527062720591et_nat @ comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F6 ) ) )
      = ( comp_s1376360633444582883et_nat @ ( image_nat_set_nat @ F6 ) @ comple7399068483239264473et_nat ) ) ).

% Union_natural
thf(fact_384_Union__natural,axiom,
    ! [F6: nat > nat] :
      ( ( comp_s174380336271864291et_nat @ comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat2 @ F6 ) ) )
      = ( comp_s6993074788030935341et_nat @ ( image_nat_nat2 @ F6 ) @ comple7399068483239264473et_nat ) ) ).

% Union_natural
thf(fact_385_UN__empty2,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : bot_bot_set_nat
          @ A3 ) )
      = bot_bot_set_nat ) ).

% UN_empty2
thf(fact_386_UN__empty,axiom,
    ! [B3: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% UN_empty
thf(fact_387_UNION__empty__conv_I1_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_388_UNION__empty__conv_I2_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_389_UN__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B3 ) ) ) ) ) ).

% UN_mono
thf(fact_390_UN__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).

% UN_mono
thf(fact_391_UN__least,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,C3: set_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A3 )
         => ( ord_less_eq_set_nat @ ( B3 @ X4 ) @ C3 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) @ C3 ) ) ).

% UN_least
thf(fact_392_UN__least,axiom,
    ! [A3: set_nat,B3: nat > set_nat,C3: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_set_nat @ ( B3 @ X4 ) @ C3 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) @ C3 ) ) ).

% UN_least
thf(fact_393_UN__upper,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_394_UN__upper,axiom,
    ! [A: nat,A3: set_nat,B3: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ).

% UN_upper
thf(fact_395_UN__subset__iff,axiom,
    ! [A3: nat > set_nat,I2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) @ B3 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ I2 )
           => ( ord_less_eq_set_nat @ ( A3 @ X ) @ B3 ) ) ) ) ).

% UN_subset_iff
thf(fact_396_case__prod__map__prod,axiom,
    ! [H3: nat > nat > $o,F6: nat > nat,G: nat > nat,X5: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ H3 @ ( produc6977886695330630970at_nat @ F6 @ G @ X5 ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [L: nat,R: nat] : ( H3 @ ( F6 @ L ) @ ( G @ R ) )
        @ X5 ) ) ).

% case_prod_map_prod
thf(fact_397_SUP__eqI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat,X5: filter_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ I3 ) @ X5 ) )
     => ( ! [Y5: filter_nat] :
            ( ! [I4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ I4 @ A3 )
               => ( ord_le2510731241096832064er_nat @ ( F6 @ I4 ) @ Y5 ) )
           => ( ord_le2510731241096832064er_nat @ X5 @ Y5 ) )
       => ( ( comple2454491259644266675er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% SUP_eqI
thf(fact_398_SUP__eqI,axiom,
    ! [A3: set_nat,F6: nat > filter_nat,X5: filter_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ I3 ) @ X5 ) )
     => ( ! [Y5: filter_nat] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_le2510731241096832064er_nat @ ( F6 @ I4 ) @ Y5 ) )
           => ( ord_le2510731241096832064er_nat @ X5 @ Y5 ) )
       => ( ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% SUP_eqI
thf(fact_399_SUP__eqI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > set_nat,X5: set_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F6 @ I3 ) @ X5 ) )
     => ( ! [Y5: set_nat] :
            ( ! [I4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F6 @ I4 ) @ Y5 ) )
           => ( ord_less_eq_set_nat @ X5 @ Y5 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% SUP_eqI
thf(fact_400_SUP__eqI,axiom,
    ! [A3: set_nat,F6: nat > set_nat,X5: set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F6 @ I3 ) @ X5 ) )
     => ( ! [Y5: set_nat] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F6 @ I4 ) @ Y5 ) )
           => ( ord_less_eq_set_nat @ X5 @ Y5 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% SUP_eqI
thf(fact_401_SUP__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat,F6: product_prod_nat_nat > set_nat,G: nat > set_nat] :
      ( ! [N: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ N @ A3 )
         => ? [X8: nat] :
              ( ( member_nat @ X8 @ B3 )
              & ( ord_less_eq_set_nat @ ( F6 @ N ) @ ( G @ X8 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_402_SUP__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ A3 )
         => ? [X8: nat] :
              ( ( member_nat @ X8 @ B3 )
              & ( ord_less_eq_set_nat @ ( F6 @ N ) @ ( G @ X8 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_403_SUP__least,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat,U: filter_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ I3 ) @ U ) )
     => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_404_SUP__least,axiom,
    ! [A3: set_nat,F6: nat > filter_nat,U: filter_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ I3 ) @ U ) )
     => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_405_SUP__least,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > set_nat,U: set_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F6 @ I3 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_406_SUP__least,axiom,
    ! [A3: set_nat,F6: nat > set_nat,U: set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F6 @ I3 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_407_SUP__mono_H,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ! [X4: nat] : ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ).

% SUP_mono'
thf(fact_408_SUP__upper,axiom,
    ! [I5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ A3 )
     => ( ord_le2510731241096832064er_nat @ ( F6 @ I5 ) @ ( comple2454491259644266675er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_409_SUP__upper,axiom,
    ! [I5: nat,A3: set_nat,F6: nat > filter_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ord_le2510731241096832064er_nat @ ( F6 @ I5 ) @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_410_SUP__upper,axiom,
    ! [I5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ A3 )
     => ( ord_less_eq_set_nat @ ( F6 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_411_SUP__upper,axiom,
    ! [I5: nat,A3: set_nat,F6: nat > set_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ord_less_eq_set_nat @ ( F6 @ I5 ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_412_SUP__le__iff,axiom,
    ! [F6: nat > set_nat,A3: set_nat,U: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ U )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X ) @ U ) ) ) ) ).

% SUP_le_iff
thf(fact_413_SUP__upper2,axiom,
    ! [I5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,U: filter_nat,F6: product_prod_nat_nat > filter_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ A3 )
     => ( ( ord_le2510731241096832064er_nat @ U @ ( F6 @ I5 ) )
       => ( ord_le2510731241096832064er_nat @ U @ ( comple2454491259644266675er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_414_SUP__upper2,axiom,
    ! [I5: nat,A3: set_nat,U: filter_nat,F6: nat > filter_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ( ord_le2510731241096832064er_nat @ U @ ( F6 @ I5 ) )
       => ( ord_le2510731241096832064er_nat @ U @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_415_SUP__upper2,axiom,
    ! [I5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,U: set_nat,F6: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F6 @ I5 ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_416_SUP__upper2,axiom,
    ! [I5: nat,A3: set_nat,U: set_nat,F6: nat > set_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F6 @ I5 ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_417_case__prod__o__map__prod,axiom,
    ! [F6: nat > nat > $o,G1: nat > nat,G2: nat > nat] :
      ( ( comp_P6344650113790022471at_nat @ ( produc6081775807080527818_nat_o @ F6 ) @ ( produc6977886695330630970at_nat @ G1 @ G2 ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [L: nat,R: nat] : ( F6 @ ( G1 @ L ) @ ( G2 @ R ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_418_SUP__subset__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat,G: product_prod_nat_nat > filter_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ord_le2510731241096832064er_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) @ ( comple2454491259644266675er_nat @ ( image_6134674671615456924er_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_419_SUP__subset__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F6: nat > filter_nat,G: nat > filter_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_le2510731241096832064er_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_420_SUP__subset__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > set_nat,G: product_prod_nat_nat > set_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_421_SUP__subset__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_422_SUP__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_423_SUP__empty,axiom,
    ! [F6: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_424_cSUP__const,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_nat
          @ ( image_nat_nat2
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_425_cSUP__const,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_426_UN__constant__eq,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > set_nat,C: set_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ( F6 @ X4 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_427_UN__constant__eq,axiom,
    ! [A: nat,A3: set_nat,F6: nat > set_nat,C: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ( F6 @ X4 )
              = C ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
          = C ) ) ) ).

% UN_constant_eq
thf(fact_428_Ex__inj__on__UNION__Sigma,axiom,
    ! [A3: nat > set_nat,I2: set_nat] :
    ? [F: nat > product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
      & ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) @ ( produc457027306803732586at_nat @ I2 @ A3 ) ) ) ).

% Ex_inj_on_UNION_Sigma
thf(fact_429_UNION__singleton__eq__range,axiom,
    ! [F6: nat > set_nat,A3: set_nat] :
      ( ( comple548664676211718543et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X: nat] : ( insert_set_nat @ ( F6 @ X ) @ bot_bot_set_set_nat )
          @ A3 ) )
      = ( image_nat_set_nat @ F6 @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_430_UNION__singleton__eq__range,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( insert_nat @ ( F6 @ X ) @ bot_bot_set_nat )
          @ A3 ) )
      = ( image_nat_nat2 @ F6 @ A3 ) ) ).

% UNION_singleton_eq_range
thf(fact_431_UN__singleton,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( insert_nat @ X @ bot_bot_set_nat )
          @ A3 ) )
      = A3 ) ).

% UN_singleton
thf(fact_432_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y7: product_prod_nat_nat,Z2: product_prod_nat_nat] : ( Y7 = Z2 )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_433_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_434_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( X = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_435_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_436_UN__simps_I1_J,axiom,
    ! [C3: set_nat,A: nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( insert_nat @ A @ ( B3 @ X ) )
              @ C3 ) )
          = bot_bot_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( insert_nat @ A @ ( B3 @ X ) )
              @ C3 ) )
          = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_437_accp__subset,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6723339807950303054prat_o @ R1 @ R22 )
     => ( ord_le2830794348966088778prat_o @ ( accp_P4765339447491148764r_prat @ R22 ) @ ( accp_P4765339447491148764r_prat @ R1 ) ) ) ).

% accp_subset
thf(fact_438_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_439_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U2: product_prod_nat_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_440_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A2: nat,B8: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A2 )
              | ( member_nat @ X @ B8 ) ) ) ) ) ).

% insert_compr
thf(fact_441_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A2: product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( X = A2 )
              | ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% insert_compr
thf(fact_442_map__prod__inj__on,axiom,
    ! [F6: nat > nat,A3: set_nat,G: nat > nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F6 @ A3 )
     => ( ( inj_on_nat_nat @ G @ B3 )
       => ( inj_on8969904277767023793at_nat @ ( produc6977886695330630970at_nat @ F6 @ G )
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% map_prod_inj_on
thf(fact_443_accp__subset__induct,axiom,
    ! [D2: ( produc1457211279475724562t_char > prat ) > $o,R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X5: produc1457211279475724562t_char > prat,P: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le2830794348966088778prat_o @ D2 @ ( accp_P4765339447491148764r_prat @ R3 ) )
     => ( ! [X4: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > prat] :
            ( ( D2 @ X4 )
           => ( ( R3 @ Z @ X4 )
             => ( D2 @ Z ) ) )
       => ( ( D2 @ X5 )
         => ( ! [X4: produc1457211279475724562t_char > prat] :
                ( ( D2 @ X4 )
               => ( ! [Z3: produc1457211279475724562t_char > prat] :
                      ( ( R3 @ Z3 @ X4 )
                     => ( P @ Z3 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X5 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_444_inj__on__convol__ident,axiom,
    ! [F6: nat > list_char,X10: set_nat] :
      ( inj_on9177509583204279951t_char
      @ ^ [X: nat] : ( produc120197956887798346t_char @ X @ ( F6 @ X ) )
      @ X10 ) ).

% inj_on_convol_ident
thf(fact_445_inj__on__convol__ident,axiom,
    ! [F6: nat > nat,X10: set_nat] :
      ( inj_on5538052773655684606at_nat
      @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ ( F6 @ X ) )
      @ X10 ) ).

% inj_on_convol_ident
thf(fact_446_SUP__Sup__eq2,axiom,
    ! [S: set_se253819897320413138t_char] :
      ( ( comple724453453808405122char_o
        @ ( image_7123945608888165160char_o
          @ ^ [I: set_Pr2507339680178222706t_char,X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ I )
          @ S ) )
      = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( comple9178978156834306527t_char @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_447_SUP__Sup__eq2,axiom,
    ! [S: set_se7855581050983116737at_nat] :
      ( ( comple3592611370556534995_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I: set_Pr1261947904930325089at_nat,X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ I )
          @ S ) )
      = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( comple5685304695842803022at_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_448_SUP__Sup__eq,axiom,
    ! [S: set_se7855581050983116737at_nat] :
      ( ( comple1901448594430626575_nat_o
        @ ( image_5253903537071781336_nat_o
          @ ^ [I: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ I )
          @ S ) )
      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( comple5685304695842803022at_nat @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_449_SUP__Sup__eq,axiom,
    ! [S: set_set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_set_nat_nat_o
          @ ^ [I: set_nat,X: nat] : ( member_nat @ X @ I )
          @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_450_Sup__set__def,axiom,
    ( comple5685304695842803022at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( complete_Sup_Sup_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_451_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A7: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A7 ) ) ) ) ) ).

% Sup_set_def
thf(fact_452_pred__subset__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R3 )
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_453_pred__subset__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R3 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_454_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_455_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A7 )
          @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_456_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_457_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_458_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_459_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_460_Sup__SUP__eq2,axiom,
    ( comple724453453808405122char_o
    = ( ^ [S3: set_nat_list_char_o,X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( comple9178978156834306527t_char @ ( image_9058305973422863104t_char @ collec2458894688891239037t_char @ ( image_1531431914690922683char_o @ produc8526122873445342073char_o @ S3 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_461_Sup__SUP__eq2,axiom,
    ( comple3592611370556534995_nat_o
    = ( ^ [S3: set_nat_nat_o,X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S3 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_462_SUP__UN__eq,axiom,
    ! [R2: nat > set_nat,S: set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_nat_nat_o
          @ ^ [I: nat,X: nat] : ( member_nat @ X @ ( R2 @ I ) )
          @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ R2 @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_463_swap__inj__on,axiom,
    ! [A3: set_Pr8803067691650843250ar_nat] :
      ( inj_on7000867008012422641t_char
      @ ( produc6166083679720228083t_char
        @ ^ [I: list_char,J: nat] : ( produc120197956887798346t_char @ J @ I ) )
      @ A3 ) ).

% swap_inj_on
thf(fact_464_swap__inj__on,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( inj_on8969904277767023793at_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
      @ A3 ) ).

% swap_inj_on
thf(fact_465_bot__empty__eq2,axiom,
    ( bot_bo8065403328713304961char_o
    = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ bot_bo1414094081888755934t_char ) ) ) ).

% bot_empty_eq2
thf(fact_466_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_467_image__constant,axiom,
    ! [X5: nat,A3: set_nat,C: set_nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( image_nat_set_nat
          @ ^ [X: nat] : C
          @ A3 )
        = ( insert_set_nat @ C @ bot_bot_set_set_nat ) ) ) ).

% image_constant
thf(fact_468_image__constant,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,C: nat] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( ( image_2486076414777270412at_nat
          @ ^ [X: product_prod_nat_nat] : C
          @ A3 )
        = ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_469_image__constant,axiom,
    ! [X5: nat,A3: set_nat,C: nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( image_nat_nat2
          @ ^ [X: nat] : C
          @ A3 )
        = ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).

% image_constant
thf(fact_470_image__constant__conv,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( image_nat_set_nat
            @ ^ [X: nat] : C
            @ A3 )
          = bot_bot_set_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( image_nat_set_nat
            @ ^ [X: nat] : C
            @ A3 )
          = ( insert_set_nat @ C @ bot_bot_set_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_471_image__constant__conv,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( image_nat_nat2
            @ ^ [X: nat] : C
            @ A3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( image_nat_nat2
            @ ^ [X: nat] : C
            @ A3 )
          = ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).

% image_constant_conv
thf(fact_472_pred__subset__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( ord_le2712619360209801397char_o
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R3 )
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ S ) )
      = ( ord_le6315511645215477266t_char @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_473_pred__subset__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 )
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_474_UN__insert__distrib,axiom,
    ! [U: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,A: nat,B3: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ U @ A3 )
     => ( ( comple7399068483239264473et_nat
          @ ( image_15824709712370754et_nat
            @ ^ [X: product_prod_nat_nat] : ( insert_nat @ A @ ( B3 @ X ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_475_UN__insert__distrib,axiom,
    ! [U: nat,A3: set_nat,A: nat,B3: nat > set_nat] :
      ( ( member_nat @ U @ A3 )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( insert_nat @ A @ ( B3 @ X ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_476_inj__on__UNION__chain,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,A3: product_prod_nat_nat > set_nat,F6: nat > nat] :
      ( ! [I3: product_prod_nat_nat,J2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ I2 )
         => ( ( member8440522571783428010at_nat @ J2 @ I2 )
           => ( ( ord_less_eq_set_nat @ ( A3 @ I3 ) @ ( A3 @ J2 ) )
              | ( ord_less_eq_set_nat @ ( A3 @ J2 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I3 @ I2 )
           => ( inj_on_nat_nat @ F6 @ ( A3 @ I3 ) ) )
       => ( inj_on_nat_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_477_inj__on__UNION__chain,axiom,
    ! [I2: set_nat,A3: nat > set_nat,F6: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( member_nat @ I3 @ I2 )
         => ( ( member_nat @ J2 @ I2 )
           => ( ( ord_less_eq_set_nat @ ( A3 @ I3 ) @ ( A3 @ J2 ) )
              | ( ord_less_eq_set_nat @ ( A3 @ J2 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I2 )
           => ( inj_on_nat_nat @ F6 @ ( A3 @ I3 ) ) )
       => ( inj_on_nat_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_478_UN__extend__simps_I1_J,axiom,
    ! [C3: set_nat,A: nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( insert_nat @ A @ ( B3 @ X ) )
              @ C3 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_479_Sigma__def,axiom,
    ( produc6030673636645808891t_char
    = ( ^ [A7: set_nat,B8: nat > set_list_char] :
          ( comple9178978156834306527t_char
          @ ( image_7918911184113263195t_char
            @ ^ [X: nat] :
                ( comple9178978156834306527t_char
                @ ( image_366428816823531018t_char
                  @ ^ [Y6: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ bot_bo1414094081888755934t_char )
                  @ ( B8 @ X ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_480_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A7: set_nat,B8: nat > set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X: nat] :
                ( comple5685304695842803022at_nat
                @ ( image_7178329752028323786at_nat
                  @ ^ [Y6: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ bot_bo2099793752762293965at_nat )
                  @ ( B8 @ X ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_481_the__elem__def,axiom,
    ( the_elem_nat
    = ( ^ [X6: set_nat] :
          ( the_nat
          @ ^ [X: nat] :
              ( X6
              = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% the_elem_def
thf(fact_482_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A7: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_483_cSUP__UNION,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,F6: nat > set_nat] :
      ( ( A3 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit5477540289124974626et_nat
            @ ( comple548664676211718543et_nat
              @ ( image_1942688902138425592et_nat
                @ ^ [X: product_prod_nat_nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) )
            = ( comple7399068483239264473et_nat
              @ ( image_15824709712370754et_nat
                @ ^ [X: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_484_cSUP__UNION,axiom,
    ! [A3: set_nat,B3: nat > set_nat,F6: nat > set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit5477540289124974626et_nat
            @ ( comple548664676211718543et_nat
              @ ( image_2194112158459175443et_nat
                @ ^ [X: nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
            = ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_485_cSUP__UNION,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,F6: nat > nat] :
      ( ( A3 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit2214826472909112428ve_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_15824709712370754et_nat
                @ ^ [X: product_prod_nat_nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) )
            = ( complete_Sup_Sup_nat
              @ ( image_2486076414777270412at_nat
                @ ^ [X: product_prod_nat_nat] : ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_486_cSUP__UNION,axiom,
    ! [A3: set_nat,B3: nat > set_nat,F6: nat > nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit2214826472909112428ve_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
            = ( complete_Sup_Sup_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_487_subset__singleton__iff__Uniq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( ? [A2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      = ( uniq_P1720416203664110854at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_488_subset__singleton__iff__Uniq,axiom,
    ! [A3: set_nat] :
      ( ( ? [A2: nat] : ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      = ( uniq_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_489_inj__on__id2,axiom,
    ! [A3: set_nat] :
      ( inj_on_nat_nat
      @ ^ [X: nat] : X
      @ A3 ) ).

% inj_on_id2
thf(fact_490_inj__on__iff__Uniq,axiom,
    ( inj_on_nat_nat
    = ( ^ [F4: nat > nat,A7: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A7 )
         => ( uniq_nat
            @ ^ [Y6: nat] :
                ( ( member_nat @ Y6 @ A7 )
                & ( ( F4 @ X )
                  = ( F4 @ Y6 ) ) ) ) ) ) ) ).

% inj_on_iff_Uniq
thf(fact_491_inj__singleton,axiom,
    ! [A3: set_nat] :
      ( inj_on_nat_set_nat
      @ ^ [X: nat] : ( insert_nat @ X @ bot_bot_set_nat )
      @ A3 ) ).

% inj_singleton
thf(fact_492_Id__on__def_H,axiom,
    ! [A3: product_prod_nat_nat > $o] :
      ( ( id_on_2554058798563519774at_nat @ ( collec3392354462482085612at_nat @ A3 ) )
      = ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X: product_prod_nat_nat,Y6: product_prod_nat_nat] :
              ( ( X = Y6 )
              & ( A3 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_493_Id__on__def_H,axiom,
    ! [A3: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A3 ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y6: nat] :
              ( ( X = Y6 )
              & ( A3 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_494_insert__Times__insert,axiom,
    ! [A: nat,A3: set_nat,B: list_char,B3: set_list_char] :
      ( ( produc6030673636645808891t_char @ ( insert_nat @ A @ A3 )
        @ ^ [Uu: nat] : ( insert_list_char @ B @ B3 ) )
      = ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B )
        @ ( sup_su7899656352754981958t_char
          @ ( produc6030673636645808891t_char @ A3
            @ ^ [Uu: nat] : ( insert_list_char @ B @ B3 ) )
          @ ( produc6030673636645808891t_char @ ( insert_nat @ A @ A3 )
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_495_insert__Times__insert,axiom,
    ! [A: nat,A3: set_nat,B: nat,B3: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A3 )
        @ ^ [Uu: nat] : ( insert_nat @ B @ B3 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : ( insert_nat @ B @ B3 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A3 )
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_496_bind__singleton__conv__image,axiom,
    ! [A3: set_nat,F6: nat > set_nat] :
      ( ( bind_nat_set_nat @ A3
        @ ^ [X: nat] : ( insert_set_nat @ ( F6 @ X ) @ bot_bot_set_set_nat ) )
      = ( image_nat_set_nat @ F6 @ A3 ) ) ).

% bind_singleton_conv_image
thf(fact_497_bind__singleton__conv__image,axiom,
    ! [A3: set_nat,F6: nat > nat] :
      ( ( bind_nat_nat @ A3
        @ ^ [X: nat] : ( insert_nat @ ( F6 @ X ) @ bot_bot_set_nat ) )
      = ( image_nat_nat2 @ F6 @ A3 ) ) ).

% bind_singleton_conv_image
thf(fact_498_less__cSUP__iff,axiom,
    ! [A3: set_nat,F6: nat > nat,A: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
       => ( ( ord_less_nat @ A @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( ord_less_nat @ A @ ( F6 @ X ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_499_range__constant,axiom,
    ! [X5: set_nat] :
      ( ( image_nat_set_nat
        @ ^ [Uu: nat] : X5
        @ top_top_set_nat )
      = ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ).

% range_constant
thf(fact_500_range__constant,axiom,
    ! [X5: nat] :
      ( ( image_nat_nat2
        @ ^ [Uu: nat] : X5
        @ top_top_set_nat )
      = ( insert_nat @ X5 @ bot_bot_set_nat ) ) ).

% range_constant
thf(fact_501_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ^ [S4: product_prod_nat_nat] : P )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ^ [S4: product_prod_nat_nat] : P )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const
thf(fact_502_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S4: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S4: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_503_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_504_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B2: nat] : P ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B2: nat] : P ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_505_UN__Un,axiom,
    ! [M: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A3 @ B3 ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B3 ) ) ) ) ).

% UN_Un
thf(fact_506_bdd__above__image__sup,axiom,
    ! [F6: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( condit2214826472909112428ve_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : ( sup_sup_nat @ ( F6 @ X ) @ ( G @ X ) )
          @ A3 ) )
      = ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
        & ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ A3 ) ) ) ) ).

% bdd_above_image_sup
thf(fact_507_bdd__above__image__sup,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( condit5477540289124974626et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( sup_sup_set_nat @ ( F6 @ X ) @ ( G @ X ) )
          @ A3 ) )
      = ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
        & ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ).

% bdd_above_image_sup
thf(fact_508_UN__simps_I3_J,axiom,
    ! [C3: set_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) )
          = bot_bot_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) )
          = ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_509_UN__simps_I2_J,axiom,
    ! [C3: set_nat,A3: nat > set_nat,B3: set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) )
          = bot_bot_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) )
          = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 ) ) ) ) ).

% UN_simps(2)
thf(fact_510_UN__insert,axiom,
    ! [B3: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( B3 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ).

% UN_insert
thf(fact_511_SUP__UNIV__bool__expand,axiom,
    ! [A3: $o > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ top_top_set_o ) )
      = ( sup_sup_set_nat @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ).

% SUP_UNIV_bool_expand
thf(fact_512_Un__eq__UN,axiom,
    ( sup_sup_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( comple7399068483239264473et_nat
          @ ( image_o_set_nat
            @ ^ [B2: $o] : ( if_set_nat @ B2 @ A7 @ B8 )
            @ top_top_set_o ) ) ) ) ).

% Un_eq_UN
thf(fact_513_UN__bool__eq,axiom,
    ! [A3: $o > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A3 @ top_top_set_o ) )
      = ( sup_sup_set_nat @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ).

% UN_bool_eq
thf(fact_514_SUP__union,axiom,
    ! [M: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A3 @ B3 ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B3 ) ) ) ) ).

% SUP_union
thf(fact_515_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_516_Collect__disj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su6327502436637775413at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_517_UNIV__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_518_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_519_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              | ( member_nat @ X @ B8 ) ) ) ) ) ).

% Un_def
thf(fact_520_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A7 )
              | ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% Un_def
thf(fact_521_insert__def,axiom,
    ( insert_nat
    = ( ^ [A2: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_522_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A2: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] : ( X = A2 ) ) ) ) ) ).

% insert_def
thf(fact_523_inj__on__add_H,axiom,
    ! [A: nat,A3: set_nat] :
      ( inj_on_nat_nat
      @ ^ [B2: nat] : ( plus_plus_nat @ B2 @ A )
      @ A3 ) ).

% inj_on_add'
thf(fact_524_rangeE,axiom,
    ! [B: set_nat,F6: nat > set_nat] :
      ( ( member_set_nat @ B @ ( image_nat_set_nat @ F6 @ top_top_set_nat ) )
     => ~ ! [X4: nat] :
            ( B
           != ( F6 @ X4 ) ) ) ).

% rangeE
thf(fact_525_rangeE,axiom,
    ! [B: product_prod_nat_nat,F6: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F6 @ top_top_set_nat ) )
     => ~ ! [X4: nat] :
            ( B
           != ( F6 @ X4 ) ) ) ).

% rangeE
thf(fact_526_rangeE,axiom,
    ! [B: nat,F6: nat > nat] :
      ( ( member_nat @ B @ ( image_nat_nat2 @ F6 @ top_top_set_nat ) )
     => ~ ! [X4: nat] :
            ( B
           != ( F6 @ X4 ) ) ) ).

% rangeE
thf(fact_527_range__composition,axiom,
    ! [F6: set_nat > nat,G: nat > set_nat] :
      ( ( image_nat_nat2
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_set_nat_nat @ F6 @ ( image_nat_set_nat @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_528_range__composition,axiom,
    ! [F6: nat > nat,G: nat > nat] :
      ( ( image_nat_nat2
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_nat_nat2 @ F6 @ ( image_nat_nat2 @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_529_range__composition,axiom,
    ! [F6: set_nat > set_nat,G: nat > set_nat] :
      ( ( image_nat_set_nat
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_7916887816326733075et_nat @ F6 @ ( image_nat_set_nat @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_530_range__composition,axiom,
    ! [F6: nat > set_nat,G: nat > nat] :
      ( ( image_nat_set_nat
        @ ^ [X: nat] : ( F6 @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_nat_set_nat @ F6 @ ( image_nat_nat2 @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_531_bind__const,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( bind_nat_nat @ A3
            @ ^ [Uu: nat] : B3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( bind_nat_nat @ A3
            @ ^ [Uu: nat] : B3 )
          = B3 ) ) ) ).

% bind_const
thf(fact_532_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [F6: nat > set_nat,A3: set_nat,G: nat > set_nat] :
      ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( sup_sup_set_nat @ ( F6 @ A2 ) @ ( G @ A2 ) )
          @ A3 ) ) ) ).

% complete_lattice_class.SUP_sup_distrib
thf(fact_533_SUP__absorb,axiom,
    ! [K: product_prod_nat_nat,I2: set_Pr1261947904930325089at_nat,A3: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_534_SUP__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_nat] :
      ( ( member_nat @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_535_Un__Union__image,axiom,
    ! [A3: nat > set_nat,B3: nat > set_nat,C3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ ( B3 @ X ) )
          @ C3 ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) ) ) ).

% Un_Union_image
thf(fact_536_UN__Un__distrib,axiom,
    ! [A3: nat > set_nat,B3: nat > set_nat,I2: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] : ( sup_sup_set_nat @ ( A3 @ I ) @ ( B3 @ I ) )
          @ I2 ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ I2 ) ) ) ) ).

% UN_Un_distrib
thf(fact_537_UN__absorb,axiom,
    ! [K: product_prod_nat_nat,I2: set_Pr1261947904930325089at_nat,A3: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_538_UN__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_nat] :
      ( ( member_nat @ K @ I2 )
     => ( ( sup_sup_set_nat @ ( A3 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% UN_absorb
thf(fact_539_SUP__lessD,axiom,
    ! [F6: product_prod_nat_nat > set_nat,A3: set_Pr1261947904930325089at_nat,Y: set_nat,I5: product_prod_nat_nat] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) @ Y )
     => ( ( member8440522571783428010at_nat @ I5 @ A3 )
       => ( ord_less_set_nat @ ( F6 @ I5 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_540_SUP__lessD,axiom,
    ! [F6: nat > set_nat,A3: set_nat,Y: set_nat,I5: nat] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ Y )
     => ( ( member_nat @ I5 @ A3 )
       => ( ord_less_set_nat @ ( F6 @ I5 ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_541_SUP__insert,axiom,
    ! [F6: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( insert_nat @ A @ A3 ) ) )
      = ( sup_sup_set_nat @ ( F6 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% SUP_insert
thf(fact_542_UN__extend__simps_I3_J,axiom,
    ! [C3: set_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = A3 ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_543_UN__extend__simps_I2_J,axiom,
    ! [C3: set_nat,A3: nat > set_nat,B3: set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
          = B3 ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
          = ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_544_cSUP__lessD,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,Y: nat,I5: product_prod_nat_nat] :
      ( ( condit2214826472909112428ve_nat @ ( image_2486076414777270412at_nat @ F6 @ A3 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_2486076414777270412at_nat @ F6 @ A3 ) ) @ Y )
       => ( ( member8440522571783428010at_nat @ I5 @ A3 )
         => ( ord_less_nat @ ( F6 @ I5 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_545_cSUP__lessD,axiom,
    ! [F6: nat > nat,A3: set_nat,Y: nat,I5: nat] :
      ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) @ Y )
       => ( ( member_nat @ I5 @ A3 )
         => ( ord_less_nat @ ( F6 @ I5 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_546_cSUP__lessD,axiom,
    ! [F6: product_prod_nat_nat > set_nat,A3: set_Pr1261947904930325089at_nat,Y: set_nat,I5: product_prod_nat_nat] :
      ( ( condit5477540289124974626et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) )
     => ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) @ Y )
       => ( ( member8440522571783428010at_nat @ I5 @ A3 )
         => ( ord_less_set_nat @ ( F6 @ I5 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_547_cSUP__lessD,axiom,
    ! [F6: nat > set_nat,A3: set_nat,Y: set_nat,I5: nat] :
      ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
     => ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ Y )
       => ( ( member_nat @ I5 @ A3 )
         => ( ord_less_set_nat @ ( F6 @ I5 ) @ Y ) ) ) ) ).

% cSUP_lessD
thf(fact_548_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_nat,F6: nat > nat,G: nat > nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
       => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ G @ A3 ) )
         => ( ( sup_sup_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ G @ A3 ) ) )
            = ( complete_Sup_Sup_nat
              @ ( image_nat_nat2
                @ ^ [A2: nat] : ( sup_sup_nat @ ( F6 @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_549_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
       => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ G @ A3 ) )
         => ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) )
            = ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [A2: nat] : ( sup_sup_set_nat @ ( F6 @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_550_inj__on__apsnd,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc3094765310956027504at_nat @ F6 )
        @ ( produc457027306803732586at_nat @ top_top_set_nat
          @ ^ [Uu: nat] : A3 ) )
      = ( inj_on_nat_nat @ F6 @ A3 ) ) ).

% inj_on_apsnd
thf(fact_551_sorted__list__of__set_Oinj__on,axiom,
    ( inj_on_nat_nat
    @ ^ [X: nat] : X
    @ top_top_set_nat ) ).

% sorted_list_of_set.inj_on
thf(fact_552_inj__on__apfst,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( inj_on8969904277767023793at_nat @ ( produc986720760941809198at_nat @ F6 )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : top_top_set_nat ) )
      = ( inj_on_nat_nat @ F6 @ A3 ) ) ).

% inj_on_apfst
thf(fact_553_fun_Oin__rel,axiom,
    ! [R3: nat > list_char > $o,A: nat > nat,B: nat > list_char] :
      ( ( bNF_re2076907181175516082t_char
        @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: nat > produc1457211279475724562t_char] :
            ( ( member3158901901048444578t_char @ Z6
              @ ( collec4820088019371311076t_char
                @ ^ [X: nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_1003422490162143483t_char @ X @ top_top_set_nat ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) ) )
            & ( ( comp_P8410057226579278353at_nat @ produc6204653154488278070t_char @ Z6 )
              = A )
            & ( ( comp_P4974907838745982882ar_nat @ produc1900778367047502200t_char @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_554_fun_Oin__rel,axiom,
    ! [R3: nat > nat > $o,A: nat > nat,B: nat > nat] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: nat > product_prod_nat_nat] :
            ( ( member6743353555990375057at_nat @ Z6
              @ ( collec8718012497556192979at_nat
                @ ^ [X: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat @ Z6 )
              = A )
            & ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_555_in__measure,axiom,
    ! [X5: nat,Y: nat,F6: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ ( measure_nat @ F6 ) )
      = ( ord_less_nat @ ( F6 @ X5 ) @ ( F6 @ Y ) ) ) ).

% in_measure
thf(fact_556_val_Ocase__transfer,axiom,
    ! [S: $o > $o > $o] :
      ( bNF_re3400349493146995663_val_o
      @ ( bNF_rel_fun_o_o_o_o
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re4064708811108098077_val_o
        @ ( bNF_re4705727531993890431at_o_o
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re6376993983822893079_val_o
          @ ( bNF_re2178160303036436095at_o_o
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re3282371258598777373al_o_o
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_o
      @ case_val_o ) ).

% val.case_transfer
thf(fact_557_val_Ocase__transfer,axiom,
    ! [S: $o > nat > $o] :
      ( bNF_re8530354864449075465al_nat
      @ ( bNF_re9072777675879608139_o_nat
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re9138709833252949743al_nat
        @ ( bNF_re7523573913982466217_o_nat
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re4353138970906064163al_nat
          @ ( bNF_re2521994545097566377_o_nat
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re8984071810037758091_o_nat
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_o
      @ case_val_nat ) ).

% val.case_transfer
thf(fact_558_val_Ocase__transfer,axiom,
    ! [S: $o > prat > $o] :
      ( bNF_re1381473823880018185l_prat
      @ ( bNF_re7136572702137213643o_prat
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re8023842771505706095l_prat
        @ ( bNF_re8284746473853242921o_prat
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re8035667688264635171l_prat
          @ ( bNF_re2805847409588092457o_prat
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re1819705386720226315o_prat
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_o
      @ case_val_prat ) ).

% val.case_transfer
thf(fact_559_val_Ocase__transfer,axiom,
    ! [S: nat > $o > $o] :
      ( bNF_re2233505157435102499_val_o
      @ ( bNF_re2952077000133017641_nat_o
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re7947119355560860307_val_o
        @ ( bNF_re1402873238235875719_nat_o
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re7761813834571909981_val_o
          @ ( bNF_re5624665906205751687_nat_o
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re2863371134291167593_nat_o
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_nat
      @ case_val_o ) ).

% val.case_transfer
thf(fact_560_val_Ocase__transfer,axiom,
    ! [S: nat > nat > $o] :
      ( bNF_re5390656012764358813al_nat
      @ ( bNF_re552426215088010623at_nat
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re3142472647042036189al_nat
        @ ( bNF_re5653821019739307937at_nat
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re6117652819335925469al_nat
          @ ( bNF_re1361929969394864033at_nat
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re478083327121898559at_nat
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_nat
      @ case_val_nat ) ).

% val.case_transfer
thf(fact_561_val_Ocase__transfer,axiom,
    ! [S: nat > prat > $o] :
      ( bNF_re5807499856486624669l_prat
      @ ( bNF_re3012592894602769663t_prat
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re3017882675219251037l_prat
        @ ( bNF_re4147878452613618977t_prat
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re5989471590236891101l_prat
          @ ( bNF_re2796068022358551841t_prat
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re462759099702918079t_prat
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_nat
      @ case_val_prat ) ).

% val.case_transfer
thf(fact_562_val_Ocase__transfer,axiom,
    ! [S: prat > $o > $o] :
      ( bNF_re1988916615843939875_val_o
      @ ( bNF_re3555829217104990121prat_o
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re5697875234658812691_val_o
        @ ( bNF_re4704002988821019399prat_o
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re1271012312234361181_val_o
          @ ( bNF_re8448475961410644743prat_o
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re7462333938542778601prat_o
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_prat
      @ case_val_o ) ).

% val.case_transfer
thf(fact_563_val_Ocase__transfer,axiom,
    ! [S: prat > nat > $o] :
      ( bNF_re3676624960930530973al_nat
      @ ( bNF_re4232132340546668031at_nat
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re3377346888436407133al_nat
        @ ( bNF_re5367417898557517345at_nat
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re4810291329183323613al_nat
          @ ( bNF_re4015607468302450209at_nat
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re1682298545646816447at_nat
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_prat
      @ case_val_nat ) ).

% val.case_transfer
thf(fact_564_val_Ocase__transfer,axiom,
    ! [S: prat > prat > $o] :
      ( bNF_re8674362571971103645l_prat
      @ ( bNF_re2544178717384787327t_prat
        @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 )
        @ S )
      @ ( bNF_re6019836462720788701l_prat
        @ ( bNF_re2294495614902143905t_prat
          @ ^ [Y7: nat,Z2: nat] : ( Y7 = Z2 )
          @ S )
        @ ( bNF_re5071352437643341533l_prat
          @ ( bNF_re2220458561309875105t_prat
            @ ^ [Y7: prat,Z2: prat] : ( Y7 = Z2 )
            @ S )
          @ ( bNF_re8661857836907285567t_prat
            @ ^ [Y7: val,Z2: val] : ( Y7 = Z2 )
            @ S ) ) )
      @ case_val_prat
      @ case_val_prat ) ).

% val.case_transfer
thf(fact_565_top__empty__eq2,axiom,
    ( top_to1537031869075220069char_o
    = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ top_to1250346684925019074t_char ) ) ) ).

% top_empty_eq2
thf(fact_566_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_567_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 )
            @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ) ).

% sup_set_def
thf(fact_568_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% sup_set_def
thf(fact_569_sup__Un__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su798857527126471912_nat_o
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R3 )
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S ) )
      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( sup_su6327502436637775413at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_570_sup__Un__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R3 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_571_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: nat > list_char > $o,P: nat > nat > $o,Q: list_char > list_char > $o,A: produc1457211279475724562t_char,A3: set_Pr2507339680178222706t_char,B: produc1457211279475724562t_char,B3: set_Pr2507339680178222706t_char] :
      ( ( bNF_re2942033592892840063char_o @ R1
        @ ( bNF_re1135979186550299152ar_o_o @ R22
          @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 ) )
        @ P
        @ Q )
     => ( ( member484235747325421115t_char @ A @ A3 )
       => ( ( member484235747325421115t_char @ B @ B3 )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R1 ) ) )
           => ( ( ord_le6315511645215477266t_char @ B3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R22 ) ) )
             => ( ( P @ ( produc6204653154488278070t_char @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q @ ( produc1900778367047502200t_char @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_572_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: nat > nat > $o,P: nat > nat > $o,Q: list_char > nat > $o,A: produc1457211279475724562t_char,A3: set_Pr2507339680178222706t_char,B: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re1113502957280211792_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 ) )
        @ P
        @ Q )
     => ( ( member484235747325421115t_char @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B3 )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P @ ( produc6204653154488278070t_char @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q @ ( produc1900778367047502200t_char @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_573_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > list_char > $o,P: nat > nat > $o,Q: nat > list_char > $o,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc1457211279475724562t_char,B3: set_Pr2507339680178222706t_char] :
      ( ( bNF_re4495586573714376046char_o @ R1
        @ ( bNF_re1135979186550299152ar_o_o @ R22
          @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 ) )
        @ P
        @ Q )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( member484235747325421115t_char @ B @ B3 )
         => ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le6315511645215477266t_char @ B3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R22 ) ) )
             => ( ( P @ ( product_fst_nat_nat @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q @ ( product_snd_nat_nat @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_574_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,P: nat > nat > $o,Q: nat > nat > $o,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re578469030762574527_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y7: $o,Z2: $o] : ( Y7 = Z2 ) )
        @ P
        @ Q )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B3 )
         => ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P @ ( product_fst_nat_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q @ ( product_snd_nat_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_575_sup__Un__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( sup_su1536542383718553321char_o
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R3 )
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ S ) )
      = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( sup_su7899656352754981958t_char @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_576_sup__Un__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 )
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ S ) )
      = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( sup_su6327502436637775413at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_577_inj__graph,axiom,
    ( inj_on3412903767820855855at_nat
    @ ^ [F4: nat > nat] :
        ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y6: nat] :
              ( Y6
              = ( F4 @ X ) ) ) )
    @ top_top_set_nat_nat ) ).

% inj_graph
thf(fact_578_inj__vimage__singleton,axiom,
    ! [F6: nat > nat,A: nat] :
      ( ( inj_on_nat_nat @ F6 @ top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F6 @ ( insert_nat @ A @ bot_bot_set_nat ) )
        @ ( insert_nat
          @ ( the_nat
            @ ^ [X: nat] :
                ( ( F6 @ X )
                = A ) )
          @ bot_bot_set_nat ) ) ) ).

% inj_vimage_singleton
thf(fact_579_mono__cSUP,axiom,
    ! [F6: nat > nat,A3: nat > nat,I2: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_nat
            @ ( complete_Sup_Sup_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_580_mono__cSUP,axiom,
    ! [F6: nat > filter_nat,A3: nat > nat,I2: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat
            @ ( comple2454491259644266675er_nat
              @ ( image_nat_filter_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_581_mono__cSUP,axiom,
    ! [F6: filter_nat > nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit4029664723990530684er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_nat
            @ ( complete_Sup_Sup_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_582_mono__cSUP,axiom,
    ! [F6: filter_nat > filter_nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit4029664723990530684er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat
            @ ( comple2454491259644266675er_nat
              @ ( image_nat_filter_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_583_mono__cSUP,axiom,
    ! [F6: set_nat > nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_nat
            @ ( complete_Sup_Sup_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_584_mono__cSUP,axiom,
    ! [F6: set_nat > filter_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto5359140060069215823er_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat
            @ ( comple2454491259644266675er_nat
              @ ( image_nat_filter_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_585_mono__cSUP,axiom,
    ! [F6: nat > set_nat,A3: nat > nat,I2: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_586_mono__cSUP,axiom,
    ! [F6: filter_nat > set_nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto6401136896278960335et_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit4029664723990530684er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_587_mono__cSUP,axiom,
    ! [F6: set_nat > set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) )
            @ ( F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ) ) ).

% mono_cSUP
thf(fact_588_mono__cSup,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) @ ( F6 @ ( complete_Sup_Sup_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_589_mono__cSup,axiom,
    ! [F6: nat > filter_nat,A3: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) @ ( F6 @ ( complete_Sup_Sup_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_590_mono__cSup,axiom,
    ! [F6: filter_nat > nat,A3: set_filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit4029664723990530684er_nat @ A3 )
       => ( ( A3 != bot_bo498966703094740906er_nat )
         => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_filter_nat_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple2454491259644266675er_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_591_mono__cSup,axiom,
    ! [F6: filter_nat > filter_nat,A3: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit4029664723990530684er_nat @ A3 )
       => ( ( A3 != bot_bo498966703094740906er_nat )
         => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_4775440762538963527er_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple2454491259644266675er_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_592_mono__cSup,axiom,
    ! [F6: set_nat > nat,A3: set_set_nat] :
      ( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit5477540289124974626et_nat @ A3 )
       => ( ( A3 != bot_bot_set_set_nat )
         => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_set_nat_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_593_mono__cSup,axiom,
    ! [F6: set_nat > filter_nat,A3: set_set_nat] :
      ( ( monoto5359140060069215823er_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit5477540289124974626et_nat @ A3 )
       => ( ( A3 != bot_bot_set_set_nat )
         => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_8573581984095792621er_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_594_mono__cSup,axiom,
    ! [F6: nat > set_nat,A3: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( F6 @ ( complete_Sup_Sup_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_595_mono__cSup,axiom,
    ! [F6: filter_nat > set_nat,A3: set_filter_nat] :
      ( ( monoto6401136896278960335et_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit4029664723990530684er_nat @ A3 )
       => ( ( A3 != bot_bo498966703094740906er_nat )
         => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_392206783450761325et_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple2454491259644266675er_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_596_mono__cSup,axiom,
    ! [F6: set_nat > set_nat,A3: set_set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit5477540289124974626et_nat @ A3 )
       => ( ( A3 != bot_bot_set_set_nat )
         => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_597_vimage__Collect__eq,axiom,
    ! [F6: nat > nat,P: nat > $o] :
      ( ( vimage_nat_nat @ F6 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [Y6: nat] : ( P @ ( F6 @ Y6 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_598_vimage__Collect__eq,axiom,
    ! [F6: product_prod_nat_nat > nat,P: nat > $o] :
      ( ( vimage4653281326611754070at_nat @ F6 @ ( collect_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [Y6: product_prod_nat_nat] : ( P @ ( F6 @ Y6 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_599_vimage__Collect__eq,axiom,
    ! [F6: nat > product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( vimage8013328719654469172at_nat @ F6 @ ( collec3392354462482085612at_nat @ P ) )
      = ( collect_nat
        @ ^ [Y6: nat] : ( P @ ( F6 @ Y6 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_600_vimage__Collect__eq,axiom,
    ! [F6: product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( vimage2449269961533847803at_nat @ F6 @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [Y6: product_prod_nat_nat] : ( P @ ( F6 @ Y6 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_601_vimage__const,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ C @ A3 )
       => ( ( vimage8013328719654469172at_nat
            @ ^ [X: nat] : C
            @ A3 )
          = top_top_set_nat ) )
      & ( ~ ( member8440522571783428010at_nat @ C @ A3 )
       => ( ( vimage8013328719654469172at_nat
            @ ^ [X: nat] : C
            @ A3 )
          = bot_bot_set_nat ) ) ) ).

% vimage_const
thf(fact_602_vimage__const,axiom,
    ! [C: nat,A3: set_nat] :
      ( ( ( member_nat @ C @ A3 )
       => ( ( vimage_nat_nat
            @ ^ [X: nat] : C
            @ A3 )
          = top_top_set_nat ) )
      & ( ~ ( member_nat @ C @ A3 )
       => ( ( vimage_nat_nat
            @ ^ [X: nat] : C
            @ A3 )
          = bot_bot_set_nat ) ) ) ).

% vimage_const
thf(fact_603_vimage__def,axiom,
    ( vimage8013328719654469172at_nat
    = ( ^ [F4: nat > product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( member8440522571783428010at_nat @ ( F4 @ X ) @ B8 ) ) ) ) ).

% vimage_def
thf(fact_604_vimage__def,axiom,
    ( vimage_nat_nat
    = ( ^ [F4: nat > nat,B8: set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( member_nat @ ( F4 @ X ) @ B8 ) ) ) ) ).

% vimage_def
thf(fact_605_vimage__def,axiom,
    ( vimage2449269961533847803at_nat
    = ( ^ [F4: product_prod_nat_nat > product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ ( F4 @ X ) @ B8 ) ) ) ) ).

% vimage_def
thf(fact_606_vimage__def,axiom,
    ( vimage4653281326611754070at_nat
    = ( ^ [F4: product_prod_nat_nat > nat,B8: set_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( member_nat @ ( F4 @ X ) @ B8 ) ) ) ) ).

% vimage_def
thf(fact_607_antimonoD,axiom,
    ! [F6: nat > nat,X5: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_less_eq_nat @ X5 @ Y )
       => ( ord_less_eq_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoD
thf(fact_608_antimonoD,axiom,
    ! [F6: nat > filter_nat,X5: nat,Y: nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_less_eq_nat @ X5 @ Y )
       => ( ord_le2510731241096832064er_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoD
thf(fact_609_antimonoD,axiom,
    ! [F6: filter_nat > nat,X5: filter_nat,Y: filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_le2510731241096832064er_nat @ X5 @ Y )
       => ( ord_less_eq_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoD
thf(fact_610_antimonoD,axiom,
    ! [F6: filter_nat > filter_nat,X5: filter_nat,Y: filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_le2510731241096832064er_nat @ X5 @ Y )
       => ( ord_le2510731241096832064er_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoD
thf(fact_611_antimonoE,axiom,
    ! [F6: nat > nat,X5: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_less_eq_nat @ X5 @ Y )
       => ( ord_less_eq_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoE
thf(fact_612_antimonoE,axiom,
    ! [F6: nat > filter_nat,X5: nat,Y: nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_less_eq_nat @ X5 @ Y )
       => ( ord_le2510731241096832064er_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoE
thf(fact_613_antimonoE,axiom,
    ! [F6: filter_nat > nat,X5: filter_nat,Y: filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_le2510731241096832064er_nat @ X5 @ Y )
       => ( ord_less_eq_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoE
thf(fact_614_antimonoE,axiom,
    ! [F6: filter_nat > filter_nat,X5: filter_nat,Y: filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( ord_le2510731241096832064er_nat @ X5 @ Y )
       => ( ord_le2510731241096832064er_nat @ ( F6 @ Y ) @ ( F6 @ X5 ) ) ) ) ).

% antimonoE
thf(fact_615_antimonoI,axiom,
    ! [F6: nat > nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
         => ( ord_less_eq_nat @ ( F6 @ Y5 ) @ ( F6 @ X4 ) ) )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 ) ) ).

% antimonoI
thf(fact_616_antimonoI,axiom,
    ! [F6: nat > filter_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ Y5 ) @ ( F6 @ X4 ) ) )
     => ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 ) ) ).

% antimonoI
thf(fact_617_antimonoI,axiom,
    ! [F6: filter_nat > nat] :
      ( ! [X4: filter_nat,Y5: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X4 @ Y5 )
         => ( ord_less_eq_nat @ ( F6 @ Y5 ) @ ( F6 @ X4 ) ) )
     => ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 ) ) ).

% antimonoI
thf(fact_618_antimonoI,axiom,
    ! [F6: filter_nat > filter_nat] :
      ( ! [X4: filter_nat,Y5: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X4 @ Y5 )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ Y5 ) @ ( F6 @ X4 ) ) )
     => ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 ) ) ).

% antimonoI
thf(fact_619_vimage__UN,axiom,
    ! [F6: nat > nat,B3: nat > set_nat,A3: set_nat] :
      ( ( vimage_nat_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( vimage_nat_nat @ F6 @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% vimage_UN
thf(fact_620_vimage__Union,axiom,
    ! [F6: nat > nat,A3: set_set_nat] :
      ( ( vimage_nat_nat @ F6 @ ( comple7399068483239264473et_nat @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( vimage_nat_nat @ F6 ) @ A3 ) ) ) ).

% vimage_Union
thf(fact_621_vimage__fst,axiom,
    ! [A3: set_nat] :
      ( ( vimage2352271050577936999ar_nat @ produc6204653154488278070t_char @ A3 )
      = ( produc6030673636645808891t_char @ A3
        @ ^ [Uu: nat] : top_to3190553582279535303t_char ) ) ).

% vimage_fst
thf(fact_622_mono__Sup,axiom,
    ! [F6: filter_nat > filter_nat,A3: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_4775440762538963527er_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple2454491259644266675er_nat @ A3 ) ) ) ) ).

% mono_Sup
thf(fact_623_mono__Sup,axiom,
    ! [F6: set_nat > filter_nat,A3: set_set_nat] :
      ( ( monoto5359140060069215823er_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ord_le2510731241096832064er_nat @ ( comple2454491259644266675er_nat @ ( image_8573581984095792621er_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% mono_Sup
thf(fact_624_mono__Sup,axiom,
    ! [F6: filter_nat > set_nat,A3: set_filter_nat] :
      ( ( monoto6401136896278960335et_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_set_nat @ F6 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_392206783450761325et_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple2454491259644266675er_nat @ A3 ) ) ) ) ).

% mono_Sup
thf(fact_625_mono__Sup,axiom,
    ! [F6: set_nat > set_nat,A3: set_set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F6 @ A3 ) ) @ ( F6 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% mono_Sup
thf(fact_626_mono__SUP,axiom,
    ! [F6: set_nat > filter_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto5359140060069215823er_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ord_le2510731241096832064er_nat
        @ ( comple2454491259644266675er_nat
          @ ( image_nat_filter_nat
            @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
            @ I2 ) )
        @ ( F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ).

% mono_SUP
thf(fact_627_mono__SUP,axiom,
    ! [F6: filter_nat > set_nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto6401136896278960335et_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_set_nat @ F6 )
     => ( ord_less_eq_set_nat
        @ ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
            @ I2 ) )
        @ ( F6 @ ( comple2454491259644266675er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) ) ) ) ).

% mono_SUP
thf(fact_628_mono__SUP,axiom,
    ! [F6: set_nat > set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ord_less_eq_set_nat
        @ ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
            @ I2 ) )
        @ ( F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ).

% mono_SUP
thf(fact_629_vimage__eq__UN,axiom,
    ( vimage_nat_nat
    = ( ^ [F4: nat > nat,B8: set_nat] :
          ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [Y6: nat] : ( vimage_nat_nat @ F4 @ ( insert_nat @ Y6 @ bot_bot_set_nat ) )
            @ B8 ) ) ) ) ).

% vimage_eq_UN
thf(fact_630_image__split__eq__Sigma,axiom,
    ! [F6: nat > set_nat,G: nat > nat,A3: set_nat] :
      ( ( image_4184609661993008672at_nat
        @ ^ [X: nat] : ( produc641871753055645167at_nat @ ( F6 @ X ) @ ( G @ X ) )
        @ A3 )
      = ( produc5318310618559172640at_nat @ ( image_nat_set_nat @ F6 @ A3 )
        @ ^ [X: set_nat] : ( image_nat_nat2 @ G @ ( inf_inf_set_nat @ ( vimage_nat_set_nat @ F6 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ A3 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_631_image__split__eq__Sigma,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( image_3168736909456693974et_nat
        @ ^ [X: nat] : ( produc4532415448927165861et_nat @ ( F6 @ X ) @ ( G @ X ) )
        @ A3 )
      = ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F6 @ A3 )
        @ ^ [X: set_nat] : ( image_nat_set_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_set_nat @ F6 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ A3 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_632_image__split__eq__Sigma,axiom,
    ! [F6: nat > nat,G: nat > set_nat,A3: set_nat] :
      ( ( image_8316717911487899424et_nat
        @ ^ [X: nat] : ( produc4207506657711014383et_nat @ ( F6 @ X ) @ ( G @ X ) )
        @ A3 )
      = ( produc8883945523214541856et_nat @ ( image_nat_nat2 @ F6 @ A3 )
        @ ^ [X: nat] : ( image_nat_set_nat @ G @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A3 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_633_image__split__eq__Sigma,axiom,
    ! [F6: nat > nat,G: nat > list_char,A3: set_nat] :
      ( ( image_1003422490162143483t_char
        @ ^ [X: nat] : ( produc120197956887798346t_char @ ( F6 @ X ) @ ( G @ X ) )
        @ A3 )
      = ( produc6030673636645808891t_char @ ( image_nat_nat2 @ F6 @ A3 )
        @ ^ [X: nat] : ( image_nat_list_char2 @ G @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A3 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_634_image__split__eq__Sigma,axiom,
    ! [F6: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( image_5846123807819985514at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ ( F6 @ X ) @ ( G @ X ) )
        @ A3 )
      = ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F6 @ A3 )
        @ ^ [X: nat] : ( image_nat_nat2 @ G @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ A3 ) ) ) ) ).

% image_split_eq_Sigma
thf(fact_635_if__image__distrib,axiom,
    ! [P: nat > $o,F6: nat > nat,G: nat > nat,S: set_nat] :
      ( ( image_nat_nat2
        @ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F6 @ X ) @ ( G @ X ) )
        @ S )
      = ( sup_sup_set_nat @ ( image_nat_nat2 @ F6 @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_nat_nat2 @ G
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X: nat] :
                  ~ ( P @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_636_if__image__distrib,axiom,
    ! [P: nat > $o,F6: nat > set_nat,G: nat > set_nat,S: set_nat] :
      ( ( image_nat_set_nat
        @ ^ [X: nat] : ( if_set_nat @ ( P @ X ) @ ( F6 @ X ) @ ( G @ X ) )
        @ S )
      = ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F6 @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_nat_set_nat @ G
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X: nat] :
                  ~ ( P @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_637_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              & ( member_nat @ X @ B8 ) ) ) ) ) ).

% Int_def
thf(fact_638_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A7 )
              & ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% Int_def
thf(fact_639_Int__Collect,axiom,
    ! [X5: nat,A3: set_nat,P: nat > $o] :
      ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X5 @ A3 )
        & ( P @ X5 ) ) ) ).

% Int_Collect
thf(fact_640_Int__Collect,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X5 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) )
      = ( ( member8440522571783428010at_nat @ X5 @ A3 )
        & ( P @ X5 ) ) ) ).

% Int_Collect
thf(fact_641_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_642_Collect__conj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_643_SUP__inf,axiom,
    ! [F6: nat > set_nat,B3: set_nat,A: set_nat] :
      ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ B3 ) ) @ A )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [B2: nat] : ( inf_inf_set_nat @ ( F6 @ B2 ) @ A )
          @ B3 ) ) ) ).

% SUP_inf
thf(fact_644_Sup__inf,axiom,
    ! [B3: set_set_nat,A: set_nat] :
      ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B3 ) @ A )
      = ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [B2: set_nat] : ( inf_inf_set_nat @ B2 @ A )
          @ B3 ) ) ) ).

% Sup_inf
thf(fact_645_inf__SUP,axiom,
    ! [A: set_nat,F6: nat > set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ B3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [B2: nat] : ( inf_inf_set_nat @ A @ ( F6 @ B2 ) )
          @ B3 ) ) ) ).

% inf_SUP
thf(fact_646_SUP__inf__distrib2,axiom,
    ! [F6: nat > set_nat,A3: set_nat,G: nat > set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [B2: nat] : ( inf_inf_set_nat @ ( F6 @ A2 ) @ ( G @ B2 ) )
                @ B3 ) )
          @ A3 ) ) ) ).

% SUP_inf_distrib2
thf(fact_647_UN__extend__simps_I5_J,axiom,
    ! [A3: set_nat,B3: nat > set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( inf_inf_set_nat @ A3 @ ( B3 @ X ) )
          @ C3 ) ) ) ).

% UN_extend_simps(5)
thf(fact_648_UN__extend__simps_I4_J,axiom,
    ! [A3: nat > set_nat,C3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( inf_inf_set_nat @ ( A3 @ X ) @ B3 )
          @ C3 ) ) ) ).

% UN_extend_simps(4)
thf(fact_649_Int__UN__distrib,axiom,
    ! [B3: set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( inf_inf_set_nat @ B3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] : ( inf_inf_set_nat @ B3 @ ( A3 @ I ) )
          @ I2 ) ) ) ).

% Int_UN_distrib
thf(fact_650_Int__UN__distrib2,axiom,
    ! [A3: nat > set_nat,I2: set_nat,B3: nat > set_nat,J3: set_nat] :
      ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ J3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [J: nat] : ( inf_inf_set_nat @ ( A3 @ I ) @ ( B3 @ J ) )
                @ J3 ) )
          @ I2 ) ) ) ).

% Int_UN_distrib2
thf(fact_651_Int__Union,axiom,
    ! [A3: set_nat,B3: set_set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( comple7399068483239264473et_nat @ B3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A3 ) @ B3 ) ) ) ).

% Int_Union
thf(fact_652_Int__Union2,axiom,
    ! [B3: set_set_nat,A3: set_nat] :
      ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B3 ) @ A3 )
      = ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [C4: set_nat] : ( inf_inf_set_nat @ C4 @ A3 )
          @ B3 ) ) ) ).

% Int_Union2
thf(fact_653_inj__on__disjoint__Un,axiom,
    ! [F6: nat > set_nat,A3: set_nat,G: nat > set_nat,B3: set_nat] :
      ( ( inj_on_nat_set_nat @ F6 @ A3 )
     => ( ( inj_on_nat_set_nat @ G @ B3 )
       => ( ( ( inf_inf_set_set_nat @ ( image_nat_set_nat @ F6 @ A3 ) @ ( image_nat_set_nat @ G @ B3 ) )
            = bot_bot_set_set_nat )
         => ( inj_on_nat_set_nat
            @ ^ [X: nat] : ( if_set_nat @ ( member_nat @ X @ A3 ) @ ( F6 @ X ) @ ( G @ X ) )
            @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_654_inj__on__disjoint__Un,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on2178005380612969504at_nat @ F6 @ A3 )
     => ( ( inj_on2178005380612969504at_nat @ G @ B3 )
       => ( ( ( inf_inf_set_nat @ ( image_2486076414777270412at_nat @ F6 @ A3 ) @ ( image_2486076414777270412at_nat @ G @ B3 ) )
            = bot_bot_set_nat )
         => ( inj_on2178005380612969504at_nat
            @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ A3 ) @ ( F6 @ X ) @ ( G @ X ) )
            @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_655_inj__on__disjoint__Un,axiom,
    ! [F6: nat > nat,A3: set_nat,G: nat > nat,B3: set_nat] :
      ( ( inj_on_nat_nat @ F6 @ A3 )
     => ( ( inj_on_nat_nat @ G @ B3 )
       => ( ( ( inf_inf_set_nat @ ( image_nat_nat2 @ F6 @ A3 ) @ ( image_nat_nat2 @ G @ B3 ) )
            = bot_bot_set_nat )
         => ( inj_on_nat_nat
            @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ A3 ) @ ( F6 @ X ) @ ( G @ X ) )
            @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ) ).

% inj_on_disjoint_Un
thf(fact_656_inj__on__vimage__singleton,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( inj_on2178005380612969504at_nat @ F6 @ A3 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ ( vimage4653281326611754070at_nat @ F6 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ A3 )
        @ ( insert8211810215607154385at_nat
          @ ( the_Pr7557018466319803784at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( ( F6 @ X )
                  = A ) ) )
          @ bot_bo2099793752762293965at_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_657_inj__on__vimage__singleton,axiom,
    ! [F6: nat > nat,A3: set_nat,A: nat] :
      ( ( inj_on_nat_nat @ F6 @ A3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F6 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ A3 )
        @ ( insert_nat
          @ ( the_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( ( F6 @ X )
                  = A ) ) )
          @ bot_bot_set_nat ) ) ) ).

% inj_on_vimage_singleton
thf(fact_658_Least__mono,axiom,
    ! [F6: nat > set_nat,S: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F6 )
     => ( ? [X8: nat] :
            ( ( member_nat @ X8 @ S )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ S )
               => ( ord_less_eq_nat @ X8 @ Xa2 ) ) )
       => ( ( ord_Least_set_nat
            @ ^ [Y6: set_nat] : ( member_set_nat @ Y6 @ ( image_nat_set_nat @ F6 @ S ) ) )
          = ( F6
            @ ( ord_Least_nat
              @ ^ [X: nat] : ( member_nat @ X @ S ) ) ) ) ) ) ).

% Least_mono
thf(fact_659_Least__mono,axiom,
    ! [F6: nat > nat,S: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F6 )
     => ( ? [X8: nat] :
            ( ( member_nat @ X8 @ S )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ S )
               => ( ord_less_eq_nat @ X8 @ Xa2 ) ) )
       => ( ( ord_Least_nat
            @ ^ [Y6: nat] : ( member_nat @ Y6 @ ( image_nat_nat2 @ F6 @ S ) ) )
          = ( F6
            @ ( ord_Least_nat
              @ ^ [X: nat] : ( member_nat @ X @ S ) ) ) ) ) ) ).

% Least_mono
thf(fact_660_Least__mono,axiom,
    ! [F6: nat > filter_nat,S: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ? [X8: nat] :
            ( ( member_nat @ X8 @ S )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ S )
               => ( ord_less_eq_nat @ X8 @ Xa2 ) ) )
       => ( ( ord_Least_filter_nat
            @ ^ [Y6: filter_nat] : ( member_filter_nat @ Y6 @ ( image_nat_filter_nat @ F6 @ S ) ) )
          = ( F6
            @ ( ord_Least_nat
              @ ^ [X: nat] : ( member_nat @ X @ S ) ) ) ) ) ) ).

% Least_mono
thf(fact_661_Least__mono,axiom,
    ! [F6: filter_nat > nat,S: set_filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_nat @ F6 )
     => ( ? [X8: filter_nat] :
            ( ( member_filter_nat @ X8 @ S )
            & ! [Xa2: filter_nat] :
                ( ( member_filter_nat @ Xa2 @ S )
               => ( ord_le2510731241096832064er_nat @ X8 @ Xa2 ) ) )
       => ( ( ord_Least_nat
            @ ^ [Y6: nat] : ( member_nat @ Y6 @ ( image_filter_nat_nat @ F6 @ S ) ) )
          = ( F6
            @ ( ord_Least_filter_nat
              @ ^ [X: filter_nat] : ( member_filter_nat @ X @ S ) ) ) ) ) ) ).

% Least_mono
thf(fact_662_Least__mono,axiom,
    ! [F6: filter_nat > filter_nat,S: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ? [X8: filter_nat] :
            ( ( member_filter_nat @ X8 @ S )
            & ! [Xa2: filter_nat] :
                ( ( member_filter_nat @ Xa2 @ S )
               => ( ord_le2510731241096832064er_nat @ X8 @ Xa2 ) ) )
       => ( ( ord_Least_filter_nat
            @ ^ [Y6: filter_nat] : ( member_filter_nat @ Y6 @ ( image_4775440762538963527er_nat @ F6 @ S ) ) )
          = ( F6
            @ ( ord_Least_filter_nat
              @ ^ [X: filter_nat] : ( member_filter_nat @ X @ S ) ) ) ) ) ) ).

% Least_mono
thf(fact_663_INF__SUP,axiom,
    ! [P: nat > nat > set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [Y6: nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( P @ X @ Y6 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( comple7399068483239264473et_nat
        @ ( image_7432509271690132940et_nat
          @ ^ [F4: nat > nat] :
              ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( P @ ( F4 @ X ) @ X )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% INF_SUP
thf(fact_664_SUP__INF,axiom,
    ! [P: nat > nat > set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y6: nat] :
              ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( P @ X @ Y6 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( comple7806235888213564991et_nat
        @ ( image_7432509271690132940et_nat
          @ ^ [X: nat > nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [Y6: nat] : ( P @ ( X @ Y6 ) @ Y6 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% SUP_INF
thf(fact_665_INT__iff,axiom,
    ! [B: nat,B3: nat > set_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( member_nat @ B @ ( B3 @ X ) ) ) ) ) ).

% INT_iff
thf(fact_666_INT__I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A3 )
         => ( member8440522571783428010at_nat @ B @ ( B3 @ X4 ) ) )
     => ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B3 @ A3 ) ) ) ) ).

% INT_I
thf(fact_667_INT__I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B: nat,B3: product_prod_nat_nat > set_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A3 )
         => ( member_nat @ B @ ( B3 @ X4 ) ) )
     => ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) ) ).

% INT_I
thf(fact_668_INT__I,axiom,
    ! [A3: set_nat,B: product_prod_nat_nat,B3: nat > set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member8440522571783428010at_nat @ B @ ( B3 @ X4 ) ) )
     => ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B3 @ A3 ) ) ) ) ).

% INT_I
thf(fact_669_INT__I,axiom,
    ! [A3: set_nat,B: nat,B3: nat > set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( member_nat @ B @ ( B3 @ X4 ) ) )
     => ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ).

% INT_I
thf(fact_670_INF__identity__eq,axiom,
    ! [A3: set_nat] :
      ( ( complete_Inf_Inf_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : X
          @ A3 ) )
      = ( complete_Inf_Inf_nat @ A3 ) ) ).

% INF_identity_eq
thf(fact_671_INF__top__conv_I2_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = top_top_set_nat ) ) ) ) ).

% INF_top_conv(2)
thf(fact_672_INF__top__conv_I1_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) )
        = top_top_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = top_top_set_nat ) ) ) ) ).

% INF_top_conv(1)
thf(fact_673_INF__top,axiom,
    ! [A3: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : top_top_set_nat
          @ A3 ) )
      = top_top_set_nat ) ).

% INF_top
thf(fact_674_cINF__const,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Inf_Inf_nat
          @ ( image_nat_nat2
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cINF_const
thf(fact_675_cINF__const,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : C
            @ A3 ) )
        = C ) ) ).

% cINF_const
thf(fact_676_INF__const,axiom,
    ! [A3: set_nat,F6: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [I: nat] : F6
            @ A3 ) )
        = F6 ) ) ).

% INF_const
thf(fact_677_INT__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = top_top_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% INT_constant
thf(fact_678_INT__insert,axiom,
    ! [B3: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ ( insert_nat @ A @ A3 ) ) )
      = ( inf_inf_set_nat @ ( B3 @ A ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ).

% INT_insert
thf(fact_679_INT__simps_I1_J,axiom,
    ! [C3: set_nat,A3: nat > set_nat,B3: set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( inf_inf_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) )
          = top_top_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( inf_inf_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) )
          = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 ) ) ) ) ).

% INT_simps(1)
thf(fact_680_INT__simps_I2_J,axiom,
    ! [C3: set_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( inf_inf_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) )
          = top_top_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( inf_inf_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) )
          = ( inf_inf_set_nat @ A3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) ) ) ) ) ).

% INT_simps(2)
thf(fact_681_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 )
            @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ) ).

% inf_set_def
thf(fact_682_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% inf_set_def
thf(fact_683_inf__Int__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( inf_in8805929356528176899char_o
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R3 )
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ S ) )
      = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( inf_in3544766959038609504t_char @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_684_inf__Int__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 )
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ S ) )
      = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( inf_in2572325071724192079at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_685_inf__Int__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in5163264567034779214_nat_o
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R3 )
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S ) )
      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_686_inf__Int__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( inf_inf_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R3 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_687_INF__commute,axiom,
    ! [F6: nat > nat > set_nat,B3: set_nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ ( F6 @ I ) @ B3 ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [J: nat] :
              ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [I: nat] : ( F6 @ I @ J )
                @ A3 ) )
          @ B3 ) ) ) ).

% INF_commute
thf(fact_688_INF__INT__eq,axiom,
    ! [R2: nat > set_nat,S: set_nat] :
      ( ( comple6214475593288795910_nat_o
        @ ( image_nat_nat_o
          @ ^ [I: nat,X: nat] : ( member_nat @ X @ ( R2 @ I ) )
          @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ R2 @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_689_Inf__set__def,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A7: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A7 ) ) ) ) ) ).

% Inf_set_def
thf(fact_690_Inf__set__def,axiom,
    ( comple1930127330929219688at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( complete_Inf_Inf_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X ) @ A7 ) ) ) ) ) ).

% Inf_set_def
thf(fact_691_INF__Int__eq,axiom,
    ! [S: set_se7855581050983116737at_nat] :
      ( ( comple6265855634338933877_nat_o
        @ ( image_5253903537071781336_nat_o
          @ ^ [I: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ I )
          @ S ) )
      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( comple1930127330929219688at_nat @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_692_INF__Int__eq,axiom,
    ! [S: set_set_nat] :
      ( ( comple6214475593288795910_nat_o
        @ ( image_set_nat_nat_o
          @ ^ [I: set_nat,X: nat] : ( member_nat @ X @ I )
          @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7806235888213564991et_nat @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_693_INT__E,axiom,
    ! [B: product_prod_nat_nat,B3: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B3 @ A3 ) ) )
     => ( ~ ( member8440522571783428010at_nat @ B @ ( B3 @ A ) )
       => ~ ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% INT_E
thf(fact_694_INT__E,axiom,
    ! [B: product_prod_nat_nat,B3: nat > set_Pr1261947904930325089at_nat,A3: set_nat,A: nat] :
      ( ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B3 @ A3 ) ) )
     => ( ~ ( member8440522571783428010at_nat @ B @ ( B3 @ A ) )
       => ~ ( member_nat @ A @ A3 ) ) ) ).

% INT_E
thf(fact_695_INT__E,axiom,
    ! [B: nat,B3: product_prod_nat_nat > set_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) )
     => ( ~ ( member_nat @ B @ ( B3 @ A ) )
       => ~ ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% INT_E
thf(fact_696_INT__E,axiom,
    ! [B: nat,B3: nat > set_nat,A3: set_nat,A: nat] :
      ( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
     => ( ~ ( member_nat @ B @ ( B3 @ A ) )
       => ~ ( member_nat @ A @ A3 ) ) ) ).

% INT_E
thf(fact_697_INT__D,axiom,
    ! [B: product_prod_nat_nat,B3: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_5423882813909314213at_nat @ B3 @ A3 ) ) )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ B @ ( B3 @ A ) ) ) ) ).

% INT_D
thf(fact_698_INT__D,axiom,
    ! [B: product_prod_nat_nat,B3: nat > set_Pr1261947904930325089at_nat,A3: set_nat,A: nat] :
      ( ( member8440522571783428010at_nat @ B @ ( comple1930127330929219688at_nat @ ( image_7178329752028323786at_nat @ B3 @ A3 ) ) )
     => ( ( member_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ B @ ( B3 @ A ) ) ) ) ).

% INT_D
thf(fact_699_INT__D,axiom,
    ! [B: nat,B3: product_prod_nat_nat > set_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member_nat @ B @ ( B3 @ A ) ) ) ) ).

% INT_D
thf(fact_700_INT__D,axiom,
    ! [B: nat,B3: nat > set_nat,A3: set_nat,A: nat] :
      ( ( member_nat @ B @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
     => ( ( member_nat @ A @ A3 )
       => ( member_nat @ B @ ( B3 @ A ) ) ) ) ).

% INT_D
thf(fact_701_INF__Int__eq2,axiom,
    ! [S: set_se253819897320413138t_char] :
      ( ( comple7993840426618028700char_o
        @ ( image_7123945608888165160char_o
          @ ^ [I: set_Pr2507339680178222706t_char,X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ I )
          @ S ) )
      = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( comple4824088763117934073t_char @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_702_INF__Int__eq2,axiom,
    ! [S: set_se7855581050983116737at_nat] :
      ( ( comple1675732732025788397_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I: set_Pr1261947904930325089at_nat,X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ I )
          @ S ) )
      = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( comple1930127330929219688at_nat @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_703_INF__greatest,axiom,
    ! [A3: set_nat,U: set_nat,F6: nat > set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ U @ ( F6 @ I3 ) ) )
     => ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% INF_greatest
thf(fact_704_INF__greatest,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,U: filter_nat,F6: product_prod_nat_nat > filter_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ U @ ( F6 @ I3 ) ) )
     => ( ord_le2510731241096832064er_nat @ U @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) ) ) ).

% INF_greatest
thf(fact_705_INF__greatest,axiom,
    ! [A3: set_nat,U: filter_nat,F6: nat > filter_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ U @ ( F6 @ I3 ) ) )
     => ( ord_le2510731241096832064er_nat @ U @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) ) ) ).

% INF_greatest
thf(fact_706_le__INF__iff,axiom,
    ! [U: set_nat,F6: nat > set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ U @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ord_less_eq_set_nat @ U @ ( F6 @ X ) ) ) ) ) ).

% le_INF_iff
thf(fact_707_INF__lower2,axiom,
    ! [I5: nat,A3: set_nat,F6: nat > set_nat,U: set_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ( ord_less_eq_set_nat @ ( F6 @ I5 ) @ U )
       => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ U ) ) ) ).

% INF_lower2
thf(fact_708_INF__lower2,axiom,
    ! [I5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat,U: filter_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ A3 )
     => ( ( ord_le2510731241096832064er_nat @ ( F6 @ I5 ) @ U )
       => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) @ U ) ) ) ).

% INF_lower2
thf(fact_709_INF__lower2,axiom,
    ! [I5: nat,A3: set_nat,F6: nat > filter_nat,U: filter_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ( ord_le2510731241096832064er_nat @ ( F6 @ I5 ) @ U )
       => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) @ U ) ) ) ).

% INF_lower2
thf(fact_710_INF__mono_H,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ! [X4: nat] : ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) )
     => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ).

% INF_mono'
thf(fact_711_INF__lower,axiom,
    ! [I5: nat,A3: set_nat,F6: nat > set_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( F6 @ I5 ) ) ) ).

% INF_lower
thf(fact_712_INF__lower,axiom,
    ! [I5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ A3 )
     => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) @ ( F6 @ I5 ) ) ) ).

% INF_lower
thf(fact_713_INF__lower,axiom,
    ! [I5: nat,A3: set_nat,F6: nat > filter_nat] :
      ( ( member_nat @ I5 @ A3 )
     => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) @ ( F6 @ I5 ) ) ) ).

% INF_lower
thf(fact_714_INF__mono,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_nat,F6: nat > set_nat,G: product_prod_nat_nat > set_nat] :
      ( ! [M2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ M2 @ B3 )
         => ? [X8: nat] :
              ( ( member_nat @ X8 @ A3 )
              & ( ord_less_eq_set_nat @ ( F6 @ X8 ) @ ( G @ M2 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ G @ B3 ) ) ) ) ).

% INF_mono
thf(fact_715_INF__mono,axiom,
    ! [B3: set_nat,A3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ! [M2: nat] :
          ( ( member_nat @ M2 @ B3 )
         => ? [X8: nat] :
              ( ( member_nat @ X8 @ A3 )
              & ( ord_less_eq_set_nat @ ( F6 @ X8 ) @ ( G @ M2 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).

% INF_mono
thf(fact_716_INF__eqI,axiom,
    ! [A3: set_nat,X5: set_nat,F6: nat > set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ X5 @ ( F6 @ I3 ) ) )
     => ( ! [Y5: set_nat] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ Y5 @ ( F6 @ I4 ) ) )
           => ( ord_less_eq_set_nat @ Y5 @ X5 ) )
       => ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% INF_eqI
thf(fact_717_INF__eqI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,X5: filter_nat,F6: product_prod_nat_nat > filter_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ X5 @ ( F6 @ I3 ) ) )
     => ( ! [Y5: filter_nat] :
            ( ! [I4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ I4 @ A3 )
               => ( ord_le2510731241096832064er_nat @ Y5 @ ( F6 @ I4 ) ) )
           => ( ord_le2510731241096832064er_nat @ Y5 @ X5 ) )
       => ( ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% INF_eqI
thf(fact_718_INF__eqI,axiom,
    ! [A3: set_nat,X5: filter_nat,F6: nat > filter_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_le2510731241096832064er_nat @ X5 @ ( F6 @ I3 ) ) )
     => ( ! [Y5: filter_nat] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_le2510731241096832064er_nat @ Y5 @ ( F6 @ I4 ) ) )
           => ( ord_le2510731241096832064er_nat @ Y5 @ X5 ) )
       => ( ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) )
          = X5 ) ) ) ).

% INF_eqI
thf(fact_719_less__INF__D,axiom,
    ! [Y: set_nat,F6: nat > set_nat,A3: set_nat,I5: nat] :
      ( ( ord_less_set_nat @ Y @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) )
     => ( ( member_nat @ I5 @ A3 )
       => ( ord_less_set_nat @ Y @ ( F6 @ I5 ) ) ) ) ).

% less_INF_D
thf(fact_720_INF__inf__distrib,axiom,
    ! [F6: nat > set_nat,A3: set_nat,G: nat > set_nat] :
      ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( inf_inf_set_nat @ ( F6 @ A2 ) @ ( G @ A2 ) )
          @ A3 ) ) ) ).

% INF_inf_distrib
thf(fact_721_INF__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_nat] :
      ( ( member_nat @ K @ I2 )
     => ( ( inf_inf_set_nat @ ( A3 @ K ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% INF_absorb
thf(fact_722_INF__sup,axiom,
    ! [F6: nat > set_nat,B3: set_nat,A: set_nat] :
      ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ B3 ) ) @ A )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [B2: nat] : ( sup_sup_set_nat @ ( F6 @ B2 ) @ A )
          @ B3 ) ) ) ).

% INF_sup
thf(fact_723_sup__INF,axiom,
    ! [A: set_nat,F6: nat > set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ B3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [B2: nat] : ( sup_sup_set_nat @ A @ ( F6 @ B2 ) )
          @ B3 ) ) ) ).

% sup_INF
thf(fact_724_INF__sup__distrib2,axiom,
    ! [F6: nat > set_nat,A3: set_nat,G: nat > set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] :
              ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [B2: nat] : ( sup_sup_set_nat @ ( F6 @ A2 ) @ ( G @ B2 ) )
                @ B3 ) )
          @ A3 ) ) ) ).

% INF_sup_distrib2
thf(fact_725_INT__extend__simps_I10_J,axiom,
    ! [B3: set_nat > set_nat,F6: nat > set_nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B3 @ ( F6 @ A2 ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat @ ( image_7916887816326733075et_nat @ B3 @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_726_INT__extend__simps_I10_J,axiom,
    ! [B3: nat > set_nat,F6: nat > nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [A2: nat] : ( B3 @ ( F6 @ A2 ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ ( image_nat_nat2 @ F6 @ A3 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_727_INTER__UNIV__conv_I2_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) )
        = top_top_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = top_top_set_nat ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_728_INTER__UNIV__conv_I1_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( ( B3 @ X )
              = top_top_set_nat ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_729_INT__subset__iff,axiom,
    ! [B3: set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ I2 )
           => ( ord_less_eq_set_nat @ B3 @ ( A3 @ X ) ) ) ) ) ).

% INT_subset_iff
thf(fact_730_INT__anti__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ B3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ) ).

% INT_anti_mono
thf(fact_731_INT__greatest,axiom,
    ! [A3: set_nat,C3: set_nat,B3: nat > set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_set_nat @ C3 @ ( B3 @ X4 ) ) )
     => ( ord_less_eq_set_nat @ C3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ).

% INT_greatest
thf(fact_732_INT__lower,axiom,
    ! [A: nat,A3: set_nat,B3: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) @ ( B3 @ A ) ) ) ).

% INT_lower
thf(fact_733_INT__extend__simps_I5_J,axiom,
    ! [A: nat,B3: nat > set_nat,C3: set_nat] :
      ( ( insert_nat @ A @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( insert_nat @ A @ ( B3 @ X ) )
          @ C3 ) ) ) ).

% INT_extend_simps(5)
thf(fact_734_INT__insert__distrib,axiom,
    ! [U: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,A: nat,B3: product_prod_nat_nat > set_nat] :
      ( ( member8440522571783428010at_nat @ U @ A3 )
     => ( ( comple7806235888213564991et_nat
          @ ( image_15824709712370754et_nat
            @ ^ [X: product_prod_nat_nat] : ( insert_nat @ A @ ( B3 @ X ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) ) ) ).

% INT_insert_distrib
thf(fact_735_INT__insert__distrib,axiom,
    ! [U: nat,A3: set_nat,A: nat,B3: nat > set_nat] :
      ( ( member_nat @ U @ A3 )
     => ( ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( insert_nat @ A @ ( B3 @ X ) )
            @ A3 ) )
        = ( insert_nat @ A @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% INT_insert_distrib
thf(fact_736_Int__Inter__image,axiom,
    ! [A3: nat > set_nat,B3: nat > set_nat,C3: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( inf_inf_set_nat @ ( A3 @ X ) @ ( B3 @ X ) )
          @ C3 ) )
      = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) ) ) ).

% Int_Inter_image
thf(fact_737_INT__Int__distrib,axiom,
    ! [A3: nat > set_nat,B3: nat > set_nat,I2: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] : ( inf_inf_set_nat @ ( A3 @ I ) @ ( B3 @ I ) )
          @ I2 ) )
      = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ I2 ) ) ) ) ).

% INT_Int_distrib
thf(fact_738_INT__absorb,axiom,
    ! [K: nat,I2: set_nat,A3: nat > set_nat] :
      ( ( member_nat @ K @ I2 )
     => ( ( inf_inf_set_nat @ ( A3 @ K ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% INT_absorb
thf(fact_739_INT__extend__simps_I7_J,axiom,
    ! [A3: set_nat,B3: nat > set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( sup_sup_set_nat @ A3 @ ( B3 @ X ) )
          @ C3 ) ) ) ).

% INT_extend_simps(7)
thf(fact_740_INT__extend__simps_I6_J,axiom,
    ! [A3: nat > set_nat,C3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( sup_sup_set_nat @ ( A3 @ X ) @ B3 )
          @ C3 ) ) ) ).

% INT_extend_simps(6)
thf(fact_741_Un__INT__distrib,axiom,
    ! [B3: set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( sup_sup_set_nat @ B3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] : ( sup_sup_set_nat @ B3 @ ( A3 @ I ) )
          @ I2 ) ) ) ).

% Un_INT_distrib
thf(fact_742_Un__INT__distrib2,axiom,
    ! [A3: nat > set_nat,I2: set_nat,B3: nat > set_nat,J3: set_nat] :
      ( ( sup_sup_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ J3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [I: nat] :
              ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [J: nat] : ( sup_sup_set_nat @ ( A3 @ I ) @ ( B3 @ J ) )
                @ J3 ) )
          @ I2 ) ) ) ).

% Un_INT_distrib2
thf(fact_743_INT__extend__simps_I9_J,axiom,
    ! [C3: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C3 @ ( B3 @ X ) ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ C3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_744_vimage__INT,axiom,
    ! [F6: nat > nat,B3: nat > set_nat,A3: set_nat] :
      ( ( vimage_nat_nat @ F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( vimage_nat_nat @ F6 @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% vimage_INT
thf(fact_745_Inf__INT__eq2,axiom,
    ( comple7993840426618028700char_o
    = ( ^ [S3: set_nat_list_char_o,X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( comple4824088763117934073t_char @ ( image_9058305973422863104t_char @ collec2458894688891239037t_char @ ( image_1531431914690922683char_o @ produc8526122873445342073char_o @ S3 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_746_Inf__INT__eq2,axiom,
    ( comple1675732732025788397_nat_o
    = ( ^ [S3: set_nat_nat_o,X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( comple1930127330929219688at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S3 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_747_INF__superset__mono,axiom,
    ! [B3: set_nat,A3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B3 )
           => ( ord_less_eq_set_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).

% INF_superset_mono
thf(fact_748_INF__superset__mono,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > filter_nat,G: product_prod_nat_nat > filter_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ B3 )
           => ( ord_le2510731241096832064er_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F6 @ A3 ) ) @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ G @ B3 ) ) ) ) ) ).

% INF_superset_mono
thf(fact_749_INF__superset__mono,axiom,
    ! [B3: set_nat,A3: set_nat,F6: nat > filter_nat,G: nat > filter_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B3 )
           => ( ord_le2510731241096832064er_nat @ ( F6 @ X4 ) @ ( G @ X4 ) ) )
       => ( ord_le2510731241096832064er_nat @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ G @ B3 ) ) ) ) ) ).

% INF_superset_mono
thf(fact_750_INF__empty,axiom,
    ! [F6: nat > set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ bot_bot_set_nat ) )
      = top_top_set_nat ) ).

% INF_empty
thf(fact_751_INF__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = top_top_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [Y6: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% INF_constant
thf(fact_752_INF__inf__const2,axiom,
    ! [I2: set_nat,F6: nat > set_nat,X5: set_nat] :
      ( ( I2 != bot_bot_set_nat )
     => ( ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [I: nat] : ( inf_inf_set_nat @ ( F6 @ I ) @ X5 )
            @ I2 ) )
        = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ I2 ) ) @ X5 ) ) ) ).

% INF_inf_const2
thf(fact_753_INF__inf__const1,axiom,
    ! [I2: set_nat,X5: set_nat,F6: nat > set_nat] :
      ( ( I2 != bot_bot_set_nat )
     => ( ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [I: nat] : ( inf_inf_set_nat @ X5 @ ( F6 @ I ) )
            @ I2 ) )
        = ( inf_inf_set_nat @ X5 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ I2 ) ) ) ) ) ).

% INF_inf_const1
thf(fact_754_INF__insert,axiom,
    ! [F6: nat > set_nat,A: nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ ( insert_nat @ A @ A3 ) ) )
      = ( inf_inf_set_nat @ ( F6 @ A ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% INF_insert
thf(fact_755_INF__union,axiom,
    ! [M: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A3 @ B3 ) ) )
      = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ M @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ M @ B3 ) ) ) ) ).

% INF_union
thf(fact_756_INT__empty,axiom,
    ! [B3: nat > set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ bot_bot_set_nat ) )
      = top_top_set_nat ) ).

% INT_empty
thf(fact_757_INT__extend__simps_I2_J,axiom,
    ! [C3: set_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( inf_inf_set_nat @ A3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = A3 ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( inf_inf_set_nat @ A3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( inf_inf_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) ) ) ) ) ).

% INT_extend_simps(2)
thf(fact_758_INT__extend__simps_I1_J,axiom,
    ! [C3: set_nat,A3: nat > set_nat,B3: set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
          = B3 ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
          = ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( inf_inf_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) ) ) ) ) ).

% INT_extend_simps(1)
thf(fact_759_inj__on__INTER,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,F6: nat > nat,A3: product_prod_nat_nat > set_nat] :
      ( ( I2 != bot_bo2099793752762293965at_nat )
     => ( ! [I3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I3 @ I2 )
           => ( inj_on_nat_nat @ F6 @ ( A3 @ I3 ) ) )
       => ( inj_on_nat_nat @ F6 @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) ) ) ) ).

% inj_on_INTER
thf(fact_760_inj__on__INTER,axiom,
    ! [I2: set_nat,F6: nat > nat,A3: nat > set_nat] :
      ( ( I2 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I2 )
           => ( inj_on_nat_nat @ F6 @ ( A3 @ I3 ) ) )
       => ( inj_on_nat_nat @ F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ) ).

% inj_on_INTER
thf(fact_761_INT__Un,axiom,
    ! [M: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A3 @ B3 ) ) )
      = ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ M @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ M @ B3 ) ) ) ) ).

% INT_Un
thf(fact_762_INT__extend__simps_I8_J,axiom,
    ! [B3: nat > set_nat,A3: set_set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [Y6: set_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ Y6 ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_763_mono__INF,axiom,
    ! [F6: set_nat > set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ord_less_eq_set_nat @ ( F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        @ ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
            @ I2 ) ) ) ) ).

% mono_INF
thf(fact_764_mono__INF,axiom,
    ! [F6: set_nat > filter_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto5359140060069215823er_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ord_le2510731241096832064er_nat @ ( F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        @ ( comple6612786467464916249er_nat
          @ ( image_nat_filter_nat
            @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
            @ I2 ) ) ) ) ).

% mono_INF
thf(fact_765_mono__INF,axiom,
    ! [F6: filter_nat > set_nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto6401136896278960335et_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_set_nat @ F6 )
     => ( ord_less_eq_set_nat @ ( F6 @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) )
        @ ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
            @ I2 ) ) ) ) ).

% mono_INF
thf(fact_766_mono__Inf,axiom,
    ! [F6: filter_nat > filter_nat,A3: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ord_le2510731241096832064er_nat @ ( F6 @ ( comple6612786467464916249er_nat @ A3 ) ) @ ( comple6612786467464916249er_nat @ ( image_4775440762538963527er_nat @ F6 @ A3 ) ) ) ) ).

% mono_Inf
thf(fact_767_image__INT,axiom,
    ! [F6: nat > set_nat,C3: set_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,J4: product_prod_nat_nat] :
      ( ( inj_on_nat_set_nat @ F6 @ C3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( B3 @ X4 ) @ C3 ) )
       => ( ( member8440522571783428010at_nat @ J4 @ A3 )
         => ( ( image_nat_set_nat @ F6 @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) )
            = ( comple1065008630642458357et_nat
              @ ( image_1942688902138425592et_nat
                @ ^ [X: product_prod_nat_nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
                @ A3 ) ) ) ) ) ) ).

% image_INT
thf(fact_768_image__INT,axiom,
    ! [F6: nat > nat,C3: set_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,J4: product_prod_nat_nat] :
      ( ( inj_on_nat_nat @ F6 @ C3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( B3 @ X4 ) @ C3 ) )
       => ( ( member8440522571783428010at_nat @ J4 @ A3 )
         => ( ( image_nat_nat2 @ F6 @ ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) )
            = ( comple7806235888213564991et_nat
              @ ( image_15824709712370754et_nat
                @ ^ [X: product_prod_nat_nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
                @ A3 ) ) ) ) ) ) ).

% image_INT
thf(fact_769_image__INT,axiom,
    ! [F6: nat > set_nat,C3: set_nat,A3: set_nat,B3: nat > set_nat,J4: nat] :
      ( ( inj_on_nat_set_nat @ F6 @ C3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( B3 @ X4 ) @ C3 ) )
       => ( ( member_nat @ J4 @ A3 )
         => ( ( image_nat_set_nat @ F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
            = ( comple1065008630642458357et_nat
              @ ( image_2194112158459175443et_nat
                @ ^ [X: nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
                @ A3 ) ) ) ) ) ) ).

% image_INT
thf(fact_770_image__INT,axiom,
    ! [F6: nat > nat,C3: set_nat,A3: set_nat,B3: nat > set_nat,J4: nat] :
      ( ( inj_on_nat_nat @ F6 @ C3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ord_less_eq_set_nat @ ( B3 @ X4 ) @ C3 ) )
       => ( ( member_nat @ J4 @ A3 )
         => ( ( image_nat_nat2 @ F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
            = ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
                @ A3 ) ) ) ) ) ) ).

% image_INT
thf(fact_771_not__less__Least,axiom,
    ! [K: nat,P: nat > $o] :
      ( ( ord_less_nat @ K @ ( ord_Least_nat @ P ) )
     => ~ ( P @ K ) ) ).

% not_less_Least
thf(fact_772_Least__le,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).

% Least_le
thf(fact_773_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S3: set_nat,P5: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S3 )
              & ( P5 @ X ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_774_LeastI,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI
thf(fact_775_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S3: set_nat,P5: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S3 )
              & ( P5 @ X ) ) ) ) ) ).

% Bleast_def
thf(fact_776_INT__simps_I4_J,axiom,
    ! [C3: set_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( minus_minus_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) )
          = top_top_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( minus_minus_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) )
          = ( minus_minus_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) ) ) ) ) ).

% INT_simps(4)
thf(fact_777_mono__cInf,axiom,
    ! [F6: nat > set_nat,A3: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat @ ( F6 @ ( complete_Inf_Inf_nat @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ) ) ).

% mono_cInf
thf(fact_778_mono__cInf,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( F6 @ ( complete_Inf_Inf_nat @ A3 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) ) ) ) ) ).

% mono_cInf
thf(fact_779_mono__cInf,axiom,
    ! [F6: nat > filter_nat,A3: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ ( complete_Inf_Inf_nat @ A3 ) ) @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) ) ) ) ) ).

% mono_cInf
thf(fact_780_mono__cInf,axiom,
    ! [F6: filter_nat > nat,A3: set_filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit3133331395780078992er_nat @ A3 )
       => ( ( A3 != bot_bo498966703094740906er_nat )
         => ( ord_less_eq_nat @ ( F6 @ ( comple6612786467464916249er_nat @ A3 ) ) @ ( complete_Inf_Inf_nat @ ( image_filter_nat_nat @ F6 @ A3 ) ) ) ) ) ) ).

% mono_cInf
thf(fact_781_mono__cInf,axiom,
    ! [F6: filter_nat > filter_nat,A3: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit3133331395780078992er_nat @ A3 )
       => ( ( A3 != bot_bo498966703094740906er_nat )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ ( comple6612786467464916249er_nat @ A3 ) ) @ ( comple6612786467464916249er_nat @ ( image_4775440762538963527er_nat @ F6 @ A3 ) ) ) ) ) ) ).

% mono_cInf
thf(fact_782_bdd__below__image__inf,axiom,
    ! [F6: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( condit1738341127787009408ow_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : ( inf_inf_nat @ ( F6 @ X ) @ ( G @ X ) )
          @ A3 ) )
      = ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
        & ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ A3 ) ) ) ) ).

% bdd_below_image_inf
thf(fact_783_bdd__below__image__inf,axiom,
    ! [F6: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( condit68592940725977398et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( inf_inf_set_nat @ ( F6 @ X ) @ ( G @ X ) )
          @ A3 ) )
      = ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
        & ( condit68592940725977398et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ).

% bdd_below_image_inf
thf(fact_784_INT__simps_I3_J,axiom,
    ! [C3: set_nat,A3: nat > set_nat,B3: set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( minus_minus_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) )
          = top_top_set_nat ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( minus_minus_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) )
          = ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 ) ) ) ) ).

% INT_simps(3)
thf(fact_785_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              & ~ ( member_nat @ X @ B8 ) ) ) ) ) ).

% set_diff_eq
thf(fact_786_set__diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A7 )
              & ~ ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% set_diff_eq
thf(fact_787_UN__extend__simps_I6_J,axiom,
    ! [A3: nat > set_nat,C3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( minus_minus_set_nat @ ( A3 @ X ) @ B3 )
          @ C3 ) ) ) ).

% UN_extend_simps(6)
thf(fact_788_less__cINF__D,axiom,
    ! [F6: nat > set_nat,A3: set_nat,Y: set_nat,I5: nat] :
      ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
     => ( ( ord_less_set_nat @ Y @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) )
       => ( ( member_nat @ I5 @ A3 )
         => ( ord_less_set_nat @ Y @ ( F6 @ I5 ) ) ) ) ) ).

% less_cINF_D
thf(fact_789_less__cINF__D,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,Y: nat,I5: product_prod_nat_nat] :
      ( ( condit1738341127787009408ow_nat @ ( image_2486076414777270412at_nat @ F6 @ A3 ) )
     => ( ( ord_less_nat @ Y @ ( complete_Inf_Inf_nat @ ( image_2486076414777270412at_nat @ F6 @ A3 ) ) )
       => ( ( member8440522571783428010at_nat @ I5 @ A3 )
         => ( ord_less_nat @ Y @ ( F6 @ I5 ) ) ) ) ) ).

% less_cINF_D
thf(fact_790_less__cINF__D,axiom,
    ! [F6: nat > nat,A3: set_nat,Y: nat,I5: nat] :
      ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
     => ( ( ord_less_nat @ Y @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) )
       => ( ( member_nat @ I5 @ A3 )
         => ( ord_less_nat @ Y @ ( F6 @ I5 ) ) ) ) ) ).

% less_cINF_D
thf(fact_791_UN__extend__simps_I7_J,axiom,
    ! [A3: set_nat,B3: nat > set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( minus_minus_set_nat @ A3 @ ( B3 @ X ) )
          @ C3 ) ) ) ).

% UN_extend_simps(7)
thf(fact_792_cINF__less__iff,axiom,
    ! [A3: set_nat,F6: nat > nat,A: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
       => ( ( ord_less_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) @ A )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( ord_less_nat @ ( F6 @ X ) @ A ) ) ) ) ) ) ).

% cINF_less_iff
thf(fact_793_cINF__inf__distrib,axiom,
    ! [A3: set_nat,F6: nat > nat,G: nat > nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
       => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ G @ A3 ) )
         => ( ( inf_inf_nat @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ G @ A3 ) ) )
            = ( complete_Inf_Inf_nat
              @ ( image_nat_nat2
                @ ^ [A2: nat] : ( inf_inf_nat @ ( F6 @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% cINF_inf_distrib
thf(fact_794_cINF__inf__distrib,axiom,
    ! [A3: set_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
       => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ G @ A3 ) )
         => ( ( inf_inf_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ A3 ) ) )
            = ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [A2: nat] : ( inf_inf_set_nat @ ( F6 @ A2 ) @ ( G @ A2 ) )
                @ A3 ) ) ) ) ) ) ).

% cINF_inf_distrib
thf(fact_795_cSUP__eq__cINF__D,axiom,
    ! [F6: nat > nat,A3: set_nat,A: nat] :
      ( ( ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
        = ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) )
     => ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
       => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
         => ( ( member_nat @ A @ A3 )
           => ( ( F6 @ A )
              = ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
thf(fact_796_cSUP__eq__cINF__D,axiom,
    ! [F6: product_prod_nat_nat > set_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) )
        = ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) )
     => ( ( condit5477540289124974626et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) )
       => ( ( condit68592940725977398et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) )
         => ( ( member8440522571783428010at_nat @ A @ A3 )
           => ( ( F6 @ A )
              = ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ F6 @ A3 ) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
thf(fact_797_cSUP__eq__cINF__D,axiom,
    ! [F6: nat > set_nat,A3: set_nat,A: nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
        = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) )
     => ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
       => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
         => ( ( member_nat @ A @ A3 )
           => ( ( F6 @ A )
              = ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
thf(fact_798_INT__extend__simps_I3_J,axiom,
    ! [C3: set_nat,A3: nat > set_nat,B3: set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
          = ( minus_minus_set_nat @ top_top_set_nat @ B3 ) ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( minus_minus_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ C3 ) ) @ B3 )
          = ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( minus_minus_set_nat @ ( A3 @ X ) @ B3 )
              @ C3 ) ) ) ) ) ).

% INT_extend_simps(3)
thf(fact_799_INT__extend__simps_I4_J,axiom,
    ! [C3: set_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( ( C3 = bot_bot_set_nat )
       => ( ( minus_minus_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = A3 ) )
      & ( ( C3 != bot_bot_set_nat )
       => ( ( minus_minus_set_nat @ A3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ C3 ) ) )
          = ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( minus_minus_set_nat @ A3 @ ( B3 @ X ) )
              @ C3 ) ) ) ) ) ).

% INT_extend_simps(4)
thf(fact_800_cINF__UNION,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,F6: nat > set_nat] :
      ( ( A3 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit68592940725977398et_nat
            @ ( comple548664676211718543et_nat
              @ ( image_1942688902138425592et_nat
                @ ^ [X: product_prod_nat_nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) )
            = ( comple7806235888213564991et_nat
              @ ( image_15824709712370754et_nat
                @ ^ [X: product_prod_nat_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cINF_UNION
thf(fact_801_cINF__UNION,axiom,
    ! [A3: set_nat,B3: nat > set_nat,F6: nat > set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit68592940725977398et_nat
            @ ( comple548664676211718543et_nat
              @ ( image_2194112158459175443et_nat
                @ ^ [X: nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
            = ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cINF_UNION
thf(fact_802_cINF__UNION,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat,F6: nat > nat] :
      ( ( A3 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit1738341127787009408ow_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_15824709712370754et_nat
                @ ^ [X: product_prod_nat_nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) )
            = ( complete_Inf_Inf_nat
              @ ( image_2486076414777270412at_nat
                @ ^ [X: product_prod_nat_nat] : ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cINF_UNION
thf(fact_803_cINF__UNION,axiom,
    ! [A3: set_nat,B3: nat > set_nat,F6: nat > nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( ( B3 @ X4 )
             != bot_bot_set_nat ) )
       => ( ( condit1738341127787009408ow_nat
            @ ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
                @ A3 ) ) )
         => ( ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
            = ( complete_Inf_Inf_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F6 @ ( B3 @ X ) ) )
                @ A3 ) ) ) ) ) ) ).

% cINF_UNION
thf(fact_804_bdd__below__image__antimono,axiom,
    ! [F6: nat > set_nat,A3: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: set_nat,Y6: set_nat] : ( ord_less_eq_set_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( condit68592940725977398et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% bdd_below_image_antimono
thf(fact_805_bdd__below__image__antimono,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) ) ) ).

% bdd_below_image_antimono
thf(fact_806_bdd__below__image__antimono,axiom,
    ! [F6: nat > filter_nat,A3: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( condit3133331395780078992er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) ) ) ).

% bdd_below_image_antimono
thf(fact_807_bdd__below__image__antimono,axiom,
    ! [F6: filter_nat > nat,A3: set_filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit4029664723990530684er_nat @ A3 )
       => ( condit1738341127787009408ow_nat @ ( image_filter_nat_nat @ F6 @ A3 ) ) ) ) ).

% bdd_below_image_antimono
thf(fact_808_bdd__below__image__antimono,axiom,
    ! [F6: filter_nat > filter_nat,A3: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit4029664723990530684er_nat @ A3 )
       => ( condit3133331395780078992er_nat @ ( image_4775440762538963527er_nat @ F6 @ A3 ) ) ) ) ).

% bdd_below_image_antimono
thf(fact_809_bdd__above__image__antimono,axiom,
    ! [F6: nat > set_nat,A3: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: set_nat,Y6: set_nat] : ( ord_less_eq_set_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F6 @ A3 ) ) ) ) ).

% bdd_above_image_antimono
thf(fact_810_bdd__above__image__antimono,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F6 @ A3 ) ) ) ) ).

% bdd_above_image_antimono
thf(fact_811_bdd__above__image__antimono,axiom,
    ! [F6: nat > filter_nat,A3: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( condit4029664723990530684er_nat @ ( image_nat_filter_nat @ F6 @ A3 ) ) ) ) ).

% bdd_above_image_antimono
thf(fact_812_bdd__above__image__antimono,axiom,
    ! [F6: filter_nat > nat,A3: set_filter_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit3133331395780078992er_nat @ A3 )
       => ( condit2214826472909112428ve_nat @ ( image_filter_nat_nat @ F6 @ A3 ) ) ) ) ).

% bdd_above_image_antimono
thf(fact_813_bdd__above__image__antimono,axiom,
    ! [F6: filter_nat > filter_nat,A3: set_filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ F6 )
     => ( ( condit3133331395780078992er_nat @ A3 )
       => ( condit4029664723990530684er_nat @ ( image_4775440762538963527er_nat @ F6 @ A3 ) ) ) ) ).

% bdd_above_image_antimono
thf(fact_814_mono__cINF,axiom,
    ! [F6: set_nat > set_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat @ ( F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
            @ ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_815_mono__cINF,axiom,
    ! [F6: set_nat > nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto2923694778811248831at_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
            @ ( complete_Inf_Inf_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_816_mono__cINF,axiom,
    ! [F6: set_nat > filter_nat,A3: nat > set_nat,I2: set_nat] :
      ( ( monoto5359140060069215823er_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit68592940725977398et_nat @ ( image_nat_set_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
            @ ( comple6612786467464916249er_nat
              @ ( image_nat_filter_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_817_mono__cINF,axiom,
    ! [F6: nat > set_nat,A3: nat > nat,I2: set_nat] :
      ( ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat @ ( F6 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ A3 @ I2 ) ) )
            @ ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_818_mono__cINF,axiom,
    ! [F6: nat > nat,A3: nat > nat,I2: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( F6 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ A3 @ I2 ) ) )
            @ ( complete_Inf_Inf_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_819_mono__cINF,axiom,
    ! [F6: nat > filter_nat,A3: nat > nat,I2: set_nat] :
      ( ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ A3 @ I2 ) ) )
            @ ( comple6612786467464916249er_nat
              @ ( image_nat_filter_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_820_mono__cINF,axiom,
    ! [F6: filter_nat > set_nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto6401136896278960335et_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( condit3133331395780078992er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_set_nat @ ( F6 @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) )
            @ ( comple7806235888213564991et_nat
              @ ( image_nat_set_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_821_mono__cINF,axiom,
    ! [F6: filter_nat > nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto756600448231510297at_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_less_eq_nat @ F6 )
     => ( ( condit3133331395780078992er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( F6 @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) )
            @ ( complete_Inf_Inf_nat
              @ ( image_nat_nat2
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_822_mono__cINF,axiom,
    ! [F6: filter_nat > filter_nat,A3: nat > filter_nat,I2: set_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
     => ( ( condit3133331395780078992er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) )
       => ( ( I2 != bot_bot_set_nat )
         => ( ord_le2510731241096832064er_nat @ ( F6 @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ A3 @ I2 ) ) )
            @ ( comple6612786467464916249er_nat
              @ ( image_nat_filter_nat
                @ ^ [X: nat] : ( F6 @ ( A3 @ X ) )
                @ I2 ) ) ) ) ) ) ).

% mono_cINF
thf(fact_823_flat__lub__def,axiom,
    ( partia4823134289123629261at_nat
    = ( ^ [B2: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
          ( if_Pro6206227464963214023at_nat @ ( ord_le3146513528884898305at_nat @ A7 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) @ B2
          @ ( the_Pr7557018466319803784at_nat
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% flat_lub_def
thf(fact_824_flat__lub__def,axiom,
    ( partial_flat_lub_nat
    = ( ^ [B2: nat,A7: set_nat] :
          ( if_nat @ ( ord_less_eq_set_nat @ A7 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ B2
          @ ( the_nat
            @ ^ [X: nat] : ( member_nat @ X @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% flat_lub_def
thf(fact_825_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_nat
    @ ^ [X: set_nat,Y6: set_nat] : ( ord_less_eq_set_nat @ Y6 @ X )
    @ ^ [X: set_nat,Y6: set_nat] : ( ord_less_set_nat @ Y6 @ X )
    @ bot_bot_set_nat ) ).

% bot.ordering_top_axioms
thf(fact_826_bot_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
    @ ^ [X: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X )
    @ bot_bot_nat ) ).

% bot.ordering_top_axioms
thf(fact_827_bot_Oordering__top__axioms,axiom,
    ( orderi2807725203697798511er_nat
    @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
    @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_less_filter_nat @ Y6 @ X )
    @ bot_bot_filter_nat ) ).

% bot.ordering_top_axioms
thf(fact_828_vimage__if,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,D: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ C @ A3 )
       => ( ( ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage2449269961533847803at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_Pro6206227464963214023at_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = top_to4669805908274784177at_nat ) )
          & ( ~ ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage2449269961533847803at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_Pro6206227464963214023at_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = B3 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ C @ A3 )
       => ( ( ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage2449269961533847803at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_Pro6206227464963214023at_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = ( uminus6524753893492686040at_nat @ B3 ) ) )
          & ( ~ ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage2449269961533847803at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_Pro6206227464963214023at_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% vimage_if
thf(fact_829_vimage__if,axiom,
    ! [C: nat,A3: set_nat,D: nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( member_nat @ C @ A3 )
       => ( ( ( member_nat @ D @ A3 )
           => ( ( vimage4653281326611754070at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = top_to4669805908274784177at_nat ) )
          & ( ~ ( member_nat @ D @ A3 )
           => ( ( vimage4653281326611754070at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = B3 ) ) ) )
      & ( ~ ( member_nat @ C @ A3 )
       => ( ( ( member_nat @ D @ A3 )
           => ( ( vimage4653281326611754070at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = ( uminus6524753893492686040at_nat @ B3 ) ) )
          & ( ~ ( member_nat @ D @ A3 )
           => ( ( vimage4653281326611754070at_nat
                @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% vimage_if
thf(fact_830_vimage__if,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,D: product_prod_nat_nat,B3: set_nat] :
      ( ( ( member8440522571783428010at_nat @ C @ A3 )
       => ( ( ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage8013328719654469172at_nat
                @ ^ [X: nat] : ( if_Pro6206227464963214023at_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = top_top_set_nat ) )
          & ( ~ ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage8013328719654469172at_nat
                @ ^ [X: nat] : ( if_Pro6206227464963214023at_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = B3 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ C @ A3 )
       => ( ( ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage8013328719654469172at_nat
                @ ^ [X: nat] : ( if_Pro6206227464963214023at_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = ( uminus5710092332889474511et_nat @ B3 ) ) )
          & ( ~ ( member8440522571783428010at_nat @ D @ A3 )
           => ( ( vimage8013328719654469172at_nat
                @ ^ [X: nat] : ( if_Pro6206227464963214023at_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = bot_bot_set_nat ) ) ) ) ) ).

% vimage_if
thf(fact_831_vimage__if,axiom,
    ! [C: nat,A3: set_nat,D: nat,B3: set_nat] :
      ( ( ( member_nat @ C @ A3 )
       => ( ( ( member_nat @ D @ A3 )
           => ( ( vimage_nat_nat
                @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = top_top_set_nat ) )
          & ( ~ ( member_nat @ D @ A3 )
           => ( ( vimage_nat_nat
                @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = B3 ) ) ) )
      & ( ~ ( member_nat @ C @ A3 )
       => ( ( ( member_nat @ D @ A3 )
           => ( ( vimage_nat_nat
                @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = ( uminus5710092332889474511et_nat @ B3 ) ) )
          & ( ~ ( member_nat @ D @ A3 )
           => ( ( vimage_nat_nat
                @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ B3 ) @ C @ D )
                @ A3 )
              = bot_bot_set_nat ) ) ) ) ) ).

% vimage_if
thf(fact_832_Compl__INT,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( uminus5710092332889474511et_nat @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% Compl_INT
thf(fact_833_Compl__UN,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( uminus5710092332889474511et_nat @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% Compl_UN
thf(fact_834_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 )
            @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ) ).

% minus_set_def
thf(fact_835_minus__set__def,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( minus_2270307095948843157_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ) ).

% minus_set_def
thf(fact_836_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_837_Collect__imp__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P ) ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_838_uminus__INF,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( uminus5710092332889474511et_nat @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% uminus_INF
thf(fact_839_uminus__SUP,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( uminus5710092332889474511et_nat @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% uminus_SUP
thf(fact_840_If__the__inv__into__in__Func,axiom,
    ! [G: nat > set_nat,C3: set_nat,B3: set_nat,X5: nat] :
      ( ( inj_on_nat_set_nat @ G @ C3 )
     => ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ B3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
       => ( member_set_nat_nat
          @ ^ [I: set_nat] : ( if_nat @ ( member_set_nat @ I @ ( image_nat_set_nat @ G @ C3 ) ) @ ( the_in5057678521256355851et_nat @ C3 @ G @ I ) @ X5 )
          @ ( bNF_We5625579243924517612at_nat @ top_top_set_set_nat @ ( sup_sup_set_nat @ B3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) ) ).

% If_the_inv_into_in_Func
thf(fact_841_If__the__inv__into__in__Func,axiom,
    ! [G: nat > product_prod_nat_nat,C3: set_nat,B3: set_nat,X5: nat] :
      ( ( inj_on5538052773655684606at_nat @ G @ C3 )
     => ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ B3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
       => ( member7868840539957676139at_nat
          @ ^ [I: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ I @ ( image_5846123807819985514at_nat @ G @ C3 ) ) @ ( the_in1052143192868928060at_nat @ C3 @ G @ I ) @ X5 )
          @ ( bNF_We3695879855305361469at_nat @ top_to4669805908274784177at_nat @ ( sup_sup_set_nat @ B3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) ) ).

% If_the_inv_into_in_Func
thf(fact_842_If__the__inv__into__in__Func,axiom,
    ! [G: nat > nat,C3: set_nat,B3: set_nat,X5: nat] :
      ( ( inj_on_nat_nat @ G @ C3 )
     => ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ B3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
       => ( member_nat_nat
          @ ^ [I: nat] : ( if_nat @ ( member_nat @ I @ ( image_nat_nat2 @ G @ C3 ) ) @ ( the_inv_into_nat_nat @ C3 @ G @ I ) @ X5 )
          @ ( bNF_We7540516734559056694at_nat @ top_top_set_nat @ ( sup_sup_set_nat @ B3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) ) ).

% If_the_inv_into_in_Func
thf(fact_843_cSup__cInf,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ( ( condit2214826472909112428ve_nat @ S )
       => ( ( complete_Sup_Sup_nat @ S )
          = ( complete_Inf_Inf_nat
            @ ( collect_nat
              @ ^ [X: nat] :
                ! [Y6: nat] :
                  ( ( member_nat @ Y6 @ S )
                 => ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ) ) ).

% cSup_cInf
thf(fact_844_cSup__cInf,axiom,
    ! [S: set_filter_nat] :
      ( ( S != bot_bo498966703094740906er_nat )
     => ( ( condit4029664723990530684er_nat @ S )
       => ( ( comple2454491259644266675er_nat @ S )
          = ( comple6612786467464916249er_nat
            @ ( collect_filter_nat
              @ ^ [X: filter_nat] :
                ! [Y6: filter_nat] :
                  ( ( member_filter_nat @ Y6 @ S )
                 => ( ord_le2510731241096832064er_nat @ Y6 @ X ) ) ) ) ) ) ) ).

% cSup_cInf
thf(fact_845_cSup__cInf,axiom,
    ! [S: set_set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ( ( condit5477540289124974626et_nat @ S )
       => ( ( comple7399068483239264473et_nat @ S )
          = ( comple7806235888213564991et_nat
            @ ( collect_set_nat
              @ ^ [X: set_nat] :
                ! [Y6: set_nat] :
                  ( ( member_set_nat @ Y6 @ S )
                 => ( ord_less_eq_set_nat @ Y6 @ X ) ) ) ) ) ) ) ).

% cSup_cInf
thf(fact_846_cInf__cSup,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ( ( condit1738341127787009408ow_nat @ S )
       => ( ( complete_Inf_Inf_nat @ S )
          = ( complete_Sup_Sup_nat
            @ ( collect_nat
              @ ^ [X: nat] :
                ! [Y6: nat] :
                  ( ( member_nat @ Y6 @ S )
                 => ( ord_less_eq_nat @ X @ Y6 ) ) ) ) ) ) ) ).

% cInf_cSup
thf(fact_847_cInf__cSup,axiom,
    ! [S: set_filter_nat] :
      ( ( S != bot_bo498966703094740906er_nat )
     => ( ( condit3133331395780078992er_nat @ S )
       => ( ( comple6612786467464916249er_nat @ S )
          = ( comple2454491259644266675er_nat
            @ ( collect_filter_nat
              @ ^ [X: filter_nat] :
                ! [Y6: filter_nat] :
                  ( ( member_filter_nat @ Y6 @ S )
                 => ( ord_le2510731241096832064er_nat @ X @ Y6 ) ) ) ) ) ) ) ).

% cInf_cSup
thf(fact_848_cInf__cSup,axiom,
    ! [S: set_set_nat] :
      ( ( S != bot_bot_set_set_nat )
     => ( ( condit68592940725977398et_nat @ S )
       => ( ( comple7806235888213564991et_nat @ S )
          = ( comple7399068483239264473et_nat
            @ ( collect_set_nat
              @ ^ [X: set_nat] :
                ! [Y6: set_nat] :
                  ( ( member_set_nat @ Y6 @ S )
                 => ( ord_less_eq_set_nat @ X @ Y6 ) ) ) ) ) ) ) ).

% cInf_cSup
thf(fact_849_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ~ ( member_nat @ X @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_850_Compl__eq,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ~ ( member8440522571783428010at_nat @ X @ A7 ) ) ) ) ).

% Compl_eq
thf(fact_851_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ~ ( P @ X ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_852_Collect__neg__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ~ ( P @ X ) )
      = ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_853_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A7: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_854_uminus__set__def,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( uminus8676089048583255045_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 ) ) ) ) ) ).

% uminus_set_def
thf(fact_855_Inter__eq,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A7: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
            ! [Y6: set_nat] :
              ( ( member_set_nat @ Y6 @ A7 )
             => ( member_nat @ X @ Y6 ) ) ) ) ) ).

% Inter_eq
thf(fact_856_Inter__eq,axiom,
    ( comple1930127330929219688at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
            ! [Y6: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y6 @ A7 )
             => ( member8440522571783428010at_nat @ X @ Y6 ) ) ) ) ) ).

% Inter_eq
thf(fact_857_INTER__eq,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) )
      = ( collect_nat
        @ ^ [Y6: nat] :
          ! [X: nat] :
            ( ( member_nat @ X @ A3 )
           => ( member_nat @ Y6 @ ( B3 @ X ) ) ) ) ) ).

% INTER_eq
thf(fact_858_Collect__ball__eq,axiom,
    ! [A3: set_nat,P: nat > nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
          ! [Y6: nat] :
            ( ( member_nat @ Y6 @ A3 )
           => ( P @ X @ Y6 ) ) )
      = ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [Y6: nat] :
              ( collect_nat
              @ ^ [X: nat] : ( P @ X @ Y6 ) )
          @ A3 ) ) ) ).

% Collect_ball_eq
thf(fact_859_Inf__eq__Sup,axiom,
    ( comple6612786467464916249er_nat
    = ( ^ [A7: set_filter_nat] :
          ( comple2454491259644266675er_nat
          @ ( collect_filter_nat
            @ ^ [B2: filter_nat] :
              ! [X: filter_nat] :
                ( ( member_filter_nat @ X @ A7 )
               => ( ord_le2510731241096832064er_nat @ B2 @ X ) ) ) ) ) ) ).

% Inf_eq_Sup
thf(fact_860_Inf__eq__Sup,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A7: set_set_nat] :
          ( comple7399068483239264473et_nat
          @ ( collect_set_nat
            @ ^ [B2: set_nat] :
              ! [X: set_nat] :
                ( ( member_set_nat @ X @ A7 )
               => ( ord_less_eq_set_nat @ B2 @ X ) ) ) ) ) ) ).

% Inf_eq_Sup
thf(fact_861_Sup__eq__Inf,axiom,
    ( comple2454491259644266675er_nat
    = ( ^ [A7: set_filter_nat] :
          ( comple6612786467464916249er_nat
          @ ( collect_filter_nat
            @ ^ [B2: filter_nat] :
              ! [X: filter_nat] :
                ( ( member_filter_nat @ X @ A7 )
               => ( ord_le2510731241096832064er_nat @ X @ B2 ) ) ) ) ) ) ).

% Sup_eq_Inf
thf(fact_862_Sup__eq__Inf,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A7: set_set_nat] :
          ( comple7806235888213564991et_nat
          @ ( collect_set_nat
            @ ^ [B2: set_nat] :
              ! [X: set_nat] :
                ( ( member_set_nat @ X @ A7 )
               => ( ord_less_eq_set_nat @ X @ B2 ) ) ) ) ) ) ).

% Sup_eq_Inf
thf(fact_863_refl__on__def_H,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
          ( ! [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ R )
             => ( produc6590410687421337004_nat_o
                @ ^ [Y6: product_prod_nat_nat,Z6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y6 @ A7 )
                    & ( member8440522571783428010at_nat @ Z6 @ A7 ) )
                @ X ) )
          & ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A7 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_864_refl__on__def_H,axiom,
    ( refl_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
          ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ R )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y6: nat,Z6: nat] :
                    ( ( member_nat @ Y6 @ A7 )
                    & ( member_nat @ Z6 @ A7 ) )
                @ X ) )
          & ! [X: nat] :
              ( ( member_nat @ X @ A7 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_865_Func__empty,axiom,
    ! [B3: set_o] :
      ( ( bNF_We709690948610045618_nat_o @ bot_bot_set_nat @ B3 )
      = ( insert_nat_o
        @ ^ [X: nat] : undefined_o
        @ bot_bot_set_nat_o ) ) ).

% Func_empty
thf(fact_866_Func__empty,axiom,
    ! [B3: set_nat] :
      ( ( bNF_We7540516734559056694at_nat @ bot_bot_set_nat @ B3 )
      = ( insert_nat_nat
        @ ^ [X: nat] : undefined_nat
        @ bot_bot_set_nat_nat ) ) ).

% Func_empty
thf(fact_867_Func__empty,axiom,
    ! [B3: set_prat] :
      ( ( bNF_We4121571284332944310t_prat @ bot_bot_set_nat @ B3 )
      = ( insert_nat_prat
        @ ^ [X: nat] : undefined_prat
        @ bot_bot_set_nat_prat ) ) ).

% Func_empty
thf(fact_868_lfp__induct2,axiom,
    ! [A: nat,B: list_char,F6: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char,P: nat > list_char > $o] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( comple3227850406620335403t_char @ F6 ) )
     => ( ( monoto7809364549836873959t_char @ top_to6005184720552305954t_char @ ord_le6315511645215477266t_char @ ord_le6315511645215477266t_char @ F6 )
       => ( ! [A5: nat,B5: list_char] :
              ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A5 @ B5 ) @ ( F6 @ ( inf_in3544766959038609504t_char @ ( comple3227850406620335403t_char @ F6 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P ) ) ) ) )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% lfp_induct2
thf(fact_869_lfp__induct2,axiom,
    ! [A: nat,B: nat,F6: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( comple1738246912304005786at_nat @ F6 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F6 )
       => ( ! [A5: nat,B5: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B5 ) @ ( F6 @ ( inf_in2572325071724192079at_nat @ ( comple1738246912304005786at_nat @ F6 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P ) ) ) ) )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% lfp_induct2
thf(fact_870_lfp__lfp,axiom,
    ! [F6: filter_nat > filter_nat > filter_nat] :
      ( ! [X4: filter_nat,Y5: filter_nat,W: filter_nat,Z: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X4 @ Y5 )
         => ( ( ord_le2510731241096832064er_nat @ W @ Z )
           => ( ord_le2510731241096832064er_nat @ ( F6 @ X4 @ W ) @ ( F6 @ Y5 @ Z ) ) ) )
     => ( ( comple7665627409031812199er_nat
          @ ^ [X: filter_nat] : ( comple7665627409031812199er_nat @ ( F6 @ X ) ) )
        = ( comple7665627409031812199er_nat
          @ ^ [X: filter_nat] : ( F6 @ X @ X ) ) ) ) ).

% lfp_lfp
thf(fact_871_lfp__induct__set,axiom,
    ! [A: nat,F6: set_nat > set_nat,P: nat > $o] :
      ( ( member_nat @ A @ ( comple7975543026063415949et_nat @ F6 ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( F6 @ ( inf_inf_set_nat @ ( comple7975543026063415949et_nat @ F6 ) @ ( collect_nat @ P ) ) ) )
             => ( P @ X4 ) )
         => ( P @ A ) ) ) ) ).

% lfp_induct_set
thf(fact_872_lfp__induct__set,axiom,
    ! [A: product_prod_nat_nat,F6: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( comple1738246912304005786at_nat @ F6 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F6 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ ( F6 @ ( inf_in2572325071724192079at_nat @ ( comple1738246912304005786at_nat @ F6 ) @ ( collec3392354462482085612at_nat @ P ) ) ) )
             => ( P @ X4 ) )
         => ( P @ A ) ) ) ) ).

% lfp_induct_set
thf(fact_873_def__lfp__induct__set,axiom,
    ! [A3: set_nat,F6: set_nat > set_nat,A: nat,P: nat > $o] :
      ( ( A3
        = ( comple7975543026063415949et_nat @ F6 ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
       => ( ( member_nat @ A @ A3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( F6 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) )
               => ( P @ X4 ) )
           => ( P @ A ) ) ) ) ) ).

% def_lfp_induct_set
thf(fact_874_def__lfp__induct__set,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( A3
        = ( comple1738246912304005786at_nat @ F6 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F6 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ ( F6 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) ) )
               => ( P @ X4 ) )
           => ( P @ A ) ) ) ) ) ).

% def_lfp_induct_set
thf(fact_875_lfp__def,axiom,
    ( comple7665627409031812199er_nat
    = ( ^ [F4: filter_nat > filter_nat] :
          ( comple6612786467464916249er_nat
          @ ( collect_filter_nat
            @ ^ [U2: filter_nat] : ( ord_le2510731241096832064er_nat @ ( F4 @ U2 ) @ U2 ) ) ) ) ) ).

% lfp_def
thf(fact_876_lfp__rolling,axiom,
    ! [G: filter_nat > filter_nat,F6: filter_nat > filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ G )
     => ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
       => ( ( G
            @ ( comple7665627409031812199er_nat
              @ ^ [X: filter_nat] : ( F6 @ ( G @ X ) ) ) )
          = ( comple7665627409031812199er_nat
            @ ^ [X: filter_nat] : ( G @ ( F6 @ X ) ) ) ) ) ) ).

% lfp_rolling
thf(fact_877_def__coinduct3,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,X10: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( comple8166066674555273823at_nat @ F6 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F6 )
       => ( ( member8440522571783428010at_nat @ A @ X10 )
         => ( ( ord_le3146513528884898305at_nat @ X10
              @ ( F6
                @ ( comple1738246912304005786at_nat
                  @ ^ [X: set_Pr1261947904930325089at_nat] : ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( F6 @ X ) @ X10 ) @ A3 ) ) ) )
           => ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% def_coinduct3
thf(fact_878_def__coinduct3,axiom,
    ! [A3: set_nat,F6: set_nat > set_nat,A: nat,X10: set_nat] :
      ( ( A3
        = ( comple1596078789208929544et_nat @ F6 ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
       => ( ( member_nat @ A @ X10 )
         => ( ( ord_less_eq_set_nat @ X10
              @ ( F6
                @ ( comple7975543026063415949et_nat
                  @ ^ [X: set_nat] : ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( F6 @ X ) @ X10 ) @ A3 ) ) ) )
           => ( member_nat @ A @ A3 ) ) ) ) ) ).

% def_coinduct3
thf(fact_879_coinduct3,axiom,
    ! [F6: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,X10: set_Pr1261947904930325089at_nat] :
      ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F6 )
     => ( ( member8440522571783428010at_nat @ A @ X10 )
       => ( ( ord_le3146513528884898305at_nat @ X10
            @ ( F6
              @ ( comple1738246912304005786at_nat
                @ ^ [X: set_Pr1261947904930325089at_nat] : ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( F6 @ X ) @ X10 ) @ ( comple8166066674555273823at_nat @ F6 ) ) ) ) )
         => ( member8440522571783428010at_nat @ A @ ( comple8166066674555273823at_nat @ F6 ) ) ) ) ) ).

% coinduct3
thf(fact_880_coinduct3,axiom,
    ! [F6: set_nat > set_nat,A: nat,X10: set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F6 )
     => ( ( member_nat @ A @ X10 )
       => ( ( ord_less_eq_set_nat @ X10
            @ ( F6
              @ ( comple7975543026063415949et_nat
                @ ^ [X: set_nat] : ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( F6 @ X ) @ X10 ) @ ( comple1596078789208929544et_nat @ F6 ) ) ) ) )
         => ( member_nat @ A @ ( comple1596078789208929544et_nat @ F6 ) ) ) ) ) ).

% coinduct3
thf(fact_881_gfp__gfp,axiom,
    ! [F6: filter_nat > filter_nat > filter_nat] :
      ( ! [X4: filter_nat,Y5: filter_nat,W: filter_nat,Z: filter_nat] :
          ( ( ord_le2510731241096832064er_nat @ X4 @ Y5 )
         => ( ( ord_le2510731241096832064er_nat @ W @ Z )
           => ( ord_le2510731241096832064er_nat @ ( F6 @ X4 @ W ) @ ( F6 @ Y5 @ Z ) ) ) )
     => ( ( comple3856628597079623266er_nat
          @ ^ [X: filter_nat] : ( comple3856628597079623266er_nat @ ( F6 @ X ) ) )
        = ( comple3856628597079623266er_nat
          @ ^ [X: filter_nat] : ( F6 @ X @ X ) ) ) ) ).

% gfp_gfp
thf(fact_882_gfp__def,axiom,
    ( comple3856628597079623266er_nat
    = ( ^ [F4: filter_nat > filter_nat] :
          ( comple2454491259644266675er_nat
          @ ( collect_filter_nat
            @ ^ [U2: filter_nat] : ( ord_le2510731241096832064er_nat @ U2 @ ( F4 @ U2 ) ) ) ) ) ) ).

% gfp_def
thf(fact_883_gfp__def,axiom,
    ( comple1596078789208929544et_nat
    = ( ^ [F4: set_nat > set_nat] :
          ( comple7399068483239264473et_nat
          @ ( collect_set_nat
            @ ^ [U2: set_nat] : ( ord_less_eq_set_nat @ U2 @ ( F4 @ U2 ) ) ) ) ) ) ).

% gfp_def
thf(fact_884_gfp__rolling,axiom,
    ! [G: filter_nat > filter_nat,F6: filter_nat > filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ G )
     => ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ F6 )
       => ( ( G
            @ ( comple3856628597079623266er_nat
              @ ^ [X: filter_nat] : ( F6 @ ( G @ X ) ) ) )
          = ( comple3856628597079623266er_nat
            @ ^ [X: filter_nat] : ( G @ ( F6 @ X ) ) ) ) ) ) ).

% gfp_rolling
thf(fact_885_def__Collect__coinduct,axiom,
    ! [A3: set_nat,P: set_nat > nat > $o,A: nat,X10: set_nat] :
      ( ( A3
        = ( comple1596078789208929544et_nat
          @ ^ [W2: set_nat] : ( collect_nat @ ( P @ W2 ) ) ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat
          @ ^ [W2: set_nat] : ( collect_nat @ ( P @ W2 ) ) )
       => ( ( member_nat @ A @ X10 )
         => ( ! [Z: nat] :
                ( ( member_nat @ Z @ X10 )
               => ( P @ ( sup_sup_set_nat @ X10 @ A3 ) @ Z ) )
           => ( member_nat @ A @ A3 ) ) ) ) ) ).

% def_Collect_coinduct
thf(fact_886_def__Collect__coinduct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,X10: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( comple8166066674555273823at_nat
          @ ^ [W2: set_Pr1261947904930325089at_nat] : ( collec3392354462482085612at_nat @ ( P @ W2 ) ) ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat
          @ ^ [W2: set_Pr1261947904930325089at_nat] : ( collec3392354462482085612at_nat @ ( P @ W2 ) ) )
       => ( ( member8440522571783428010at_nat @ A @ X10 )
         => ( ! [Z: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Z @ X10 )
               => ( P @ ( sup_su6327502436637775413at_nat @ X10 @ A3 ) @ Z ) )
           => ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% def_Collect_coinduct
thf(fact_887_bij__betw__disjoint__Un,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,C3: set_nat,G: product_prod_nat_nat > nat,B3: set_Pr1261947904930325089at_nat,D2: set_nat] :
      ( ( bij_be5333170631980326235at_nat @ F6 @ A3 @ C3 )
     => ( ( bij_be5333170631980326235at_nat @ G @ B3 @ D2 )
       => ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
            = bot_bo2099793752762293965at_nat )
         => ( ( ( inf_inf_set_nat @ C3 @ D2 )
              = bot_bot_set_nat )
           => ( bij_be5333170631980326235at_nat
              @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ A3 ) @ ( F6 @ X ) @ ( G @ X ) )
              @ ( sup_su6327502436637775413at_nat @ A3 @ B3 )
              @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ) ) ).

% bij_betw_disjoint_Un
thf(fact_888_bij__betw__disjoint__Un,axiom,
    ! [F6: nat > nat,A3: set_nat,C3: set_nat,G: nat > nat,B3: set_nat,D2: set_nat] :
      ( ( bij_betw_nat_nat @ F6 @ A3 @ C3 )
     => ( ( bij_betw_nat_nat @ G @ B3 @ D2 )
       => ( ( ( inf_inf_set_nat @ A3 @ B3 )
            = bot_bot_set_nat )
         => ( ( ( inf_inf_set_nat @ C3 @ D2 )
              = bot_bot_set_nat )
           => ( bij_betw_nat_nat
              @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ A3 ) @ ( F6 @ X ) @ ( G @ X ) )
              @ ( sup_sup_set_nat @ A3 @ B3 )
              @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ) ) ).

% bij_betw_disjoint_Un
thf(fact_889_bij__betw__UNION__chain,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,A3: product_prod_nat_nat > set_nat,F6: nat > nat,A8: product_prod_nat_nat > set_nat] :
      ( ! [I3: product_prod_nat_nat,J2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ I2 )
         => ( ( member8440522571783428010at_nat @ J2 @ I2 )
           => ( ( ord_less_eq_set_nat @ ( A3 @ I3 ) @ ( A3 @ J2 ) )
              | ( ord_less_eq_set_nat @ ( A3 @ J2 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I3 @ I2 )
           => ( bij_betw_nat_nat @ F6 @ ( A3 @ I3 ) @ ( A8 @ I3 ) ) )
       => ( bij_betw_nat_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A3 @ I2 ) ) @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ A8 @ I2 ) ) ) ) ) ).

% bij_betw_UNION_chain
thf(fact_890_bij__betw__UNION__chain,axiom,
    ! [I2: set_nat,A3: nat > set_nat,F6: nat > nat,A8: nat > set_nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( member_nat @ I3 @ I2 )
         => ( ( member_nat @ J2 @ I2 )
           => ( ( ord_less_eq_set_nat @ ( A3 @ I3 ) @ ( A3 @ J2 ) )
              | ( ord_less_eq_set_nat @ ( A3 @ J2 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I2 )
           => ( bij_betw_nat_nat @ F6 @ ( A3 @ I3 ) @ ( A8 @ I3 ) ) )
       => ( bij_betw_nat_nat @ F6 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A8 @ I2 ) ) ) ) ) ).

% bij_betw_UNION_chain
thf(fact_891_bij__image__INT,axiom,
    ! [F6: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( bij_betw_nat_set_nat @ F6 @ top_top_set_nat @ top_top_set_set_nat )
     => ( ( image_nat_set_nat @ F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
        = ( comple1065008630642458357et_nat
          @ ( image_2194112158459175443et_nat
            @ ^ [X: nat] : ( image_nat_set_nat @ F6 @ ( B3 @ X ) )
            @ A3 ) ) ) ) ).

% bij_image_INT
thf(fact_892_bij__image__INT,axiom,
    ! [F6: nat > nat,B3: nat > set_nat,A3: set_nat] :
      ( ( bij_betw_nat_nat @ F6 @ top_top_set_nat @ top_top_set_nat )
     => ( ( image_nat_nat2 @ F6 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
        = ( comple7806235888213564991et_nat
          @ ( image_nat_set_nat
            @ ^ [X: nat] : ( image_nat_nat2 @ F6 @ ( B3 @ X ) )
            @ A3 ) ) ) ) ).

% bij_image_INT
thf(fact_893_INF__SUP__set,axiom,
    ! [G: product_prod_nat_nat > set_nat,A3: set_se7855581050983116737at_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_5853224409075402146et_nat
          @ ^ [B8: set_Pr1261947904930325089at_nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ B8 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_5853224409075402146et_nat
          @ ^ [B8: set_Pr1261947904930325089at_nat] : ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ G @ B8 ) )
          @ ( collec5514110066124741708at_nat
            @ ^ [Uu: set_Pr1261947904930325089at_nat] :
              ? [F4: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
                ( ( Uu
                  = ( image_3778355557410424805at_nat @ F4 @ A3 ) )
                & ! [X: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X @ A3 )
                   => ( member8440522571783428010at_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_894_INF__SUP__set,axiom,
    ! [G: nat > set_nat,A3: set_set_nat] :
      ( ( comple7806235888213564991et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [B8: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B8 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [B8: set_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ B8 ) )
          @ ( collect_set_nat
            @ ^ [Uu: set_nat] :
              ? [F4: set_nat > nat] :
                ( ( Uu
                  = ( image_set_nat_nat @ F4 @ A3 ) )
                & ! [X: set_nat] :
                    ( ( member_set_nat @ X @ A3 )
                   => ( member_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_895_SUP__INF__set,axiom,
    ! [G: product_prod_nat_nat > set_nat,A3: set_se7855581050983116737at_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_5853224409075402146et_nat
          @ ^ [X: set_Pr1261947904930325089at_nat] : ( comple7806235888213564991et_nat @ ( image_15824709712370754et_nat @ G @ X ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat
        @ ( image_5853224409075402146et_nat
          @ ^ [X: set_Pr1261947904930325089at_nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ G @ X ) )
          @ ( collec5514110066124741708at_nat
            @ ^ [Uu: set_Pr1261947904930325089at_nat] :
              ? [F4: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
                ( ( Uu
                  = ( image_3778355557410424805at_nat @ F4 @ A3 ) )
                & ! [X: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X @ A3 )
                   => ( member8440522571783428010at_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_896_SUP__INF__set,axiom,
    ! [G: nat > set_nat,A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [X: set_nat] : ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ G @ X ) )
          @ A3 ) )
      = ( comple7806235888213564991et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [X: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ X ) )
          @ ( collect_set_nat
            @ ^ [Uu: set_nat] :
              ? [F4: set_nat > nat] :
                ( ( Uu
                  = ( image_set_nat_nat @ F4 @ A3 ) )
                & ! [X: set_nat] :
                    ( ( member_set_nat @ X @ A3 )
                   => ( member_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_897_Inf__Sup__le,axiom,
    ! [A3: set_set_set_nat] :
      ( ord_less_eq_set_nat @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ comple7399068483239264473et_nat @ A3 ) )
      @ ( comple7399068483239264473et_nat
        @ ( image_5842784325960735177et_nat @ comple7806235888213564991et_nat
          @ ( collect_set_set_nat
            @ ^ [Uu: set_set_nat] :
              ? [F4: set_set_nat > set_nat] :
                ( ( Uu
                  = ( image_5842784325960735177et_nat @ F4 @ A3 ) )
                & ! [X: set_set_nat] :
                    ( ( member_set_set_nat @ X @ A3 )
                   => ( member_set_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% Inf_Sup_le
thf(fact_898_Setcompr__eq__image,axiom,
    ! [F6: nat > set_nat,A3: set_nat] :
      ( ( collect_set_nat
        @ ^ [Uu: set_nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( member_nat @ X @ A3 ) ) )
      = ( image_nat_set_nat @ F6 @ A3 ) ) ).

% Setcompr_eq_image
thf(fact_899_Setcompr__eq__image,axiom,
    ! [F6: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: product_prod_nat_nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( member8440522571783428010at_nat @ X @ A3 ) ) )
      = ( image_2486076414777270412at_nat @ F6 @ A3 ) ) ).

% Setcompr_eq_image
thf(fact_900_Setcompr__eq__image,axiom,
    ! [F6: nat > nat,A3: set_nat] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( member_nat @ X @ A3 ) ) )
      = ( image_nat_nat2 @ F6 @ A3 ) ) ).

% Setcompr_eq_image
thf(fact_901_Setcompr__eq__image,axiom,
    ! [F6: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [Uu: product_prod_nat_nat] :
          ? [X: product_prod_nat_nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( member8440522571783428010at_nat @ X @ A3 ) ) )
      = ( image_5168914502847457605at_nat @ F6 @ A3 ) ) ).

% Setcompr_eq_image
thf(fact_902_Setcompr__eq__image,axiom,
    ! [F6: nat > product_prod_nat_nat,A3: set_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [Uu: product_prod_nat_nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( member_nat @ X @ A3 ) ) )
      = ( image_5846123807819985514at_nat @ F6 @ A3 ) ) ).

% Setcompr_eq_image
thf(fact_903_setcompr__eq__image,axiom,
    ! [F6: nat > set_nat,P: nat > $o] :
      ( ( collect_set_nat
        @ ^ [Uu: set_nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_set_nat @ F6 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_904_setcompr__eq__image,axiom,
    ! [F6: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_nat2 @ F6 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_905_setcompr__eq__image,axiom,
    ! [F6: product_prod_nat_nat > nat,P: product_prod_nat_nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: product_prod_nat_nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( P @ X ) ) )
      = ( image_2486076414777270412at_nat @ F6 @ ( collec3392354462482085612at_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_906_setcompr__eq__image,axiom,
    ! [F6: nat > product_prod_nat_nat,P: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [Uu: product_prod_nat_nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( P @ X ) ) )
      = ( image_5846123807819985514at_nat @ F6 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_907_setcompr__eq__image,axiom,
    ! [F6: product_prod_nat_nat > product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [Uu: product_prod_nat_nat] :
          ? [X: product_prod_nat_nat] :
            ( ( Uu
              = ( F6 @ X ) )
            & ( P @ X ) ) )
      = ( image_5168914502847457605at_nat @ F6 @ ( collec3392354462482085612at_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_908_finite_Omono,axiom,
    ( monoto1084075169776741543_nat_o @ top_to28523156410905465_nat_o @ ord_le3964352015994296041_nat_o @ ord_le3964352015994296041_nat_o
    @ ^ [P2: set_nat > $o,X: set_nat] :
        ( ( X = bot_bot_set_nat )
        | ? [A7: set_nat,A2: nat] :
            ( ( X
              = ( insert_nat @ A2 @ A7 ) )
            & ( P2 @ A7 ) ) ) ) ).

% finite.mono
thf(fact_909_full__SetCompr__eq,axiom,
    ! [F6: nat > set_nat] :
      ( ( collect_set_nat
        @ ^ [U2: set_nat] :
          ? [X: nat] :
            ( U2
            = ( F6 @ X ) ) )
      = ( image_nat_set_nat @ F6 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_910_full__SetCompr__eq,axiom,
    ! [F6: nat > nat] :
      ( ( collect_nat
        @ ^ [U2: nat] :
          ? [X: nat] :
            ( U2
            = ( F6 @ X ) ) )
      = ( image_nat_nat2 @ F6 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_911_full__SetCompr__eq,axiom,
    ! [F6: nat > product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [U2: product_prod_nat_nat] :
          ? [X: nat] :
            ( U2
            = ( F6 @ X ) ) )
      = ( image_5846123807819985514at_nat @ F6 @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_912_Gr__def,axiom,
    ( bNF_Gr_nat_list_char
    = ( ^ [A7: set_nat,F4: nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ A2 @ ( F4 @ A2 ) ) )
              & ( member_nat @ A2 @ A7 ) ) ) ) ) ).

% Gr_def
thf(fact_913_Gr__def,axiom,
    ( bNF_Gr_nat_nat
    = ( ^ [A7: set_nat,F4: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A2 @ ( F4 @ A2 ) ) )
              & ( member_nat @ A2 @ A7 ) ) ) ) ) ).

% Gr_def
thf(fact_914_Collect__ex__eq,axiom,
    ! [P: product_prod_nat_nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
          ? [X6: nat] : ( P @ X @ X6 ) )
      = ( comple5685304695842803022at_nat
        @ ( image_7178329752028323786at_nat
          @ ^ [Y6: nat] :
              ( collec3392354462482085612at_nat
              @ ^ [X: product_prod_nat_nat] : ( P @ X @ Y6 ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_915_Collect__ex__eq,axiom,
    ! [P: nat > nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
          ? [X6: nat] : ( P @ X @ X6 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y6: nat] :
              ( collect_nat
              @ ^ [X: nat] : ( P @ X @ Y6 ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_916_Sup__Inf,axiom,
    ! [A3: set_set_set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ comple7806235888213564991et_nat @ A3 ) )
      = ( comple7806235888213564991et_nat
        @ ( image_5842784325960735177et_nat @ comple7399068483239264473et_nat
          @ ( collect_set_set_nat
            @ ^ [Uu: set_set_nat] :
              ? [F4: set_set_nat > set_nat] :
                ( ( Uu
                  = ( image_5842784325960735177et_nat @ F4 @ A3 ) )
                & ! [X: set_set_nat] :
                    ( ( member_set_set_nat @ X @ A3 )
                   => ( member_set_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% Sup_Inf
thf(fact_917_Inf__Sup,axiom,
    ! [A3: set_set_set_nat] :
      ( ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ comple7399068483239264473et_nat @ A3 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_5842784325960735177et_nat @ comple7806235888213564991et_nat
          @ ( collect_set_set_nat
            @ ^ [Uu: set_set_nat] :
              ? [F4: set_set_nat > set_nat] :
                ( ( Uu
                  = ( image_5842784325960735177et_nat @ F4 @ A3 ) )
                & ! [X: set_set_nat] :
                    ( ( member_set_set_nat @ X @ A3 )
                   => ( member_set_nat @ ( F4 @ X ) @ X ) ) ) ) ) ) ) ).

% Inf_Sup
thf(fact_918_Sup__Inf__le,axiom,
    ! [A3: set_set_filter_nat] :
      ( ord_le2510731241096832064er_nat
      @ ( comple2454491259644266675er_nat
        @ ( image_2688478650529214589er_nat @ comple6612786467464916249er_nat
          @ ( collec1743139820085972843er_nat
            @ ^ [Uu: set_filter_nat] :
              ? [F4: set_filter_nat > filter_nat] :
                ( ( Uu
                  = ( image_2688478650529214589er_nat @ F4 @ A3 ) )
                & ! [X: set_filter_nat] :
                    ( ( member8572727008022913325er_nat @ X @ A3 )
                   => ( member_filter_nat @ ( F4 @ X ) @ X ) ) ) ) ) )
      @ ( comple6612786467464916249er_nat @ ( image_2688478650529214589er_nat @ comple2454491259644266675er_nat @ A3 ) ) ) ).

% Sup_Inf_le
thf(fact_919_Sup__Inf__le,axiom,
    ! [A3: set_set_set_nat] :
      ( ord_less_eq_set_nat
      @ ( comple7399068483239264473et_nat
        @ ( image_5842784325960735177et_nat @ comple7806235888213564991et_nat
          @ ( collect_set_set_nat
            @ ^ [Uu: set_set_nat] :
              ? [F4: set_set_nat > set_nat] :
                ( ( Uu
                  = ( image_5842784325960735177et_nat @ F4 @ A3 ) )
                & ! [X: set_set_nat] :
                    ( ( member_set_set_nat @ X @ A3 )
                   => ( member_set_nat @ ( F4 @ X ) @ X ) ) ) ) ) )
      @ ( comple7806235888213564991et_nat @ ( image_5842784325960735177et_nat @ comple7399068483239264473et_nat @ A3 ) ) ) ).

% Sup_Inf_le
thf(fact_920_iteratesp_Omono,axiom,
    ! [F6: filter_nat > filter_nat] :
      ( monoto585190391252645927_nat_o @ top_to8962845839458735263_nat_o @ ord_le2350800233650606735_nat_o @ ord_le2350800233650606735_nat_o
      @ ^ [P2: filter_nat > $o,X: filter_nat] :
          ( ? [Y6: filter_nat] :
              ( ( X
                = ( F6 @ Y6 ) )
              & ( P2 @ Y6 ) )
          | ? [M3: set_filter_nat] :
              ( ( X
                = ( comple2454491259644266675er_nat @ M3 ) )
              & ( comple7182774465050373296er_nat @ ord_le2510731241096832064er_nat @ M3 )
              & ! [Y6: filter_nat] :
                  ( ( member_filter_nat @ Y6 @ M3 )
                 => ( P2 @ Y6 ) ) ) ) ) ).

% iteratesp.mono
thf(fact_921_iteratesp_Omono,axiom,
    ! [F6: set_nat > set_nat] :
      ( monoto1084075169776741543_nat_o @ top_to28523156410905465_nat_o @ ord_le3964352015994296041_nat_o @ ord_le3964352015994296041_nat_o
      @ ^ [P2: set_nat > $o,X: set_nat] :
          ( ? [Y6: set_nat] :
              ( ( X
                = ( F6 @ Y6 ) )
              & ( P2 @ Y6 ) )
          | ? [M3: set_set_nat] :
              ( ( X
                = ( comple7399068483239264473et_nat @ M3 ) )
              & ( comple675287286122868822et_nat @ ord_less_eq_set_nat @ M3 )
              & ! [Y6: set_nat] :
                  ( ( member_set_nat @ Y6 @ M3 )
                 => ( P2 @ Y6 ) ) ) ) ) ).

% iteratesp.mono
thf(fact_922_image2__def,axiom,
    ( bNF_Gr3738443398878377306t_char
    = ( ^ [A7: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > nat,G3: product_prod_nat_nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: product_prod_nat_nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F4 @ A2 ) @ ( G3 @ A2 ) ) )
              & ( member8440522571783428010at_nat @ A2 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_923_image2__def,axiom,
    ( bNF_Gr7397543130437154909t_char
    = ( ^ [A7: set_nat,F4: nat > nat,G3: nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F4 @ A2 ) @ ( G3 @ A2 ) ) )
              & ( member_nat @ A2 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_924_image2__def,axiom,
    ( bNF_Gr8954129895687480649at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > nat,G3: product_prod_nat_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: product_prod_nat_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F4 @ A2 ) @ ( G3 @ A2 ) ) )
              & ( member8440522571783428010at_nat @ A2 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_925_image2__def,axiom,
    ( bNF_Gr7318156773085685964at_nat
    = ( ^ [A7: set_nat,F4: nat > nat,G3: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F4 @ A2 ) @ ( G3 @ A2 ) ) )
              & ( member_nat @ A2 @ A7 ) ) ) ) ) ).

% image2_def
thf(fact_926_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F4: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y6: nat] :
                ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y6 ) )
                | ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y6 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_927_chain__compr,axiom,
    ! [Ord: nat > nat > $o,A3: set_nat,P: nat > $o] :
      ( ( comple7016393980872852640in_nat @ Ord @ A3 )
     => ( comple7016393980872852640in_nat @ Ord
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( P @ X ) ) ) ) ) ).

% chain_compr
thf(fact_928_chain__compr,axiom,
    ! [Ord: product_prod_nat_nat > product_prod_nat_nat > $o,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( comple3588573705683770417at_nat @ Ord @ A3 )
     => ( comple3588573705683770417at_nat @ Ord
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( P @ X ) ) ) ) ) ).

% chain_compr
thf(fact_929_mlex__leq,axiom,
    ! [F6: nat > nat,X5: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F6 @ X5 ) @ ( F6 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ ( mlex_prod_nat @ F6 @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_930_mlex__less,axiom,
    ! [F6: nat > nat,X5: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F6 @ X5 ) @ ( F6 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ ( mlex_prod_nat @ F6 @ R3 ) ) ) ).

% mlex_less
thf(fact_931_mlex__iff,axiom,
    ! [X5: nat,Y: nat,F6: nat > nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ ( mlex_prod_nat @ F6 @ R3 ) )
      = ( ( ord_less_nat @ ( F6 @ X5 ) @ ( F6 @ Y ) )
        | ( ( ( F6 @ X5 )
            = ( F6 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_932_iteratesp__def,axiom,
    ( comple787645200385910945er_nat
    = ( ^ [F4: filter_nat > filter_nat] :
          ( comple6377367993240666408_nat_o
          @ ^ [P2: filter_nat > $o,X: filter_nat] :
              ( ? [Y6: filter_nat] :
                  ( ( X
                    = ( F4 @ Y6 ) )
                  & ( P2 @ Y6 ) )
              | ? [M3: set_filter_nat] :
                  ( ( X
                    = ( comple2454491259644266675er_nat @ M3 ) )
                  & ( comple7182774465050373296er_nat @ ord_le2510731241096832064er_nat @ M3 )
                  & ! [Y6: filter_nat] :
                      ( ( member_filter_nat @ Y6 @ M3 )
                     => ( P2 @ Y6 ) ) ) ) ) ) ) ).

% iteratesp_def
thf(fact_933_iteratesp__def,axiom,
    ( comple8270374135192184263et_nat
    = ( ^ [F4: set_nat > set_nat] :
          ( comple7308811355306212098_nat_o
          @ ^ [P2: set_nat > $o,X: set_nat] :
              ( ? [Y6: set_nat] :
                  ( ( X
                    = ( F4 @ Y6 ) )
                  & ( P2 @ Y6 ) )
              | ? [M3: set_set_nat] :
                  ( ( X
                    = ( comple7399068483239264473et_nat @ M3 ) )
                  & ( comple675287286122868822et_nat @ ord_less_eq_set_nat @ M3 )
                  & ! [Y6: set_nat] :
                      ( ( member_set_nat @ Y6 @ M3 )
                     => ( P2 @ Y6 ) ) ) ) ) ) ) ).

% iteratesp_def
thf(fact_934_relImage__def,axiom,
    ( bNF_Gr3178947973406375814at_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,F4: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A12: nat,A23: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_935_mlex__prod__def,axiom,
    ( mlex_prod_list_char
    = ( ^ [F4: list_char > nat,R4: set_Pr4480099295215233283t_char] :
          ( inv_im6437862472226580897t_char @ ( lex_pr1681056561245511266t_char @ less_than @ R4 )
          @ ^ [X: list_char] : ( produc120197956887798346t_char @ ( F4 @ X ) @ X ) ) ) ) ).

% mlex_prod_def
thf(fact_936_mlex__prod__def,axiom,
    ( mlex_prod_nat
    = ( ^ [F4: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( inv_im8721129524585065343at_nat @ ( lex_prod_nat_nat @ less_than @ R4 )
          @ ^ [X: nat] : ( product_Pair_nat_nat @ ( F4 @ X ) @ X ) ) ) ) ).

% mlex_prod_def
thf(fact_937_less__than__iff,axiom,
    ! [X5: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ less_than )
      = ( ord_less_nat @ X5 @ Y ) ) ).

% less_than_iff
thf(fact_938_relImage__relInvImage,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F6: nat > nat,A3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R3
        @ ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F6 @ A3 )
          @ ^ [Uu: nat] : ( image_nat_nat2 @ F6 @ A3 ) ) )
     => ( ( bNF_Gr3178947973406375814at_nat @ ( bNF_Gr8020361688147858275at_nat @ A3 @ R3 @ F6 ) @ F6 )
        = R3 ) ) ).

% relImage_relInvImage
thf(fact_939_relImage__relInvImage,axiom,
    ! [R3: set_Pr5488025237498180813et_nat,F6: nat > set_nat,A3: set_nat] :
      ( ( ord_le4402255868550298733et_nat @ R3
        @ ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F6 @ A3 )
          @ ^ [Uu: set_nat] : ( image_nat_set_nat @ F6 @ A3 ) ) )
     => ( ( bNF_Gr553682781689516860et_nat @ ( bNF_Gr7866296307673166617et_nat @ A3 @ R3 @ F6 ) @ F6 )
        = R3 ) ) ).

% relImage_relInvImage
thf(fact_940_finite__def,axiom,
    ( finite_finite_nat
    = ( comple7308811355306212098_nat_o
      @ ^ [P2: set_nat > $o,X: set_nat] :
          ( ( X = bot_bot_set_nat )
          | ? [A7: set_nat,A2: nat] :
              ( ( X
                = ( insert_nat @ A2 @ A7 ) )
              & ( P2 @ A7 ) ) ) ) ) ).

% finite_def
thf(fact_941_relInvImage__def,axiom,
    ( bNF_Gr4292214848008494032at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > nat] :
          ( collec7088162979684241874at_nat
          @ ^ [Uu: produc859450856879609959at_nat] :
            ? [A12: product_prod_nat_nat,A23: product_prod_nat_nat] :
              ( ( Uu
                = ( produc6161850002892822231at_nat @ A12 @ A23 ) )
              & ( member8440522571783428010at_nat @ A12 @ A7 )
              & ( member8440522571783428010at_nat @ A23 @ A7 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_942_relInvImage__def,axiom,
    ( bNF_Gr8020361688147858275at_nat
    = ( ^ [A7: set_nat,R4: set_Pr1261947904930325089at_nat,F4: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A12: nat,A23: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A12 @ A23 ) )
              & ( member_nat @ A12 @ A7 )
              & ( member_nat @ A23 @ A7 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_943_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_944_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_945_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_946_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_947_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_948_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_949_finite__Collect__not,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ~ ( P @ X ) ) )
        = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Collect_not
thf(fact_950_finite__Collect__not,axiom,
    ! [P: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_951_finite__Collect__bounded__ex,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
              ? [Y6: product_prod_nat_nat] :
                ( ( P @ Y6 )
                & ( Q @ X @ Y6 ) ) ) )
        = ( ! [Y6: product_prod_nat_nat] :
              ( ( P @ Y6 )
             => ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y6 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_952_finite__Collect__bounded__ex,axiom,
    ! [P: product_prod_nat_nat > $o,Q: nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y6: product_prod_nat_nat] :
                ( ( P @ Y6 )
                & ( Q @ X @ Y6 ) ) ) )
        = ( ! [Y6: product_prod_nat_nat] :
              ( ( P @ Y6 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y6 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_953_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: product_prod_nat_nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
              ? [Y6: nat] :
                ( ( P @ Y6 )
                & ( Q @ X @ Y6 ) ) ) )
        = ( ! [Y6: nat] :
              ( ( P @ Y6 )
             => ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [X: product_prod_nat_nat] : ( Q @ X @ Y6 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_954_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y6: nat] :
                ( ( P @ Y6 )
                & ( Q @ X @ Y6 ) ) ) )
        = ( ! [Y6: nat] :
              ( ( P @ Y6 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y6 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_955_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B8: set_nat] : ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_956_finite__SigmaI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A5 ) ) )
       => ( finite8785817246233100311at_nat @ ( produc7672662199629908489at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_957_finite__SigmaI,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A5 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_958_finite__UN__I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B3 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_959_finite__UN__I,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_960_finite__INT,axiom,
    ! [I2: set_nat,A3: nat > set_nat] :
      ( ? [X8: nat] :
          ( ( member_nat @ X8 @ I2 )
          & ( finite_finite_nat @ ( A3 @ X8 ) ) )
     => ( finite_finite_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% finite_INT
thf(fact_961_bdd__below__UN,axiom,
    ! [I2: set_nat,A3: nat > set_nat] :
      ( ( finite_finite_nat @ I2 )
     => ( ( condit1738341127787009408ow_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ I2 )
             => ( condit1738341127787009408ow_nat @ ( A3 @ X ) ) ) ) ) ) ).

% bdd_below_UN
thf(fact_962_bdd__above__UN,axiom,
    ! [I2: set_nat,A3: nat > set_nat] :
      ( ( finite_finite_nat @ I2 )
     => ( ( condit2214826472909112428ve_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ I2 )
             => ( condit2214826472909112428ve_nat @ ( A3 @ X ) ) ) ) ) ) ).

% bdd_above_UN
thf(fact_963_finite__image__set2,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F6: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X: product_prod_nat_nat,Y6: product_prod_nat_nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_964_finite__image__set2,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F6: product_prod_nat_nat > product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: product_prod_nat_nat,Y6: product_prod_nat_nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_965_finite__image__set2,axiom,
    ! [P: product_prod_nat_nat > $o,Q: nat > $o,F6: product_prod_nat_nat > nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X: product_prod_nat_nat,Y6: nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_966_finite__image__set2,axiom,
    ! [P: product_prod_nat_nat > $o,Q: nat > $o,F6: product_prod_nat_nat > nat > nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: product_prod_nat_nat,Y6: nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_967_finite__image__set2,axiom,
    ! [P: nat > $o,Q: product_prod_nat_nat > $o,F6: nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X: nat,Y6: product_prod_nat_nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_968_finite__image__set2,axiom,
    ! [P: nat > $o,Q: product_prod_nat_nat > $o,F6: nat > product_prod_nat_nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y6: product_prod_nat_nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_969_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F6: nat > nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X: nat,Y6: nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_970_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F6: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y6: nat] :
                ( ( Uu
                  = ( F6 @ X @ Y6 ) )
                & ( P @ X )
                & ( Q @ Y6 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_971_finite__image__set,axiom,
    ! [P: product_prod_nat_nat > $o,F6: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [X: product_prod_nat_nat] :
              ( ( Uu
                = ( F6 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_972_finite__image__set,axiom,
    ! [P: product_prod_nat_nat > $o,F6: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: product_prod_nat_nat] :
              ( ( Uu
                = ( F6 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_973_finite__image__set,axiom,
    ! [P: nat > $o,F6: nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F6 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_974_finite__image__set,axiom,
    ! [P: nat > $o,F6: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F6 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_975_finite__SigmaI2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A3 )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A5 ) ) )
       => ( finite8785817246233100311at_nat @ ( produc7672662199629908489at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_976_finite__SigmaI2,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A3 )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A5 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_977_finite__cartesian__productD1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
     => ( ( B3 != bot_bot_set_nat )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_cartesian_productD1
thf(fact_978_finite__cartesian__productD2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
     => ( ( A3 != bot_bot_set_nat )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_cartesian_productD2
thf(fact_979_finite__cartesian__product__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ( A3 = bot_bot_set_nat )
        | ( B3 = bot_bot_set_nat )
        | ( ( finite_finite_nat @ A3 )
          & ( finite_finite_nat @ B3 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_980_finite__inverse__image__gen,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > product_prod_nat_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( inj_on8969904277767023793at_nat @ F6 @ D2 )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ J @ D2 )
                & ( member8440522571783428010at_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_981_finite__inverse__image__gen,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: nat > product_prod_nat_nat,D2: set_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( inj_on5538052773655684606at_nat @ F6 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D2 )
                & ( member8440522571783428010at_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_982_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F6: product_prod_nat_nat > nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on2178005380612969504at_nat @ F6 @ D2 )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ J @ D2 )
                & ( member_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_983_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F6: nat > nat,D2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F6 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D2 )
                & ( member_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_984_finite__cartesian__product,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_985_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat,R3: product_prod_nat_nat > nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A3 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B3 )
                  & ( R3 @ X4 @ Xa3 ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A2 @ A3 )
                        & ( R3 @ A2 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_986_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B3 )
                  & ( R3 @ X4 @ Xa3 ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat @ A2 @ A3 )
                        & ( R3 @ A2 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_987_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_12: product_prod_nat_nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_988_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_12: nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_989_pigeonhole__infinite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > nat] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F6 @ A3 ) )
       => ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
            & ~ ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [A2: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ A2 @ A3 )
                      & ( ( F6 @ A2 )
                        = ( F6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_990_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F6: nat > set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F6 @ A3 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A2: nat] :
                      ( ( member_nat @ A2 @ A3 )
                      & ( ( F6 @ A2 )
                        = ( F6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_991_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F6: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat2 @ F6 @ A3 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A2: nat] :
                      ( ( member_nat @ A2 @ A3 )
                      & ( ( F6 @ A2 )
                        = ( F6 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_992_nat__seg__image__imp__finite,axiom,
    ! [A3: set_set_nat,F6: nat > set_nat,N3: nat] :
      ( ( A3
        = ( image_nat_set_nat @ F6
          @ ( collect_nat
            @ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_993_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F6: nat > nat,N3: nat] :
      ( ( A3
        = ( image_nat_nat2 @ F6
          @ ( collect_nat
            @ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) ) )
     => ( finite_finite_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_994_finite__conv__nat__seg__image,axiom,
    ( finite1152437895449049373et_nat
    = ( ^ [A7: set_set_nat] :
        ? [N2: nat,F4: nat > set_nat] :
          ( A7
          = ( image_nat_set_nat @ F4
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_995_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A7: set_nat] :
        ? [N2: nat,F4: nat > nat] :
          ( A7
          = ( image_nat_nat2 @ F4
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_996_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [F: nat > nat,N: nat] :
          ( ( ( image_nat_nat2 @ F @ A3 )
            = ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
          & ( inj_on_nat_nat @ F @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_997_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ? [N: nat,F: nat > set_nat] :
          ( ( A3
            = ( image_nat_set_nat @ F
              @ ( collect_nat
                @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) )
          & ( inj_on_nat_set_nat @ F
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_998_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [N: nat,F: nat > nat] :
          ( ( A3
            = ( image_nat_nat2 @ F
              @ ( collect_nat
                @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) )
          & ( inj_on_nat_nat @ F
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_999_finite__range__imageI,axiom,
    ! [G: nat > set_nat,F6: set_nat > set_nat] :
      ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ G @ top_top_set_nat ) )
     => ( finite1152437895449049373et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( F6 @ ( G @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_1000_finite__range__imageI,axiom,
    ! [G: nat > set_nat,F6: set_nat > nat] :
      ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ G @ top_top_set_nat ) )
     => ( finite_finite_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : ( F6 @ ( G @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_1001_finite__range__imageI,axiom,
    ! [G: nat > nat,F6: nat > set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat2 @ G @ top_top_set_nat ) )
     => ( finite1152437895449049373et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( F6 @ ( G @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_1002_finite__range__imageI,axiom,
    ! [G: nat > nat,F6: nat > nat] :
      ( ( finite_finite_nat @ ( image_nat_nat2 @ G @ top_top_set_nat ) )
     => ( finite_finite_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : ( F6 @ ( G @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_1003_finite__inverse__image,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( inj_on8969904277767023793at_nat @ F6 @ top_to4669805908274784177at_nat )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J: product_prod_nat_nat] : ( member8440522571783428010at_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1004_finite__inverse__image,axiom,
    ! [A3: set_nat,F6: product_prod_nat_nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on2178005380612969504at_nat @ F6 @ top_to4669805908274784177at_nat )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J: product_prod_nat_nat] : ( member_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1005_finite__inverse__image,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F6: nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( inj_on5538052773655684606at_nat @ F6 @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member8440522571783428010at_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1006_finite__inverse__image,axiom,
    ! [A3: set_nat,F6: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F6 @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( F6 @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1007_in__finite__psubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B3 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A3 @ B3 )
        & ( finite_finite_nat @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_1008_infinite__cartesian__product,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ~ ( finite_finite_nat @ B3 )
       => ~ ( finite6177210948735845034at_nat
            @ ( produc457027306803732586at_nat @ A3
              @ ^ [Uu: nat] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_1009_INF__filter__bot__base,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,F7: product_prod_nat_nat > filter_nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ I2 )
         => ! [J2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ J2 @ I2 )
             => ? [X8: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X8 @ I2 )
                  & ( ord_le2510731241096832064er_nat @ ( F7 @ X8 ) @ ( inf_inf_filter_nat @ ( F7 @ I3 ) @ ( F7 @ J2 ) ) ) ) ) )
     => ( ( ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F7 @ I2 ) )
          = bot_bot_filter_nat )
        = ( ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ I2 )
              & ( ( F7 @ X )
                = bot_bot_filter_nat ) ) ) ) ) ).

% INF_filter_bot_base
thf(fact_1010_INF__filter__bot__base,axiom,
    ! [I2: set_nat,F7: nat > filter_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I2 )
         => ! [J2: nat] :
              ( ( member_nat @ J2 @ I2 )
             => ? [X8: nat] :
                  ( ( member_nat @ X8 @ I2 )
                  & ( ord_le2510731241096832064er_nat @ ( F7 @ X8 ) @ ( inf_inf_filter_nat @ ( F7 @ I3 ) @ ( F7 @ J2 ) ) ) ) ) )
     => ( ( ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F7 @ I2 ) )
          = bot_bot_filter_nat )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ I2 )
              & ( ( F7 @ X )
                = bot_bot_filter_nat ) ) ) ) ) ).

% INF_filter_bot_base
thf(fact_1011_Inf__filter__def,axiom,
    ( comple6612786467464916249er_nat
    = ( ^ [S3: set_filter_nat] :
          ( comple2454491259644266675er_nat
          @ ( collect_filter_nat
            @ ^ [F8: filter_nat] :
              ! [X: filter_nat] :
                ( ( member_filter_nat @ X @ S3 )
               => ( ord_le2510731241096832064er_nat @ F8 @ X ) ) ) ) ) ) ).

% Inf_filter_def
thf(fact_1012_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A7 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ).

% less_set_def
thf(fact_1013_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B8: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A7 )
          @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ).

% less_set_def
thf(fact_1014_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A7: set_nat,B8: set_nat] :
            ( ( ord_less_set_nat @ A7 @ B8 )
            & ( finite_finite_nat @ B8 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_1015_finite__less__ub,axiom,
    ! [F6: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F6 @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F6 @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_1016_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I5: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I5 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_1017_fold__union__pair,axiom,
    ! [B3: set_list_char,X5: nat,A3: set_Pr2507339680178222706t_char] :
      ( ( finite3619747586194699384t_char @ B3 )
     => ( ( sup_su7899656352754981958t_char
          @ ( comple9178978156834306527t_char
            @ ( image_366428816823531018t_char
              @ ^ [Y6: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X5 @ Y6 ) @ bot_bo1414094081888755934t_char )
              @ B3 ) )
          @ A3 )
        = ( finite3391364926417624327t_char
          @ ^ [Y6: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X5 @ Y6 ) )
          @ A3
          @ B3 ) ) ) ).

% fold_union_pair
thf(fact_1018_fold__union__pair,axiom,
    ! [B3: set_nat,X5: nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( sup_su6327502436637775413at_nat
          @ ( comple5685304695842803022at_nat
            @ ( image_7178329752028323786at_nat
              @ ^ [Y6: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) @ bot_bo2099793752762293965at_nat )
              @ B3 ) )
          @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [Y6: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ Y6 ) )
          @ A3
          @ B3 ) ) ) ).

% fold_union_pair
thf(fact_1019_SUP__principal,axiom,
    ! [A3: nat > set_nat,I2: set_nat] :
      ( ( comple2454491259644266675er_nat
        @ ( image_nat_filter_nat
          @ ^ [I: nat] : ( principal_nat @ ( A3 @ I ) )
          @ I2 ) )
      = ( principal_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) ) ) ).

% SUP_principal
thf(fact_1020_image__fold__insert,axiom,
    ! [A3: set_nat,F6: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_set_nat @ F6 @ A3 )
        = ( finite4178521680790401110et_nat
          @ ^ [K2: nat] : ( insert_set_nat @ ( F6 @ K2 ) )
          @ bot_bot_set_set_nat
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_1021_image__fold__insert,axiom,
    ! [A3: set_nat,F6: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( image_nat_nat2 @ F6 @ A3 )
        = ( finite5529483035118572448et_nat
          @ ^ [K2: nat] : ( insert_nat @ ( F6 @ K2 ) )
          @ bot_bot_set_nat
          @ A3 ) ) ) ).

% image_fold_insert
thf(fact_1022_product__fold,axiom,
    ! [A3: set_nat,B3: set_list_char] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite3619747586194699384t_char @ B3 )
       => ( ( produc6030673636645808891t_char @ A3
            @ ^ [Uu: nat] : B3 )
          = ( finite290769237276952792t_char
            @ ^ [X: nat,Z6: set_Pr2507339680178222706t_char] :
                ( finite3391364926417624327t_char
                @ ^ [Y6: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X @ Y6 ) )
                @ Z6
                @ B3 )
            @ bot_bo1414094081888755934t_char
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_1023_product__fold,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : B3 )
          = ( finite3745491028973389255at_nat
            @ ^ [X: nat,Z6: set_Pr1261947904930325089at_nat] :
                ( finite3745491028973389255at_nat
                @ ^ [Y6: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) )
                @ Z6
                @ B3 )
            @ bot_bo2099793752762293965at_nat
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_1024_Id__on__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( id_on_nat @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) )
          @ bot_bo2099793752762293965at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_1025_INF__principal__finite,axiom,
    ! [X10: set_nat,F6: nat > set_nat] :
      ( ( finite_finite_nat @ X10 )
     => ( ( comple6612786467464916249er_nat
          @ ( image_nat_filter_nat
            @ ^ [X: nat] : ( principal_nat @ ( F6 @ X ) )
            @ X10 ) )
        = ( principal_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F6 @ X10 ) ) ) ) ) ).

% INF_principal_finite
thf(fact_1026_Set__filter__fold,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( filter_nat2 @ P @ A3 )
        = ( finite5529483035118572448et_nat
          @ ^ [X: nat,A9: set_nat] : ( if_set_nat @ ( P @ X ) @ ( insert_nat @ X @ A9 ) @ A9 )
          @ bot_bot_set_nat
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_1027_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr4480099295215233283t_char,X5: produc1457211279475724562t_char,X10: set_Pr2507339680178222706t_char] :
      ( ( finite3801884896073088588t_char @ S )
     => ( ( sup_su7899656352754981958t_char @ ( relcom5956200836249630357t_char @ ( insert3372640069787401954t_char @ X5 @ bot_bo1414094081888755934t_char ) @ S ) @ X10 )
        = ( finite859346576728541399t_char
          @ ( produc6181602235716527061t_char
            @ ^ [W2: list_char,Z6: list_char,A9: set_Pr2507339680178222706t_char] :
                ( if_set1248378584462554936t_char
                @ ( ( produc1900778367047502200t_char @ X5 )
                  = W2 )
                @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ X5 ) @ Z6 ) @ A9 )
                @ A9 ) )
          @ X10
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1028_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr8803067691650843250ar_nat,X5: produc1457211279475724562t_char,X10: set_Pr1261947904930325089at_nat] :
      ( ( finite187957507792297275ar_nat @ S )
     => ( ( sup_su6327502436637775413at_nat @ ( relcom5812064743826168580ar_nat @ ( insert3372640069787401954t_char @ X5 @ bot_bo1414094081888755934t_char ) @ S ) @ X10 )
        = ( finite6062518653932963223at_nat
          @ ( produc179999148950717060at_nat
            @ ^ [W2: list_char,Z6: nat,A9: set_Pr1261947904930325089at_nat] :
                ( if_set4744415642439091239at_nat
                @ ( ( produc1900778367047502200t_char @ X5 )
                  = W2 )
                @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ ( produc6204653154488278070t_char @ X5 ) @ Z6 ) @ A9 )
                @ A9 ) )
          @ X10
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1029_Set_Ofilter__def,axiom,
    ( filter_nat2
    = ( ^ [P5: nat > $o,A7: set_nat] :
          ( collect_nat
          @ ^ [A2: nat] :
              ( ( member_nat @ A2 @ A7 )
              & ( P5 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1030_Set_Ofilter__def,axiom,
    ( filter5640266504077782706at_nat
    = ( ^ [P5: product_prod_nat_nat > $o,A7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [A2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A2 @ A7 )
              & ( P5 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1031_relcomp__unfold,axiom,
    ( relcom6453288877151117700t_char
    = ( ^ [R: set_Pr1261947904930325089at_nat,S4: set_Pr2507339680178222706t_char] :
          ( collec2458894688891239037t_char
          @ ( produc8526122873445342073char_o
            @ ^ [X: nat,Z6: list_char] :
              ? [Y6: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R )
                & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y6 @ Z6 ) @ S4 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_1032_relcomp__unfold,axiom,
    ( relcom5812064743826168580ar_nat
    = ( ^ [R: set_Pr2507339680178222706t_char,S4: set_Pr8803067691650843250ar_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Z6: nat] :
              ? [Y6: list_char] :
                ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R )
                & ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ Y6 @ Z6 ) @ S4 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_1033_relcomp__unfold,axiom,
    ( relcomp_nat_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Z6: nat] :
              ? [Y6: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Z6 ) @ S4 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_1034_inter__Set__filter,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = ( filter5640266504077782706at_nat
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 )
          @ B3 ) ) ) ).

% inter_Set_filter
thf(fact_1035_inter__Set__filter,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = ( filter_nat2
          @ ^ [X: nat] : ( member_nat @ X @ A3 )
          @ B3 ) ) ) ).

% inter_Set_filter
thf(fact_1036_insert__relcomp__fold,axiom,
    ! [S: set_Pr4480099295215233283t_char,X5: produc1457211279475724562t_char,R3: set_Pr2507339680178222706t_char] :
      ( ( finite3801884896073088588t_char @ S )
     => ( ( relcom5956200836249630357t_char @ ( insert3372640069787401954t_char @ X5 @ R3 ) @ S )
        = ( finite859346576728541399t_char
          @ ( produc6181602235716527061t_char
            @ ^ [W2: list_char,Z6: list_char,A9: set_Pr2507339680178222706t_char] :
                ( if_set1248378584462554936t_char
                @ ( ( produc1900778367047502200t_char @ X5 )
                  = W2 )
                @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ X5 ) @ Z6 ) @ A9 )
                @ A9 ) )
          @ ( relcom5956200836249630357t_char @ R3 @ S )
          @ S ) ) ) ).

% insert_relcomp_fold
thf(fact_1037_insert__relcomp__fold,axiom,
    ! [S: set_Pr8803067691650843250ar_nat,X5: produc1457211279475724562t_char,R3: set_Pr2507339680178222706t_char] :
      ( ( finite187957507792297275ar_nat @ S )
     => ( ( relcom5812064743826168580ar_nat @ ( insert3372640069787401954t_char @ X5 @ R3 ) @ S )
        = ( finite6062518653932963223at_nat
          @ ( produc179999148950717060at_nat
            @ ^ [W2: list_char,Z6: nat,A9: set_Pr1261947904930325089at_nat] :
                ( if_set4744415642439091239at_nat
                @ ( ( produc1900778367047502200t_char @ X5 )
                  = W2 )
                @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ ( produc6204653154488278070t_char @ X5 ) @ Z6 ) @ A9 )
                @ A9 ) )
          @ ( relcom5812064743826168580ar_nat @ R3 @ S )
          @ S ) ) ) ).

% insert_relcomp_fold
thf(fact_1038_comp__fun__commute__product__fold,axiom,
    ! [B3: set_list_char] :
      ( ( finite3619747586194699384t_char @ B3 )
     => ( finite1002329793627884815t_char
        @ ^ [X: nat,Z6: set_Pr2507339680178222706t_char] :
            ( finite3391364926417624327t_char
            @ ^ [Y6: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X @ Y6 ) )
            @ Z6
            @ B3 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1039_comp__fun__commute__product__fold,axiom,
    ! [B3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( finite1174777150170982526at_nat
        @ ^ [X: nat,Z6: set_Pr1261947904930325089at_nat] :
            ( finite3745491028973389255at_nat
            @ ^ [Y6: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) )
            @ Z6
            @ B3 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1040_comp__fun__commute__filter__fold,axiom,
    ! [P: nat > $o] :
      ( finite4743618823921050025et_nat
      @ ^ [X: nat,A9: set_nat] : ( if_set_nat @ ( P @ X ) @ ( insert_nat @ X @ A9 ) @ A9 ) ) ).

% comp_fun_commute_filter_fold
thf(fact_1041_comp__fun__commute__Image__fold,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( finite8114906667978294233et_nat
      @ ( produc7950369827622487762et_nat
        @ ^ [X: product_prod_nat_nat,Y6: nat,A7: set_nat] : ( if_set_nat @ ( member8440522571783428010at_nat @ X @ S ) @ ( insert_nat @ Y6 @ A7 ) @ A7 ) ) ) ).

% comp_fun_commute_Image_fold
thf(fact_1042_comp__fun__commute__Image__fold,axiom,
    ! [S: set_nat] :
      ( finite3235644144709805302et_nat
      @ ( produc9076025845751757049et_nat
        @ ^ [X: nat,Y6: nat,A7: set_nat] : ( if_set_nat @ ( member_nat @ X @ S ) @ ( insert_nat @ Y6 @ A7 ) @ A7 ) ) ) ).

% comp_fun_commute_Image_fold
thf(fact_1043_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_1044_max__ext_Ocases,axiom,
    ! [A13: set_Pr1261947904930325089at_nat,A24: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A13 @ A24 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A13 )
         => ( ( finite6177210948735845034at_nat @ A24 )
           => ( ( A24 != bot_bo2099793752762293965at_nat )
             => ~ ! [X8: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X8 @ A13 )
                   => ? [Xa2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa2 @ A24 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X8 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1045_max__ext_Ocases,axiom,
    ! [A13: set_nat,A24: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A13 @ A24 ) @ ( max_ext_nat @ R3 ) )
     => ~ ( ( finite_finite_nat @ A13 )
         => ( ( finite_finite_nat @ A24 )
           => ( ( A24 != bot_bot_set_nat )
             => ~ ! [X8: nat] :
                    ( ( member_nat @ X8 @ A13 )
                   => ? [Xa2: nat] :
                        ( ( member_nat @ Xa2 @ A24 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X8 @ Xa2 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1046_comp__fun__commute__Pow__fold,axiom,
    ( finite7859065814662781535et_nat
    @ ^ [X: nat,A7: set_set_nat] : ( sup_sup_set_set_nat @ A7 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ A7 ) ) ) ).

% comp_fun_commute_Pow_fold
thf(fact_1047_max__ext_Omax__extI,axiom,
    ! [X10: set_Pr1261947904930325089at_nat,Y9: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X10 )
     => ( ( finite6177210948735845034at_nat @ Y9 )
       => ( ( Y9 != bot_bo2099793752762293965at_nat )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X10 )
               => ? [Xa3: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa3 @ Y9 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Xa3 ) @ R3 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X10 @ Y9 ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1048_max__ext_Omax__extI,axiom,
    ! [X10: set_nat,Y9: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X10 )
     => ( ( finite_finite_nat @ Y9 )
       => ( ( Y9 != bot_bot_set_nat )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ X10 )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ Y9 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Xa3 ) @ R3 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X10 @ Y9 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1049_max__ext_Osimps,axiom,
    ! [A13: set_Pr1261947904930325089at_nat,A24: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A13 @ A24 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
      = ( ( finite6177210948735845034at_nat @ A13 )
        & ( finite6177210948735845034at_nat @ A24 )
        & ( A24 != bot_bo2099793752762293965at_nat )
        & ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A13 )
           => ? [Y6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y6 @ A24 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y6 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1050_max__ext_Osimps,axiom,
    ! [A13: set_nat,A24: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A13 @ A24 ) @ ( max_ext_nat @ R3 ) )
      = ( ( finite_finite_nat @ A13 )
        & ( finite_finite_nat @ A24 )
        & ( A24 != bot_bot_set_nat )
        & ! [X: nat] :
            ( ( member_nat @ X @ A13 )
           => ? [Y6: nat] :
                ( ( member_nat @ Y6 @ A24 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1051_max__ext__def,axiom,
    ( max_ext_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ( max_extp_nat
              @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R4 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_1052_max__extp__eq,axiom,
    ( max_extp_nat
    = ( ^ [R: nat > nat > $o,X: set_nat,Y6: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y6 ) @ ( max_ext_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_1053_max__extp__max__ext__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 ) )
      = ( ^ [X: set_nat,Y6: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y6 ) @ ( max_ext_nat @ R3 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1054_Pow__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( pow_nat @ A3 )
        = ( finite4178521680790401110et_nat
          @ ^ [X: nat,A7: set_set_nat] : ( sup_sup_set_set_nat @ A7 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ A7 ) )
          @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
          @ A3 ) ) ) ).

% Pow_fold
thf(fact_1055_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [F6: nat > set_nat,S: set_nat] :
      ( ! [I3: nat] : ( ord_less_eq_set_nat @ ( F6 @ I3 ) @ S )
     => ( ( finite_finite_nat @ S )
       => ( ? [N4: nat] :
              ( ! [N: nat] :
                  ( ( ord_less_eq_nat @ N @ N4 )
                 => ! [M2: nat] :
                      ( ( ord_less_eq_nat @ M2 @ N4 )
                     => ( ( ord_less_nat @ M2 @ N )
                       => ( ord_less_set_nat @ ( F6 @ M2 ) @ ( F6 @ N ) ) ) ) )
              & ! [N: nat] :
                  ( ( ord_less_eq_nat @ N4 @ N )
                 => ( ( F6 @ N4 )
                    = ( F6 @ N ) ) ) )
         => ( ( F6 @ ( finite_card_nat @ S ) )
            = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ top_top_set_nat ) ) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_1056_card__Collect__less__nat,axiom,
    ! [N3: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) )
      = N3 ) ).

% card_Collect_less_nat
thf(fact_1057_UN__Pow__subset,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ord_le6893508408891458716et_nat
      @ ( comple548664676211718543et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X: nat] : ( pow_nat @ ( B3 @ X ) )
          @ A3 ) )
      @ ( pow_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ).

% UN_Pow_subset
thf(fact_1058_Pow__INT__eq,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( pow_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple1065008630642458357et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X: nat] : ( pow_nat @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% Pow_INT_eq
thf(fact_1059_card__cartesian__product__singleton,axiom,
    ! [X5: nat,A3: set_nat] :
      ( ( finite711546835091564841at_nat
        @ ( produc457027306803732586at_nat @ ( insert_nat @ X5 @ bot_bot_set_nat )
          @ ^ [Uu: nat] : A3 ) )
      = ( finite_card_nat @ A3 ) ) ).

% card_cartesian_product_singleton
thf(fact_1060_converse__unfold,axiom,
    ( conver1319806385908312430t_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collec8608152508653127293ar_nat
          @ ( produc2492928673396900345_nat_o
            @ ^ [Y6: list_char,X: nat] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R ) ) ) ) ) ).

% converse_unfold
thf(fact_1061_converse__unfold,axiom,
    ( converse_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [Y6: nat,X: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R ) ) ) ) ) ).

% converse_unfold
thf(fact_1062_conversep__converse__eq,axiom,
    ! [R2: set_Pr8803067691650843250ar_nat] :
      ( ( conver3568184245214596648ar_nat
        @ ^ [X: list_char,Y6: nat] : ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y6 ) @ R2 ) )
      = ( ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( conver678582252583363310ar_nat @ R2 ) ) ) ) ).

% conversep_converse_eq
thf(fact_1063_conversep__converse__eq,axiom,
    ! [R2: set_Pr2507339680178222706t_char] :
      ( ( conver4209408378539545768t_char
        @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R2 ) )
      = ( ^ [X: list_char,Y6: nat] : ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y6 ) @ ( conver1319806385908312430t_char @ R2 ) ) ) ) ).

% conversep_converse_eq
thf(fact_1064_conversep__converse__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( conversep_nat_nat
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R2 ) )
      = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( converse_nat_nat @ R2 ) ) ) ) ).

% conversep_converse_eq
thf(fact_1065_converse__def,axiom,
    ( conver1319806385908312430t_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collec8608152508653127293ar_nat
          @ ( produc2492928673396900345_nat_o
            @ ( conver4209408378539545768t_char
              @ ^ [X: nat,Y6: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ R ) ) ) ) ) ) ).

% converse_def
thf(fact_1066_converse__def,axiom,
    ( converse_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( conversep_nat_nat
              @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R ) ) ) ) ) ) ).

% converse_def
thf(fact_1067_Image__fold,axiom,
    ! [R3: set_Pr2539167527615954998at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( finite8785817246233100311at_nat @ R3 )
     => ( ( image_4770140012798214206at_nat @ R3 @ S )
        = ( finite8077199247998863184et_nat
          @ ( produc7950369827622487762et_nat
            @ ^ [X: product_prod_nat_nat,Y6: nat,A7: set_nat] : ( if_set_nat @ ( member8440522571783428010at_nat @ X @ S ) @ ( insert_nat @ Y6 @ A7 ) @ A7 ) )
          @ bot_bot_set_nat
          @ R3 ) ) ) ).

% Image_fold
thf(fact_1068_Image__fold,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_nat] :
      ( ( finite6177210948735845034at_nat @ R3 )
     => ( ( image_nat_nat @ R3 @ S )
        = ( finite5806358023512212031et_nat
          @ ( produc9076025845751757049et_nat
            @ ^ [X: nat,Y6: nat,A7: set_nat] : ( if_set_nat @ ( member_nat @ X @ S ) @ ( insert_nat @ Y6 @ A7 ) @ A7 ) )
          @ bot_bot_set_nat
          @ R3 ) ) ) ).

% Image_fold
thf(fact_1069_Image__UN,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( image_nat_nat @ R2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( image_nat_nat @ R2 @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% Image_UN
thf(fact_1070_Image__singleton,axiom,
    ! [R2: set_Pr7717912310451564380at_nat,A: nat] :
      ( ( image_8130187405840929308at_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( collec3392354462482085612at_nat
        @ ^ [B2: product_prod_nat_nat] : ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ A @ B2 ) @ R2 ) ) ) ).

% Image_singleton
thf(fact_1071_Image__singleton,axiom,
    ! [R2: set_Pr2507339680178222706t_char,A: nat] :
      ( ( image_nat_list_char @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( collect_list_char
        @ ^ [B2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B2 ) @ R2 ) ) ) ).

% Image_singleton
thf(fact_1072_Image__singleton,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( collect_nat
        @ ^ [B2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R2 ) ) ) ).

% Image_singleton
thf(fact_1073_Image__INT__subset,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B3: nat > set_nat,A3: set_nat] :
      ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      @ ( comple7806235888213564991et_nat
        @ ( image_nat_set_nat
          @ ^ [X: nat] : ( image_nat_nat @ R2 @ ( B3 @ X ) )
          @ A3 ) ) ) ).

% Image_INT_subset
thf(fact_1074_Image__eq__UN,axiom,
    ( image_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,B8: set_nat] :
          ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [Y6: nat] : ( image_nat_nat @ R @ ( insert_nat @ Y6 @ bot_bot_set_nat ) )
            @ B8 ) ) ) ) ).

% Image_eq_UN
thf(fact_1075_Sigma__Image,axiom,
    ! [A3: set_nat,B3: nat > set_nat,X10: set_nat] :
      ( ( image_nat_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) @ X10 )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( inf_inf_set_nat @ X10 @ A3 ) ) ) ) ).

% Sigma_Image
thf(fact_1076_quotient__def,axiom,
    ( equiv_quotient_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
          ( comple548664676211718543et_nat
          @ ( image_2194112158459175443et_nat
            @ ^ [X: nat] : ( insert_set_nat @ ( image_nat_nat @ R @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ bot_bot_set_set_nat )
            @ A7 ) ) ) ) ).

% quotient_def
thf(fact_1077_Mask_Onull__def,axiom,
    null_nat = zero_zero_nat ).

% Mask.null_def
thf(fact_1078_sum_Ofinite__Collect__op,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,X5: product_prod_nat_nat > nat,Y: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [I: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ I @ I2 )
              & ( ( X5 @ I )
               != zero_zero_nat ) ) ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [I: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ I @ I2 )
                & ( ( Y @ I )
                 != zero_zero_nat ) ) ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [I: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ I @ I2 )
                & ( ( plus_plus_nat @ ( X5 @ I ) @ ( Y @ I ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_1079_sum_Ofinite__Collect__op,axiom,
    ! [I2: set_nat,X5: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I2 )
              & ( ( X5 @ I )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I2 )
                & ( ( Y @ I )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I2 )
                & ( ( plus_plus_nat @ ( X5 @ I ) @ ( Y @ I ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_1080_quotient__diff1,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( inj_on5702713390224345969at_nat
        @ ^ [A2: product_prod_nat_nat] : ( equiv_3811336339175339080at_nat @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) @ R2 )
        @ A3 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( equiv_3811336339175339080at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ R2 )
          = ( minus_4207664762107033000at_nat @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) @ ( equiv_3811336339175339080at_nat @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) @ R2 ) ) ) ) ) ).

% quotient_diff1
thf(fact_1081_quotient__diff1,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat,A: nat] :
      ( ( inj_on8105003582846801791et_nat
        @ ^ [A2: nat] : ( equiv_quotient_nat @ ( insert_nat @ A2 @ bot_bot_set_nat ) @ R2 )
        @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ( ( equiv_quotient_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ R2 )
          = ( minus_2163939370556025621et_nat @ ( equiv_quotient_nat @ A3 @ R2 ) @ ( equiv_quotient_nat @ ( insert_nat @ A @ bot_bot_set_nat ) @ R2 ) ) ) ) ) ).

% quotient_diff1
thf(fact_1082_finite__equiv__class,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X10: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ R2
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : A3 ) )
       => ( ( member_set_nat @ X10 @ ( equiv_quotient_nat @ A3 @ R2 ) )
         => ( finite_finite_nat @ X10 ) ) ) ) ).

% finite_equiv_class
thf(fact_1083_card__quotient__disjoint,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on8105003582846801791et_nat
          @ ^ [X: nat] : ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
          @ A3 )
       => ( ( finite_card_set_nat @ ( equiv_quotient_nat @ A3 @ R2 ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_quotient_disjoint
thf(fact_1084_finite__quotient,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ R2
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : A3 ) )
       => ( finite1152437895449049373et_nat @ ( equiv_quotient_nat @ A3 @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_1085_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X )
    @ ^ [X: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_1086_trancl__finite__eq__relpow,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R3 )
     => ( ( transi6264000038957366511cl_nat @ R3 )
        = ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [N2: nat] : ( compow1820118003163928468at_nat @ N2 @ R3 )
            @ ( collect_nat
              @ ^ [N2: nat] :
                  ( ( ord_less_nat @ zero_zero_nat @ N2 )
                  & ( ord_less_eq_nat @ N2 @ ( finite711546835091564841at_nat @ R3 ) ) ) ) ) ) ) ) ).

% trancl_finite_eq_relpow
thf(fact_1087_trancl__subset__Sigma,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R2
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : A3 ) )
     => ( ord_le3146513528884898305at_nat @ ( transi6264000038957366511cl_nat @ R2 )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : A3 ) ) ) ).

% trancl_subset_Sigma
thf(fact_1088_trancl__insert2,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( transi6264000038957366511cl_nat @ R2 )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y6: nat] :
                ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                  | ( X = A ) )
                & ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ Y6 ) @ ( transi6264000038957366511cl_nat @ R2 ) )
                  | ( Y6 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_1089_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ N5 )
         => ( ord_less_eq_nat @ K @ N ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_1090_Image__INT__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat,B3: nat > set_nat] :
      ( ( single5507084439971879973at_nat @ ( converse_nat_nat @ R2 ) )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( image_nat_nat @ R2 @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
          = ( comple7806235888213564991et_nat
            @ ( image_nat_set_nat
              @ ^ [X: nat] : ( image_nat_nat @ R2 @ ( B3 @ X ) )
              @ A3 ) ) ) ) ) ).

% Image_INT_eq
thf(fact_1091_sum_Oinsert_H,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,P3: product_prod_nat_nat > nat,I5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ I2 )
              & ( ( P3 @ X )
               != zero_zero_nat ) ) ) )
     => ( ( ( member8440522571783428010at_nat @ I5 @ I2 )
         => ( ( groups3963206862555359278at_nat @ P3 @ ( insert8211810215607154385at_nat @ I5 @ I2 ) )
            = ( groups3963206862555359278at_nat @ P3 @ I2 ) ) )
        & ( ~ ( member8440522571783428010at_nat @ I5 @ I2 )
         => ( ( groups3963206862555359278at_nat @ P3 @ ( insert8211810215607154385at_nat @ I5 @ I2 ) )
            = ( plus_plus_nat @ ( P3 @ I5 ) @ ( groups3963206862555359278at_nat @ P3 @ I2 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_1092_sum_Oinsert_H,axiom,
    ! [I2: set_nat,P3: nat > nat,I5: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I2 )
              & ( ( P3 @ X )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_nat @ I5 @ I2 )
         => ( ( groups1986416967739987077at_nat @ P3 @ ( insert_nat @ I5 @ I2 ) )
            = ( groups1986416967739987077at_nat @ P3 @ I2 ) ) )
        & ( ~ ( member_nat @ I5 @ I2 )
         => ( ( groups1986416967739987077at_nat @ P3 @ ( insert_nat @ I5 @ I2 ) )
            = ( plus_plus_nat @ ( P3 @ I5 ) @ ( groups1986416967739987077at_nat @ P3 @ I2 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_1093_sum_Onon__neutral_H,axiom,
    ! [G: nat > nat,I2: set_nat] :
      ( ( groups1986416967739987077at_nat @ G
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I2 )
              & ( ( G @ X )
               != zero_zero_nat ) ) ) )
      = ( groups1986416967739987077at_nat @ G @ I2 ) ) ).

% sum.non_neutral'
thf(fact_1094_sum_Onon__neutral_H,axiom,
    ! [G: product_prod_nat_nat > nat,I2: set_Pr1261947904930325089at_nat] :
      ( ( groups3963206862555359278at_nat @ G
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ I2 )
              & ( ( G @ X )
               != zero_zero_nat ) ) ) )
      = ( groups3963206862555359278at_nat @ G @ I2 ) ) ).

% sum.non_neutral'
thf(fact_1095_sum_Odistrib__triv_H,axiom,
    ! [I2: set_nat,G: nat > nat,H3: nat > nat] :
      ( ( finite_finite_nat @ I2 )
     => ( ( groups1986416967739987077at_nat
          @ ^ [I: nat] : ( plus_plus_nat @ ( G @ I ) @ ( H3 @ I ) )
          @ I2 )
        = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I2 ) @ ( groups1986416967739987077at_nat @ H3 @ I2 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_1096_sum_Odistrib_H,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,H3: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ I2 )
              & ( ( G @ X )
               != zero_zero_nat ) ) ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ I2 )
                & ( ( H3 @ X )
                 != zero_zero_nat ) ) ) )
       => ( ( groups3963206862555359278at_nat
            @ ^ [I: product_prod_nat_nat] : ( plus_plus_nat @ ( G @ I ) @ ( H3 @ I ) )
            @ I2 )
          = ( plus_plus_nat @ ( groups3963206862555359278at_nat @ G @ I2 ) @ ( groups3963206862555359278at_nat @ H3 @ I2 ) ) ) ) ) ).

% sum.distrib'
thf(fact_1097_sum_Odistrib_H,axiom,
    ! [I2: set_nat,G: nat > nat,H3: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ I2 )
              & ( ( G @ X )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ I2 )
                & ( ( H3 @ X )
                 != zero_zero_nat ) ) ) )
       => ( ( groups1986416967739987077at_nat
            @ ^ [I: nat] : ( plus_plus_nat @ ( G @ I ) @ ( H3 @ I ) )
            @ I2 )
          = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I2 ) @ ( groups1986416967739987077at_nat @ H3 @ I2 ) ) ) ) ) ).

% sum.distrib'
thf(fact_1098_val_Osize__gen_I3_J,axiom,
    ! [X3: prat] :
      ( ( size_val @ ( rat @ X3 ) )
      = zero_zero_nat ) ).

% val.size_gen(3)
thf(fact_1099_val_Osize__gen_I2_J,axiom,
    ! [X2: nat] :
      ( ( size_val @ ( address @ X2 ) )
      = zero_zero_nat ) ).

% val.size_gen(2)
thf(fact_1100_val_Osize__gen_I1_J,axiom,
    ! [X1: $o] :
      ( ( size_val @ ( bool @ X1 ) )
      = zero_zero_nat ) ).

% val.size_gen(1)
thf(fact_1101_rtrancl__finite__eq__relpow,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R3 )
     => ( ( transi2905341329935302413cl_nat @ R3 )
        = ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [N2: nat] : ( compow1820118003163928468at_nat @ N2 @ R3 )
            @ ( collect_nat
              @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ ( finite711546835091564841at_nat @ R3 ) ) ) ) ) ) ) ).

% rtrancl_finite_eq_relpow
thf(fact_1102_antimono__funpow,axiom,
    ! [Q: set_nat > set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ Q )
     => ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: set_nat,Y6: set_nat] : ( ord_less_eq_set_nat @ Y6 @ X )
        @ ^ [I: nat] : ( compow8708494347934031032et_nat @ I @ Q @ top_top_set_nat ) ) ) ).

% antimono_funpow
thf(fact_1103_antimono__funpow,axiom,
    ! [Q: filter_nat > filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ Q )
     => ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: filter_nat,Y6: filter_nat] : ( ord_le2510731241096832064er_nat @ Y6 @ X )
        @ ^ [I: nat] : ( compow8527595912636063980er_nat @ I @ Q @ top_top_filter_nat ) ) ) ).

% antimono_funpow
thf(fact_1104_relpowp__relpow__eq,axiom,
    ! [N3: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( compow_nat_nat_o @ N3
        @ ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ R3 ) )
      = ( ^ [X: nat,Y6: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( compow1820118003163928468at_nat @ N3 @ R3 ) ) ) ) ).

% relpowp_relpow_eq
thf(fact_1105_acc__downwards__aux,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ ( transi8609417484261754244at_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ B @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc_downwards_aux
thf(fact_1106_acc__downwards__aux,axiom,
    ! [B: nat,A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( transi2905341329935302413cl_nat @ R2 ) )
     => ( ( member_nat @ A @ ( acc_nat @ R2 ) )
       => ( member_nat @ B @ ( acc_nat @ R2 ) ) ) ) ).

% acc_downwards_aux
thf(fact_1107_acc__downwards,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ ( transi8609417484261754244at_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ B @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc_downwards
thf(fact_1108_acc__downwards,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( transi2905341329935302413cl_nat @ R2 ) )
       => ( member_nat @ B @ ( acc_nat @ R2 ) ) ) ) ).

% acc_downwards
thf(fact_1109_trancl__subset__Sigma__aux,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R2 ) )
     => ( ( ord_le3000389064537975527at_nat @ R2
          @ ( produc2761391749766926216at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : A3 ) )
       => ( ( A = B )
          | ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1110_trancl__subset__Sigma__aux,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
     => ( ( ord_le3146513528884898305at_nat @ R2
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : A3 ) )
       => ( ( A = B )
          | ( member_nat @ A @ A3 ) ) ) ) ).

% trancl_subset_Sigma_aux
thf(fact_1111_rtrancl__insert,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( transi2905341329935302413cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( transi2905341329935302413cl_nat @ R2 )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y6: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ ( transi2905341329935302413cl_nat @ R2 ) )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ Y6 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ) ) ).

% rtrancl_insert
thf(fact_1112_rtrancl__is__UN__relpow,axiom,
    ( transi2905341329935302413cl_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [N2: nat] : ( compow1820118003163928468at_nat @ N2 @ R4 )
            @ top_top_set_nat ) ) ) ) ).

% rtrancl_is_UN_relpow
thf(fact_1113_rtrancl__imp__UN__relpow,axiom,
    ! [P3: product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ P3 @ ( transi2905341329935302413cl_nat @ R3 ) )
     => ( member8440522571783428010at_nat @ P3
        @ ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [N2: nat] : ( compow1820118003163928468at_nat @ N2 @ R3 )
            @ top_top_set_nat ) ) ) ) ).

% rtrancl_imp_UN_relpow
thf(fact_1114_funpow__inj__finite,axiom,
    ! [P3: product_prod_nat_nat > product_prod_nat_nat,X5: product_prod_nat_nat] :
      ( ( inj_on8969904277767023793at_nat @ P3 @ top_to4669805908274784177at_nat )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Y6: product_prod_nat_nat] :
              ? [N2: nat] :
                ( Y6
                = ( compow6694791279094763314at_nat @ N2 @ P3 @ X5 ) ) ) )
       => ~ ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( compow6694791279094763314at_nat @ N @ P3 @ X5 )
               != X5 ) ) ) ) ).

% funpow_inj_finite
thf(fact_1115_funpow__inj__finite,axiom,
    ! [P3: nat > nat,X5: nat] :
      ( ( inj_on_nat_nat @ P3 @ top_top_set_nat )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Y6: nat] :
              ? [N2: nat] :
                ( Y6
                = ( compow_nat_nat @ N2 @ P3 @ X5 ) ) ) )
       => ~ ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( compow_nat_nat @ N @ P3 @ X5 )
               != X5 ) ) ) ) ).

% funpow_inj_finite
thf(fact_1116_trancl__insert,axiom,
    ! [Y: nat,X5: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X5 ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( transi6264000038957366511cl_nat @ R2 )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ B2 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ) ) ).

% trancl_insert
thf(fact_1117_mono__funpow,axiom,
    ! [Q: set_nat > set_nat] :
      ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ Q )
     => ( monoto6489329683466618047et_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_set_nat
        @ ^ [I: nat] : ( compow8708494347934031032et_nat @ I @ Q @ bot_bot_set_nat ) ) ) ).

% mono_funpow
thf(fact_1118_mono__funpow,axiom,
    ! [Q: nat > nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ Q )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat
        @ ^ [I: nat] : ( compow_nat_nat @ I @ Q @ bot_bot_nat ) ) ) ).

% mono_funpow
thf(fact_1119_mono__funpow,axiom,
    ! [Q: filter_nat > filter_nat] :
      ( ( monoto767375909635968937er_nat @ top_to4409321577746088390er_nat @ ord_le2510731241096832064er_nat @ ord_le2510731241096832064er_nat @ Q )
     => ( monoto910217884302365593er_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le2510731241096832064er_nat
        @ ^ [I: nat] : ( compow8527595912636063980er_nat @ I @ Q @ bot_bot_filter_nat ) ) ) ).

% mono_funpow
thf(fact_1120_card__UN__disjoint,axiom,
    ! [I2: set_nat,A3: nat > set_nat] :
      ( ( finite_finite_nat @ I2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ I2 )
           => ( finite_finite_nat @ ( A3 @ X4 ) ) )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ I2 )
             => ! [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ I2 )
                 => ( ( X4 != Xa2 )
                   => ( ( inf_inf_set_nat @ ( A3 @ X4 ) @ ( A3 @ Xa2 ) )
                      = bot_bot_set_nat ) ) ) )
         => ( ( finite_card_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
            = ( groups3542108847815614940at_nat
              @ ^ [I: nat] : ( finite_card_nat @ ( A3 @ I ) )
              @ I2 ) ) ) ) ) ).

% card_UN_disjoint
thf(fact_1121_sum_Oneutral__const,axiom,
    ! [A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu: nat] : zero_zero_nat
        @ A3 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_1122_sum_Odelta_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( ( member8440522571783428010at_nat @ A @ S )
         => ( ( groups977919841031483927at_nat
              @ ^ [K2: product_prod_nat_nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member8440522571783428010at_nat @ A @ S )
         => ( ( groups977919841031483927at_nat
              @ ^ [K2: product_prod_nat_nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_1123_sum_Odelta_H,axiom,
    ! [S: set_nat,A: nat,B: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups3542108847815614940at_nat
              @ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups3542108847815614940at_nat
              @ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta'
thf(fact_1124_sum_Odelta,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( ( member8440522571783428010at_nat @ A @ S )
         => ( ( groups977919841031483927at_nat
              @ ^ [K2: product_prod_nat_nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member8440522571783428010at_nat @ A @ S )
         => ( ( groups977919841031483927at_nat
              @ ^ [K2: product_prod_nat_nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_1125_sum_Odelta,axiom,
    ! [S: set_nat,A: nat,B: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups3542108847815614940at_nat
              @ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups3542108847815614940at_nat
              @ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
              @ S )
            = zero_zero_nat ) ) ) ) ).

% sum.delta
thf(fact_1126_card__SigmaI,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( finite_finite_nat @ ( B3 @ X4 ) ) )
       => ( ( finite711546835091564841at_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [A2: nat] : ( finite_card_nat @ ( B3 @ A2 ) )
            @ A3 ) ) ) ) ).

% card_SigmaI
thf(fact_1127_sum__subtractf__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F6: product_prod_nat_nat > nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F6 @ X4 ) ) )
     => ( ( groups977919841031483927at_nat
          @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F6 @ X ) @ ( G @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F6 @ A3 ) @ ( groups977919841031483927at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_1128_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: nat > nat,F6: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F6 @ X4 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( F6 @ X ) @ ( G @ X ) )
          @ A3 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F6 @ A3 ) @ ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_1129_sum_Oimage__eq,axiom,
    ! [G: nat > nat,A3: set_nat] :
      ( ( inj_on_nat_nat @ G @ A3 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( image_nat_nat2 @ G @ A3 ) )
        = ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ).

% sum.image_eq
thf(fact_1130_sum_OSigma,axiom,
    ! [A3: set_nat,B3: nat > set_nat,G: nat > nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( finite_finite_nat @ ( B3 @ X4 ) ) )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X: nat] : ( groups3542108847815614940at_nat @ ( G @ X ) @ ( B3 @ X ) )
            @ A3 )
          = ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G ) @ ( produc457027306803732586at_nat @ A3 @ B3 ) ) ) ) ) ).

% sum.Sigma
thf(fact_1131_sum_Ocartesian__product,axiom,
    ! [G: nat > nat > nat,B3: set_nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( groups3542108847815614940at_nat @ ( G @ X ) @ B3 )
        @ A3 )
      = ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) ) ) ).

% sum.cartesian_product
thf(fact_1132_sum__mono,axiom,
    ! [K3: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
      ( ! [I3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I3 @ K3 )
         => ( ord_less_eq_nat @ ( F6 @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups977919841031483927at_nat @ F6 @ K3 ) @ ( groups977919841031483927at_nat @ G @ K3 ) ) ) ).

% sum_mono
thf(fact_1133_sum__mono,axiom,
    ! [K3: set_nat,F6: nat > nat,G: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K3 )
         => ( ord_less_eq_nat @ ( F6 @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F6 @ K3 ) @ ( groups3542108847815614940at_nat @ G @ K3 ) ) ) ).

% sum_mono
thf(fact_1134_sum_Oreindex__bij__betw,axiom,
    ! [H3: nat > nat,S: set_nat,T: set_nat,G: nat > nat] :
      ( ( bij_betw_nat_nat @ H3 @ S @ T )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( G @ ( H3 @ X ) )
          @ S )
        = ( groups3542108847815614940at_nat @ G @ T ) ) ) ).

% sum.reindex_bij_betw
thf(fact_1135_sum_Odistrib,axiom,
    ! [G: nat > nat,H3: nat > nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H3 @ X ) )
        @ A3 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A3 ) @ ( groups3542108847815614940at_nat @ H3 @ A3 ) ) ) ).

% sum.distrib
thf(fact_1136_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B3: set_nat,A3: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B3 )
        @ A3 )
      = ( groups3542108847815614940at_nat
        @ ^ [J: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( G @ I @ J )
            @ A3 )
        @ B3 ) ) ).

% sum.swap
thf(fact_1137_sum_Oswap__restrict,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat,G: product_prod_nat_nat > nat > nat,R3: product_prod_nat_nat > nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups977919841031483927at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y6: nat] :
                      ( ( member_nat @ Y6 @ B3 )
                      & ( R3 @ X @ Y6 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y6: nat] :
                ( groups977919841031483927at_nat
                @ ^ [X: product_prod_nat_nat] : ( G @ X @ Y6 )
                @ ( collec3392354462482085612at_nat
                  @ ^ [X: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ X @ A3 )
                      & ( R3 @ X @ Y6 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_1138_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_Pr1261947904930325089at_nat,G: nat > product_prod_nat_nat > nat,R3: nat > product_prod_nat_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X: nat] :
                ( groups977919841031483927at_nat @ ( G @ X )
                @ ( collec3392354462482085612at_nat
                  @ ^ [Y6: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ Y6 @ B3 )
                      & ( R3 @ X @ Y6 ) ) ) )
            @ A3 )
          = ( groups977919841031483927at_nat
            @ ^ [Y6: product_prod_nat_nat] :
                ( groups3542108847815614940at_nat
                @ ^ [X: nat] : ( G @ X @ Y6 )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A3 )
                      & ( R3 @ X @ Y6 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_1139_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B3: set_nat,G: nat > nat > nat,R3: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X: nat] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y6: nat] :
                      ( ( member_nat @ Y6 @ B3 )
                      & ( R3 @ X @ Y6 ) ) ) )
            @ A3 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y6: nat] :
                ( groups3542108847815614940at_nat
                @ ^ [X: nat] : ( G @ X @ Y6 )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A3 )
                      & ( R3 @ X @ Y6 ) ) ) )
            @ B3 ) ) ) ) ).

% sum.swap_restrict
thf(fact_1140_filterlim__compose,axiom,
    ! [G: set_nat > set_nat,F33: filter_set_nat,F23: filter_set_nat,F6: nat > set_nat,F13: filter_nat] :
      ( ( filter3356702471237901762et_nat @ G @ F33 @ F23 )
     => ( ( filter3212408913953519116et_nat @ F6 @ F23 @ F13 )
       => ( filter3212408913953519116et_nat
          @ ^ [X: nat] : ( G @ ( F6 @ X ) )
          @ F33
          @ F13 ) ) ) ).

% filterlim_compose
thf(fact_1141_filterlim__compose,axiom,
    ! [G: set_nat > nat,F33: filter_nat,F23: filter_set_nat,F6: nat > set_nat,F13: filter_nat] :
      ( ( filter8870146046152925708at_nat @ G @ F33 @ F23 )
     => ( ( filter3212408913953519116et_nat @ F6 @ F23 @ F13 )
       => ( filterlim_nat_nat
          @ ^ [X: nat] : ( G @ ( F6 @ X ) )
          @ F33
          @ F13 ) ) ) ).

% filterlim_compose
thf(fact_1142_filterlim__compose,axiom,
    ! [G: nat > set_nat,F33: filter_set_nat,F23: filter_nat,F6: nat > nat,F13: filter_nat] :
      ( ( filter3212408913953519116et_nat @ G @ F33 @ F23 )
     => ( ( filterlim_nat_nat @ F6 @ F23 @ F13 )
       => ( filter3212408913953519116et_nat
          @ ^ [X: nat] : ( G @ ( F6 @ X ) )
          @ F33
          @ F13 ) ) ) ).

% filterlim_compose
thf(fact_1143_filterlim__compose,axiom,
    ! [G: nat > nat,F33: filter_nat,F23: filter_nat,F6: nat > nat,F13: filter_nat] :
      ( ( filterlim_nat_nat @ G @ F33 @ F23 )
     => ( ( filterlim_nat_nat @ F6 @ F23 @ F13 )
       => ( filterlim_nat_nat
          @ ^ [X: nat] : ( G @ ( F6 @ X ) )
          @ F33
          @ F13 ) ) ) ).

% filterlim_compose
thf(fact_1144_filterlim__ident,axiom,
    ! [F7: filter_nat] :
      ( filterlim_nat_nat
      @ ^ [X: nat] : X
      @ F7
      @ F7 ) ).

% filterlim_ident
thf(fact_1145_filterlim__inf,axiom,
    ! [F6: nat > set_nat,F23: filter_set_nat,F33: filter_set_nat,F13: filter_nat] :
      ( ( filter3212408913953519116et_nat @ F6 @ ( inf_in335088806127669288et_nat @ F23 @ F33 ) @ F13 )
      = ( ( filter3212408913953519116et_nat @ F6 @ F23 @ F13 )
        & ( filter3212408913953519116et_nat @ F6 @ F33 @ F13 ) ) ) ).

% filterlim_inf
thf(fact_1146_filterlim__inf,axiom,
    ! [F6: nat > nat,F23: filter_nat,F33: filter_nat,F13: filter_nat] :
      ( ( filterlim_nat_nat @ F6 @ ( inf_inf_filter_nat @ F23 @ F33 ) @ F13 )
      = ( ( filterlim_nat_nat @ F6 @ F23 @ F13 )
        & ( filterlim_nat_nat @ F6 @ F33 @ F13 ) ) ) ).

% filterlim_inf
thf(fact_1147_sum_Oimage__gen,axiom,
    ! [S: set_Pr1261947904930325089at_nat,H3: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( groups977919841031483927at_nat @ H3 @ S )
        = ( groups3542108847815614940at_nat
          @ ^ [Y6: nat] :
              ( groups977919841031483927at_nat @ H3
              @ ( collec3392354462482085612at_nat
                @ ^ [X: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X @ S )
                    & ( ( G @ X )
                      = Y6 ) ) ) )
          @ ( image_2486076414777270412at_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_1148_sum_Oimage__gen,axiom,
    ! [S: set_nat,H3: nat > nat,G: nat > set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( groups3542108847815614940at_nat @ H3 @ S )
        = ( groups8294997508430121362at_nat
          @ ^ [Y6: set_nat] :
              ( groups3542108847815614940at_nat @ H3
              @ ( collect_nat
                @ ^ [X: nat] :
                    ( ( member_nat @ X @ S )
                    & ( ( G @ X )
                      = Y6 ) ) ) )
          @ ( image_nat_set_nat @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_1149_sum_Oimage__gen,axiom,
    ! [S: set_nat,H3: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( groups3542108847815614940at_nat @ H3 @ S )
        = ( groups3542108847815614940at_nat
          @ ^ [Y6: nat] :
              ( groups3542108847815614940at_nat @ H3
              @ ( collect_nat
                @ ^ [X: nat] :
                    ( ( member_nat @ X @ S )
                    & ( ( G @ X )
                      = Y6 ) ) ) )
          @ ( image_nat_nat2 @ G @ S ) ) ) ) ).

% sum.image_gen
thf(fact_1150_sum__multicount__gen,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > product_prod_nat_nat > $o,K: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( finite6177210948735845034at_nat @ T2 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ T2 )
             => ( ( finite711546835091564841at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [I: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ I @ S2 )
                        & ( R3 @ I @ X4 ) ) ) )
                = ( K @ X4 ) ) )
         => ( ( groups977919841031483927at_nat
              @ ^ [I: product_prod_nat_nat] :
                  ( finite711546835091564841at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [J: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ J @ T2 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups977919841031483927at_nat @ K @ T2 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1151_sum__multicount__gen,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,T2: set_nat,R3: product_prod_nat_nat > nat > $o,K: nat > nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ T2 )
             => ( ( finite711546835091564841at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [I: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ I @ S2 )
                        & ( R3 @ I @ X4 ) ) ) )
                = ( K @ X4 ) ) )
         => ( ( groups977919841031483927at_nat
              @ ^ [I: product_prod_nat_nat] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T2 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K @ T2 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1152_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_Pr1261947904930325089at_nat,R3: nat > product_prod_nat_nat > $o,K: product_prod_nat_nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite6177210948735845034at_nat @ T2 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ T2 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S2 )
                        & ( R3 @ I @ X4 ) ) ) )
                = ( K @ X4 ) ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite711546835091564841at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [J: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ J @ T2 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups977919841031483927at_nat @ K @ T2 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1153_sum__multicount__gen,axiom,
    ! [S2: set_nat,T2: set_nat,R3: nat > nat > $o,K: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ T2 )
             => ( ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [I: nat] :
                        ( ( member_nat @ I @ S2 )
                        & ( R3 @ I @ X4 ) ) ) )
                = ( K @ X4 ) ) )
         => ( ( groups3542108847815614940at_nat
              @ ^ [I: nat] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T2 )
                        & ( R3 @ I @ J ) ) ) )
              @ S2 )
            = ( groups3542108847815614940at_nat @ K @ T2 ) ) ) ) ) ).

% sum_multicount_gen
thf(fact_1154_sum_Ointer__filter,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( groups977919841031483927at_nat @ G
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups977919841031483927at_nat
          @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_1155_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3542108847815614940at_nat @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_filter
thf(fact_1156_filterlim__INF_H,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: nat > set_nat,F7: filter_set_nat,G4: product_prod_nat_nat > filter_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( ( filter3212408913953519116et_nat @ F6 @ F7 @ ( G4 @ X5 ) )
       => ( filter3212408913953519116et_nat @ F6 @ F7 @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ G4 @ A3 ) ) ) ) ) ).

% filterlim_INF'
thf(fact_1157_filterlim__INF_H,axiom,
    ! [X5: nat,A3: set_nat,F6: nat > set_nat,F7: filter_set_nat,G4: nat > filter_nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( filter3212408913953519116et_nat @ F6 @ F7 @ ( G4 @ X5 ) )
       => ( filter3212408913953519116et_nat @ F6 @ F7 @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ G4 @ A3 ) ) ) ) ) ).

% filterlim_INF'
thf(fact_1158_filterlim__INF_H,axiom,
    ! [X5: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F6: nat > nat,F7: filter_nat,G4: product_prod_nat_nat > filter_nat] :
      ( ( member8440522571783428010at_nat @ X5 @ A3 )
     => ( ( filterlim_nat_nat @ F6 @ F7 @ ( G4 @ X5 ) )
       => ( filterlim_nat_nat @ F6 @ F7 @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ G4 @ A3 ) ) ) ) ) ).

% filterlim_INF'
thf(fact_1159_filterlim__INF_H,axiom,
    ! [X5: nat,A3: set_nat,F6: nat > nat,F7: filter_nat,G4: nat > filter_nat] :
      ( ( member_nat @ X5 @ A3 )
     => ( ( filterlim_nat_nat @ F6 @ F7 @ ( G4 @ X5 ) )
       => ( filterlim_nat_nat @ F6 @ F7 @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ G4 @ A3 ) ) ) ) ) ).

% filterlim_INF'
thf(fact_1160_filterlim__If,axiom,
    ! [F6: nat > set_nat,G4: filter_set_nat,F7: filter_nat,P: nat > $o,G: nat > set_nat] :
      ( ( filter3212408913953519116et_nat @ F6 @ G4 @ ( inf_inf_filter_nat @ F7 @ ( principal_nat @ ( collect_nat @ P ) ) ) )
     => ( ( filter3212408913953519116et_nat @ G @ G4
          @ ( inf_inf_filter_nat @ F7
            @ ( principal_nat
              @ ( collect_nat
                @ ^ [X: nat] :
                    ~ ( P @ X ) ) ) ) )
       => ( filter3212408913953519116et_nat
          @ ^ [X: nat] : ( if_set_nat @ ( P @ X ) @ ( F6 @ X ) @ ( G @ X ) )
          @ G4
          @ F7 ) ) ) ).

% filterlim_If
thf(fact_1161_filterlim__If,axiom,
    ! [F6: nat > nat,G4: filter_nat,F7: filter_nat,P: nat > $o,G: nat > nat] :
      ( ( filterlim_nat_nat @ F6 @ G4 @ ( inf_inf_filter_nat @ F7 @ ( principal_nat @ ( collect_nat @ P ) ) ) )
     => ( ( filterlim_nat_nat @ G @ G4
          @ ( inf_inf_filter_nat @ F7
            @ ( principal_nat
              @ ( collect_nat
                @ ^ [X: nat] :
                    ~ ( P @ X ) ) ) ) )
       => ( filterlim_nat_nat
          @ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F6 @ X ) @ ( G @ X ) )
          @ G4
          @ F7 ) ) ) ).

% filterlim_If
thf(fact_1162_sum__nonneg__leq__bound,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > nat,B3: nat,I5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ! [I3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F6 @ I3 ) ) )
       => ( ( ( groups977919841031483927at_nat @ F6 @ S2 )
            = B3 )
         => ( ( member8440522571783428010at_nat @ I5 @ S2 )
           => ( ord_less_eq_nat @ ( F6 @ I5 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_1163_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F6: nat > nat,B3: nat,I5: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F6 @ I3 ) ) )
       => ( ( ( groups3542108847815614940at_nat @ F6 @ S2 )
            = B3 )
         => ( ( member_nat @ I5 @ S2 )
           => ( ord_less_eq_nat @ ( F6 @ I5 ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_1164_sum__nonneg__0,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > nat,I5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ! [I3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F6 @ I3 ) ) )
       => ( ( ( groups977919841031483927at_nat @ F6 @ S2 )
            = zero_zero_nat )
         => ( ( member8440522571783428010at_nat @ I5 @ S2 )
           => ( ( F6 @ I5 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_1165_sum__nonneg__0,axiom,
    ! [S2: set_nat,F6: nat > nat,I5: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F6 @ I3 ) ) )
       => ( ( ( groups3542108847815614940at_nat @ F6 @ S2 )
            = zero_zero_nat )
         => ( ( member_nat @ I5 @ S2 )
           => ( ( F6 @ I5 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_1166_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_set_nat,G: nat > set_nat,H3: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite1152437895449049373et_nat @ T )
       => ( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ G @ S ) @ T )
         => ( ( groups8294997508430121362at_nat
              @ ^ [Y6: set_nat] :
                  ( groups3542108847815614940at_nat @ H3
                  @ ( collect_nat
                    @ ^ [X: nat] :
                        ( ( member_nat @ X @ S )
                        & ( ( G @ X )
                          = Y6 ) ) ) )
              @ T )
            = ( groups3542108847815614940at_nat @ H3 @ S ) ) ) ) ) ).

% sum.group
thf(fact_1167_sum_Ogroup,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_nat,G: product_prod_nat_nat > nat,H3: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ G @ S ) @ T )
         => ( ( groups3542108847815614940at_nat
              @ ^ [Y6: nat] :
                  ( groups977919841031483927at_nat @ H3
                  @ ( collec3392354462482085612at_nat
                    @ ^ [X: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X @ S )
                        & ( ( G @ X )
                          = Y6 ) ) ) )
              @ T )
            = ( groups977919841031483927at_nat @ H3 @ S ) ) ) ) ) ).

% sum.group
thf(fact_1168_sum_Ogroup,axiom,
    ! [S: set_nat,T: set_nat,G: nat > nat,H3: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ G @ S ) @ T )
         => ( ( groups3542108847815614940at_nat
              @ ^ [Y6: nat] :
                  ( groups3542108847815614940at_nat @ H3
                  @ ( collect_nat
                    @ ^ [X: nat] :
                        ( ( member_nat @ X @ S )
                        & ( ( G @ X )
                          = Y6 ) ) ) )
              @ T )
            = ( groups3542108847815614940at_nat @ H3 @ S ) ) ) ) ) ).

% sum.group
thf(fact_1169_sum_Ointer__restrict,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( groups977919841031483927at_nat @ G @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
        = ( groups977919841031483927at_nat
          @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( member8440522571783428010at_nat @ X @ B3 ) @ ( G @ X ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_1170_sum_Ointer__restrict,axiom,
    ! [A3: set_nat,G: nat > nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A3 @ B3 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( if_nat @ ( member_nat @ X @ B3 ) @ ( G @ X ) @ zero_zero_nat )
          @ A3 ) ) ) ).

% sum.inter_restrict
thf(fact_1171_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( groups977919841031483927at_nat @ G
          @ ( minus_1356011639430497352at_nat @ A3
            @ ( collec3392354462482085612at_nat
              @ ^ [X: product_prod_nat_nat] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups977919841031483927at_nat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_1172_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3542108847815614940at_nat @ G
          @ ( minus_minus_set_nat @ A3
            @ ( collect_nat
              @ ^ [X: nat] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups3542108847815614940at_nat @ G @ A3 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_1173_sum_OG__def,axiom,
    ( groups3963206862555359278at_nat
    = ( ^ [P2: product_prod_nat_nat > nat,I6: set_Pr1261947904930325089at_nat] :
          ( if_nat
          @ ( finite6177210948735845034at_nat
            @ ( collec3392354462482085612at_nat
              @ ^ [X: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X @ I6 )
                  & ( ( P2 @ X )
                   != zero_zero_nat ) ) ) )
          @ ( groups977919841031483927at_nat @ P2
            @ ( collec3392354462482085612at_nat
              @ ^ [X: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X @ I6 )
                  & ( ( P2 @ X )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_1174_sum_OG__def,axiom,
    ( groups1986416967739987077at_nat
    = ( ^ [P2: nat > nat,I6: set_nat] :
          ( if_nat
          @ ( finite_finite_nat
            @ ( collect_nat
              @ ^ [X: nat] :
                  ( ( member_nat @ X @ I6 )
                  & ( ( P2 @ X )
                   != zero_zero_nat ) ) ) )
          @ ( groups3542108847815614940at_nat @ P2
            @ ( collect_nat
              @ ^ [X: nat] :
                  ( ( member_nat @ X @ I6 )
                  & ( ( P2 @ X )
                   != zero_zero_nat ) ) ) )
          @ zero_zero_nat ) ) ) ).

% sum.G_def
thf(fact_1175_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S5: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat,H3: product_prod_nat_nat > product_prod_nat_nat,S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S5 )
     => ( ( finite6177210948735845034at_nat @ T3 )
       => ( ( bij_be1923491781212474806at_nat @ H3 @ ( minus_1356011639430497352at_nat @ S @ S5 ) @ ( minus_1356011639430497352at_nat @ T @ T3 ) )
         => ( ! [A5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ A5 @ S5 )
               => ( ( G @ ( H3 @ A5 ) )
                  = zero_zero_nat ) )
           => ( ! [B5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ B5 @ T3 )
                 => ( ( G @ B5 )
                    = zero_zero_nat ) )
             => ( ( groups977919841031483927at_nat
                  @ ^ [X: product_prod_nat_nat] : ( G @ ( H3 @ X ) )
                  @ S )
                = ( groups977919841031483927at_nat @ G @ T ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_1176_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S5: set_Pr1261947904930325089at_nat,T3: set_nat,H3: product_prod_nat_nat > nat,S: set_Pr1261947904930325089at_nat,T: set_nat,G: nat > nat] :
      ( ( finite6177210948735845034at_nat @ S5 )
     => ( ( finite_finite_nat @ T3 )
       => ( ( bij_be5333170631980326235at_nat @ H3 @ ( minus_1356011639430497352at_nat @ S @ S5 ) @ ( minus_minus_set_nat @ T @ T3 ) )
         => ( ! [A5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ A5 @ S5 )
               => ( ( G @ ( H3 @ A5 ) )
                  = zero_zero_nat ) )
           => ( ! [B5: nat] :
                  ( ( member_nat @ B5 @ T3 )
                 => ( ( G @ B5 )
                    = zero_zero_nat ) )
             => ( ( groups977919841031483927at_nat
                  @ ^ [X: product_prod_nat_nat] : ( G @ ( H3 @ X ) )
                  @ S )
                = ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_1177_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S5: set_nat,T3: set_Pr1261947904930325089at_nat,H3: nat > product_prod_nat_nat,S: set_nat,T: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat] :
      ( ( finite_finite_nat @ S5 )
     => ( ( finite6177210948735845034at_nat @ T3 )
       => ( ( bij_be8693218025023041337at_nat @ H3 @ ( minus_minus_set_nat @ S @ S5 ) @ ( minus_1356011639430497352at_nat @ T @ T3 ) )
         => ( ! [A5: nat] :
                ( ( member_nat @ A5 @ S5 )
               => ( ( G @ ( H3 @ A5 ) )
                  = zero_zero_nat ) )
           => ( ! [B5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ B5 @ T3 )
                 => ( ( G @ B5 )
                    = zero_zero_nat ) )
             => ( ( groups3542108847815614940at_nat
                  @ ^ [X: nat] : ( G @ ( H3 @ X ) )
                  @ S )
                = ( groups977919841031483927at_nat @ G @ T ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_1178_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S5: set_nat,T3: set_nat,H3: nat > nat,S: set_nat,T: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ S5 )
     => ( ( finite_finite_nat @ T3 )
       => ( ( bij_betw_nat_nat @ H3 @ ( minus_minus_set_nat @ S @ S5 ) @ ( minus_minus_set_nat @ T @ T3 ) )
         => ( ! [A5: nat] :
                ( ( member_nat @ A5 @ S5 )
               => ( ( G @ ( H3 @ A5 ) )
                  = zero_zero_nat ) )
           => ( ! [B5: nat] :
                  ( ( member_nat @ B5 @ T3 )
                 => ( ( G @ B5 )
                    = zero_zero_nat ) )
             => ( ( groups3542108847815614940at_nat
                  @ ^ [X: nat] : ( G @ ( H3 @ X ) )
                  @ S )
                = ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_1179_sum_OIf__cases,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,H3: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( groups977919841031483927at_nat
          @ ^ [X: product_prod_nat_nat] : ( if_nat @ ( P @ X ) @ ( H3 @ X ) @ ( G @ X ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups977919841031483927at_nat @ H3 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) ) @ ( groups977919841031483927at_nat @ G @ ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_1180_sum_OIf__cases,axiom,
    ! [A3: set_nat,P: nat > $o,H3: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( H3 @ X ) @ ( G @ X ) )
          @ A3 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ H3 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_1181_sum_Odelta__remove,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat > nat,C: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( ( member8440522571783428010at_nat @ A @ S )
         => ( ( groups977919841031483927at_nat
              @ ^ [K2: product_prod_nat_nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups977919841031483927at_nat @ C @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
        & ( ~ ( member8440522571783428010at_nat @ A @ S )
         => ( ( groups977919841031483927at_nat
              @ ^ [K2: product_prod_nat_nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S )
            = ( groups977919841031483927at_nat @ C @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_1182_sum_Odelta__remove,axiom,
    ! [S: set_nat,A: nat,B: nat > nat,C: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( member_nat @ A @ S )
         => ( ( groups3542108847815614940at_nat
              @ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups3542108847815614940at_nat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S )
         => ( ( groups3542108847815614940at_nat
              @ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S )
            = ( groups3542108847815614940at_nat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_1183_sum_OUNION__disjoint,axiom,
    ! [I2: set_nat,A3: nat > set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ I2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ I2 )
           => ( finite_finite_nat @ ( A3 @ X4 ) ) )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ I2 )
             => ! [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ I2 )
                 => ( ( X4 != Xa2 )
                   => ( ( inf_inf_set_nat @ ( A3 @ X4 ) @ ( A3 @ Xa2 ) )
                      = bot_bot_set_nat ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
            = ( groups3542108847815614940at_nat
              @ ^ [X: nat] : ( groups3542108847815614940at_nat @ G @ ( A3 @ X ) )
              @ I2 ) ) ) ) ) ).

% sum.UNION_disjoint
thf(fact_1184_card__UN__le,axiom,
    ! [I2: set_nat,A3: nat > set_nat] :
      ( ( finite_finite_nat @ I2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ I2 ) ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( finite_card_nat @ ( A3 @ I ) )
          @ I2 ) ) ) ).

% card_UN_le
thf(fact_1185_pred__nat__trancl__eq__le,axiom,
    ! [M4: nat,N3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M4 @ N3 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M4 @ N3 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_1186_less__eq,axiom,
    ! [M4: nat,N3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M4 @ N3 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M4 @ N3 ) ) ).

% less_eq
thf(fact_1187_less__than__def,axiom,
    ( less_than
    = ( transi6264000038957366511cl_nat @ pred_nat ) ) ).

% less_than_def
thf(fact_1188_UN__equiv__class,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,F6: product_prod_nat_nat > set_nat,A: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( equiv_7243916823904213361et_nat @ R2 @ F6 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
         => ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) )
            = ( F6 @ A ) ) ) ) ) ).

% UN_equiv_class
thf(fact_1189_UN__equiv__class,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,F6: nat > set_nat,A: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( equiv_4704879099140956910et_nat @ R2 @ F6 )
       => ( ( member_nat @ A @ A3 )
         => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) )
            = ( F6 @ A ) ) ) ) ) ).

% UN_equiv_class
thf(fact_1190_equiv__Eps__in,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X10: set_Pr1261947904930325089at_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X10 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( member8440522571783428010at_nat
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ X10 ) )
          @ X10 ) ) ) ).

% equiv_Eps_in
thf(fact_1191_equiv__Eps__in,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X10: set_nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X10 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( member_nat
          @ ( fChoice_nat
            @ ^ [X: nat] : ( member_nat @ X @ X10 ) )
          @ X10 ) ) ) ).

% equiv_Eps_in
thf(fact_1192_equiv__Eps__in,axiom,
    ! [A3: set_prat,R2: set_Pr3298805672376727137t_prat,X10: set_prat] :
      ( ( equiv_equiv_prat @ A3 @ R2 )
     => ( ( member_set_prat @ X10 @ ( equiv_quotient_prat @ A3 @ R2 ) )
       => ( member_prat
          @ ( fChoice_prat
            @ ^ [X: prat] : ( member_prat @ X @ X10 ) )
          @ X10 ) ) ) ).

% equiv_Eps_in
thf(fact_1193_equiv__Eps__preserves,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X10: set_Pr1261947904930325089at_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X10 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( member8440522571783428010at_nat
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ X10 ) )
          @ A3 ) ) ) ).

% equiv_Eps_preserves
thf(fact_1194_equiv__Eps__preserves,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X10: set_nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X10 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( member_nat
          @ ( fChoice_nat
            @ ^ [X: nat] : ( member_nat @ X @ X10 ) )
          @ A3 ) ) ) ).

% equiv_Eps_preserves
thf(fact_1195_equiv__Eps__preserves,axiom,
    ! [A3: set_prat,R2: set_Pr3298805672376727137t_prat,X10: set_prat] :
      ( ( equiv_equiv_prat @ A3 @ R2 )
     => ( ( member_set_prat @ X10 @ ( equiv_quotient_prat @ A3 @ R2 ) )
       => ( member_prat
          @ ( fChoice_prat
            @ ^ [X: prat] : ( member_prat @ X @ X10 ) )
          @ A3 ) ) ) ).

% equiv_Eps_preserves
thf(fact_1196_UN__equiv__class__type,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,F6: product_prod_nat_nat > set_nat,X10: set_Pr1261947904930325089at_nat,B3: set_set_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( equiv_7243916823904213361et_nat @ R2 @ F6 )
       => ( ( member2643936169264416010at_nat @ X10 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A3 )
               => ( member_set_nat @ ( F6 @ X4 ) @ B3 ) )
           => ( member_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ X10 ) ) @ B3 ) ) ) ) ) ).

% UN_equiv_class_type
thf(fact_1197_UN__equiv__class__type,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,F6: nat > set_nat,X10: set_nat,B3: set_set_nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( equiv_4704879099140956910et_nat @ R2 @ F6 )
       => ( ( member_set_nat @ X10 @ ( equiv_quotient_nat @ A3 @ R2 ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
               => ( member_set_nat @ ( F6 @ X4 ) @ B3 ) )
           => ( member_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ X10 ) ) @ B3 ) ) ) ) ) ).

% UN_equiv_class_type
thf(fact_1198_UN__equiv__class__type2,axiom,
    ! [A14: set_Pr1261947904930325089at_nat,R12: set_Pr8693737435421807431at_nat,A25: set_Pr1261947904930325089at_nat,R23: set_Pr8693737435421807431at_nat,F6: product_prod_nat_nat > product_prod_nat_nat > set_nat,X14: set_Pr1261947904930325089at_nat,X24: set_Pr1261947904930325089at_nat,B3: set_set_nat] :
      ( ( equiv_5407799809235660215at_nat @ A14 @ R12 )
     => ( ( equiv_5407799809235660215at_nat @ A25 @ R23 )
       => ( ( equiv_4637926981076332931et_nat @ R12 @ R23 @ F6 )
         => ( ( member2643936169264416010at_nat @ X14 @ ( equiv_3811336339175339080at_nat @ A14 @ R12 ) )
           => ( ( member2643936169264416010at_nat @ X24 @ ( equiv_3811336339175339080at_nat @ A25 @ R23 ) )
             => ( ! [X12: product_prod_nat_nat,X22: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X12 @ A14 )
                   => ( ( member8440522571783428010at_nat @ X22 @ A25 )
                     => ( member_set_nat @ ( F6 @ X12 @ X22 ) @ B3 ) ) )
               => ( member_set_nat
                  @ ( comple7399068483239264473et_nat
                    @ ( image_15824709712370754et_nat
                      @ ^ [X13: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F6 @ X13 ) @ X24 ) )
                      @ X14 ) )
                  @ B3 ) ) ) ) ) ) ) ).

% UN_equiv_class_type2
thf(fact_1199_UN__equiv__class__type2,axiom,
    ! [A14: set_Pr1261947904930325089at_nat,R12: set_Pr8693737435421807431at_nat,A25: set_nat,R23: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > nat > set_nat,X14: set_Pr1261947904930325089at_nat,X24: set_nat,B3: set_set_nat] :
      ( ( equiv_5407799809235660215at_nat @ A14 @ R12 )
     => ( ( equiv_equiv_nat @ A25 @ R23 )
       => ( ( equiv_6930685768344799580et_nat @ R12 @ R23 @ F6 )
         => ( ( member2643936169264416010at_nat @ X14 @ ( equiv_3811336339175339080at_nat @ A14 @ R12 ) )
           => ( ( member_set_nat @ X24 @ ( equiv_quotient_nat @ A25 @ R23 ) )
             => ( ! [X12: product_prod_nat_nat,X22: nat] :
                    ( ( member8440522571783428010at_nat @ X12 @ A14 )
                   => ( ( member_nat @ X22 @ A25 )
                     => ( member_set_nat @ ( F6 @ X12 @ X22 ) @ B3 ) ) )
               => ( member_set_nat
                  @ ( comple7399068483239264473et_nat
                    @ ( image_15824709712370754et_nat
                      @ ^ [X13: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F6 @ X13 ) @ X24 ) )
                      @ X14 ) )
                  @ B3 ) ) ) ) ) ) ) ).

% UN_equiv_class_type2
thf(fact_1200_UN__equiv__class__type2,axiom,
    ! [A14: set_nat,R12: set_Pr1261947904930325089at_nat,A25: set_Pr1261947904930325089at_nat,R23: set_Pr8693737435421807431at_nat,F6: nat > product_prod_nat_nat > set_nat,X14: set_nat,X24: set_Pr1261947904930325089at_nat,B3: set_set_nat] :
      ( ( equiv_equiv_nat @ A14 @ R12 )
     => ( ( equiv_5407799809235660215at_nat @ A25 @ R23 )
       => ( ( equiv_2960386318186113850et_nat @ R12 @ R23 @ F6 )
         => ( ( member_set_nat @ X14 @ ( equiv_quotient_nat @ A14 @ R12 ) )
           => ( ( member2643936169264416010at_nat @ X24 @ ( equiv_3811336339175339080at_nat @ A25 @ R23 ) )
             => ( ! [X12: nat,X22: product_prod_nat_nat] :
                    ( ( member_nat @ X12 @ A14 )
                   => ( ( member8440522571783428010at_nat @ X22 @ A25 )
                     => ( member_set_nat @ ( F6 @ X12 @ X22 ) @ B3 ) ) )
               => ( member_set_nat
                  @ ( comple7399068483239264473et_nat
                    @ ( image_nat_set_nat
                      @ ^ [X13: nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F6 @ X13 ) @ X24 ) )
                      @ X14 ) )
                  @ B3 ) ) ) ) ) ) ) ).

% UN_equiv_class_type2
thf(fact_1201_UN__equiv__class__type2,axiom,
    ! [A14: set_nat,R12: set_Pr1261947904930325089at_nat,A25: set_nat,R23: set_Pr1261947904930325089at_nat,F6: nat > nat > set_nat,X14: set_nat,X24: set_nat,B3: set_set_nat] :
      ( ( equiv_equiv_nat @ A14 @ R12 )
     => ( ( equiv_equiv_nat @ A25 @ R23 )
       => ( ( equiv_4377714518549004389et_nat @ R12 @ R23 @ F6 )
         => ( ( member_set_nat @ X14 @ ( equiv_quotient_nat @ A14 @ R12 ) )
           => ( ( member_set_nat @ X24 @ ( equiv_quotient_nat @ A25 @ R23 ) )
             => ( ! [X12: nat,X22: nat] :
                    ( ( member_nat @ X12 @ A14 )
                   => ( ( member_nat @ X22 @ A25 )
                     => ( member_set_nat @ ( F6 @ X12 @ X22 ) @ B3 ) ) )
               => ( member_set_nat
                  @ ( comple7399068483239264473et_nat
                    @ ( image_nat_set_nat
                      @ ^ [X13: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F6 @ X13 ) @ X24 ) )
                      @ X14 ) )
                  @ B3 ) ) ) ) ) ) ) ).

% UN_equiv_class_type2
thf(fact_1202_UN__equiv__class__inject,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,F6: product_prod_nat_nat > set_nat,X10: set_Pr1261947904930325089at_nat,Y9: set_Pr1261947904930325089at_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( equiv_7243916823904213361et_nat @ R2 @ F6 )
       => ( ( ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ X10 ) )
            = ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F6 @ Y9 ) ) )
         => ( ( member2643936169264416010at_nat @ X10 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
           => ( ( member2643936169264416010at_nat @ Y9 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
             => ( ! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ A3 )
                   => ( ( member8440522571783428010at_nat @ Y5 @ A3 )
                     => ( ( ( F6 @ X4 )
                          = ( F6 @ Y5 ) )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ R2 ) ) ) )
               => ( X10 = Y9 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
thf(fact_1203_UN__equiv__class__inject,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,F6: nat > set_nat,X10: set_nat,Y9: set_nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( equiv_4704879099140956910et_nat @ R2 @ F6 )
       => ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ X10 ) )
            = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F6 @ Y9 ) ) )
         => ( ( member_set_nat @ X10 @ ( equiv_quotient_nat @ A3 @ R2 ) )
           => ( ( member_set_nat @ Y9 @ ( equiv_quotient_nat @ A3 @ R2 ) )
             => ( ! [X4: nat,Y5: nat] :
                    ( ( member_nat @ X4 @ A3 )
                   => ( ( member_nat @ Y5 @ A3 )
                     => ( ( ( F6 @ X4 )
                          = ( F6 @ Y5 ) )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 ) ) ) )
               => ( X10 = Y9 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
thf(fact_1204_UN__equiv__class2,axiom,
    ! [A14: set_Pr1261947904930325089at_nat,R12: set_Pr8693737435421807431at_nat,A25: set_Pr1261947904930325089at_nat,R23: set_Pr8693737435421807431at_nat,F6: product_prod_nat_nat > product_prod_nat_nat > set_nat,A13: product_prod_nat_nat,A24: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A14 @ R12 )
     => ( ( equiv_5407799809235660215at_nat @ A25 @ R23 )
       => ( ( equiv_4637926981076332931et_nat @ R12 @ R23 @ F6 )
         => ( ( member8440522571783428010at_nat @ A13 @ A14 )
           => ( ( member8440522571783428010at_nat @ A24 @ A25 )
             => ( ( comple7399068483239264473et_nat
                  @ ( image_15824709712370754et_nat
                    @ ^ [X13: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F6 @ X13 ) @ ( image_2484338115069272851at_nat @ R23 @ ( insert8211810215607154385at_nat @ A24 @ bot_bo2099793752762293965at_nat ) ) ) )
                    @ ( image_2484338115069272851at_nat @ R12 @ ( insert8211810215607154385at_nat @ A13 @ bot_bo2099793752762293965at_nat ) ) ) )
                = ( F6 @ A13 @ A24 ) ) ) ) ) ) ) ).

% UN_equiv_class2
thf(fact_1205_UN__equiv__class2,axiom,
    ! [A14: set_nat,R12: set_Pr1261947904930325089at_nat,A25: set_Pr1261947904930325089at_nat,R23: set_Pr8693737435421807431at_nat,F6: nat > product_prod_nat_nat > set_nat,A13: nat,A24: product_prod_nat_nat] :
      ( ( equiv_equiv_nat @ A14 @ R12 )
     => ( ( equiv_5407799809235660215at_nat @ A25 @ R23 )
       => ( ( equiv_2960386318186113850et_nat @ R12 @ R23 @ F6 )
         => ( ( member_nat @ A13 @ A14 )
           => ( ( member8440522571783428010at_nat @ A24 @ A25 )
             => ( ( comple7399068483239264473et_nat
                  @ ( image_nat_set_nat
                    @ ^ [X13: nat] : ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ ( F6 @ X13 ) @ ( image_2484338115069272851at_nat @ R23 @ ( insert8211810215607154385at_nat @ A24 @ bot_bo2099793752762293965at_nat ) ) ) )
                    @ ( image_nat_nat @ R12 @ ( insert_nat @ A13 @ bot_bot_set_nat ) ) ) )
                = ( F6 @ A13 @ A24 ) ) ) ) ) ) ) ).

% UN_equiv_class2
thf(fact_1206_UN__equiv__class2,axiom,
    ! [A14: set_Pr1261947904930325089at_nat,R12: set_Pr8693737435421807431at_nat,A25: set_nat,R23: set_Pr1261947904930325089at_nat,F6: product_prod_nat_nat > nat > set_nat,A13: product_prod_nat_nat,A24: nat] :
      ( ( equiv_5407799809235660215at_nat @ A14 @ R12 )
     => ( ( equiv_equiv_nat @ A25 @ R23 )
       => ( ( equiv_6930685768344799580et_nat @ R12 @ R23 @ F6 )
         => ( ( member8440522571783428010at_nat @ A13 @ A14 )
           => ( ( member_nat @ A24 @ A25 )
             => ( ( comple7399068483239264473et_nat
                  @ ( image_15824709712370754et_nat
                    @ ^ [X13: product_prod_nat_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F6 @ X13 ) @ ( image_nat_nat @ R23 @ ( insert_nat @ A24 @ bot_bot_set_nat ) ) ) )
                    @ ( image_2484338115069272851at_nat @ R12 @ ( insert8211810215607154385at_nat @ A13 @ bot_bo2099793752762293965at_nat ) ) ) )
                = ( F6 @ A13 @ A24 ) ) ) ) ) ) ) ).

% UN_equiv_class2
thf(fact_1207_UN__equiv__class2,axiom,
    ! [A14: set_nat,R12: set_Pr1261947904930325089at_nat,A25: set_nat,R23: set_Pr1261947904930325089at_nat,F6: nat > nat > set_nat,A13: nat,A24: nat] :
      ( ( equiv_equiv_nat @ A14 @ R12 )
     => ( ( equiv_equiv_nat @ A25 @ R23 )
       => ( ( equiv_4377714518549004389et_nat @ R12 @ R23 @ F6 )
         => ( ( member_nat @ A13 @ A14 )
           => ( ( member_nat @ A24 @ A25 )
             => ( ( comple7399068483239264473et_nat
                  @ ( image_nat_set_nat
                    @ ^ [X13: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F6 @ X13 ) @ ( image_nat_nat @ R23 @ ( insert_nat @ A24 @ bot_bot_set_nat ) ) ) )
                    @ ( image_nat_nat @ R12 @ ( insert_nat @ A13 @ bot_bot_set_nat ) ) ) )
                = ( F6 @ A13 @ A24 ) ) ) ) ) ) ) ).

% UN_equiv_class2
thf(fact_1208_proj__Eps,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X10: set_Pr1261947904930325089at_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X10 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( ( equiv_7265450603036132554at_nat @ R2
            @ ( fChoic6978938873391328853at_nat
              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ X10 ) ) )
          = X10 ) ) ) ).

% proj_Eps
thf(fact_1209_proj__Eps,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X10: set_nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X10 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( ( equiv_proj_nat_nat @ R2
            @ ( fChoice_nat
              @ ^ [X: nat] : ( member_nat @ X @ X10 ) ) )
          = X10 ) ) ) ).

% proj_Eps
thf(fact_1210_proj__Eps,axiom,
    ! [A3: set_prat,R2: set_Pr3298805672376727137t_prat,X10: set_prat] :
      ( ( equiv_equiv_prat @ A3 @ R2 )
     => ( ( member_set_prat @ X10 @ ( equiv_quotient_prat @ A3 @ R2 ) )
       => ( ( equiv_proj_prat_prat @ R2
            @ ( fChoice_prat
              @ ^ [X: prat] : ( member_prat @ X @ X10 ) ) )
          = X10 ) ) ) ).

% proj_Eps
thf(fact_1211_UN__finite2__subset,axiom,
    ! [A3: nat > set_nat,B3: nat > set_nat,K: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ top_top_set_nat ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ top_top_set_nat ) ) ) ) ).

% UN_finite2_subset
thf(fact_1212_image__add__atLeastLessThan_H,axiom,
    ! [K: nat,I5: nat,J4: nat] :
      ( ( image_nat_nat2
        @ ^ [N2: nat] : ( plus_plus_nat @ N2 @ K )
        @ ( set_or4665077453230672383an_nat @ I5 @ J4 ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I5 @ K ) @ ( plus_plus_nat @ J4 @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_1213_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M4: nat,K: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N3 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
        @ ( set_or4665077453230672383an_nat @ M4 @ N3 ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_1214_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X5: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat2
            @ ^ [I: nat] : ( minus_minus_nat @ I @ C )
            @ ( set_or4665077453230672383an_nat @ X5 @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X5 @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X5 @ Y )
           => ( ( image_nat_nat2
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C )
                @ ( set_or4665077453230672383an_nat @ X5 @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X5 @ Y )
           => ( ( image_nat_nat2
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C )
                @ ( set_or4665077453230672383an_nat @ X5 @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_1215_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S ) ) ).

% card_sum_le_nat_sum
thf(fact_1216_UN__UN__finite__eq,axiom,
    ! [A3: nat > set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [N2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ top_top_set_nat ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ top_top_set_nat ) ) ) ).

% UN_UN_finite_eq
thf(fact_1217_UN__finite2__eq,axiom,
    ! [A3: nat > set_nat,B3: nat > set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
     => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ top_top_set_nat ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ top_top_set_nat ) ) ) ) ).

% UN_finite2_eq
thf(fact_1218_inj__on__funpow__least,axiom,
    ! [N3: nat,F6: nat > nat,S2: nat] :
      ( ( ( compow_nat_nat @ N3 @ F6 @ S2 )
        = S2 )
     => ( ! [M2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ M2 )
           => ( ( ord_less_nat @ M2 @ N3 )
             => ( ( compow_nat_nat @ M2 @ F6 @ S2 )
               != S2 ) ) )
       => ( inj_on_nat_nat
          @ ^ [K2: nat] : ( compow_nat_nat @ K2 @ F6 @ S2 )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% inj_on_funpow_least
thf(fact_1219_UN__finite__subset,axiom,
    ! [A3: nat > set_nat,C3: set_nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ C3 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A3 @ top_top_set_nat ) ) @ C3 ) ) ).

% UN_finite_subset
thf(fact_1220_eventually__INF__base,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,F7: product_prod_nat_nat > filter_nat,P: nat > $o] :
      ( ( B3 != bot_bo2099793752762293965at_nat )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ B3 )
           => ! [B5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B5 @ B3 )
               => ? [X8: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X8 @ B3 )
                    & ( ord_le2510731241096832064er_nat @ ( F7 @ X8 ) @ ( inf_inf_filter_nat @ ( F7 @ A5 ) @ ( F7 @ B5 ) ) ) ) ) )
       => ( ( eventually_nat @ P @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F7 @ B3 ) ) )
          = ( ? [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ B3 )
                & ( eventually_nat @ P @ ( F7 @ X ) ) ) ) ) ) ) ).

% eventually_INF_base
thf(fact_1221_eventually__INF__base,axiom,
    ! [B3: set_nat,F7: nat > filter_nat,P: nat > $o] :
      ( ( B3 != bot_bot_set_nat )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ B3 )
           => ! [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
               => ? [X8: nat] :
                    ( ( member_nat @ X8 @ B3 )
                    & ( ord_le2510731241096832064er_nat @ ( F7 @ X8 ) @ ( inf_inf_filter_nat @ ( F7 @ A5 ) @ ( F7 @ B5 ) ) ) ) ) )
       => ( ( eventually_nat @ P @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F7 @ B3 ) ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ B3 )
                & ( eventually_nat @ P @ ( F7 @ X ) ) ) ) ) ) ) ).

% eventually_INF_base
thf(fact_1222_val_Osize_I6_J,axiom,
    ! [X3: prat] :
      ( ( size_size_val @ ( rat @ X3 ) )
      = zero_zero_nat ) ).

% val.size(6)
thf(fact_1223_eventually__const,axiom,
    ! [F7: filter_nat,P: $o] :
      ( ( F7 != bot_bot_filter_nat )
     => ( ( eventually_nat
          @ ^ [X: nat] : P
          @ F7 )
        = P ) ) ).

% eventually_const
thf(fact_1224_eventually__inf__principal,axiom,
    ! [P: product_prod_nat_nat > $o,F7: filter1242075044329608583at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( eventu1038000079068216329at_nat @ P @ ( inf_in7187749990401937269at_nat @ F7 @ ( princi5835078837122344214at_nat @ S2 ) ) )
      = ( eventu1038000079068216329at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ S2 )
           => ( P @ X ) )
        @ F7 ) ) ).

% eventually_inf_principal
thf(fact_1225_eventually__inf__principal,axiom,
    ! [P: nat > $o,F7: filter_nat,S2: set_nat] :
      ( ( eventually_nat @ P @ ( inf_inf_filter_nat @ F7 @ ( principal_nat @ S2 ) ) )
      = ( eventually_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ S2 )
           => ( P @ X ) )
        @ F7 ) ) ).

% eventually_inf_principal
thf(fact_1226_le__principal,axiom,
    ! [F7: filter1242075044329608583at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le5085569704386948391at_nat @ F7 @ ( princi5835078837122344214at_nat @ A3 ) )
      = ( eventu1038000079068216329at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 )
        @ F7 ) ) ).

% le_principal
thf(fact_1227_le__principal,axiom,
    ! [F7: filter_nat,A3: set_nat] :
      ( ( ord_le2510731241096832064er_nat @ F7 @ ( principal_nat @ A3 ) )
      = ( eventually_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 )
        @ F7 ) ) ).

% le_principal
thf(fact_1228_not__eventually__impI,axiom,
    ! [P: nat > $o,F7: filter_nat,Q: nat > $o] :
      ( ( eventually_nat @ P @ F7 )
     => ( ~ ( eventually_nat @ Q @ F7 )
       => ~ ( eventually_nat
            @ ^ [X: nat] :
                ( ( P @ X )
               => ( Q @ X ) )
            @ F7 ) ) ) ).

% not_eventually_impI
thf(fact_1229_eventually__conj__iff,axiom,
    ! [P: nat > $o,Q: nat > $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) )
        @ F7 )
      = ( ( eventually_nat @ P @ F7 )
        & ( eventually_nat @ Q @ F7 ) ) ) ).

% eventually_conj_iff
thf(fact_1230_eventually__rev__mp,axiom,
    ! [P: nat > $o,F7: filter_nat,Q: nat > $o] :
      ( ( eventually_nat @ P @ F7 )
     => ( ( eventually_nat
          @ ^ [X: nat] :
              ( ( P @ X )
             => ( Q @ X ) )
          @ F7 )
       => ( eventually_nat @ Q @ F7 ) ) ) ).

% eventually_rev_mp
thf(fact_1231_eventually__subst,axiom,
    ! [P: nat > $o,Q: nat > $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] :
            ( ( P @ N2 )
            = ( Q @ N2 ) )
        @ F7 )
     => ( ( eventually_nat @ P @ F7 )
        = ( eventually_nat @ Q @ F7 ) ) ) ).

% eventually_subst
thf(fact_1232_eventually__elim2,axiom,
    ! [P: nat > $o,F7: filter_nat,Q: nat > $o,R3: nat > $o] :
      ( ( eventually_nat @ P @ F7 )
     => ( ( eventually_nat @ Q @ F7 )
       => ( ! [I3: nat] :
              ( ( P @ I3 )
             => ( ( Q @ I3 )
               => ( R3 @ I3 ) ) )
         => ( eventually_nat @ R3 @ F7 ) ) ) ) ).

% eventually_elim2
thf(fact_1233_eventually__conj,axiom,
    ! [P: nat > $o,F7: filter_nat,Q: nat > $o] :
      ( ( eventually_nat @ P @ F7 )
     => ( ( eventually_nat @ Q @ F7 )
       => ( eventually_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) )
          @ F7 ) ) ) ).

% eventually_conj
thf(fact_1234_eventually__True,axiom,
    ! [F7: filter_nat] :
      ( eventually_nat
      @ ^ [X: nat] : $true
      @ F7 ) ).

% eventually_True
thf(fact_1235_eventually__mp,axiom,
    ! [P: nat > $o,Q: nat > $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) )
        @ F7 )
     => ( ( eventually_nat @ P @ F7 )
       => ( eventually_nat @ Q @ F7 ) ) ) ).

% eventually_mp
thf(fact_1236_eventually__frequently__const__simps_I3_J,axiom,
    ! [P: nat > $o,C3: $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            | C3 )
        @ F7 )
      = ( ( eventually_nat @ P @ F7 )
        | C3 ) ) ).

% eventually_frequently_const_simps(3)
thf(fact_1237_eventually__frequently__const__simps_I4_J,axiom,
    ! [C3: $o,P: nat > $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [X: nat] :
            ( C3
            | ( P @ X ) )
        @ F7 )
      = ( C3
        | ( eventually_nat @ P @ F7 ) ) ) ).

% eventually_frequently_const_simps(4)
thf(fact_1238_eventually__frequently__const__simps_I6_J,axiom,
    ! [C3: $o,P: nat > $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [X: nat] :
            ( C3
           => ( P @ X ) )
        @ F7 )
      = ( C3
       => ( eventually_nat @ P @ F7 ) ) ) ).

% eventually_frequently_const_simps(6)
thf(fact_1239_False__imp__not__eventually,axiom,
    ! [P: nat > $o,Net: filter_nat] :
      ( ! [X4: nat] :
          ~ ( P @ X4 )
     => ( ( Net != bot_bot_filter_nat )
       => ~ ( eventually_nat @ P @ Net ) ) ) ).

% False_imp_not_eventually
thf(fact_1240_eventually__const__iff,axiom,
    ! [P: $o,F7: filter_nat] :
      ( ( eventually_nat
        @ ^ [X: nat] : P
        @ F7 )
      = ( P
        | ( F7 = bot_bot_filter_nat ) ) ) ).

% eventually_const_iff
thf(fact_1241_trivial__limit__def,axiom,
    ! [F7: filter_nat] :
      ( ( F7 = bot_bot_filter_nat )
      = ( eventually_nat
        @ ^ [X: nat] : $false
        @ F7 ) ) ).

% trivial_limit_def
thf(fact_1242_eventually__ball__finite__distrib,axiom,
    ! [A3: set_nat,P: nat > nat > $o,Net: filter_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( eventually_nat
          @ ^ [X: nat] :
            ! [Y6: nat] :
              ( ( member_nat @ Y6 @ A3 )
             => ( P @ X @ Y6 ) )
          @ Net )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A3 )
             => ( eventually_nat
                @ ^ [Y6: nat] : ( P @ Y6 @ X )
                @ Net ) ) ) ) ) ).

% eventually_ball_finite_distrib
thf(fact_1243_eventually__ball__finite,axiom,
    ! [A3: set_nat,P: nat > nat > $o,Net: filter_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
           => ( eventually_nat
              @ ^ [Y6: nat] : ( P @ Y6 @ X4 )
              @ Net ) )
       => ( eventually_nat
          @ ^ [X: nat] :
            ! [Y6: nat] :
              ( ( member_nat @ Y6 @ A3 )
             => ( P @ X @ Y6 ) )
          @ Net ) ) ) ).

% eventually_ball_finite
thf(fact_1244_eventually__INF1,axiom,
    ! [I5: product_prod_nat_nat,I2: set_Pr1261947904930325089at_nat,P: nat > $o,F7: product_prod_nat_nat > filter_nat] :
      ( ( member8440522571783428010at_nat @ I5 @ I2 )
     => ( ( eventually_nat @ P @ ( F7 @ I5 ) )
       => ( eventually_nat @ P @ ( comple6612786467464916249er_nat @ ( image_6134674671615456924er_nat @ F7 @ I2 ) ) ) ) ) ).

% eventually_INF1
thf(fact_1245_eventually__INF1,axiom,
    ! [I5: nat,I2: set_nat,P: nat > $o,F7: nat > filter_nat] :
      ( ( member_nat @ I5 @ I2 )
     => ( ( eventually_nat @ P @ ( F7 @ I5 ) )
       => ( eventually_nat @ P @ ( comple6612786467464916249er_nat @ ( image_nat_filter_nat @ F7 @ I2 ) ) ) ) ) ).

% eventually_INF1
thf(fact_1246_filterlim__iff,axiom,
    ( filter3212408913953519116et_nat
    = ( ^ [F4: nat > set_nat,F24: filter_set_nat,F14: filter_nat] :
        ! [P5: set_nat > $o] :
          ( ( eventually_set_nat @ P5 @ F24 )
         => ( eventually_nat
            @ ^ [X: nat] : ( P5 @ ( F4 @ X ) )
            @ F14 ) ) ) ) ).

% filterlim_iff
thf(fact_1247_filterlim__iff,axiom,
    ( filterlim_nat_nat
    = ( ^ [F4: nat > nat,F24: filter_nat,F14: filter_nat] :
        ! [P5: nat > $o] :
          ( ( eventually_nat @ P5 @ F24 )
         => ( eventually_nat
            @ ^ [X: nat] : ( P5 @ ( F4 @ X ) )
            @ F14 ) ) ) ) ).

% filterlim_iff
thf(fact_1248_filterlim__cong,axiom,
    ! [F13: filter_set_nat,F15: filter_set_nat,F23: filter_nat,F25: filter_nat,F6: nat > set_nat,G: nat > set_nat] :
      ( ( F13 = F15 )
     => ( ( F23 = F25 )
       => ( ( eventually_nat
            @ ^ [X: nat] :
                ( ( F6 @ X )
                = ( G @ X ) )
            @ F23 )
         => ( ( filter3212408913953519116et_nat @ F6 @ F13 @ F23 )
            = ( filter3212408913953519116et_nat @ G @ F15 @ F25 ) ) ) ) ) ).

% filterlim_cong
thf(fact_1249_filterlim__cong,axiom,
    ! [F13: filter_nat,F15: filter_nat,F23: filter_nat,F25: filter_nat,F6: nat > nat,G: nat > nat] :
      ( ( F13 = F15 )
     => ( ( F23 = F25 )
       => ( ( eventually_nat
            @ ^ [X: nat] :
                ( ( F6 @ X )
                = ( G @ X ) )
            @ F23 )
         => ( ( filterlim_nat_nat @ F6 @ F13 @ F23 )
            = ( filterlim_nat_nat @ G @ F15 @ F25 ) ) ) ) ) ).

% filterlim_cong
thf(fact_1250_eventually__compose__filterlim,axiom,
    ! [P: set_nat > $o,F7: filter_set_nat,F6: nat > set_nat,G4: filter_nat] :
      ( ( eventually_set_nat @ P @ F7 )
     => ( ( filter3212408913953519116et_nat @ F6 @ F7 @ G4 )
       => ( eventually_nat
          @ ^ [X: nat] : ( P @ ( F6 @ X ) )
          @ G4 ) ) ) ).

% eventually_compose_filterlim
thf(fact_1251_eventually__compose__filterlim,axiom,
    ! [P: nat > $o,F7: filter_nat,F6: nat > nat,G4: filter_nat] :
      ( ( eventually_nat @ P @ F7 )
     => ( ( filterlim_nat_nat @ F6 @ F7 @ G4 )
       => ( eventually_nat
          @ ^ [X: nat] : ( P @ ( F6 @ X ) )
          @ G4 ) ) ) ).

% eventually_compose_filterlim
thf(fact_1252_val_Osize_I4_J,axiom,
    ! [X1: $o] :
      ( ( size_size_val @ ( bool @ X1 ) )
      = zero_zero_nat ) ).

% val.size(4)
thf(fact_1253_val_Osize_I5_J,axiom,
    ! [X2: nat] :
      ( ( size_size_val @ ( address @ X2 ) )
      = zero_zero_nat ) ).

% val.size(5)
thf(fact_1254_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_1255_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_1256_filterlim__atMost__at__top,axiom,
    filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).

% filterlim_atMost_at_top
thf(fact_1257_filterlim__minus__const__nat__at__top,axiom,
    ! [C: nat] :
      ( filterlim_nat_nat
      @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ C )
      @ at_top_nat
      @ at_top_nat ) ).

% filterlim_minus_const_nat_at_top
thf(fact_1258_filterlim__lessThan__at__top,axiom,
    filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).

% filterlim_lessThan_at_top
thf(fact_1259_filterlim__add__const__nat__at__top,axiom,
    ! [C: nat] :
      ( filterlim_nat_nat
      @ ^ [N2: nat] : ( plus_plus_nat @ N2 @ C )
      @ at_top_nat
      @ at_top_nat ) ).

% filterlim_add_const_nat_at_top
thf(fact_1260_le__sequentially,axiom,
    ! [F7: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F7 @ at_top_nat )
      = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F7 ) ) ) ).

% le_sequentially
thf(fact_1261_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_1262_range__mod,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( image_nat_nat2
          @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N3 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).

% range_mod
thf(fact_1263_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N2: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_1264_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_1265_card__Collect__le__nat,axiom,
    ! [N3: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N3 ) ) )
      = ( suc @ N3 ) ) ).

% card_Collect_le_nat
thf(fact_1266_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I: nat] : ( P @ ( suc @ I ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_1267_Least__Suc,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ N3 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_1268_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M5: nat,N2: nat] :
            ( N2
            = ( suc @ M5 ) ) ) ) ) ).

% pred_nat_def
thf(fact_1269_card__less,axiom,
    ! [M: set_nat,I5: nat] :
      ( ( member_nat @ zero_zero_nat @ M )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M )
                & ( ord_less_nat @ K2 @ ( suc @ I5 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_1270_card__less__Suc,axiom,
    ! [M: set_nat,I5: nat] :
      ( ( member_nat @ zero_zero_nat @ M )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M )
                  & ( ord_less_nat @ K2 @ I5 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M )
                & ( ord_less_nat @ K2 @ ( suc @ I5 ) ) ) ) ) ) ) ).

% card_less_Suc

% Helper facts (17)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X5: nat,Y: nat] :
      ( ( if_nat @ $false @ X5 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X5: nat,Y: nat] :
      ( ( if_nat @ $true @ X5 @ Y )
      = X5 ) ).

thf(help_If_2_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X5: prat,Y: prat] :
      ( ( if_prat @ $false @ X5 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X5: prat,Y: prat] :
      ( ( if_prat @ $true @ X5 @ Y )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X6: nat] : ( P @ X6 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__PosRat__Oprat_T,axiom,
    ! [P: prat > $o] :
      ( ( P @ ( fChoice_prat @ P ) )
      = ( ? [X6: prat] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X5: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X5 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X5: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X5 @ Y )
      = X5 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X5 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X5: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X5 @ Y )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( P @ ( fChoic6978938873391328853at_nat @ P ) )
      = ( ? [X6: product_prod_nat_nat] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( if_set4744415642439091239at_nat @ $false @ X5 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( if_set4744415642439091239at_nat @ $true @ X5 @ Y )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_T,axiom,
    ! [P: produc1457211279475724562t_char > $o] :
      ( ( P @ ( fChoic2476838209665704038t_char @ P ) )
      = ( ? [X6: produc1457211279475724562t_char] : ( P @ X6 ) ) ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_T,axiom,
    ! [X5: set_Pr2507339680178222706t_char,Y: set_Pr2507339680178222706t_char] :
      ( ( if_set1248378584462554936t_char @ $false @ X5 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_T,axiom,
    ! [X5: set_Pr2507339680178222706t_char,Y: set_Pr2507339680178222706t_char] :
      ( ( if_set1248378584462554936t_char @ $true @ X5 @ Y )
      = X5 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( equal_on_mask_a_b @ pi @ a2 @ b2 )
    = ( equal_on_mask_a_b @ pi @ b2 @ a2 ) ) ).

%------------------------------------------------------------------------------