TPTP Problem File: SLH0291^1.p

%------------------------------------------------------------------------------
% 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_00155_004256__7109840_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2251 ( 739 unt; 951 typ;   0 def)
%            Number of atoms       : 3751 (1424 equ;   0 cnn)
%            Maximal formula atoms :   10 (   2 avg)
%            Number of connectives : 11431 ( 253   ~;  51   |; 414   &;9321   @)
%                                         (   0 <=>;1392  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :  107 ( 106 usr)
%            Number of type conns  : 4810 (4810   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  848 ( 845 usr;  46 con; 0-4 aty)
%            Number of variables   : 4677 (1646   ^;2827   !; 204   ?;4677   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:05:12.559
%------------------------------------------------------------------------------
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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__Product____Type__Oprod_It__Set__Oset_It__PosRat__Oprat_J_Mt__Set__Oset_It__PosRat__Oprat_J_J,type,
    produc3531106461118509933t_prat: $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__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr6370437063884598352at_nat: $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_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    set_Pr9093778441882193744at_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__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_M_062_Itf__a_Mt__PosRat__Oprat_J_J,type,
    produc7047118689089425383a_prat: $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_Itf__a_M_062_Itf__a_Mt__PosRat__Oprat_J_J_J,type,
    set_Pr4163352480945123784a_prat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J_J,type,
    set_Pr2748872423075033990prat_a: $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__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc85711943791777264at_nat: $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__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    set_na7240141749078679150od_o_o: $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__Product____Type__Oprod_Itf__a_M_062_Itf__a_Mt__PosRat__Oprat_J_J,type,
    produc4459976518580069096a_prat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J,type,
    produc3375937593958228902prat_a: $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_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    set_Product_prod_a_a: $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_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    set_Product_prod_o_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__PosRat__Oprat_J,type,
    product_prod_a_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    set_nat_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_I_062_Itf__a_Mt__PosRat__Oprat_J_J,type,
    set_a_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__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    product_prod_a_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_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__Product____Type__Ounit,type,
    product_unit: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $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__Nat__Onat,type,
    nat: $tType ).

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

% Explicit typings (845)
thf(sy_c_BNF__Def_OGr_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,
    bNF_Gr6062854646325018466r_prat: set_Pr3433046674199179443r_prat > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ) > set_Pr1685704261205283783r_prat ).

thf(sy_c_BNF__Def_OGr_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,
    bNF_Gr7494250897068838581t_char: set_Pr3433046674199179443r_prat > ( ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char ) > set_Pr1935176096852982554t_char ).

thf(sy_c_BNF__Def_OGr_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,
    bNF_Gr4667860031443508641t_char: set_Pr3433046674199179443r_prat > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char ) > set_Pr7632435056502277254t_char ).

thf(sy_c_BNF__Def_OGr_001_062_Itf__a_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J,type,
    bNF_Gr6139830065717436617prat_a: set_a_prat > ( ( a > prat ) > produc3375937593958228902prat_a ) > set_Pr9117357208957936942prat_a ).

thf(sy_c_BNF__Def_OGr_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
    bNF_Gr_a_prat_a: set_a_prat > ( ( a > prat ) > a ) > set_Pr2748872423075033990prat_a ).

thf(sy_c_BNF__Def_OGr_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr2197830230412165350t_char: set_list_char > ( list_char > list_char ) > set_Pr4480099295215233283t_char ).

thf(sy_c_BNF__Def_OGr_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    bNF_Gr_list_char_nat: set_list_char > ( list_char > nat ) > set_Pr8803067691650843250ar_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr4088865321246080482at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_M_Eo_J_J,type,
    bNF_Gr1522612872074377213char_o: set_nat_nat_o > ( ( nat > nat > $o ) > nat > list_char > $o ) > ( nat > nat > $o ) > ( nat > list_char > $o ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bNF_Gr5554451822430810830_nat_o: set_nat_nat_o > ( ( nat > nat > $o ) > nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_M_Eo_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    bNF_Gr5836264614112020157char_o: set_nat_o > ( ( nat > $o ) > list_char > $o ) > ( nat > $o ) > ( list_char > $o ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_Grp_nat_o_nat_o: set_nat_o > ( ( nat > $o ) > nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    bNF_Gr7768808254035958895t_char: set_Pr3900080669421301403t_char > ( ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > produc7606469099237612818ar_nat > list_char ) > ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > ( produc7606469099237612818ar_nat > list_char ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    bNF_Gr7503386797330181086at_nat: set_Pr3900080669421301403t_char > ( ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > produc7606469099237612818ar_nat > nat ) > ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > ( produc7606469099237612818ar_nat > nat ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Gr589112650214279513at_nat: set_Pr4092731227417792307ar_nat > ( ( produc1457211279475724562t_char > nat ) > produc1457211279475724562t_char > product_prod_nat_nat ) > ( produc1457211279475724562t_char > nat ) > ( produc1457211279475724562t_char > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Nat__Onat_J,type,
    bNF_Gr298557646745593647ar_nat: set_Pr6539244058741650442at_nat > ( ( produc1457211279475724562t_char > product_prod_nat_nat ) > produc1457211279475724562t_char > nat ) > ( produc1457211279475724562t_char > product_prod_nat_nat ) > ( produc1457211279475724562t_char > nat ) > $o ).

thf(sy_c_BNF__Def_OGrp_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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_Gr2209344289654512049_nat_o: set_Pr2458342521480944603at_nat > ( ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_Eo_001_Eo,type,
    bNF_Grp_o_o: set_o > ( $o > $o ) > $o > $o > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr2905446856833220225t_char: set_nat > ( nat > list_char ) > nat > list_char > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Grp_nat_nat: set_nat > ( nat > nat ) > nat > nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr3553472063123884001at_nat: set_nat > ( nat > product_prod_nat_nat ) > nat > product_prod_nat_nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr3750019412944930853t_char: set_Pr2507339680178222706t_char > ( produc1457211279475724562t_char > list_char ) > produc1457211279475724562t_char > list_char > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat,type,
    bNF_Gr3073014387538510996ar_nat: set_Pr2507339680178222706t_char > ( produc1457211279475724562t_char > nat ) > produc1457211279475724562t_char > nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
    bNF_Gr8859634586176591525_nat_o: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_Gr193424670081168899at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat > $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_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    bNF_re6527413442372678928ar_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( list_char > nat > $o ) > ( ( nat > nat ) > list_char ) > ( ( nat > nat ) > nat ) > $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_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re3320998365863425151at_nat: ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > $o ) > ( ( nat > nat ) > nat ) > ( ( nat > nat ) > nat ) > $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_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    bNF_re1435683047850566962ar_nat: ( nat > nat > $o ) > ( list_char > nat > $o ) > ( nat > list_char ) > ( nat > nat ) > $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__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_re7653730848126131926t_char: ( produc7606469099237612818ar_nat > produc7606469099237612818ar_nat > $o ) > ( produc1457211279475724562t_char > list_char > $o ) > ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > ( produc7606469099237612818ar_nat > list_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat,type,
    bNF_re742088632225134789ar_nat: ( produc7606469099237612818ar_nat > produc7606469099237612818ar_nat > $o ) > ( produc1457211279475724562t_char > nat > $o ) > ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > ( produc7606469099237612818ar_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5269148346131715199at_nat: ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > ( nat > nat > $o ) > ( produc1457211279475724562t_char > nat ) > ( produc1457211279475724562t_char > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2049273209911613842at_nat: ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > ( nat > product_prod_nat_nat > $o ) > ( produc1457211279475724562t_char > nat ) > ( produc1457211279475724562t_char > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    bNF_re7912597853723674548at_nat: ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > ( product_prod_nat_nat > nat > $o ) > ( produc1457211279475724562t_char > product_prod_nat_nat ) > ( produc1457211279475724562t_char > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo,type,
    bNF_re6701202036956822900_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    bNF_re6814798851205064350ar_nat: ( set_nat > set_nat > $o ) > ( list_char > nat > $o ) > ( set_nat > list_char ) > ( set_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_re7456022984530013470t_char: ( set_nat > set_nat > $o ) > ( nat > list_char > $o ) > ( set_nat > nat ) > ( set_nat > list_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re2508443332600976141at_nat: ( set_nat > set_nat > $o ) > ( nat > nat > $o ) > ( set_nat > nat ) > ( set_nat > nat ) > $o ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr8575425969231580588t_char: set_Pr3433046674199179443r_prat > ( ( produc1457211279475724562t_char > prat ) > nat ) > ( ( produc1457211279475724562t_char > prat ) > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001_062_Itf__a_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J,type,
    bNF_Gr6386687543108125413prat_a: set_o > ( $o > a > prat ) > ( $o > produc3375937593958228902prat_a ) > set_Pr9117357208957936942prat_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
    bNF_Gr8379640946404278973prat_a: set_o > ( $o > a > prat ) > ( $o > a ) > set_Pr2748872423075033990prat_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr7618826833391781745t_char: set_o > ( $o > nat ) > ( $o > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr1946814795223983584at_nat: set_o > ( $o > nat ) > ( $o > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr5170369590546435438t_char: set_list_char > ( list_char > nat ) > ( list_char > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr5258588615557311325at_nat: set_list_char > ( list_char > nat ) > ( list_char > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
    bNF_Gr4706834691868246441prat_a: set_nat > ( nat > a > prat ) > ( nat > a ) > set_Pr2748872423075033990prat_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr689107047223432174t_char: set_nat > ( nat > list_char ) > ( nat > list_char ) > set_Pr4480099295215233283t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat,type,
    bNF_Gr6756318997112205789ar_nat: set_nat > ( nat > list_char ) > ( nat > nat ) > set_Pr8803067691650843250ar_nat ).

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__Set__Oset_It__Nat__Onat_J_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
    bNF_Gr8551326311310947167prat_a: set_set_nat > ( set_nat > a > prat ) > ( set_nat > a ) > set_Pr2748872423075033990prat_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr3129696893582918419t_char: set_set_nat > ( set_nat > nat ) > ( set_nat > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr6681173980307635330at_nat: set_set_nat > ( set_nat > nat ) > ( set_nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__PosRat__Oprat_J_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
    bNF_Gr3930513041845697759prat_a: set_set_prat > ( set_prat > a > prat ) > ( set_prat > a ) > set_Pr2748872423075033990prat_a ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__PosRat__Oprat_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr7301876082586963091t_char: set_set_prat > ( set_prat > nat ) > ( set_prat > list_char ) > set_Pr2507339680178222706t_char ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Set__Oset_It__PosRat__Oprat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Gr6835880997613667842at_nat: set_set_prat > ( set_prat > nat ) > ( set_prat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_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,
    bNF_Gr2145404550722348580r_prat: set_Pr1685704261205283783r_prat > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ) > set_Pr1685704261205283783r_prat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelImage_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Nat__Onat,type,
    bNF_Gr5060297752514649919at_nat: set_Pr1685704261205283783r_prat > ( ( produc1457211279475724562t_char > prat ) > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_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,
    bNF_Gr7532869937377485441r_prat: set_Pr3433046674199179443r_prat > set_Pr1685704261205283783r_prat > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ) > set_Pr1685704261205283783r_prat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001_Eo_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    bNF_Gr7259518749458858748r_prat: set_o > set_Pr1685704261205283783r_prat > ( $o > produc1457211279475724562t_char > prat ) > set_Product_prod_o_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    bNF_Gr3267885531569300736r_prat: set_nat > set_Pr1685704261205283783r_prat > ( nat > produc1457211279475724562t_char > prat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    bNF_Gr8791779402478003658r_prat: set_set_nat > set_Pr1685704261205283783r_prat > ( set_nat > produc1457211279475724562t_char > prat ) > set_Pr5488025237498180813et_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage_001t__Set__Oset_It__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,
    bNF_Gr2010402745563153482r_prat: set_set_prat > set_Pr1685704261205283783r_prat > ( set_prat > produc1457211279475724562t_char > prat ) > set_Pr141576177145560269t_prat ).

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_Eo,type,
    complete_Inf_Inf_o: set_o > $o ).

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_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_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

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__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_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__below_001t__Nat__Onat,type,
    condit1738341127787009408ow_nat: set_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_Oproj_001t__PosRat__Oprat_001t__PosRat__Oprat,type,
    equiv_proj_prat_prat: set_Pr3298805672376727137t_prat > prat > set_prat ).

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_Filter_Oat__bot_001t__Nat__Onat,type,
    at_bot_nat: filter_nat ).

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

thf(sy_c_Finite__Set_Ocomp__fun__commute_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    finite1002329793627884815t_char: ( nat > set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char ) > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite2115694454571419734at_nat: set_nat_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__String__Ochar_J,type,
    finite3619747586194699384t_char: set_list_char > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
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    comp_P559749852281925063et_nat: ( produc7606469099237612818ar_nat > nat ) > ( set_nat > produc7606469099237612818ar_nat ) > set_nat > nat ).

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    comp_P4974907838745982882ar_nat: ( produc1457211279475724562t_char > list_char ) > ( nat > produc1457211279475724562t_char ) > nat > list_char ).

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    comp_P26140635296239040ar_nat: ( produc1457211279475724562t_char > list_char ) > ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > produc7606469099237612818ar_nat > list_char ).

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    comp_P7799612233565674328et_nat: ( produc1457211279475724562t_char > list_char ) > ( set_nat > produc1457211279475724562t_char ) > set_nat > list_char ).

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    comp_P8410057226579278353at_nat: ( produc1457211279475724562t_char > nat ) > ( nat > produc1457211279475724562t_char ) > nat > nat ).

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    comp_P3187634502937433873ar_nat: ( produc1457211279475724562t_char > nat ) > ( produc7606469099237612818ar_nat > produc1457211279475724562t_char ) > produc7606469099237612818ar_nat > nat ).

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    comp_P488745826847564231et_nat: ( produc1457211279475724562t_char > nat ) > ( set_nat > produc1457211279475724562t_char ) > set_nat > nat ).

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    comp_P6240816377000802031at_nat: ( product_prod_nat_nat > nat ) > ( ( nat > nat ) > product_prod_nat_nat ) > ( nat > nat ) > nat ).

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    comp_P1955610211387027840at_nat: ( product_prod_nat_nat > nat ) > ( nat > product_prod_nat_nat ) > nat > nat ).

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    comp_P8271837224898685794ar_nat: ( product_prod_nat_nat > nat ) > ( produc7606469099237612818ar_nat > product_prod_nat_nat ) > produc7606469099237612818ar_nat > nat ).

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    comp_P2122579405136797538t_char: ( product_prod_nat_nat > nat ) > ( produc1457211279475724562t_char > product_prod_nat_nat ) > produc1457211279475724562t_char > nat ).

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    comp_P1567445206330693457at_nat: ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > nat ).

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    fun_up9120501182914450918t_prat: ( ( produc1457211279475724562t_char > prat ) > prat ) > ( produc1457211279475724562t_char > prat ) > prat > ( produc1457211279475724562t_char > prat ) > prat ).

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thf(sy_c_Fun_Ofun__upd_001tf__a_001t__PosRat__Oprat,type,
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
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    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 ).

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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
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    the_Pr7520103294981519086r_prat: ( produc3336119888008106215r_prat > $o ) > produc3336119888008106215r_prat ).

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    the_Pr6337852224591121453t_char: ( produc2489117125269924006t_char > $o ) > produc2489117125269924006t_char ).

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thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_PosRat_Opmax,type,
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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,
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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_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__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Nat__Onat,type,
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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_Transitive__Closure_Oacyclic_001t__Nat__Onat,type,
    transi3139058337892597825ic_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wellfounded_Oacc_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    acc_nat_nat: set_Pr7682762132356531903at_nat > set_nat_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,
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thf(sy_c_Wellfounded_Oacc_001_Eo,type,
    acc_o: set_Product_prod_o_o > set_o ).

thf(sy_c_Wellfounded_Oacc_001t__List__Olist_It__String__Ochar_J,type,
    acc_list_char: set_Pr4480099295215233283t_char > set_list_char ).

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_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,
    acc_Pr8150405688010203948r_prat: set_Pr113032449794829511r_prat > set_Pr1685704261205283783r_prat ).

thf(sy_c_Wellfounded_Oacc_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,
    acc_Pr5366037250316522495t_char: set_Pr7915859021238449415t_char > set_Pr1935176096852982554t_char ).

thf(sy_c_Wellfounded_Oacc_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J_J,type,
    acc_Pr1587061423467440147prat_a: set_Pr2088549235032396295prat_a > set_Pr9117357208957936942prat_a ).

thf(sy_c_Wellfounded_Oacc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Wellfounded_Oacc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Oacc_001t__Set__Oset_It__Nat__Onat_J,type,
    acc_set_nat: set_Pr5488025237498180813et_nat > set_set_nat ).

thf(sy_c_Wellfounded_Oacc_001t__Set__Oset_It__PosRat__Oprat_J,type,
    acc_set_prat: set_Pr141576177145560269t_prat > set_set_prat ).

thf(sy_c_Wellfounded_Oaccp_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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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,
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thf(sy_c_Wellfounded_Oaccp_001_Eo,type,
    accp_o: ( $o > $o > $o ) > $o > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__String__Ochar_J,type,
    accp_list_char: ( list_char > list_char > $o ) > list_char > $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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    accp_P8393741129312856816r_prat: ( produc3336119888008106215r_prat > produc3336119888008106215r_prat > $o ) > produc3336119888008106215r_prat > $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__PosRat__Oprat_J,type,
    accp_P6822921472154139145t_prat: ( produc4561108331071084498t_prat > produc4561108331071084498t_prat > $o ) > produc4561108331071084498t_prat > $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_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__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    accp_P5529878251199943754r_prat: ( produc6235724684372434707r_prat > produc6235724684372434707r_prat > $o ) > produc6235724684372434707r_prat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J_J,type,
    accp_P8439977412596685783prat_a: ( produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o ) > produc4654305426090421966prat_a > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_Itf__a_Mt__PosRat__Oprat_J_J,type,
    accp_P6402918970019748386a_prat: ( produc4032483947639269611a_prat > produc4032483947639269611a_prat > $o ) > produc4032483947639269611a_prat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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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,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__PosRat__Oprat_J,type,
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thf(sy_c_Wellfounded_Oaccp_001t__Set__Oset_It__Nat__Onat_J,type,
    accp_set_nat: ( set_nat > set_nat > $o ) > set_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Set__Oset_It__PosRat__Oprat_J,type,
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thf(sy_c_Wellfounded_Oaccp_001tf__a,type,
    accp_a: ( a > a > $o ) > a > $o ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Olex__prod_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,
    lex_pr789199185301838959r_prat: set_Pr1685704261205283783r_prat > set_Pr1685704261205283783r_prat > set_Pr113032449794829511r_prat ).

thf(sy_c_Wellfounded_Olex__prod_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,
    lex_pr5953235755497499842t_char: set_Pr1685704261205283783r_prat > set_Pr6148874766616261511t_char > set_Pr7915859021238449415t_char ).

thf(sy_c_Wellfounded_Olex__prod_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,
    lex_pr7982885678635892526t_char: set_Pr1685704261205283783r_prat > set_Pr3996677194857090567t_char > set_Pr6148874766616261511t_char ).

thf(sy_c_Wellfounded_Olex__prod_001_062_Itf__a_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J,type,
    lex_pr7488954505079209174prat_a: set_Pr7218575088661268167a_prat > set_Pr6010151575929665415prat_a > set_Pr2088549235032396295prat_a ).

thf(sy_c_Wellfounded_Olex__prod_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
    lex_prod_a_prat_a: set_Pr7218575088661268167a_prat > set_Product_prod_a_a > set_Pr6010151575929665415prat_a ).

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__extp_001t__Nat__Onat,type,
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thf(sy_c_Wellfounded_Omax__extp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Omin__ext_001t__Nat__Onat,type,
    min_ext_nat: set_Pr1261947904930325089at_nat > set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
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thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Wellfounded_Owf_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Wfrec_Osame__fst_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,
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thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_Wfrec_Osame__fst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_fChoice_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
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thf(sy_c_fChoice_001_Eo,type,
    fChoice_o: ( $o > $o ) > $o ).

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

thf(sy_c_fChoice_001t__PosRat__Oprat,type,
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thf(sy_c_fChoice_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,
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thf(sy_c_fChoice_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,
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thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_fChoice_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_member_001t__Nat__Onat,type,
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thf(sy_c_member_001t__PosRat__Oprat,type,
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    member3618324141103465936t_char: produc8069765938759941799t_char > set_Pr6148874766616261511t_char > $o ).

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    member7650881556348516432prat_a: produc3797646013676224551prat_a > set_Pr2088549235032396295prat_a > $o ).

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    member5179433394215856592prat_a: produc54085900832316071prat_a > set_Pr6010151575929665415prat_a > $o ).

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    member151853725258021968t_char: produc6071853487833700903t_char > set_Pr3996677194857090567t_char > $o ).

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    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

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    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

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    member413714312206344214t_prat: produc3531106461118509933t_prat > set_Pr141576177145560269t_prat > $o ).

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    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $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_001tf__a,type,
    member_a: a > set_a > $o ).

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

thf(sy_v__092_060pi_062_H,type,
    pi2: a > prat ).

thf(sy_v_hl____,type,
    hl: a ).

% Relevant facts (1272)
thf(fact_0__092_060open_062_092_060pi_062_H_Ahl_A_061_Apadd_A_I_092_060pi_062_Ahl_J_A_ISOME_Ap_O_A_092_060pi_062_H_Ahl_A_061_Apadd_A_I_092_060pi_062_Ahl_J_Ap_J_092_060close_062,axiom,
    ( ( pi2 @ hl )
    = ( padd @ ( pi @ hl )
      @ ( fChoice_prat
        @ ^ [P: prat] :
            ( ( pi2 @ hl )
            = ( padd @ ( pi @ hl ) @ P ) ) ) ) ) ).

% \<open>\<pi>' hl = padd (\<pi> hl) (SOME p. \<pi>' hl = padd (\<pi> hl) p)\<close>
thf(fact_1__092_060open_062_092_060forall_062hl_O_Apgte_A_I_092_060pi_062_H_Ahl_J_A_I_092_060pi_062_Ahl_J_092_060close_062,axiom,
    ! [Hl: a] : ( pgte @ ( pi2 @ Hl ) @ ( pi @ Hl ) ) ).

% \<open>\<forall>hl. pgte (\<pi>' hl) (\<pi> hl)\<close>
thf(fact_2_add__masks_Oelims,axiom,
    ! [X: a > prat,Xa: a > prat,Xb: a,Y: prat] :
      ( ( ( add_masks_a @ X @ Xa @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_3_add__masks_Oelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_4_add__masks_Osimps,axiom,
    ( add_masks_a
    = ( ^ [Pi: a > prat,Pi2: a > prat,Hl2: a] : ( padd @ ( Pi @ Hl2 ) @ ( Pi2 @ Hl2 ) ) ) ) ).

% add_masks.simps
thf(fact_5_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( padd @ ( Pi @ Hl2 ) @ ( Pi2 @ Hl2 ) ) ) ) ).

% add_masks.simps
thf(fact_6_add__masks__asso,axiom,
    ! [A: a > prat,B: a > prat,C: a > prat] :
      ( ( add_masks_a @ ( add_masks_a @ A @ B ) @ C )
      = ( add_masks_a @ A @ ( add_masks_a @ B @ C ) ) ) ).

% add_masks_asso
thf(fact_7_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_8_add__masks__comm,axiom,
    ( add_masks_a
    = ( ^ [A2: a > prat,B2: a > prat] : ( add_masks_a @ B2 @ A2 ) ) ) ).

% add_masks_comm
thf(fact_9_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_10_some__equality,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( P2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ( fChoic6978938873391328853at_nat @ P2 )
          = A ) ) ) ).

% some_equality
thf(fact_11_some__equality,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
      ( ( P2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ( fChoic2476838209665704038t_char @ P2 )
          = A ) ) ) ).

% some_equality
thf(fact_12_some__equality,axiom,
    ! [P2: prat > $o,A: prat] :
      ( ( P2 @ A )
     => ( ! [X2: prat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ( fChoice_prat @ P2 )
          = A ) ) ) ).

% some_equality
thf(fact_13_some__eq__trivial,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( fChoic6978938873391328853at_nat
        @ ^ [Y2: product_prod_nat_nat] : ( Y2 = X ) )
      = X ) ).

% some_eq_trivial
thf(fact_14_some__eq__trivial,axiom,
    ! [X: produc1457211279475724562t_char] :
      ( ( fChoic2476838209665704038t_char
        @ ^ [Y2: produc1457211279475724562t_char] : ( Y2 = X ) )
      = X ) ).

% some_eq_trivial
thf(fact_15_some__eq__trivial,axiom,
    ! [X: prat] :
      ( ( fChoice_prat
        @ ^ [Y2: prat] : ( Y2 = X ) )
      = X ) ).

% some_eq_trivial
thf(fact_16_some__sym__eq__trivial,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( fChoic6978938873391328853at_nat
        @ ( ^ [Y3: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y3 = Z )
          @ X ) )
      = X ) ).

% some_sym_eq_trivial
thf(fact_17_some__sym__eq__trivial,axiom,
    ! [X: produc1457211279475724562t_char] :
      ( ( fChoic2476838209665704038t_char
        @ ( ^ [Y3: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] : ( Y3 = Z )
          @ X ) )
      = X ) ).

% some_sym_eq_trivial
thf(fact_18_some__sym__eq__trivial,axiom,
    ! [X: prat] :
      ( ( fChoice_prat
        @ ( ^ [Y3: prat,Z: prat] : ( Y3 = Z )
          @ X ) )
      = X ) ).

% some_sym_eq_trivial
thf(fact_19__092_060open_062greater__mask_A_092_060pi_062_H_A_092_060pi_062_A_092_060Longrightarrow_062_A_092_060forall_062hl_O_Apgte_A_I_092_060pi_062_H_Ahl_J_A_I_092_060pi_062_Ahl_J_092_060close_062,axiom,
    ( ( greater_mask_a @ pi2 @ pi )
   => ! [Hl: a] : ( pgte @ ( pi2 @ Hl ) @ ( pi @ Hl ) ) ) ).

% \<open>greater_mask \<pi>' \<pi> \<Longrightarrow> \<forall>hl. pgte (\<pi>' hl) (\<pi> hl)\<close>
thf(fact_20_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
        ? [R: produc1457211279475724562t_char > prat] :
          ( Pi
          = ( add_ma6586698345353345811t_char @ Pi2 @ R ) ) ) ) ).

% greater_mask_def
thf(fact_21_greater__mask__def,axiom,
    ( greater_mask_a
    = ( ^ [Pi: a > prat,Pi2: a > prat] :
        ? [R: a > prat] :
          ( Pi
          = ( add_masks_a @ Pi2 @ R ) ) ) ) ).

% greater_mask_def
thf(fact_22_verit__sko__ex_H,axiom,
    ! [P2: product_prod_nat_nat > $o,A3: $o] :
      ( ( ( P2 @ ( fChoic6978938873391328853at_nat @ P2 ) )
        = A3 )
     => ( ( ? [X3: product_prod_nat_nat] : ( P2 @ X3 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_23_verit__sko__ex_H,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A3: $o] :
      ( ( ( P2 @ ( fChoic2476838209665704038t_char @ P2 ) )
        = A3 )
     => ( ( ? [X3: produc1457211279475724562t_char] : ( P2 @ X3 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_24_verit__sko__ex_H,axiom,
    ! [P2: prat > $o,A3: $o] :
      ( ( ( P2 @ ( fChoice_prat @ P2 ) )
        = A3 )
     => ( ( ? [X3: prat] : ( P2 @ X3 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_25_verit__sko__forall,axiom,
    ( ( ^ [P3: product_prod_nat_nat > $o] :
        ! [X4: product_prod_nat_nat] : ( P3 @ X4 ) )
    = ( ^ [P4: product_prod_nat_nat > $o] :
          ( P4
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ~ ( P4 @ X5 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_26_verit__sko__forall,axiom,
    ( ( ^ [P3: produc1457211279475724562t_char > $o] :
        ! [X4: produc1457211279475724562t_char] : ( P3 @ X4 ) )
    = ( ^ [P4: produc1457211279475724562t_char > $o] :
          ( P4
          @ ( fChoic2476838209665704038t_char
            @ ^ [X5: produc1457211279475724562t_char] :
                ~ ( P4 @ X5 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_27_verit__sko__forall,axiom,
    ( ( ^ [P3: prat > $o] :
        ! [X4: prat] : ( P3 @ X4 ) )
    = ( ^ [P4: prat > $o] :
          ( P4
          @ ( fChoice_prat
            @ ^ [X5: prat] :
                ~ ( P4 @ X5 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_28_someI2,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat,Q: product_prod_nat_nat > $o] :
      ( ( P2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( fChoic6978938873391328853at_nat @ P2 ) ) ) ) ).

% someI2
thf(fact_29_someI2,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,Q: produc1457211279475724562t_char > $o] :
      ( ( P2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( fChoic2476838209665704038t_char @ P2 ) ) ) ) ).

% someI2
thf(fact_30_someI2,axiom,
    ! [P2: prat > $o,A: prat,Q: prat > $o] :
      ( ( P2 @ A )
     => ( ! [X2: prat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( fChoice_prat @ P2 ) ) ) ) ).

% someI2
thf(fact_31_verit__sko__forall_H,axiom,
    ! [P2: product_prod_nat_nat > $o,A3: $o] :
      ( ( ( P2
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ~ ( P2 @ X5 ) ) )
        = A3 )
     => ( ( ! [X3: product_prod_nat_nat] : ( P2 @ X3 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_32_verit__sko__forall_H,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A3: $o] :
      ( ( ( P2
          @ ( fChoic2476838209665704038t_char
            @ ^ [X5: produc1457211279475724562t_char] :
                ~ ( P2 @ X5 ) ) )
        = A3 )
     => ( ( ! [X3: produc1457211279475724562t_char] : ( P2 @ X3 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_33_verit__sko__forall_H,axiom,
    ! [P2: prat > $o,A3: $o] :
      ( ( ( P2
          @ ( fChoice_prat
            @ ^ [X5: prat] :
                ~ ( P2 @ X5 ) ) )
        = A3 )
     => ( ( ! [X3: prat] : ( P2 @ X3 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_34_verit__sko__forall_H_H,axiom,
    ! [B3: product_prod_nat_nat,A3: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( B3 = A3 )
     => ( ( ( fChoic6978938873391328853at_nat @ P2 )
          = A3 )
        = ( ( fChoic6978938873391328853at_nat @ P2 )
          = B3 ) ) ) ).

% verit_sko_forall''
thf(fact_35_verit__sko__forall_H_H,axiom,
    ! [B3: produc1457211279475724562t_char,A3: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o] :
      ( ( B3 = A3 )
     => ( ( ( fChoic2476838209665704038t_char @ P2 )
          = A3 )
        = ( ( fChoic2476838209665704038t_char @ P2 )
          = B3 ) ) ) ).

% verit_sko_forall''
thf(fact_36_verit__sko__forall_H_H,axiom,
    ! [B3: prat,A3: prat,P2: prat > $o] :
      ( ( B3 = A3 )
     => ( ( ( fChoice_prat @ P2 )
          = A3 )
        = ( ( fChoice_prat @ P2 )
          = B3 ) ) ) ).

% verit_sko_forall''
thf(fact_37_someI__ex,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ? [X_1: product_prod_nat_nat] : ( P2 @ X_1 )
     => ( P2 @ ( fChoic6978938873391328853at_nat @ P2 ) ) ) ).

% someI_ex
thf(fact_38_someI__ex,axiom,
    ! [P2: produc1457211279475724562t_char > $o] :
      ( ? [X_1: produc1457211279475724562t_char] : ( P2 @ X_1 )
     => ( P2 @ ( fChoic2476838209665704038t_char @ P2 ) ) ) ).

% someI_ex
thf(fact_39_someI__ex,axiom,
    ! [P2: prat > $o] :
      ( ? [X_1: prat] : ( P2 @ X_1 )
     => ( P2 @ ( fChoice_prat @ P2 ) ) ) ).

% someI_ex
thf(fact_40_some__eq__imp,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( fChoic6978938873391328853at_nat @ P2 )
        = A )
     => ( ( P2 @ B )
       => ( P2 @ A ) ) ) ).

% some_eq_imp
thf(fact_41_some__eq__imp,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,B: produc1457211279475724562t_char] :
      ( ( ( fChoic2476838209665704038t_char @ P2 )
        = A )
     => ( ( P2 @ B )
       => ( P2 @ A ) ) ) ).

% some_eq_imp
thf(fact_42_some__eq__imp,axiom,
    ! [P2: prat > $o,A: prat,B: prat] :
      ( ( ( fChoice_prat @ P2 )
        = A )
     => ( ( P2 @ B )
       => ( P2 @ A ) ) ) ).

% some_eq_imp
thf(fact_43_tfl__some,axiom,
    ! [P5: product_prod_nat_nat > $o,X6: product_prod_nat_nat] :
      ( ( P5 @ X6 )
     => ( P5 @ ( fChoic6978938873391328853at_nat @ P5 ) ) ) ).

% tfl_some
thf(fact_44_tfl__some,axiom,
    ! [P5: produc1457211279475724562t_char > $o,X6: produc1457211279475724562t_char] :
      ( ( P5 @ X6 )
     => ( P5 @ ( fChoic2476838209665704038t_char @ P5 ) ) ) ).

% tfl_some
thf(fact_45_tfl__some,axiom,
    ! [P5: prat > $o,X6: prat] :
      ( ( P5 @ X6 )
     => ( P5 @ ( fChoice_prat @ P5 ) ) ) ).

% tfl_some
thf(fact_46_Eps__cong,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( fChoic6978938873391328853at_nat @ P2 )
        = ( fChoic6978938873391328853at_nat @ Q ) ) ) ).

% Eps_cong
thf(fact_47_Eps__cong,axiom,
    ! [P2: produc1457211279475724562t_char > $o,Q: produc1457211279475724562t_char > $o] :
      ( ! [X2: produc1457211279475724562t_char] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( fChoic2476838209665704038t_char @ P2 )
        = ( fChoic2476838209665704038t_char @ Q ) ) ) ).

% Eps_cong
thf(fact_48_Eps__cong,axiom,
    ! [P2: prat > $o,Q: prat > $o] :
      ( ! [X2: prat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( fChoice_prat @ P2 )
        = ( fChoice_prat @ Q ) ) ) ).

% Eps_cong
thf(fact_49_someI,axiom,
    ! [P2: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ( P2 @ X )
     => ( P2 @ ( fChoic6978938873391328853at_nat @ P2 ) ) ) ).

% someI
thf(fact_50_someI,axiom,
    ! [P2: produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char] :
      ( ( P2 @ X )
     => ( P2 @ ( fChoic2476838209665704038t_char @ P2 ) ) ) ).

% someI
thf(fact_51_someI,axiom,
    ! [P2: prat > $o,X: prat] :
      ( ( P2 @ X )
     => ( P2 @ ( fChoice_prat @ P2 ) ) ) ).

% someI
thf(fact_52_verit__sko__forall__indirect2,axiom,
    ! [X: product_prod_nat_nat,P2: product_prod_nat_nat > $o,P6: product_prod_nat_nat > $o] :
      ( ( X
        = ( fChoic6978938873391328853at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ~ ( P2 @ X5 ) ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
            = ( P6 @ X2 ) )
       => ( ( ! [X3: product_prod_nat_nat] : ( P6 @ X3 ) )
          = ( P2 @ X ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_53_verit__sko__forall__indirect2,axiom,
    ! [X: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o,P6: produc1457211279475724562t_char > $o] :
      ( ( X
        = ( fChoic2476838209665704038t_char
          @ ^ [X5: produc1457211279475724562t_char] :
              ~ ( P2 @ X5 ) ) )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
            = ( P6 @ X2 ) )
       => ( ( ! [X3: produc1457211279475724562t_char] : ( P6 @ X3 ) )
          = ( P2 @ X ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_54_verit__sko__forall__indirect2,axiom,
    ! [X: prat,P2: prat > $o,P6: prat > $o] :
      ( ( X
        = ( fChoice_prat
          @ ^ [X5: prat] :
              ~ ( P2 @ X5 ) ) )
     => ( ! [X2: prat] :
            ( ( P2 @ X2 )
            = ( P6 @ X2 ) )
       => ( ( ! [X3: prat] : ( P6 @ X3 ) )
          = ( P2 @ X ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_55_verit__sko__forall__indirect,axiom,
    ! [X: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( X
        = ( fChoic6978938873391328853at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ~ ( P2 @ X5 ) ) )
     => ( ( ! [X3: product_prod_nat_nat] : ( P2 @ X3 ) )
        = ( P2 @ X ) ) ) ).

% verit_sko_forall_indirect
thf(fact_56_verit__sko__forall__indirect,axiom,
    ! [X: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o] :
      ( ( X
        = ( fChoic2476838209665704038t_char
          @ ^ [X5: produc1457211279475724562t_char] :
              ~ ( P2 @ X5 ) ) )
     => ( ( ! [X3: produc1457211279475724562t_char] : ( P2 @ X3 ) )
        = ( P2 @ X ) ) ) ).

% verit_sko_forall_indirect
thf(fact_57_verit__sko__forall__indirect,axiom,
    ! [X: prat,P2: prat > $o] :
      ( ( X
        = ( fChoice_prat
          @ ^ [X5: prat] :
              ~ ( P2 @ X5 ) ) )
     => ( ( ! [X3: prat] : ( P2 @ X3 ) )
        = ( P2 @ X ) ) ) ).

% verit_sko_forall_indirect
thf(fact_58_some1__equality,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( P2 @ X6 )
          & ! [Y4: product_prod_nat_nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P2 @ A )
       => ( ( fChoic6978938873391328853at_nat @ P2 )
          = A ) ) ) ).

% some1_equality
thf(fact_59_some1__equality,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
      ( ? [X6: produc1457211279475724562t_char] :
          ( ( P2 @ X6 )
          & ! [Y4: produc1457211279475724562t_char] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P2 @ A )
       => ( ( fChoic2476838209665704038t_char @ P2 )
          = A ) ) ) ).

% some1_equality
thf(fact_60_some1__equality,axiom,
    ! [P2: prat > $o,A: prat] :
      ( ? [X6: prat] :
          ( ( P2 @ X6 )
          & ! [Y4: prat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P2 @ A )
       => ( ( fChoice_prat @ P2 )
          = A ) ) ) ).

% some1_equality
thf(fact_61_verit__sko__ex__indirect2,axiom,
    ! [X: product_prod_nat_nat,P2: product_prod_nat_nat > $o,P6: product_prod_nat_nat > $o] :
      ( ( X
        = ( fChoic6978938873391328853at_nat @ P2 ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
            = ( P6 @ X2 ) )
       => ( ( ? [X3: product_prod_nat_nat] : ( P6 @ X3 ) )
          = ( P2 @ X ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_62_verit__sko__ex__indirect2,axiom,
    ! [X: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o,P6: produc1457211279475724562t_char > $o] :
      ( ( X
        = ( fChoic2476838209665704038t_char @ P2 ) )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
            = ( P6 @ X2 ) )
       => ( ( ? [X3: produc1457211279475724562t_char] : ( P6 @ X3 ) )
          = ( P2 @ X ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_63_verit__sko__ex__indirect2,axiom,
    ! [X: prat,P2: prat > $o,P6: prat > $o] :
      ( ( X
        = ( fChoice_prat @ P2 ) )
     => ( ! [X2: prat] :
            ( ( P2 @ X2 )
            = ( P6 @ X2 ) )
       => ( ( ? [X3: prat] : ( P6 @ X3 ) )
          = ( P2 @ X ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_64_verit__sko__ex__indirect,axiom,
    ! [X: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( X
        = ( fChoic6978938873391328853at_nat @ P2 ) )
     => ( ( ? [X3: product_prod_nat_nat] : ( P2 @ X3 ) )
        = ( P2 @ X ) ) ) ).

% verit_sko_ex_indirect
thf(fact_65_verit__sko__ex__indirect,axiom,
    ! [X: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o] :
      ( ( X
        = ( fChoic2476838209665704038t_char @ P2 ) )
     => ( ( ? [X3: produc1457211279475724562t_char] : ( P2 @ X3 ) )
        = ( P2 @ X ) ) ) ).

% verit_sko_ex_indirect
thf(fact_66_verit__sko__ex__indirect,axiom,
    ! [X: prat,P2: prat > $o] :
      ( ( X
        = ( fChoice_prat @ P2 ) )
     => ( ( ? [X3: prat] : ( P2 @ X3 ) )
        = ( P2 @ X ) ) ) ).

% verit_sko_ex_indirect
thf(fact_67_some__eq__ex,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( P2 @ ( fChoic6978938873391328853at_nat @ P2 ) )
      = ( ? [X3: product_prod_nat_nat] : ( P2 @ X3 ) ) ) ).

% some_eq_ex
thf(fact_68_some__eq__ex,axiom,
    ! [P2: produc1457211279475724562t_char > $o] :
      ( ( P2 @ ( fChoic2476838209665704038t_char @ P2 ) )
      = ( ? [X3: produc1457211279475724562t_char] : ( P2 @ X3 ) ) ) ).

% some_eq_ex
thf(fact_69_some__eq__ex,axiom,
    ! [P2: prat > $o] :
      ( ( P2 @ ( fChoice_prat @ P2 ) )
      = ( ? [X3: prat] : ( P2 @ X3 ) ) ) ).

% some_eq_ex
thf(fact_70_someI2__bex,axiom,
    ! [A3: set_o,P2: $o > $o,Q: $o > $o] :
      ( ? [X6: $o] :
          ( ( member_o @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: $o] :
            ( ( ( member_o @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_71_someI2__bex,axiom,
    ! [A3: set_set_prat,P2: set_prat > $o,Q: set_prat > $o] :
      ( ? [X6: set_prat] :
          ( ( member_set_prat @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: set_prat] :
            ( ( ( member_set_prat @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_set_prat
            @ ^ [X5: set_prat] :
                ( ( member_set_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_72_someI2__bex,axiom,
    ! [A3: set_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
      ( ? [X6: set_nat] :
          ( ( member_set_nat @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: set_nat] :
            ( ( ( member_set_nat @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_73_someI2__bex,axiom,
    ! [A3: set_nat,P2: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: nat] :
            ( ( ( member_nat @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_74_someI2__bex,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ? [X6: produc1457211279475724562t_char > prat] :
          ( ( member2336342582916103932r_prat @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: produc1457211279475724562t_char > prat] :
            ( ( ( member2336342582916103932r_prat @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoic3782516149722485159r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat] :
                ( ( member2336342582916103932r_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_75_someI2__bex,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_76_someI2__bex,axiom,
    ! [A3: set_Pr2507339680178222706t_char,P2: produc1457211279475724562t_char > $o,Q: produc1457211279475724562t_char > $o] :
      ( ? [X6: produc1457211279475724562t_char] :
          ( ( member484235747325421115t_char @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( ( member484235747325421115t_char @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoic2476838209665704038t_char
            @ ^ [X5: produc1457211279475724562t_char] :
                ( ( member484235747325421115t_char @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_77_someI2__bex,axiom,
    ! [A3: set_prat,P2: prat > $o,Q: prat > $o] :
      ( ? [X6: prat] :
          ( ( member_prat @ X6 @ A3 )
          & ( P2 @ X6 ) )
     => ( ! [X2: prat] :
            ( ( ( member_prat @ X2 @ A3 )
              & ( P2 @ X2 ) )
           => ( Q @ X2 ) )
       => ( Q
          @ ( fChoice_prat
            @ ^ [X5: prat] :
                ( ( member_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_78_someI2__ex,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ? [X_1: product_prod_nat_nat] : ( P2 @ X_1 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( fChoic6978938873391328853at_nat @ P2 ) ) ) ) ).

% someI2_ex
thf(fact_79_someI2__ex,axiom,
    ! [P2: produc1457211279475724562t_char > $o,Q: produc1457211279475724562t_char > $o] :
      ( ? [X_1: produc1457211279475724562t_char] : ( P2 @ X_1 )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( fChoic2476838209665704038t_char @ P2 ) ) ) ) ).

% someI2_ex
thf(fact_80_someI2__ex,axiom,
    ! [P2: prat > $o,Q: prat > $o] :
      ( ? [X_1: prat] : ( P2 @ X_1 )
     => ( ! [X2: prat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( fChoice_prat @ P2 ) ) ) ) ).

% someI2_ex
thf(fact_81_upper__bounded__def,axiom,
    ( upper_bounded
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,P: prat] :
        ! [Hl2: produc1457211279475724562t_char] : ( pgte @ P @ ( Pi2 @ Hl2 ) ) ) ) ).

% upper_bounded_def
thf(fact_82_p__greater__exists,axiom,
    ( pgte
    = ( ^ [A2: prat,B2: prat] :
        ? [R: prat] :
          ( A2
          = ( padd @ B2 @ R ) ) ) ) ).

% p_greater_exists
thf(fact_83_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_84_padd__cancellative,axiom,
    ! [A: prat,X: prat,B: prat,Y: prat] :
      ( ( A
        = ( padd @ X @ B ) )
     => ( ( A
          = ( padd @ Y @ B ) )
       => ( X = Y ) ) ) ).

% padd_cancellative
thf(fact_85_padd__comm,axiom,
    ( padd
    = ( ^ [A2: prat,B2: prat] : ( padd @ B2 @ A2 ) ) ) ).

% padd_comm
thf(fact_86_padd__asso,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( padd @ ( padd @ A @ B ) @ C )
      = ( padd @ A @ ( padd @ B @ C ) ) ) ).

% padd_asso
thf(fact_87_pgte__antisym,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pgte @ B @ A )
       => ( A = B ) ) ) ).

% pgte_antisym
thf(fact_88_Nitpick_OEps__psimp,axiom,
    ! [P2: product_prod_nat_nat > $o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( P2 @ X )
     => ( ~ ( P2 @ Y )
       => ( ( ( fChoic6978938873391328853at_nat @ P2 )
            = Y )
         => ( ( fChoic6978938873391328853at_nat @ P2 )
            = X ) ) ) ) ).

% Nitpick.Eps_psimp
thf(fact_89_Nitpick_OEps__psimp,axiom,
    ! [P2: produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char,Y: produc1457211279475724562t_char] :
      ( ( P2 @ X )
     => ( ~ ( P2 @ Y )
       => ( ( ( fChoic2476838209665704038t_char @ P2 )
            = Y )
         => ( ( fChoic2476838209665704038t_char @ P2 )
            = X ) ) ) ) ).

% Nitpick.Eps_psimp
thf(fact_90_Nitpick_OEps__psimp,axiom,
    ! [P2: prat > $o,X: prat,Y: prat] :
      ( ( P2 @ X )
     => ( ~ ( P2 @ Y )
       => ( ( ( fChoice_prat @ P2 )
            = Y )
         => ( ( fChoice_prat @ P2 )
            = X ) ) ) ) ).

% Nitpick.Eps_psimp
thf(fact_91_minus__empty,axiom,
    ! [Pi3: ( produc1457211279475724562t_char > prat ) > prat] :
      ( Pi3
      = ( add_ma5081279659257173972r_prat @ Pi3 @ empty_6616431780912394577r_prat ) ) ).

% minus_empty
thf(fact_92_minus__empty,axiom,
    ! [Pi3: a > prat] :
      ( Pi3
      = ( add_masks_a @ Pi3 @ empty_mask_a ) ) ).

% minus_empty
thf(fact_93_minus__empty,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat] :
      ( Pi3
      = ( add_ma6586698345353345811t_char @ Pi3 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_94_comp__one__def,axiom,
    ( comp_one
    = ( ^ [P: prat] :
          ( fChoice_prat
          @ ^ [R: prat] :
              ( ( padd @ P @ R )
              = pwrite ) ) ) ) ).

% comp_one_def
thf(fact_95_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_96_pgt__implies__pgte,axiom,
    ! [A: prat,B: prat] :
      ( ( pgt @ A @ B )
     => ( pgte @ A @ B ) ) ).

% pgt_implies_pgte
thf(fact_97_not__pgte__charact,axiom,
    ! [A: prat,B: prat] :
      ( ( ~ ( pgte @ A @ B ) )
      = ( pgt @ B @ A ) ) ).

% not_pgte_charact
thf(fact_98_padd__comp__one,axiom,
    ! [X: prat] :
      ( ( pgte @ pwrite @ X )
     => ( ( padd @ X @ ( comp_one @ X ) )
        = pwrite ) ) ).

% padd_comp_one
thf(fact_99_mem__Collect__eq,axiom,
    ! [A: $o,P2: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_100_mem__Collect__eq,axiom,
    ! [A: set_prat,P2: set_prat > $o] :
      ( ( member_set_prat @ A @ ( collect_set_prat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
    ! [A: produc3336119888008106215r_prat,P2: produc3336119888008106215r_prat > $o] :
      ( ( member1381939458143960592r_prat @ A @ ( collec1156307336577696338r_prat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
    ! [A: list_char,P2: list_char > $o] :
      ( ( member_list_char @ A @ ( collect_list_char @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
    ! [A: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( member2336342582916103932r_prat @ A @ ( collec7093475260167521854r_prat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
    ! [A: nat > nat,P2: ( nat > nat ) > $o] :
      ( ( member_nat_nat @ A @ ( collect_nat_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_108_Collect__mem__eq,axiom,
    ! [A3: set_o] :
      ( ( collect_o
        @ ^ [X5: $o] : ( member_o @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
    ! [A3: set_set_prat] :
      ( ( collect_set_prat
        @ ^ [X5: set_prat] : ( member_set_prat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
    ! [A3: set_Pr1685704261205283783r_prat] :
      ( ( collec1156307336577696338r_prat
        @ ^ [X5: produc3336119888008106215r_prat] : ( member1381939458143960592r_prat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
    ! [A3: set_list_char] :
      ( ( collect_list_char
        @ ^ [X5: list_char] : ( member_list_char @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_112_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
    ! [A3: set_Pr3433046674199179443r_prat] :
      ( ( collec7093475260167521854r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A3: set_nat_nat] :
      ( ( collect_nat_nat
        @ ^ [X5: nat > nat] : ( member_nat_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_117_Collect__cong,axiom,
    ! [P2: produc3336119888008106215r_prat > $o,Q: produc3336119888008106215r_prat > $o] :
      ( ! [X2: produc3336119888008106215r_prat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec1156307336577696338r_prat @ P2 )
        = ( collec1156307336577696338r_prat @ Q ) ) ) ).

% Collect_cong
thf(fact_118_Collect__cong,axiom,
    ! [P2: list_char > $o,Q: list_char > $o] :
      ( ! [X2: list_char] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_list_char @ P2 )
        = ( collect_list_char @ Q ) ) ) ).

% Collect_cong
thf(fact_119_Collect__cong,axiom,
    ! [P2: set_nat > $o,Q: set_nat > $o] :
      ( ! [X2: set_nat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_set_nat @ P2 )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_120_Collect__cong,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ! [X2: produc1457211279475724562t_char > prat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec7093475260167521854r_prat @ P2 )
        = ( collec7093475260167521854r_prat @ Q ) ) ) ).

% Collect_cong
thf(fact_121_Collect__cong,axiom,
    ! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
      ( ! [X2: nat > nat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_nat_nat @ P2 )
        = ( collect_nat_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_122_Collect__cong,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_123_Collect__cong,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( P2 @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec3392354462482085612at_nat @ P2 )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_124_half__plus__half,axiom,
    ( ( padd @ half @ half )
    = pwrite ) ).

% half_plus_half
thf(fact_125_padd__one__ineq__sum,axiom,
    ! [A: prat,B: prat,X: prat,Aa: prat,Bb: prat] :
      ( ( ( padd @ A @ B )
        = pwrite )
     => ( ( pgte @ X @ Aa )
       => ( ( pgte @ X @ Bb )
         => ( pgte @ X @ ( padd @ ( pmult @ A @ Aa ) @ ( pmult @ B @ Bb ) ) ) ) ) ) ).

% padd_one_ineq_sum
thf(fact_126_multiply__smaller__pwrite,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ pwrite @ A )
     => ( ( pgte @ pwrite @ B )
       => ( pgte @ pwrite @ ( pmult @ A @ B ) ) ) ) ).

% multiply_smaller_pwrite
thf(fact_127_pinv__pwrite,axiom,
    ( ( pinv @ pwrite )
    = pwrite ) ).

% pinv_pwrite
thf(fact_128_pmin__greater,axiom,
    ! [A: prat,B: prat] : ( pgte @ A @ ( pmin @ A @ B ) ) ).

% pmin_greater
thf(fact_129_pmin__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmin @ A @ B )
        = B ) ) ).

% pmin_is
thf(fact_130_empty__mask_Osimps,axiom,
    ( empty_3446695950879338768t_char
    = ( ^ [Hl2: produc1457211279475724562t_char] : pnone ) ) ).

% empty_mask.simps
thf(fact_131_empty__mask_Osimps,axiom,
    ( empty_mask_a
    = ( ^ [Hl2: a] : pnone ) ) ).

% empty_mask.simps
thf(fact_132_empty__mask_Osimps,axiom,
    ( empty_6616431780912394577r_prat
    = ( ^ [Hl2: produc1457211279475724562t_char > prat] : pnone ) ) ).

% empty_mask.simps
thf(fact_133_empty__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_134_empty__mask_Oelims,axiom,
    ! [X: a,Y: prat] :
      ( ( ( empty_mask_a @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_135_empty__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: prat] :
      ( ( ( empty_6616431780912394577r_prat @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_136_pmax__is__smaller,axiom,
    ! [X: prat,A: prat,B: prat] :
      ( ( pgte @ X @ A )
     => ( ( pgte @ X @ B )
       => ( pgte @ X @ ( pmax @ A @ B ) ) ) ) ).

% pmax_is_smaller
thf(fact_137_pmax__smaller,axiom,
    ! [A: prat,B: prat] : ( pgte @ ( pmax @ A @ B ) @ A ) ).

% pmax_smaller
thf(fact_138_pmult__comm,axiom,
    ( pmult
    = ( ^ [A2: prat,B2: prat] : ( pmult @ B2 @ A2 ) ) ) ).

% pmult_comm
thf(fact_139_pmult__special_I2_J,axiom,
    ! [X: prat] :
      ( ( pmult @ pnone @ X )
      = pnone ) ).

% pmult_special(2)
thf(fact_140_pmax__comm,axiom,
    ( pmax
    = ( ^ [A2: prat,B2: prat] : ( pmax @ B2 @ A2 ) ) ) ).

% pmax_comm
thf(fact_141_pmin__comm,axiom,
    ( pmin
    = ( ^ [A2: prat,B2: prat] : ( pmin @ B2 @ A2 ) ) ) ).

% pmin_comm
thf(fact_142_pmin__pmax,axiom,
    ! [X: prat,A: prat,B: prat] :
      ( ( pgte @ X @ ( pmin @ A @ B ) )
     => ( X
        = ( pmin @ ( pmax @ X @ A ) @ ( pmax @ X @ B ) ) ) ) ).

% pmin_pmax
thf(fact_143_pmult__order,axiom,
    ! [A: prat,B: prat,P7: prat] :
      ( ( pgte @ A @ B )
     => ( pgte @ ( pmult @ P7 @ A ) @ ( pmult @ B @ P7 ) ) ) ).

% pmult_order
thf(fact_144_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_145_pmult__padd,axiom,
    ! [A: prat,B: prat,X: prat,C: prat,Y: prat] :
      ( ( pmult @ A @ ( padd @ ( pmult @ B @ X ) @ ( pmult @ C @ Y ) ) )
      = ( padd @ ( pmult @ ( pmult @ A @ B ) @ X ) @ ( pmult @ ( pmult @ A @ C ) @ Y ) ) ) ).

% pmult_padd
thf(fact_146_pmult__special_I1_J,axiom,
    ! [X: prat] :
      ( ( pmult @ pwrite @ X )
      = X ) ).

% pmult_special(1)
thf(fact_147_padd__zero,axiom,
    ! [A: prat,B: prat] :
      ( ( pnone
        = ( padd @ A @ B ) )
      = ( ( A = pnone )
        & ( B = pnone ) ) ) ).

% padd_zero
thf(fact_148_pmax__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmax @ A @ B )
        = A ) ) ).

% pmax_is
thf(fact_149_multiply__mask__def,axiom,
    ( multiply_mask
    = ( ^ [P: prat,Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( pmult @ P @ ( Pi2 @ Hl2 ) ) ) ) ).

% multiply_mask_def
thf(fact_150_pinv__double__half,axiom,
    ! [P7: prat] :
      ( ( ppos @ P7 )
     => ( ( pmult @ half @ ( pinv @ P7 ) )
        = ( pinv @ ( padd @ P7 @ P7 ) ) ) ) ).

% pinv_double_half
thf(fact_151_half__between__0__1,axiom,
    ( ( ppos @ half )
    & ( pgt @ pwrite @ half ) ) ).

% half_between_0_1
thf(fact_152_pinv__pmult__ok,axiom,
    ! [P7: prat] :
      ( ( ppos @ P7 )
     => ( ( pmult @ P7 @ ( pinv @ P7 ) )
        = pwrite ) ) ).

% pinv_pmult_ok
thf(fact_153_pinv__def,axiom,
    ( pinv
    = ( pdiv @ pwrite ) ) ).

% pinv_def
thf(fact_154_pinv__inverts,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ B )
       => ( pgte @ ( pinv @ B ) @ ( pinv @ A ) ) ) ) ).

% pinv_inverts
thf(fact_155_empty__mask_Opelims,axiom,
    ! [X: a,Y: prat] :
      ( ( ( empty_mask_a @ X )
        = Y )
     => ( ( accp_a @ empty_mask_rel_a @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_a @ empty_mask_rel_a @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_156_empty__mask_Opelims,axiom,
    ! [X: produc5803078220529002682t_char,Y: prat] :
      ( ( ( empty_1134373636806348472t_char @ X )
        = Y )
     => ( ( accp_P224314924008452035t_char @ empty_5292206184366095105t_char @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P224314924008452035t_char @ empty_5292206184366095105t_char @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_157_empty__mask_Opelims,axiom,
    ! [X: produc4654305426090421966prat_a,Y: prat] :
      ( ( ( empty_2833097053203474636prat_a @ X )
        = Y )
     => ( ( accp_P8439977412596685783prat_a @ empty_7473894104531696405prat_a @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P8439977412596685783prat_a @ empty_7473894104531696405prat_a @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_158_empty__mask_Opelims,axiom,
    ! [X: product_prod_nat_nat,Y: prat] :
      ( ( ( empty_8898153353432991231at_nat @ X )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ empty_1894168904691715784at_nat @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P4275260045618599050at_nat @ empty_1894168904691715784at_nat @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_159_empty__mask_Opelims,axiom,
    ! [X: nat,Y: prat] :
      ( ( ( empty_mask_nat @ X )
        = Y )
     => ( ( accp_nat @ empty_mask_rel_nat @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_nat @ empty_mask_rel_nat @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_160_empty__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_161_empty__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: prat] :
      ( ( ( empty_6616431780912394577r_prat @ X )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_162_ppos__inv,axiom,
    ! [P7: prat] :
      ( ( ppos @ P7 )
     => ( ppos @ ( pinv @ P7 ) ) ) ).

% ppos_inv
thf(fact_163_ppos__eq__pnone,axiom,
    ( ppos
    = ( ^ [P: prat] : ( P != pnone ) ) ) ).

% ppos_eq_pnone
thf(fact_164_pmult__pdiv__cancel,axiom,
    ! [A: prat,X: prat] :
      ( ( ppos @ A )
     => ( ( pmult @ A @ ( pdiv @ X @ A ) )
        = X ) ) ).

% pmult_pdiv_cancel
thf(fact_165_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_166_pdiv__smaller,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ A )
       => ( pgte @ pwrite @ ( pdiv @ B @ A ) ) ) ) ).

% pdiv_smaller
thf(fact_167_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_168_pmult__ppos,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ppos @ ( pmult @ A @ B ) ) ) ) ).

% pmult_ppos
thf(fact_169_accp__induct__rule,axiom,
    ! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A: produc5803078220529002682t_char,P2: produc5803078220529002682t_char > $o] :
      ( ( accp_P224314924008452035t_char @ R2 @ A )
     => ( ! [X2: produc5803078220529002682t_char] :
            ( ( accp_P224314924008452035t_char @ R2 @ X2 )
           => ( ! [Y5: produc5803078220529002682t_char] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_170_accp__induct__rule,axiom,
    ! [R2: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,A: produc4654305426090421966prat_a,P2: produc4654305426090421966prat_a > $o] :
      ( ( accp_P8439977412596685783prat_a @ R2 @ A )
     => ( ! [X2: produc4654305426090421966prat_a] :
            ( ( accp_P8439977412596685783prat_a @ R2 @ X2 )
           => ( ! [Y5: produc4654305426090421966prat_a] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_171_accp__induct__rule,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( accp_P4275260045618599050at_nat @ R2 @ X2 )
           => ( ! [Y5: product_prod_nat_nat] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_172_accp__induct__rule,axiom,
    ! [R2: nat > nat > $o,A: nat,P2: nat > $o] :
      ( ( accp_nat @ R2 @ A )
     => ( ! [X2: nat] :
            ( ( accp_nat @ R2 @ X2 )
           => ( ! [Y5: nat] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_173_accp__induct__rule,axiom,
    ! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o] :
      ( ( accp_P8892882183480857371t_char @ R2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( accp_P8892882183480857371t_char @ R2 @ X2 )
           => ( ! [Y5: produc1457211279475724562t_char] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_174_accp__induct__rule,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X2: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X2 )
           => ( ! [Y5: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_175_not__accp__down,axiom,
    ! [R3: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X: produc5803078220529002682t_char] :
      ( ~ ( accp_P224314924008452035t_char @ R3 @ X )
     => ~ ! [Z2: produc5803078220529002682t_char] :
            ( ( R3 @ Z2 @ X )
           => ( accp_P224314924008452035t_char @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_176_not__accp__down,axiom,
    ! [R3: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,X: produc4654305426090421966prat_a] :
      ( ~ ( accp_P8439977412596685783prat_a @ R3 @ X )
     => ~ ! [Z2: produc4654305426090421966prat_a] :
            ( ( R3 @ Z2 @ X )
           => ( accp_P8439977412596685783prat_a @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_177_not__accp__down,axiom,
    ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ~ ( accp_P4275260045618599050at_nat @ R3 @ X )
     => ~ ! [Z2: product_prod_nat_nat] :
            ( ( R3 @ Z2 @ X )
           => ( accp_P4275260045618599050at_nat @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_178_not__accp__down,axiom,
    ! [R3: nat > nat > $o,X: nat] :
      ( ~ ( accp_nat @ R3 @ X )
     => ~ ! [Z2: nat] :
            ( ( R3 @ Z2 @ X )
           => ( accp_nat @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_179_not__accp__down,axiom,
    ! [R3: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char] :
      ( ~ ( accp_P8892882183480857371t_char @ R3 @ X )
     => ~ ! [Z2: produc1457211279475724562t_char] :
            ( ( R3 @ Z2 @ X )
           => ( accp_P8892882183480857371t_char @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_180_not__accp__down,axiom,
    ! [R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
      ( ~ ( accp_P4765339447491148764r_prat @ R3 @ X )
     => ~ ! [Z2: produc1457211279475724562t_char > prat] :
            ( ( R3 @ Z2 @ X )
           => ( accp_P4765339447491148764r_prat @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_181_accp__downward,axiom,
    ! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,B: produc5803078220529002682t_char,A: produc5803078220529002682t_char] :
      ( ( accp_P224314924008452035t_char @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P224314924008452035t_char @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_182_accp__downward,axiom,
    ! [R2: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,B: produc4654305426090421966prat_a,A: produc4654305426090421966prat_a] :
      ( ( accp_P8439977412596685783prat_a @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P8439977412596685783prat_a @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_183_accp__downward,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P4275260045618599050at_nat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_184_accp__downward,axiom,
    ! [R2: nat > nat > $o,B: nat,A: nat] :
      ( ( accp_nat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_nat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_185_accp__downward,axiom,
    ! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,B: produc1457211279475724562t_char,A: produc1457211279475724562t_char] :
      ( ( accp_P8892882183480857371t_char @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P8892882183480857371t_char @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_186_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_187_accp__induct,axiom,
    ! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A: produc5803078220529002682t_char,P2: produc5803078220529002682t_char > $o] :
      ( ( accp_P224314924008452035t_char @ R2 @ A )
     => ( ! [X2: produc5803078220529002682t_char] :
            ( ( accp_P224314924008452035t_char @ R2 @ X2 )
           => ( ! [Y5: produc5803078220529002682t_char] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_188_accp__induct,axiom,
    ! [R2: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,A: produc4654305426090421966prat_a,P2: produc4654305426090421966prat_a > $o] :
      ( ( accp_P8439977412596685783prat_a @ R2 @ A )
     => ( ! [X2: produc4654305426090421966prat_a] :
            ( ( accp_P8439977412596685783prat_a @ R2 @ X2 )
           => ( ! [Y5: produc4654305426090421966prat_a] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_189_accp__induct,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( accp_P4275260045618599050at_nat @ R2 @ X2 )
           => ( ! [Y5: product_prod_nat_nat] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_190_accp__induct,axiom,
    ! [R2: nat > nat > $o,A: nat,P2: nat > $o] :
      ( ( accp_nat @ R2 @ A )
     => ( ! [X2: nat] :
            ( ( accp_nat @ R2 @ X2 )
           => ( ! [Y5: nat] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_191_accp__induct,axiom,
    ! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o] :
      ( ( accp_P8892882183480857371t_char @ R2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( accp_P8892882183480857371t_char @ R2 @ X2 )
           => ( ! [Y5: produc1457211279475724562t_char] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_192_accp__induct,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X2: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X2 )
           => ( ! [Y5: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y5 @ X2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_193_accpI,axiom,
    ! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X: produc5803078220529002682t_char] :
      ( ! [Y4: produc5803078220529002682t_char] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P224314924008452035t_char @ R2 @ Y4 ) )
     => ( accp_P224314924008452035t_char @ R2 @ X ) ) ).

% accpI
thf(fact_194_accpI,axiom,
    ! [R2: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,X: produc4654305426090421966prat_a] :
      ( ! [Y4: produc4654305426090421966prat_a] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P8439977412596685783prat_a @ R2 @ Y4 ) )
     => ( accp_P8439977412596685783prat_a @ R2 @ X ) ) ).

% accpI
thf(fact_195_accpI,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [Y4: product_prod_nat_nat] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P4275260045618599050at_nat @ R2 @ Y4 ) )
     => ( accp_P4275260045618599050at_nat @ R2 @ X ) ) ).

% accpI
thf(fact_196_accpI,axiom,
    ! [R2: nat > nat > $o,X: nat] :
      ( ! [Y4: nat] :
          ( ( R2 @ Y4 @ X )
         => ( accp_nat @ R2 @ Y4 ) )
     => ( accp_nat @ R2 @ X ) ) ).

% accpI
thf(fact_197_accpI,axiom,
    ! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char] :
      ( ! [Y4: produc1457211279475724562t_char] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P8892882183480857371t_char @ R2 @ Y4 ) )
     => ( accp_P8892882183480857371t_char @ R2 @ X ) ) ).

% accpI
thf(fact_198_accpI,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
      ( ! [Y4: produc1457211279475724562t_char > prat] :
          ( ( R2 @ Y4 @ X )
         => ( accp_P4765339447491148764r_prat @ R2 @ Y4 ) )
     => ( accp_P4765339447491148764r_prat @ R2 @ X ) ) ).

% accpI
thf(fact_199_accp_Osimps,axiom,
    ( accp_P224314924008452035t_char
    = ( ^ [R: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A2: produc5803078220529002682t_char] :
        ? [X5: produc5803078220529002682t_char] :
          ( ( A2 = X5 )
          & ! [Y2: produc5803078220529002682t_char] :
              ( ( R @ Y2 @ X5 )
             => ( accp_P224314924008452035t_char @ R @ Y2 ) ) ) ) ) ).

% accp.simps
thf(fact_200_accp_Osimps,axiom,
    ( accp_P8439977412596685783prat_a
    = ( ^ [R: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,A2: produc4654305426090421966prat_a] :
        ? [X5: produc4654305426090421966prat_a] :
          ( ( A2 = X5 )
          & ! [Y2: produc4654305426090421966prat_a] :
              ( ( R @ Y2 @ X5 )
             => ( accp_P8439977412596685783prat_a @ R @ Y2 ) ) ) ) ) ).

% accp.simps
thf(fact_201_accp_Osimps,axiom,
    ( accp_P4275260045618599050at_nat
    = ( ^ [R: product_prod_nat_nat > product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
        ? [X5: product_prod_nat_nat] :
          ( ( A2 = X5 )
          & ! [Y2: product_prod_nat_nat] :
              ( ( R @ Y2 @ X5 )
             => ( accp_P4275260045618599050at_nat @ R @ Y2 ) ) ) ) ) ).

% accp.simps
thf(fact_202_accp_Osimps,axiom,
    ( accp_nat
    = ( ^ [R: nat > nat > $o,A2: nat] :
        ? [X5: nat] :
          ( ( A2 = X5 )
          & ! [Y2: nat] :
              ( ( R @ Y2 @ X5 )
             => ( accp_nat @ R @ Y2 ) ) ) ) ) ).

% accp.simps
thf(fact_203_accp_Osimps,axiom,
    ( accp_P8892882183480857371t_char
    = ( ^ [R: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A2: produc1457211279475724562t_char] :
        ? [X5: produc1457211279475724562t_char] :
          ( ( A2 = X5 )
          & ! [Y2: produc1457211279475724562t_char] :
              ( ( R @ Y2 @ X5 )
             => ( accp_P8892882183480857371t_char @ R @ Y2 ) ) ) ) ) ).

% accp.simps
thf(fact_204_accp_Osimps,axiom,
    ( accp_P4765339447491148764r_prat
    = ( ^ [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A2: produc1457211279475724562t_char > prat] :
        ? [X5: produc1457211279475724562t_char > prat] :
          ( ( A2 = X5 )
          & ! [Y2: produc1457211279475724562t_char > prat] :
              ( ( R @ Y2 @ X5 )
             => ( accp_P4765339447491148764r_prat @ R @ Y2 ) ) ) ) ) ).

% accp.simps
thf(fact_205_accp_Ocases,axiom,
    ! [R2: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,A: produc5803078220529002682t_char] :
      ( ( accp_P224314924008452035t_char @ R2 @ A )
     => ! [Y5: produc5803078220529002682t_char] :
          ( ( R2 @ Y5 @ A )
         => ( accp_P224314924008452035t_char @ R2 @ Y5 ) ) ) ).

% accp.cases
thf(fact_206_accp_Ocases,axiom,
    ! [R2: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,A: produc4654305426090421966prat_a] :
      ( ( accp_P8439977412596685783prat_a @ R2 @ A )
     => ! [Y5: produc4654305426090421966prat_a] :
          ( ( R2 @ Y5 @ A )
         => ( accp_P8439977412596685783prat_a @ R2 @ Y5 ) ) ) ).

% accp.cases
thf(fact_207_accp_Ocases,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ A )
     => ! [Y5: product_prod_nat_nat] :
          ( ( R2 @ Y5 @ A )
         => ( accp_P4275260045618599050at_nat @ R2 @ Y5 ) ) ) ).

% accp.cases
thf(fact_208_accp_Ocases,axiom,
    ! [R2: nat > nat > $o,A: nat] :
      ( ( accp_nat @ R2 @ A )
     => ! [Y5: nat] :
          ( ( R2 @ Y5 @ A )
         => ( accp_nat @ R2 @ Y5 ) ) ) ).

% accp.cases
thf(fact_209_accp_Ocases,axiom,
    ! [R2: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
      ( ( accp_P8892882183480857371t_char @ R2 @ A )
     => ! [Y5: produc1457211279475724562t_char] :
          ( ( R2 @ Y5 @ A )
         => ( accp_P8892882183480857371t_char @ R2 @ Y5 ) ) ) ).

% accp.cases
thf(fact_210_accp_Ocases,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ! [Y5: produc1457211279475724562t_char > prat] :
          ( ( R2 @ Y5 @ A )
         => ( accp_P4765339447491148764r_prat @ R2 @ Y5 ) ) ) ).

% accp.cases
thf(fact_211_add__acc__uni__mask,axiom,
    ( add_acc_a
    = ( ^ [Pi2: a > prat,Hl2: a,P: prat] : ( add_masks_a @ Pi2 @ ( uni_mask_a @ Hl2 @ P ) ) ) ) ).

% add_acc_uni_mask
thf(fact_212_add__acc__uni__mask,axiom,
    ( add_ac6043276672051819257t_char
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char,P: prat] : ( add_ma6586698345353345811t_char @ Pi2 @ ( uni_ma6578582744724537101t_char @ Hl2 @ P ) ) ) ) ).

% add_acc_uni_mask
thf(fact_213_some__in__eq,axiom,
    ! [A3: set_o] :
      ( ( member_o
        @ ( fChoice_o
          @ ^ [X5: $o] : ( member_o @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_o ) ) ).

% some_in_eq
thf(fact_214_some__in__eq,axiom,
    ! [A3: set_set_prat] :
      ( ( member_set_prat
        @ ( fChoice_set_prat
          @ ^ [X5: set_prat] : ( member_set_prat @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_set_prat ) ) ).

% some_in_eq
thf(fact_215_some__in__eq,axiom,
    ! [A3: set_Pr3433046674199179443r_prat] :
      ( ( member2336342582916103932r_prat
        @ ( fChoic3782516149722485159r_prat
          @ ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo3164680191535668511r_prat ) ) ).

% some_in_eq
thf(fact_216_some__in__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat
        @ ( fChoic6978938873391328853at_nat
          @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo2099793752762293965at_nat ) ) ).

% some_in_eq
thf(fact_217_some__in__eq,axiom,
    ! [A3: set_Pr2507339680178222706t_char] :
      ( ( member484235747325421115t_char
        @ ( fChoic2476838209665704038t_char
          @ ^ [X5: produc1457211279475724562t_char] : ( member484235747325421115t_char @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo1414094081888755934t_char ) ) ).

% some_in_eq
thf(fact_218_some__in__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( member_set_nat
        @ ( fChoice_set_nat
          @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_set_nat ) ) ).

% some_in_eq
thf(fact_219_some__in__eq,axiom,
    ! [A3: set_nat] :
      ( ( member_nat
        @ ( fChoice_nat
          @ ^ [X5: nat] : ( member_nat @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_nat ) ) ).

% some_in_eq
thf(fact_220_some__in__eq,axiom,
    ! [A3: set_prat] :
      ( ( member_prat
        @ ( fChoice_prat
          @ ^ [X5: prat] : ( member_prat @ X5 @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_prat ) ) ).

% some_in_eq
thf(fact_221_valid__maskI,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat] :
      ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi3 @ Hl3 ) )
     => ( ! [F: list_char] :
            ( ( Pi3 @ ( produc120197956887798346t_char @ null_nat @ F ) )
            = pnone )
       => ( valid_mask @ Pi3 ) ) ) ).

% valid_maskI
thf(fact_222_valid__mask_Osimps,axiom,
    ( valid_mask
    = ( ^ [Pi2: produc1457211279475724562t_char > prat] :
          ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi2 @ Hl2 ) )
          & ! [F2: list_char] :
              ( ( Pi2 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
              = pnone ) ) ) ) ).

% valid_mask.simps
thf(fact_223_valid__mask_Oelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X )
        = Y )
     => ( Y
        = ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
          & ! [F2: list_char] :
              ( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
              = pnone ) ) ) ) ).

% valid_mask.elims(1)
thf(fact_224_valid__mask_Oelims_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X )
     => ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl ) )
        & ! [F3: list_char] :
            ( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
            = pnone ) ) ) ).

% valid_mask.elims(2)
thf(fact_225_valid__mask_Oelims_I3_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X )
     => ~ ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
          & ! [F: list_char] :
              ( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
              = pnone ) ) ) ).

% valid_mask.elims(3)
thf(fact_226_valid__mask_Opelims_I3_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
           => ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
              & ! [F: list_char] :
                  ( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
                  = pnone ) ) ) ) ) ).

% valid_mask.pelims(3)
thf(fact_227_valid__mask_Opelims_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
           => ~ ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl ) )
                & ! [F3: list_char] :
                    ( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
                    = pnone ) ) ) ) ) ).

% valid_mask.pelims(2)
thf(fact_228_valid__mask_Opelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( Y
              = ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
                & ! [F2: list_char] :
                    ( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
                    = pnone ) ) )
           => ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X ) ) ) ) ).

% valid_mask.pelims(1)
thf(fact_229_valid__null__def,axiom,
    ( valid_null
    = ( ^ [Pi2: produc1457211279475724562t_char > prat] :
        ! [F2: list_char] :
          ( ( Pi2 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
          = pnone ) ) ) ).

% valid_null_def
thf(fact_230_empty__def,axiom,
    ( bot_bo6182153949361364531r_prat
    = ( collec1156307336577696338r_prat
      @ ^ [X5: produc3336119888008106215r_prat] : $false ) ) ).

% empty_def
thf(fact_231_empty__def,axiom,
    ( bot_bo8503570744482963627t_char
    = ( collect_list_char
      @ ^ [X5: list_char] : $false ) ) ).

% empty_def
thf(fact_232_empty__def,axiom,
    ( bot_bo3164680191535668511r_prat
    = ( collec7093475260167521854r_prat
      @ ^ [X5: produc1457211279475724562t_char > prat] : $false ) ) ).

% empty_def
thf(fact_233_empty__def,axiom,
    ( bot_bot_set_nat_nat
    = ( collect_nat_nat
      @ ^ [X5: nat > nat] : $false ) ) ).

% empty_def
thf(fact_234_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X5: set_nat] : $false ) ) ).

% empty_def
thf(fact_235_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X5: nat] : $false ) ) ).

% empty_def
thf(fact_236_empty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X5: product_prod_nat_nat] : $false ) ) ).

% empty_def
thf(fact_237_pred__equals__eq2,axiom,
    ! [R3: set_Pr7632435056502277254t_char,S: set_Pr7632435056502277254t_char] :
      ( ( ( ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R3 ) )
        = ( ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_238_pred__equals__eq2,axiom,
    ! [R3: set_Pr1935176096852982554t_char,S: set_Pr1935176096852982554t_char] :
      ( ( ( ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R3 ) )
        = ( ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_239_pred__equals__eq2,axiom,
    ! [R3: set_Pr1685704261205283783r_prat,S: set_Pr1685704261205283783r_prat] :
      ( ( ( ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R3 ) )
        = ( ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_240_pred__equals__eq2,axiom,
    ! [R3: set_Pr9117357208957936942prat_a,S: set_Pr9117357208957936942prat_a] :
      ( ( ( ^ [X5: a > prat,Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ R3 ) )
        = ( ^ [X5: a > prat,Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_241_pred__equals__eq2,axiom,
    ! [R3: set_Pr2748872423075033990prat_a,S: set_Pr2748872423075033990prat_a] :
      ( ( ( ^ [X5: a > prat,Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ R3 ) )
        = ( ^ [X5: a > prat,Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_242_pred__equals__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R3 ) )
        = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_243_uni__mask_Osimps,axiom,
    ( uni_ma6578582744724537101t_char
    = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char ) ) ).

% uni_mask.simps
thf(fact_244_uni__mask_Osimps,axiom,
    ( uni_mask_a
    = ( fun_upd_a_prat @ empty_mask_a ) ) ).

% uni_mask.simps
thf(fact_245_uni__mask_Osimps,axiom,
    ( uni_ma8757825773229499086r_prat
    = ( fun_up9120501182914450918t_prat @ empty_6616431780912394577r_prat ) ) ).

% uni_mask.simps
thf(fact_246_uni__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( uni_ma6578582744724537101t_char @ X @ Xa )
        = Y )
     => ( Y
        = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X @ Xa ) ) ) ).

% uni_mask.elims
thf(fact_247_uni__mask_Oelims,axiom,
    ! [X: a,Xa: prat,Y: a > prat] :
      ( ( ( uni_mask_a @ X @ Xa )
        = Y )
     => ( Y
        = ( fun_upd_a_prat @ empty_mask_a @ X @ Xa ) ) ) ).

% uni_mask.elims
thf(fact_248_uni__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: prat,Y: ( produc1457211279475724562t_char > prat ) > prat] :
      ( ( ( uni_ma8757825773229499086r_prat @ X @ Xa )
        = Y )
     => ( Y
        = ( fun_up9120501182914450918t_prat @ empty_6616431780912394577r_prat @ X @ Xa ) ) ) ).

% uni_mask.elims
thf(fact_249_image2__eqI,axiom,
    ! [B: nat,F4: $o > nat,X: $o,C: list_char,G: $o > list_char,A3: set_o] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_o @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr7618826833391781745t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_250_image2__eqI,axiom,
    ! [B: nat,F4: nat > nat,X: nat,C: list_char,G: nat > list_char,A3: set_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_nat @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr7397543130437154909t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_251_image2__eqI,axiom,
    ! [B: nat,F4: set_prat > nat,X: set_prat,C: list_char,G: set_prat > list_char,A3: set_set_prat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_set_prat @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr7301876082586963091t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_252_image2__eqI,axiom,
    ! [B: nat,F4: set_nat > nat,X: set_nat,C: list_char,G: set_nat > list_char,A3: set_set_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_set_nat @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr3129696893582918419t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_253_image2__eqI,axiom,
    ! [B: a > prat,F4: $o > a > prat,X: $o,C: a,G: $o > a,A3: set_o] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_o @ X @ A3 )
         => ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ B @ C ) @ ( bNF_Gr8379640946404278973prat_a @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_254_image2__eqI,axiom,
    ! [B: a > prat,F4: nat > a > prat,X: nat,C: a,G: nat > a,A3: set_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_nat @ X @ A3 )
         => ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ B @ C ) @ ( bNF_Gr4706834691868246441prat_a @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_255_image2__eqI,axiom,
    ! [B: a > prat,F4: set_prat > a > prat,X: set_prat,C: a,G: set_prat > a,A3: set_set_prat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_set_prat @ X @ A3 )
         => ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ B @ C ) @ ( bNF_Gr3930513041845697759prat_a @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_256_image2__eqI,axiom,
    ! [B: a > prat,F4: set_nat > a > prat,X: set_nat,C: a,G: set_nat > a,A3: set_set_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_set_nat @ X @ A3 )
         => ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ B @ C ) @ ( bNF_Gr8551326311310947167prat_a @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_257_image2__eqI,axiom,
    ! [B: nat,F4: ( produc1457211279475724562t_char > prat ) > nat,X: produc1457211279475724562t_char > prat,C: list_char,G: ( produc1457211279475724562t_char > prat ) > list_char,A3: set_Pr3433046674199179443r_prat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member2336342582916103932r_prat @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr8575425969231580588t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_258_image2__eqI,axiom,
    ! [B: a > prat,F4: $o > a > prat,X: $o,C: produc3375937593958228902prat_a,G: $o > produc3375937593958228902prat_a,A3: set_o] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member_o @ X @ A3 )
         => ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ B @ C ) @ ( bNF_Gr6386687543108125413prat_a @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_259_add__acc_Oelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char,Xb: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( add_ac6043276672051819257t_char @ X @ Xa @ Xb )
        = Y )
     => ( Y
        = ( fun_up829850848796697765r_prat @ X @ Xa @ ( padd @ ( X @ Xa ) @ Xb ) ) ) ) ).

% add_acc.elims
thf(fact_260_add__acc_Oelims,axiom,
    ! [X: a > prat,Xa: a,Xb: prat,Y: a > prat] :
      ( ( ( add_acc_a @ X @ Xa @ Xb )
        = Y )
     => ( Y
        = ( fun_upd_a_prat @ X @ Xa @ ( padd @ ( X @ Xa ) @ Xb ) ) ) ) ).

% add_acc.elims
thf(fact_261_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_262_add__acc_Osimps,axiom,
    ( add_ac6043276672051819257t_char
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char,P: prat] : ( fun_up829850848796697765r_prat @ Pi2 @ Hl2 @ ( padd @ ( Pi2 @ Hl2 ) @ P ) ) ) ) ).

% add_acc.simps
thf(fact_263_add__acc_Osimps,axiom,
    ( add_acc_a
    = ( ^ [Pi2: a > prat,Hl2: a,P: prat] : ( fun_upd_a_prat @ Pi2 @ Hl2 @ ( padd @ ( Pi2 @ Hl2 ) @ P ) ) ) ) ).

% add_acc.simps
thf(fact_264_uni__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( uni_ma6578582744724537101t_char @ X @ Xa )
        = Y )
     => ( ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X @ Xa ) )
       => ~ ( ( Y
              = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X @ Xa ) )
           => ~ ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X @ Xa ) ) ) ) ) ).

% uni_mask.pelims
thf(fact_265_uni__mask_Opelims,axiom,
    ! [X: a,Xa: prat,Y: a > prat] :
      ( ( ( uni_mask_a @ X @ Xa )
        = Y )
     => ( ( accp_P8961225421893513396a_prat @ uni_mask_rel_a @ ( product_Pair_a_prat @ X @ Xa ) )
       => ~ ( ( Y
              = ( fun_upd_a_prat @ empty_mask_a @ X @ Xa ) )
           => ~ ( accp_P8961225421893513396a_prat @ uni_mask_rel_a @ ( product_Pair_a_prat @ X @ Xa ) ) ) ) ) ).

% uni_mask.pelims
thf(fact_266_uni__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: prat,Y: ( produc1457211279475724562t_char > prat ) > prat] :
      ( ( ( uni_ma8757825773229499086r_prat @ X @ Xa )
        = Y )
     => ( ( accp_P6822921472154139145t_prat @ uni_ma4504258969416521949r_prat @ ( produc5508603645230141260t_prat @ X @ Xa ) )
       => ~ ( ( Y
              = ( fun_up9120501182914450918t_prat @ empty_6616431780912394577r_prat @ X @ Xa ) )
           => ~ ( accp_P6822921472154139145t_prat @ uni_ma4504258969416521949r_prat @ ( produc5508603645230141260t_prat @ X @ Xa ) ) ) ) ) ).

% uni_mask.pelims
thf(fact_267_add__acc_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char,Xb: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( add_ac6043276672051819257t_char @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P5529878251199943754r_prat @ add_ac7939326059656485296t_char @ ( produc5928798746291773453r_prat @ X @ ( produc2920179824973321483r_prat @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( fun_up829850848796697765r_prat @ X @ Xa @ ( padd @ ( X @ Xa ) @ Xb ) ) )
           => ~ ( accp_P5529878251199943754r_prat @ add_ac7939326059656485296t_char @ ( produc5928798746291773453r_prat @ X @ ( produc2920179824973321483r_prat @ Xa @ Xb ) ) ) ) ) ) ).

% add_acc.pelims
thf(fact_268_add__acc_Opelims,axiom,
    ! [X: a > prat,Xa: a,Xb: prat,Y: a > prat] :
      ( ( ( add_acc_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6402918970019748386a_prat @ add_acc_rel_a @ ( produc3933618890892774373a_prat @ X @ ( product_Pair_a_prat @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( fun_upd_a_prat @ X @ Xa @ ( padd @ ( X @ Xa ) @ Xb ) ) )
           => ~ ( accp_P6402918970019748386a_prat @ add_acc_rel_a @ ( produc3933618890892774373a_prat @ X @ ( product_Pair_a_prat @ Xa @ Xb ) ) ) ) ) ) ).

% add_acc.pelims
thf(fact_269_add__masks_Opelims,axiom,
    ! [X: a > prat,Xa: a > prat,Xb: a,Y: prat] :
      ( ( ( add_masks_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8439977412596685783prat_a @ add_masks_rel_a @ ( produc2856268368059332798prat_a @ X @ ( produc3557287861060643862prat_a @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P8439977412596685783prat_a @ add_masks_rel_a @ ( produc2856268368059332798prat_a @ X @ ( produc3557287861060643862prat_a @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_270_add__masks_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_271_full__mask__def,axiom,
    ( full_mask
    = ( ^ [Hl2: produc1457211279475724562t_char] :
          ( if_prat
          @ ( ( produc6204653154488278070t_char @ Hl2 )
            = null_nat )
          @ pnone
          @ pwrite ) ) ) ).

% full_mask_def
thf(fact_272_accp__acc__eq,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( accp_o
        @ ^ [X5: $o,Y2: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: $o] : ( member_o @ X5 @ ( acc_o @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_273_accp__acc__eq,axiom,
    ! [R2: set_Pr141576177145560269t_prat] :
      ( ( accp_set_prat
        @ ^ [X5: set_prat,Y2: set_prat] : ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: set_prat] : ( member_set_prat @ X5 @ ( acc_set_prat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_274_accp__acc__eq,axiom,
    ! [R2: set_Pr5488025237498180813et_nat] :
      ( ( accp_set_nat
        @ ^ [X5: set_nat,Y2: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: set_nat] : ( member_set_nat @ X5 @ ( acc_set_nat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_275_accp__acc__eq,axiom,
    ! [R2: set_Pr7915859021238449415t_char] :
      ( ( accp_P224314924008452035t_char
        @ ^ [X5: produc5803078220529002682t_char,Y2: produc5803078220529002682t_char] : ( member2120560544256518992t_char @ ( produc5760621871510326551t_char @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc5803078220529002682t_char] : ( member5171123587969721059t_char @ X5 @ ( acc_Pr5366037250316522495t_char @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_276_accp__acc__eq,axiom,
    ! [R2: set_Pr2088549235032396295prat_a] :
      ( ( accp_P8439977412596685783prat_a
        @ ^ [X5: produc4654305426090421966prat_a,Y2: produc4654305426090421966prat_a] : ( member7650881556348516432prat_a @ ( produc446452354332447767prat_a @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc4654305426090421966prat_a] : ( member5370254448528625399prat_a @ X5 @ ( acc_Pr1587061423467440147prat_a @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_277_accp__acc__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( accp_P4275260045618599050at_nat
        @ ^ [X5: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_278_accp__acc__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( accp_nat
        @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: nat] : ( member_nat @ X5 @ ( acc_nat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_279_accp__acc__eq,axiom,
    ! [R2: set_Pr3996677194857090567t_char] :
      ( ( accp_P8892882183480857371t_char
        @ ^ [X5: produc1457211279475724562t_char,Y2: produc1457211279475724562t_char] : ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc1457211279475724562t_char] : ( member484235747325421115t_char @ X5 @ ( acc_Pr3913195418537732951t_char @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_280_accp__acc__eq,axiom,
    ! [R2: set_Pr1685704261205283783r_prat] :
      ( ( accp_P4765339447491148764r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) ) ) ).

% accp_acc_eq
thf(fact_281_add__masks_Ocases,axiom,
    ! [X: produc5803078220529002682t_char] :
      ~ ! [Pi4: produc1457211279475724562t_char > prat,Pi5: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char] :
          ( X
         != ( produc2957999048406202538t_char @ Pi4 @ ( produc2591414526589101846t_char @ Pi5 @ Hl3 ) ) ) ).

% add_masks.cases
thf(fact_282_add__masks_Ocases,axiom,
    ! [X: produc4654305426090421966prat_a] :
      ~ ! [Pi4: a > prat,Pi5: a > prat,Hl3: a] :
          ( X
         != ( produc2856268368059332798prat_a @ Pi4 @ ( produc3557287861060643862prat_a @ Pi5 @ Hl3 ) ) ) ).

% add_masks.cases
thf(fact_283_acc_Ocases,axiom,
    ! [A: $o,R2: set_Product_prod_o_o] :
      ( ( member_o @ A @ ( acc_o @ R2 ) )
     => ! [Y5: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ A ) @ R2 )
         => ( member_o @ Y5 @ ( acc_o @ R2 ) ) ) ) ).

% acc.cases
thf(fact_284_acc_Ocases,axiom,
    ! [A: set_prat,R2: set_Pr141576177145560269t_prat] :
      ( ( member_set_prat @ A @ ( acc_set_prat @ R2 ) )
     => ! [Y5: set_prat] :
          ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Y5 @ A ) @ R2 )
         => ( member_set_prat @ Y5 @ ( acc_set_prat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_285_acc_Ocases,axiom,
    ! [A: set_nat,R2: set_Pr5488025237498180813et_nat] :
      ( ( member_set_nat @ A @ ( acc_set_nat @ R2 ) )
     => ! [Y5: set_nat] :
          ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y5 @ A ) @ R2 )
         => ( member_set_nat @ Y5 @ ( acc_set_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_286_acc_Ocases,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ! [Y5: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ A ) @ R2 )
         => ( member8440522571783428010at_nat @ Y5 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_287_acc_Ocases,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ! [Y5: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ A ) @ R2 )
         => ( member_nat @ Y5 @ ( acc_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_288_acc_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat] :
      ( ( member2336342582916103932r_prat @ A @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
     => ! [Y5: produc1457211279475724562t_char > prat] :
          ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y5 @ A ) @ R2 )
         => ( member2336342582916103932r_prat @ Y5 @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_289_acc_Osimps,axiom,
    ! [A: $o,R2: set_Product_prod_o_o] :
      ( ( member_o @ A @ ( acc_o @ R2 ) )
      = ( ? [X5: $o] :
            ( ( A = X5 )
            & ! [Y2: $o] :
                ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y2 @ X5 ) @ R2 )
               => ( member_o @ Y2 @ ( acc_o @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_290_acc_Osimps,axiom,
    ! [A: set_prat,R2: set_Pr141576177145560269t_prat] :
      ( ( member_set_prat @ A @ ( acc_set_prat @ R2 ) )
      = ( ? [X5: set_prat] :
            ( ( A = X5 )
            & ! [Y2: set_prat] :
                ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Y2 @ X5 ) @ R2 )
               => ( member_set_prat @ Y2 @ ( acc_set_prat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_291_acc_Osimps,axiom,
    ! [A: set_nat,R2: set_Pr5488025237498180813et_nat] :
      ( ( member_set_nat @ A @ ( acc_set_nat @ R2 ) )
      = ( ? [X5: set_nat] :
            ( ( A = X5 )
            & ! [Y2: set_nat] :
                ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y2 @ X5 ) @ R2 )
               => ( member_set_nat @ Y2 @ ( acc_set_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_292_acc_Osimps,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
      = ( ? [X5: product_prod_nat_nat] :
            ( ( A = X5 )
            & ! [Y2: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y2 @ X5 ) @ R2 )
               => ( member8440522571783428010at_nat @ Y2 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_293_acc_Osimps,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
      = ( ? [X5: nat] :
            ( ( A = X5 )
            & ! [Y2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X5 ) @ R2 )
               => ( member_nat @ Y2 @ ( acc_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_294_acc_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat] :
      ( ( member2336342582916103932r_prat @ A @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
      = ( ? [X5: produc1457211279475724562t_char > prat] :
            ( ( A = X5 )
            & ! [Y2: produc1457211279475724562t_char > prat] :
                ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y2 @ X5 ) @ R2 )
               => ( member2336342582916103932r_prat @ Y2 @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_295_acc_Ointros,axiom,
    ! [X: $o,R2: set_Product_prod_o_o] :
      ( ! [Y4: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X ) @ R2 )
         => ( member_o @ Y4 @ ( acc_o @ R2 ) ) )
     => ( member_o @ X @ ( acc_o @ R2 ) ) ) ).

% acc.intros
thf(fact_296_acc_Ointros,axiom,
    ! [X: set_prat,R2: set_Pr141576177145560269t_prat] :
      ( ! [Y4: set_prat] :
          ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Y4 @ X ) @ R2 )
         => ( member_set_prat @ Y4 @ ( acc_set_prat @ R2 ) ) )
     => ( member_set_prat @ X @ ( acc_set_prat @ R2 ) ) ) ).

% acc.intros
thf(fact_297_acc_Ointros,axiom,
    ! [X: set_nat,R2: set_Pr5488025237498180813et_nat] :
      ( ! [Y4: set_nat] :
          ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y4 @ X ) @ R2 )
         => ( member_set_nat @ Y4 @ ( acc_set_nat @ R2 ) ) )
     => ( member_set_nat @ X @ ( acc_set_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_298_acc_Ointros,axiom,
    ! [X: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X ) @ R2 )
         => ( member8440522571783428010at_nat @ Y4 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) )
     => ( member8440522571783428010at_nat @ X @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_299_acc_Ointros,axiom,
    ! [X: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X ) @ R2 )
         => ( member_nat @ Y4 @ ( acc_nat @ R2 ) ) )
     => ( member_nat @ X @ ( acc_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_300_acc_Ointros,axiom,
    ! [X: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat] :
      ( ! [Y4: produc1457211279475724562t_char > prat] :
          ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y4 @ X ) @ R2 )
         => ( member2336342582916103932r_prat @ Y4 @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) )
     => ( member2336342582916103932r_prat @ X @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) ) ).

% acc.intros
thf(fact_301_acc__induct,axiom,
    ! [A: $o,R2: set_Product_prod_o_o,P2: $o > $o] :
      ( ( member_o @ A @ ( acc_o @ R2 ) )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( acc_o @ R2 ) )
           => ( ! [Y5: $o] :
                  ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_302_acc__induct,axiom,
    ! [A: set_prat,R2: set_Pr141576177145560269t_prat,P2: set_prat > $o] :
      ( ( member_set_prat @ A @ ( acc_set_prat @ R2 ) )
     => ( ! [X2: set_prat] :
            ( ( member_set_prat @ X2 @ ( acc_set_prat @ R2 ) )
           => ( ! [Y5: set_prat] :
                  ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_303_acc__induct,axiom,
    ! [A: set_nat,R2: set_Pr5488025237498180813et_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A @ ( acc_set_nat @ R2 ) )
     => ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ ( acc_set_nat @ R2 ) )
           => ( ! [Y5: set_nat] :
                  ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_304_acc__induct,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
           => ( ! [Y5: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_305_acc__induct,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( acc_nat @ R2 ) )
           => ( ! [Y5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_306_acc__induct,axiom,
    ! [A: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( member2336342582916103932r_prat @ A @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
     => ( ! [X2: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X2 @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
           => ( ! [Y5: produc1457211279475724562t_char > prat] :
                  ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_307_acc__downward,axiom,
    ! [B: $o,R2: set_Product_prod_o_o,A: $o] :
      ( ( member_o @ B @ ( acc_o @ R2 ) )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 )
       => ( member_o @ A @ ( acc_o @ R2 ) ) ) ) ).

% acc_downward
thf(fact_308_acc__downward,axiom,
    ! [B: set_prat,R2: set_Pr141576177145560269t_prat,A: set_prat] :
      ( ( member_set_prat @ B @ ( acc_set_prat @ R2 ) )
     => ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ A @ B ) @ R2 )
       => ( member_set_prat @ A @ ( acc_set_prat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_309_acc__downward,axiom,
    ! [B: set_nat,R2: set_Pr5488025237498180813et_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( acc_set_nat @ R2 ) )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ R2 )
       => ( member_set_nat @ A @ ( acc_set_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_310_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_311_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_312_acc__downward,axiom,
    ! [B: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat,A: produc1457211279475724562t_char > prat] :
      ( ( member2336342582916103932r_prat @ B @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
     => ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ B ) @ R2 )
       => ( member2336342582916103932r_prat @ A @ ( acc_Pr1479980713244937752r_prat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_313_not__acc__down,axiom,
    ! [X: $o,R3: set_Product_prod_o_o] :
      ( ~ ( member_o @ X @ ( acc_o @ R3 ) )
     => ~ ! [Z2: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z2 @ X ) @ R3 )
           => ( member_o @ Z2 @ ( acc_o @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_314_not__acc__down,axiom,
    ! [X: set_prat,R3: set_Pr141576177145560269t_prat] :
      ( ~ ( member_set_prat @ X @ ( acc_set_prat @ R3 ) )
     => ~ ! [Z2: set_prat] :
            ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Z2 @ X ) @ R3 )
           => ( member_set_prat @ Z2 @ ( acc_set_prat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_315_not__acc__down,axiom,
    ! [X: set_nat,R3: set_Pr5488025237498180813et_nat] :
      ( ~ ( member_set_nat @ X @ ( acc_set_nat @ R3 ) )
     => ~ ! [Z2: set_nat] :
            ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Z2 @ X ) @ R3 )
           => ( member_set_nat @ Z2 @ ( acc_set_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_316_not__acc__down,axiom,
    ! [X: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( acc_Pr2806007504492836550at_nat @ R3 ) )
     => ~ ! [Z2: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z2 @ X ) @ R3 )
           => ( member8440522571783428010at_nat @ Z2 @ ( acc_Pr2806007504492836550at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_317_not__acc__down,axiom,
    ! [X: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member_nat @ X @ ( acc_nat @ R3 ) )
     => ~ ! [Z2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z2 @ X ) @ R3 )
           => ( member_nat @ Z2 @ ( acc_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_318_not__acc__down,axiom,
    ! [X: produc1457211279475724562t_char > prat,R3: set_Pr1685704261205283783r_prat] :
      ( ~ ( member2336342582916103932r_prat @ X @ ( acc_Pr1479980713244937752r_prat @ R3 ) )
     => ~ ! [Z2: produc1457211279475724562t_char > prat] :
            ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Z2 @ X ) @ R3 )
           => ( member2336342582916103932r_prat @ Z2 @ ( acc_Pr1479980713244937752r_prat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_319_acc__induct__rule,axiom,
    ! [A: $o,R2: set_Product_prod_o_o,P2: $o > $o] :
      ( ( member_o @ A @ ( acc_o @ R2 ) )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( acc_o @ R2 ) )
           => ( ! [Y5: $o] :
                  ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_320_acc__induct__rule,axiom,
    ! [A: set_prat,R2: set_Pr141576177145560269t_prat,P2: set_prat > $o] :
      ( ( member_set_prat @ A @ ( acc_set_prat @ R2 ) )
     => ( ! [X2: set_prat] :
            ( ( member_set_prat @ X2 @ ( acc_set_prat @ R2 ) )
           => ( ! [Y5: set_prat] :
                  ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_321_acc__induct__rule,axiom,
    ! [A: set_nat,R2: set_Pr5488025237498180813et_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A @ ( acc_set_nat @ R2 ) )
     => ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ ( acc_set_nat @ R2 ) )
           => ( ! [Y5: set_nat] :
                  ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_322_acc__induct__rule,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
           => ( ! [Y5: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_323_acc__induct__rule,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( acc_nat @ R2 ) )
           => ( ! [Y5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_324_acc__induct__rule,axiom,
    ! [A: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( member2336342582916103932r_prat @ A @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
     => ( ! [X2: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X2 @ ( acc_Pr1479980713244937752r_prat @ R2 ) )
           => ( ! [Y5: produc1457211279475724562t_char > prat] :
                  ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y5 @ X2 ) @ R2 )
                 => ( P2 @ Y5 ) )
             => ( P2 @ X2 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_325_acc__def,axiom,
    ( acc_Pr8150405688010203948r_prat
    = ( ^ [R: set_Pr113032449794829511r_prat] :
          ( collec1156307336577696338r_prat
          @ ( accp_P8393741129312856816r_prat
            @ ^ [X5: produc3336119888008106215r_prat,Y2: produc3336119888008106215r_prat] : ( member4925443958475225872r_prat @ ( produc3047153517949909847r_prat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_326_acc__def,axiom,
    ( acc_list_char
    = ( ^ [R: set_Pr4480099295215233283t_char] :
          ( collect_list_char
          @ ( accp_list_char
            @ ^ [X5: list_char,Y2: list_char] : ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_327_acc__def,axiom,
    ( acc_set_nat
    = ( ^ [R: set_Pr5488025237498180813et_nat] :
          ( collect_set_nat
          @ ( accp_set_nat
            @ ^ [X5: set_nat,Y2: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_328_acc__def,axiom,
    ( acc_nat_nat
    = ( ^ [R: set_Pr7682762132356531903at_nat] :
          ( collect_nat_nat
          @ ( accp_nat_nat
            @ ^ [X5: nat > nat,Y2: nat > nat] : ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_329_acc__def,axiom,
    ( acc_Pr5366037250316522495t_char
    = ( ^ [R: set_Pr7915859021238449415t_char] :
          ( collec1912346231567511845t_char
          @ ( accp_P224314924008452035t_char
            @ ^ [X5: produc5803078220529002682t_char,Y2: produc5803078220529002682t_char] : ( member2120560544256518992t_char @ ( produc5760621871510326551t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_330_acc__def,axiom,
    ( acc_Pr1587061423467440147prat_a
    = ( ^ [R: set_Pr2088549235032396295prat_a] :
          ( collec8956631807754062137prat_a
          @ ( accp_P8439977412596685783prat_a
            @ ^ [X5: produc4654305426090421966prat_a,Y2: produc4654305426090421966prat_a] : ( member7650881556348516432prat_a @ ( produc446452354332447767prat_a @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_331_acc__def,axiom,
    ( acc_Pr3913195418537732951t_char
    = ( ^ [R: set_Pr3996677194857090567t_char] :
          ( collec2458894688891239037t_char
          @ ( accp_P8892882183480857371t_char
            @ ^ [X5: produc1457211279475724562t_char,Y2: produc1457211279475724562t_char] : ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_332_acc__def,axiom,
    ( acc_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ( accp_nat
            @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_333_acc__def,axiom,
    ( acc_Pr2806007504492836550at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat] :
          ( collec3392354462482085612at_nat
          @ ( accp_P4275260045618599050at_nat
            @ ^ [X5: product_prod_nat_nat,Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_334_acc__def,axiom,
    ( acc_Pr1479980713244937752r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat] :
          ( collec7093475260167521854r_prat
          @ ( accp_P4765339447491148764r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% acc_def
thf(fact_335_pre__get__m_Osimps,axiom,
    pre_get_m_nat_nat = product_fst_nat_nat ).

% pre_get_m.simps
thf(fact_336_pre__get__m_Osimps,axiom,
    pre_ge2099197569584961301ar_nat = produc5563429021163328950ar_nat ).

% pre_get_m.simps
thf(fact_337_pre__get__m_Osimps,axiom,
    pre_ge2740421702909910421t_char = produc6204653154488278070t_char ).

% pre_get_m.simps
thf(fact_338_pre__get__m_Oelims,axiom,
    ! [X: product_prod_nat_nat,Y: nat] :
      ( ( ( pre_get_m_nat_nat @ X )
        = Y )
     => ( Y
        = ( product_fst_nat_nat @ X ) ) ) ).

% pre_get_m.elims
thf(fact_339_pre__get__m_Oelims,axiom,
    ! [X: produc7606469099237612818ar_nat,Y: list_char] :
      ( ( ( pre_ge2099197569584961301ar_nat @ X )
        = Y )
     => ( Y
        = ( produc5563429021163328950ar_nat @ X ) ) ) ).

% pre_get_m.elims
thf(fact_340_pre__get__m_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X )
        = Y )
     => ( Y
        = ( produc6204653154488278070t_char @ X ) ) ) ).

% pre_get_m.elims
thf(fact_341_pre__get__m_Opelims,axiom,
    ! [X: produc5803078220529002682t_char,Y: produc1457211279475724562t_char > prat] :
      ( ( ( pre_ge1302051351345738485t_char @ X )
        = Y )
     => ( ( accp_P224314924008452035t_char @ pre_ge6804097903176389034t_char @ X )
       => ~ ( ( Y
              = ( produc583991874214702742t_char @ X ) )
           => ~ ( accp_P224314924008452035t_char @ pre_ge6804097903176389034t_char @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_342_pre__get__m_Opelims,axiom,
    ! [X: produc4654305426090421966prat_a,Y: a > prat] :
      ( ( ( pre_ge8710741304189273353prat_a @ X )
        = Y )
     => ( ( accp_P8439977412596685783prat_a @ pre_ge8191486456481419710prat_a @ X )
       => ~ ( ( Y
              = ( produc725351007466518186prat_a @ X ) )
           => ~ ( accp_P8439977412596685783prat_a @ pre_ge8191486456481419710prat_a @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_343_pre__get__m_Opelims,axiom,
    ! [X: product_prod_nat_nat,Y: nat] :
      ( ( ( pre_get_m_nat_nat @ X )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ pre_ge2650188337464070969at_nat @ X )
       => ~ ( ( Y
              = ( product_fst_nat_nat @ X ) )
           => ~ ( accp_P4275260045618599050at_nat @ pre_ge2650188337464070969at_nat @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_344_pre__get__m_Opelims,axiom,
    ! [X: produc7606469099237612818ar_nat,Y: list_char] :
      ( ( ( pre_ge2099197569584961301ar_nat @ X )
        = Y )
     => ( ( accp_P5818767966387969819ar_nat @ pre_ge8107634837601432778ar_nat @ X )
       => ~ ( ( Y
              = ( produc5563429021163328950ar_nat @ X ) )
           => ~ ( accp_P5818767966387969819ar_nat @ pre_ge8107634837601432778ar_nat @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_345_pre__get__m_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X )
       => ~ ( ( Y
              = ( produc6204653154488278070t_char @ X ) )
           => ~ ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_346_image2__def,axiom,
    ( bNF_Gr1946814795223983584at_nat
    = ( ^ [A4: set_o,F2: $o > nat,G2: $o > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: $o] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_o @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_347_image2__def,axiom,
    ( bNF_Gr7318156773085685964at_nat
    = ( ^ [A4: set_nat,F2: nat > nat,G2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_nat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_348_image2__def,axiom,
    ( bNF_Gr6835880997613667842at_nat
    = ( ^ [A4: set_set_prat,F2: set_prat > nat,G2: set_prat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: set_prat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_set_prat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_349_image2__def,axiom,
    ( bNF_Gr6681173980307635330at_nat
    = ( ^ [A4: set_set_nat,F2: set_nat > nat,G2: set_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: set_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_set_nat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_350_image2__def,axiom,
    ( bNF_Gr7618826833391781745t_char
    = ( ^ [A4: set_o,F2: $o > nat,G2: $o > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: $o] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_o @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_351_image2__def,axiom,
    ( bNF_Gr7397543130437154909t_char
    = ( ^ [A4: set_nat,F2: nat > nat,G2: nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_nat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_352_image2__def,axiom,
    ( bNF_Gr7301876082586963091t_char
    = ( ^ [A4: set_set_prat,F2: set_prat > nat,G2: set_prat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: set_prat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_set_prat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_353_image2__def,axiom,
    ( bNF_Gr3129696893582918419t_char
    = ( ^ [A4: set_set_nat,F2: set_nat > nat,G2: set_nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: set_nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_set_nat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_354_image2__def,axiom,
    ( bNF_Gr8379640946404278973prat_a
    = ( ^ [A4: set_o,F2: $o > a > prat,G2: $o > a] :
          ( collec2652489992548837521prat_a
          @ ^ [Uu: produc3375937593958228902prat_a] :
            ? [A2: $o] :
              ( ( Uu
                = ( produc3557287861060643862prat_a @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_o @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_355_image2__def,axiom,
    ( bNF_Gr4706834691868246441prat_a
    = ( ^ [A4: set_nat,F2: nat > a > prat,G2: nat > a] :
          ( collec2652489992548837521prat_a
          @ ^ [Uu: produc3375937593958228902prat_a] :
            ? [A2: nat] :
              ( ( Uu
                = ( produc3557287861060643862prat_a @ ( F2 @ A2 ) @ ( G2 @ A2 ) ) )
              & ( member_nat @ A2 @ A4 ) ) ) ) ) ).

% image2_def
thf(fact_356_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char,A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char,R2: set_Pr1685704261205283783r_prat,S2: set_Pr3996677194857090567t_char] :
      ( ( member3618324141103465936t_char @ ( produc2405850776935319959t_char @ ( produc2591414526589101846t_char @ A @ B ) @ ( produc2591414526589101846t_char @ A5 @ B4 ) ) @ ( lex_pr7982885678635892526t_char @ R2 @ S2 ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_357_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc2489117125269924006t_char,A5: produc1457211279475724562t_char > prat,B4: produc2489117125269924006t_char,R2: set_Pr1685704261205283783r_prat,S2: set_Pr6148874766616261511t_char] :
      ( ( member2120560544256518992t_char @ ( produc5760621871510326551t_char @ ( produc2957999048406202538t_char @ A @ B ) @ ( produc2957999048406202538t_char @ A5 @ B4 ) ) @ ( lex_pr5953235755497499842t_char @ R2 @ S2 ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member3618324141103465936t_char @ ( produc2405850776935319959t_char @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_358_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,A5: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat,R2: set_Pr1685704261205283783r_prat,S2: set_Pr1685704261205283783r_prat] :
      ( ( member4925443958475225872r_prat @ ( produc3047153517949909847r_prat @ ( produc1164665353665699927r_prat @ A @ B ) @ ( produc1164665353665699927r_prat @ A5 @ B4 ) ) @ ( lex_pr789199185301838959r_prat @ R2 @ S2 ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_359_in__lex__prod,axiom,
    ! [A: a > prat,B: produc3375937593958228902prat_a,A5: a > prat,B4: produc3375937593958228902prat_a,R2: set_Pr7218575088661268167a_prat,S2: set_Pr6010151575929665415prat_a] :
      ( ( member7650881556348516432prat_a @ ( produc446452354332447767prat_a @ ( produc2856268368059332798prat_a @ A @ B ) @ ( produc2856268368059332798prat_a @ A5 @ B4 ) ) @ ( lex_pr7488954505079209174prat_a @ R2 @ S2 ) )
      = ( ( member5727982501243699472a_prat @ ( produc328766861932669271a_prat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member5179433394215856592prat_a @ ( produc8912224561598320023prat_a @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_360_in__lex__prod,axiom,
    ! [A: a > prat,B: a,A5: a > prat,B4: a,R2: set_Pr7218575088661268167a_prat,S2: set_Product_prod_a_a] :
      ( ( member5179433394215856592prat_a @ ( produc8912224561598320023prat_a @ ( produc3557287861060643862prat_a @ A @ B ) @ ( produc3557287861060643862prat_a @ A5 @ B4 ) ) @ ( lex_prod_a_prat_a @ R2 @ S2 ) )
      = ( ( member5727982501243699472a_prat @ ( produc328766861932669271a_prat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_361_in__lex__prod,axiom,
    ! [A: nat,B: nat,A5: 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 @ A5 @ B4 ) ) @ ( lex_prod_nat_nat @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_362_in__lex__prod,axiom,
    ! [A: nat,B: list_char,A5: nat,B4: list_char,R2: set_Pr1261947904930325089at_nat,S2: set_Pr4480099295215233283t_char] :
      ( ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc120197956887798346t_char @ A5 @ B4 ) ) @ ( lex_pr1681056561245511266t_char @ R2 @ S2 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A5 ) @ R2 )
        | ( ( A = A5 )
          & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B @ B4 ) @ S2 ) ) ) ) ).

% in_lex_prod
thf(fact_363_accp__eq__acc,axiom,
    ( accp_o
    = ( ^ [R: $o > $o > $o,X5: $o] : ( member_o @ X5 @ ( acc_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_364_accp__eq__acc,axiom,
    ( accp_set_prat
    = ( ^ [R: set_prat > set_prat > $o,X5: set_prat] : ( member_set_prat @ X5 @ ( acc_set_prat @ ( collec8766186246528183128t_prat @ ( produc7606350445391071582prat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_365_accp__eq__acc,axiom,
    ( accp_set_nat
    = ( ^ [R: set_nat > set_nat > $o,X5: set_nat] : ( member_set_nat @ X5 @ ( acc_set_nat @ ( collec6662362479098859352et_nat @ ( produc6247414631856714078_nat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_366_accp__eq__acc,axiom,
    ( accp_P224314924008452035t_char
    = ( ^ [R: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X5: produc5803078220529002682t_char] : ( member5171123587969721059t_char @ X5 @ ( acc_Pr5366037250316522495t_char @ ( collec6882854600143928722t_char @ ( produc2026754670181186412char_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_367_accp__eq__acc,axiom,
    ( accp_P8439977412596685783prat_a
    = ( ^ [R: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,X5: produc4654305426090421966prat_a] : ( member5370254448528625399prat_a @ X5 @ ( acc_Pr1587061423467440147prat_a @ ( collec5955733625780314770prat_a @ ( produc3044637790421222508at_a_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_368_accp__eq__acc,axiom,
    ( accp_P4275260045618599050at_nat
    = ( ^ [R: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ ( acc_Pr2806007504492836550at_nat @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_369_accp__eq__acc,axiom,
    ( accp_P8892882183480857371t_char
    = ( ^ [R: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X5: produc1457211279475724562t_char] : ( member484235747325421115t_char @ X5 @ ( acc_Pr3913195418537732951t_char @ ( collec1162968953386343058t_char @ ( produc755220117336157804char_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_370_accp__eq__acc,axiom,
    ( accp_P4765339447491148764r_prat
    = ( ^ [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ ( acc_Pr1479980713244937752r_prat @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_371_accp__eq__acc,axiom,
    ( accp_nat
    = ( ^ [R: nat > nat > $o,X5: nat] : ( member_nat @ X5 @ ( acc_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R ) ) ) ) ) ) ).

% accp_eq_acc
thf(fact_372_acc__subset__induct,axiom,
    ! [D2: set_o,R3: set_Product_prod_o_o,X: $o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ D2 @ ( acc_o @ R3 ) )
     => ( ! [X2: $o,Z2: $o] :
            ( ( member_o @ X2 @ D2 )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z2 @ X2 ) @ R3 )
             => ( member_o @ Z2 @ D2 ) ) )
       => ( ( member_o @ X @ D2 )
         => ( ! [X2: $o] :
                ( ( member_o @ X2 @ D2 )
               => ( ! [Z3: $o] :
                      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Z3 @ X2 ) @ R3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_373_acc__subset__induct,axiom,
    ! [D2: set_set_prat,R3: set_Pr141576177145560269t_prat,X: set_prat,P2: set_prat > $o] :
      ( ( ord_le8696906415146163484t_prat @ D2 @ ( acc_set_prat @ R3 ) )
     => ( ! [X2: set_prat,Z2: set_prat] :
            ( ( member_set_prat @ X2 @ D2 )
           => ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Z2 @ X2 ) @ R3 )
             => ( member_set_prat @ Z2 @ D2 ) ) )
       => ( ( member_set_prat @ X @ D2 )
         => ( ! [X2: set_prat] :
                ( ( member_set_prat @ X2 @ D2 )
               => ( ! [Z3: set_prat] :
                      ( ( member413714312206344214t_prat @ ( produc1046168772652838309t_prat @ Z3 @ X2 ) @ R3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_374_acc__subset__induct,axiom,
    ! [D2: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ D2 @ ( acc_Pr2806007504492836550at_nat @ R3 ) )
     => ( ! [X2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ D2 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z2 @ X2 ) @ R3 )
             => ( member8440522571783428010at_nat @ Z2 @ D2 ) ) )
       => ( ( member8440522571783428010at_nat @ X @ D2 )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ D2 )
               => ( ! [Z3: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z3 @ X2 ) @ R3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_375_acc__subset__induct,axiom,
    ! [D2: set_Pr3433046674199179443r_prat,R3: set_Pr1685704261205283783r_prat,X: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6641467840663156819r_prat @ D2 @ ( acc_Pr1479980713244937752r_prat @ R3 ) )
     => ( ! [X2: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X2 @ D2 )
           => ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Z2 @ X2 ) @ R3 )
             => ( member2336342582916103932r_prat @ Z2 @ D2 ) ) )
       => ( ( member2336342582916103932r_prat @ X @ D2 )
         => ( ! [X2: produc1457211279475724562t_char > prat] :
                ( ( member2336342582916103932r_prat @ X2 @ D2 )
               => ( ! [Z3: produc1457211279475724562t_char > prat] :
                      ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Z3 @ X2 ) @ R3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_376_acc__subset__induct,axiom,
    ! [D2: set_set_nat,R3: set_Pr5488025237498180813et_nat,X: set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ D2 @ ( acc_set_nat @ R3 ) )
     => ( ! [X2: set_nat,Z2: set_nat] :
            ( ( member_set_nat @ X2 @ D2 )
           => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Z2 @ X2 ) @ R3 )
             => ( member_set_nat @ Z2 @ D2 ) ) )
       => ( ( member_set_nat @ X @ D2 )
         => ( ! [X2: set_nat] :
                ( ( member_set_nat @ X2 @ D2 )
               => ( ! [Z3: set_nat] :
                      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ Z3 @ X2 ) @ R3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_377_acc__subset__induct,axiom,
    ! [D2: set_nat,R3: set_Pr1261947904930325089at_nat,X: nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ D2 @ ( acc_nat @ R3 ) )
     => ( ! [X2: nat,Z2: nat] :
            ( ( member_nat @ X2 @ D2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z2 @ X2 ) @ R3 )
             => ( member_nat @ Z2 @ D2 ) ) )
       => ( ( member_nat @ X @ D2 )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ D2 )
               => ( ! [Z3: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z3 @ X2 ) @ R3 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_378_double__inclusion,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% double_inclusion
thf(fact_379_double__inclusion,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% double_inclusion
thf(fact_380_Eps__case__prod__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char] :
      ( ( fChoic1284671975056783866t_char
        @ ( produc186087365799766445char_o
          @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc2591414526589101846t_char @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_381_Eps__case__prod__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc2489117125269924006t_char] :
      ( ( fChoic3450237333247147022t_char
        @ ( produc8988401804721015065char_o
          @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc2489117125269924006t_char] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc2957999048406202538t_char @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_382_Eps__case__prod__eq,axiom,
    ! [X: a > prat,Y: produc3375937593958228902prat_a] :
      ( ( fChoic9085910172109011490prat_a
        @ ( produc1272685208547636229at_a_o
          @ ^ [X7: a > prat,Y6: produc3375937593958228902prat_a] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc2856268368059332798prat_a @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_383_Eps__case__prod__eq,axiom,
    ! [X: a > prat,Y: a] :
      ( ( fChoic952938374269289722prat_a
        @ ( produc7462986796979360941at_a_o
          @ ^ [X7: a > prat,Y6: a] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc3557287861060643862prat_a @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_384_Eps__case__prod__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > prat] :
      ( ( fChoic869448611404120123r_prat
        @ ( produc5069634436926473260prat_o
          @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > prat] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc1164665353665699927r_prat @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_385_Eps__case__prod__eq,axiom,
    ! [X: nat,Y: list_char] :
      ( ( fChoic2476838209665704038t_char
        @ ( produc8526122873445342073char_o
          @ ^ [X7: nat,Y6: list_char] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc120197956887798346t_char @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_386_Eps__case__prod__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( fChoic6978938873391328853at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X7: nat,Y6: nat] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_387_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_388_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_389_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_390_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat > list_char > $o] : ( ord_le2712619360209801397char_o @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_391_verit__comp__simplify1_I2_J,axiom,
    ! [A: ( produc1457211279475724562t_char > prat ) > $o] : ( ord_le2830794348966088778prat_o @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_392_verit__comp__simplify1_I2_J,axiom,
    ! [A: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] : ( ord_le6723339807950303054prat_o @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_393_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_394_Collect__subset,axiom,
    ! [A3: set_o,P2: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_395_Collect__subset,axiom,
    ! [A3: set_set_prat,P2: set_prat > $o] :
      ( ord_le8696906415146163484t_prat
      @ ( collect_set_prat
        @ ^ [X5: set_prat] :
            ( ( member_set_prat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_396_Collect__subset,axiom,
    ! [A3: set_Pr1685704261205283783r_prat,P2: produc3336119888008106215r_prat > $o] :
      ( ord_le295572976866519911r_prat
      @ ( collec1156307336577696338r_prat
        @ ^ [X5: produc3336119888008106215r_prat] :
            ( ( member1381939458143960592r_prat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_397_Collect__subset,axiom,
    ! [A3: set_list_char,P2: list_char > $o] :
      ( ord_le1207710660481778807t_char
      @ ( collect_list_char
        @ ^ [X5: list_char] :
            ( ( member_list_char @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_398_Collect__subset,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ord_le6641467840663156819r_prat
      @ ( collec7093475260167521854r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_399_Collect__subset,axiom,
    ! [A3: set_nat_nat,P2: ( nat > nat ) > $o] :
      ( ord_le9059583361652607317at_nat
      @ ( collect_nat_nat
        @ ^ [X5: nat > nat] :
            ( ( member_nat_nat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_400_Collect__subset,axiom,
    ! [A3: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_401_Collect__subset,axiom,
    ! [A3: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_402_Collect__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ A3 )
            & ( P2 @ X5 ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_403_subset__CollectI,axiom,
    ! [B3: set_o,A3: set_o,Q: $o > $o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ B3 @ A3 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_less_eq_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_404_subset__CollectI,axiom,
    ! [B3: set_set_prat,A3: set_set_prat,Q: set_prat > $o,P2: set_prat > $o] :
      ( ( ord_le8696906415146163484t_prat @ B3 @ A3 )
     => ( ! [X2: set_prat] :
            ( ( member_set_prat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le8696906415146163484t_prat
          @ ( collect_set_prat
            @ ^ [X5: set_prat] :
                ( ( member_set_prat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collect_set_prat
            @ ^ [X5: set_prat] :
                ( ( member_set_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_405_subset__CollectI,axiom,
    ! [B3: set_Pr1685704261205283783r_prat,A3: set_Pr1685704261205283783r_prat,Q: produc3336119888008106215r_prat > $o,P2: produc3336119888008106215r_prat > $o] :
      ( ( ord_le295572976866519911r_prat @ B3 @ A3 )
     => ( ! [X2: produc3336119888008106215r_prat] :
            ( ( member1381939458143960592r_prat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le295572976866519911r_prat
          @ ( collec1156307336577696338r_prat
            @ ^ [X5: produc3336119888008106215r_prat] :
                ( ( member1381939458143960592r_prat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collec1156307336577696338r_prat
            @ ^ [X5: produc3336119888008106215r_prat] :
                ( ( member1381939458143960592r_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_406_subset__CollectI,axiom,
    ! [B3: set_list_char,A3: set_list_char,Q: list_char > $o,P2: list_char > $o] :
      ( ( ord_le1207710660481778807t_char @ B3 @ A3 )
     => ( ! [X2: list_char] :
            ( ( member_list_char @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le1207710660481778807t_char
          @ ( collect_list_char
            @ ^ [X5: list_char] :
                ( ( member_list_char @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collect_list_char
            @ ^ [X5: list_char] :
                ( ( member_list_char @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_407_subset__CollectI,axiom,
    ! [B3: set_Pr3433046674199179443r_prat,A3: set_Pr3433046674199179443r_prat,Q: ( produc1457211279475724562t_char > prat ) > $o,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6641467840663156819r_prat @ B3 @ A3 )
     => ( ! [X2: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le6641467840663156819r_prat
          @ ( collec7093475260167521854r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat] :
                ( ( member2336342582916103932r_prat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collec7093475260167521854r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat] :
                ( ( member2336342582916103932r_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_408_subset__CollectI,axiom,
    ! [B3: set_nat_nat,A3: set_nat_nat,Q: ( nat > nat ) > $o,P2: ( nat > nat ) > $o] :
      ( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
     => ( ! [X2: nat > nat] :
            ( ( member_nat_nat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le9059583361652607317at_nat
          @ ( collect_nat_nat
            @ ^ [X5: nat > nat] :
                ( ( member_nat_nat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collect_nat_nat
            @ ^ [X5: nat > nat] :
                ( ( member_nat_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_409_subset__CollectI,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,Q: set_nat > $o,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
     => ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_410_subset__CollectI,axiom,
    ! [B3: set_nat,A3: set_nat,Q: nat > $o,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_411_subset__CollectI,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ B3 )
           => ( ( Q @ X2 )
             => ( P2 @ X2 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ B3 )
                & ( Q @ X5 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_412_subset__Collect__iff,axiom,
    ! [B3: set_o,A3: set_o,P2: $o > $o] :
      ( ( ord_less_eq_set_o @ B3 @ A3 )
     => ( ( ord_less_eq_set_o @ B3
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: $o] :
              ( ( member_o @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_413_subset__Collect__iff,axiom,
    ! [B3: set_set_prat,A3: set_set_prat,P2: set_prat > $o] :
      ( ( ord_le8696906415146163484t_prat @ B3 @ A3 )
     => ( ( ord_le8696906415146163484t_prat @ B3
          @ ( collect_set_prat
            @ ^ [X5: set_prat] :
                ( ( member_set_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: set_prat] :
              ( ( member_set_prat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_414_subset__Collect__iff,axiom,
    ! [B3: set_Pr1685704261205283783r_prat,A3: set_Pr1685704261205283783r_prat,P2: produc3336119888008106215r_prat > $o] :
      ( ( ord_le295572976866519911r_prat @ B3 @ A3 )
     => ( ( ord_le295572976866519911r_prat @ B3
          @ ( collec1156307336577696338r_prat
            @ ^ [X5: produc3336119888008106215r_prat] :
                ( ( member1381939458143960592r_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: produc3336119888008106215r_prat] :
              ( ( member1381939458143960592r_prat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_415_subset__Collect__iff,axiom,
    ! [B3: set_list_char,A3: set_list_char,P2: list_char > $o] :
      ( ( ord_le1207710660481778807t_char @ B3 @ A3 )
     => ( ( ord_le1207710660481778807t_char @ B3
          @ ( collect_list_char
            @ ^ [X5: list_char] :
                ( ( member_list_char @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: list_char] :
              ( ( member_list_char @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_416_subset__Collect__iff,axiom,
    ! [B3: set_Pr3433046674199179443r_prat,A3: set_Pr3433046674199179443r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6641467840663156819r_prat @ B3 @ A3 )
     => ( ( ord_le6641467840663156819r_prat @ B3
          @ ( collec7093475260167521854r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat] :
                ( ( member2336342582916103932r_prat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: produc1457211279475724562t_char > prat] :
              ( ( member2336342582916103932r_prat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_417_subset__Collect__iff,axiom,
    ! [B3: set_nat_nat,A3: set_nat_nat,P2: ( nat > nat ) > $o] :
      ( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
     => ( ( ord_le9059583361652607317at_nat @ B3
          @ ( collect_nat_nat
            @ ^ [X5: nat > nat] :
                ( ( member_nat_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: nat > nat] :
              ( ( member_nat_nat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_418_subset__Collect__iff,axiom,
    ! [B3: set_set_nat,A3: set_set_nat,P2: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ B3
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: set_nat] :
              ( ( member_set_nat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_419_subset__Collect__iff,axiom,
    ! [B3: set_nat,A3: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( ord_less_eq_set_nat @ B3
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_420_subset__Collect__iff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ B3
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ A3 )
                & ( P2 @ X5 ) ) ) )
        = ( ! [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ B3 )
             => ( P2 @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_421_acc__subset,axiom,
    ! [R1: set_Pr1685704261205283783r_prat,R22: set_Pr1685704261205283783r_prat] :
      ( ( ord_le295572976866519911r_prat @ R1 @ R22 )
     => ( ord_le6641467840663156819r_prat @ ( acc_Pr1479980713244937752r_prat @ R22 ) @ ( acc_Pr1479980713244937752r_prat @ R1 ) ) ) ).

% acc_subset
thf(fact_422_acc__subset,axiom,
    ! [R1: set_Pr8693737435421807431at_nat,R22: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R1 @ R22 )
     => ( ord_le3146513528884898305at_nat @ ( acc_Pr2806007504492836550at_nat @ R22 ) @ ( acc_Pr2806007504492836550at_nat @ R1 ) ) ) ).

% acc_subset
thf(fact_423_acc__subset,axiom,
    ! [R1: set_Pr5488025237498180813et_nat,R22: set_Pr5488025237498180813et_nat] :
      ( ( ord_le4402255868550298733et_nat @ R1 @ R22 )
     => ( ord_le6893508408891458716et_nat @ ( acc_set_nat @ R22 ) @ ( acc_set_nat @ R1 ) ) ) ).

% acc_subset
thf(fact_424_acc__subset,axiom,
    ! [R1: set_Pr1261947904930325089at_nat,R22: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ R1 @ R22 )
     => ( ord_less_eq_set_nat @ ( acc_nat @ R22 ) @ ( acc_nat @ R1 ) ) ) ).

% acc_subset
thf(fact_425_ssubsetI,axiom,
    ! [A3: set_Pr7632435056502277254t_char,B3: set_Pr7632435056502277254t_char] :
      ( ! [Pi5: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char] :
          ( ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ Pi5 @ H ) @ A3 )
         => ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ Pi5 @ H ) @ B3 ) )
     => ( ord_le3803256517986266150t_char @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_426_ssubsetI,axiom,
    ! [A3: set_Pr1935176096852982554t_char,B3: set_Pr1935176096852982554t_char] :
      ( ! [Pi5: produc1457211279475724562t_char > prat,H: produc2489117125269924006t_char] :
          ( ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ Pi5 @ H ) @ A3 )
         => ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ Pi5 @ H ) @ B3 ) )
     => ( ord_le5742415681470231226t_char @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_427_ssubsetI,axiom,
    ! [A3: set_Pr1685704261205283783r_prat,B3: set_Pr1685704261205283783r_prat] :
      ( ! [Pi5: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > prat] :
          ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Pi5 @ H ) @ A3 )
         => ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Pi5 @ H ) @ B3 ) )
     => ( ord_le295572976866519911r_prat @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_428_ssubsetI,axiom,
    ! [A3: set_Pr9117357208957936942prat_a,B3: set_Pr9117357208957936942prat_a] :
      ( ! [Pi5: a > prat,H: produc3375937593958228902prat_a] :
          ( ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ Pi5 @ H ) @ A3 )
         => ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ Pi5 @ H ) @ B3 ) )
     => ( ord_le7550862147281444046prat_a @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_429_ssubsetI,axiom,
    ! [A3: set_Pr2748872423075033990prat_a,B3: set_Pr2748872423075033990prat_a] :
      ( ! [Pi5: a > prat,H: a] :
          ( ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ Pi5 @ H ) @ A3 )
         => ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ Pi5 @ H ) @ B3 ) )
     => ( ord_le7405015385136159526prat_a @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_430_ssubsetI,axiom,
    ! [A3: set_Pr2507339680178222706t_char,B3: set_Pr2507339680178222706t_char] :
      ( ! [Pi5: nat,H: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi5 @ H ) @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi5 @ H ) @ B3 ) )
     => ( ord_le6315511645215477266t_char @ A3 @ B3 ) ) ).

% ssubsetI
thf(fact_431_exE__some,axiom,
    ! [P2: product_prod_nat_nat > $o,C: product_prod_nat_nat] :
      ( ? [X_1: product_prod_nat_nat] : ( P2 @ X_1 )
     => ( ( C
          = ( fChoic6978938873391328853at_nat @ P2 ) )
       => ( P2 @ C ) ) ) ).

% exE_some
thf(fact_432_exE__some,axiom,
    ! [P2: produc1457211279475724562t_char > $o,C: produc1457211279475724562t_char] :
      ( ? [X_1: produc1457211279475724562t_char] : ( P2 @ X_1 )
     => ( ( C
          = ( fChoic2476838209665704038t_char @ P2 ) )
       => ( P2 @ C ) ) ) ).

% exE_some
thf(fact_433_exE__some,axiom,
    ! [P2: prat > $o,C: prat] :
      ( ? [X_1: prat] : ( P2 @ X_1 )
     => ( ( C
          = ( fChoice_prat @ P2 ) )
       => ( P2 @ C ) ) ) ).

% exE_some
thf(fact_434_split__paired__Eps,axiom,
    ( fChoic1284671975056783866t_char
    = ( ^ [P4: produc2489117125269924006t_char > $o] :
          ( fChoic1284671975056783866t_char
          @ ( produc186087365799766445char_o
            @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char] : ( P4 @ ( produc2591414526589101846t_char @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_435_split__paired__Eps,axiom,
    ( fChoic3450237333247147022t_char
    = ( ^ [P4: produc5803078220529002682t_char > $o] :
          ( fChoic3450237333247147022t_char
          @ ( produc8988401804721015065char_o
            @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc2489117125269924006t_char] : ( P4 @ ( produc2957999048406202538t_char @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_436_split__paired__Eps,axiom,
    ( fChoic9085910172109011490prat_a
    = ( ^ [P4: produc4654305426090421966prat_a > $o] :
          ( fChoic9085910172109011490prat_a
          @ ( produc1272685208547636229at_a_o
            @ ^ [A2: a > prat,B2: produc3375937593958228902prat_a] : ( P4 @ ( produc2856268368059332798prat_a @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_437_split__paired__Eps,axiom,
    ( fChoic952938374269289722prat_a
    = ( ^ [P4: produc3375937593958228902prat_a > $o] :
          ( fChoic952938374269289722prat_a
          @ ( produc7462986796979360941at_a_o
            @ ^ [A2: a > prat,B2: a] : ( P4 @ ( produc3557287861060643862prat_a @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_438_split__paired__Eps,axiom,
    ( fChoic869448611404120123r_prat
    = ( ^ [P4: produc3336119888008106215r_prat > $o] :
          ( fChoic869448611404120123r_prat
          @ ( produc5069634436926473260prat_o
            @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : ( P4 @ ( produc1164665353665699927r_prat @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_439_split__paired__Eps,axiom,
    ( fChoic2476838209665704038t_char
    = ( ^ [P4: produc1457211279475724562t_char > $o] :
          ( fChoic2476838209665704038t_char
          @ ( produc8526122873445342073char_o
            @ ^ [A2: nat,B2: list_char] : ( P4 @ ( produc120197956887798346t_char @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_440_split__paired__Eps,axiom,
    ( fChoic6978938873391328853at_nat
    = ( ^ [P4: product_prod_nat_nat > $o] :
          ( fChoic6978938873391328853at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] : ( P4 @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_441_bind__const,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ( A3 = bot_bot_set_set_nat )
       => ( ( bind_set_nat_set_nat @ A3
            @ ^ [Uu: set_nat] : B3 )
          = bot_bot_set_set_nat ) )
      & ( ( A3 != bot_bot_set_set_nat )
       => ( ( bind_set_nat_set_nat @ A3
            @ ^ [Uu: set_nat] : B3 )
          = B3 ) ) ) ).

% bind_const
thf(fact_442_bind__const,axiom,
    ! [A3: set_set_nat,B3: set_nat] :
      ( ( ( A3 = bot_bot_set_set_nat )
       => ( ( bind_set_nat_nat @ A3
            @ ^ [Uu: set_nat] : B3 )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_set_nat )
       => ( ( bind_set_nat_nat @ A3
            @ ^ [Uu: set_nat] : B3 )
          = B3 ) ) ) ).

% bind_const
thf(fact_443_bind__const,axiom,
    ! [A3: set_nat,B3: set_set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( bind_nat_set_nat @ A3
            @ ^ [Uu: nat] : B3 )
          = bot_bot_set_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( bind_nat_set_nat @ A3
            @ ^ [Uu: nat] : B3 )
          = B3 ) ) ) ).

% bind_const
thf(fact_444_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_445_case__prodI,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char] :
      ( ( F4 @ A @ B )
     => ( produc186087365799766445char_o @ F4 @ ( produc2591414526589101846t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_446_case__prodI,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > $o,A: produc1457211279475724562t_char > prat,B: produc2489117125269924006t_char] :
      ( ( F4 @ A @ B )
     => ( produc8988401804721015065char_o @ F4 @ ( produc2957999048406202538t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_447_case__prodI,axiom,
    ! [F4: ( a > prat ) > produc3375937593958228902prat_a > $o,A: a > prat,B: produc3375937593958228902prat_a] :
      ( ( F4 @ A @ B )
     => ( produc1272685208547636229at_a_o @ F4 @ ( produc2856268368059332798prat_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_448_case__prodI,axiom,
    ! [F4: ( a > prat ) > a > $o,A: a > prat,B: a] :
      ( ( F4 @ A @ B )
     => ( produc7462986796979360941at_a_o @ F4 @ ( produc3557287861060643862prat_a @ A @ B ) ) ) ).

% case_prodI
thf(fact_449_case__prodI,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( F4 @ A @ B )
     => ( produc5069634436926473260prat_o @ F4 @ ( produc1164665353665699927r_prat @ A @ B ) ) ) ).

% case_prodI
thf(fact_450_case__prodI,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( F4 @ A @ B )
     => ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_451_case__prodI,axiom,
    ! [F4: nat > nat > $o,A: nat,B: nat] :
      ( ( F4 @ A @ B )
     => ( produc6081775807080527818_nat_o @ F4 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_452_case__prodI2,axiom,
    ! [P7: produc2489117125269924006t_char,C: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > $o] :
      ( ! [A6: produc1457211279475724562t_char > prat,B5: produc1457211279475724562t_char] :
          ( ( P7
            = ( produc2591414526589101846t_char @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc186087365799766445char_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_453_case__prodI2,axiom,
    ! [P7: produc5803078220529002682t_char,C: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > $o] :
      ( ! [A6: produc1457211279475724562t_char > prat,B5: produc2489117125269924006t_char] :
          ( ( P7
            = ( produc2957999048406202538t_char @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc8988401804721015065char_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_454_case__prodI2,axiom,
    ! [P7: produc4654305426090421966prat_a,C: ( a > prat ) > produc3375937593958228902prat_a > $o] :
      ( ! [A6: a > prat,B5: produc3375937593958228902prat_a] :
          ( ( P7
            = ( produc2856268368059332798prat_a @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc1272685208547636229at_a_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_455_case__prodI2,axiom,
    ! [P7: produc3375937593958228902prat_a,C: ( a > prat ) > a > $o] :
      ( ! [A6: a > prat,B5: a] :
          ( ( P7
            = ( produc3557287861060643862prat_a @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc7462986796979360941at_a_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_456_case__prodI2,axiom,
    ! [P7: produc3336119888008106215r_prat,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ! [A6: produc1457211279475724562t_char > prat,B5: produc1457211279475724562t_char > prat] :
          ( ( P7
            = ( produc1164665353665699927r_prat @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc5069634436926473260prat_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_457_case__prodI2,axiom,
    ! [P7: produc1457211279475724562t_char,C: nat > list_char > $o] :
      ( ! [A6: nat,B5: list_char] :
          ( ( P7
            = ( produc120197956887798346t_char @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc8526122873445342073char_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_458_case__prodI2,axiom,
    ! [P7: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A6: nat,B5: nat] :
          ( ( P7
            = ( product_Pair_nat_nat @ A6 @ B5 ) )
         => ( C @ A6 @ B5 ) )
     => ( produc6081775807080527818_nat_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_459_case__prod__conv,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_460_case__prod__conv,axiom,
    ! [F4: nat > list_char > nat,A: nat,B: list_char] :
      ( ( produc9163108120003956271ar_nat @ F4 @ ( produc120197956887798346t_char @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_461_case__prod__conv,axiom,
    ! [F4: nat > nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc2626176000494625587at_nat @ F4 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_462_case__prod__conv,axiom,
    ! [F4: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F4 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_463_case__prod__conv,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( produc5069634436926473260prat_o @ F4 @ ( produc1164665353665699927r_prat @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_464_case__prod__conv,axiom,
    ! [F4: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F4 @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F4 @ A @ B ) ) ).

% case_prod_conv
thf(fact_465_split__part,axiom,
    ! [P2: $o,Q: nat > list_char > $o] :
      ( ( produc8526122873445342073char_o
        @ ^ [A2: nat,B2: list_char] :
            ( P2
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab: produc1457211279475724562t_char] :
            ( P2
            & ( produc8526122873445342073char_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_466_split__part,axiom,
    ! [P2: $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( produc5069634436926473260prat_o
        @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] :
            ( P2
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab: produc3336119888008106215r_prat] :
            ( P2
            & ( produc5069634436926473260prat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_467_split__part,axiom,
    ! [P2: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A2: nat,B2: nat] :
            ( P2
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P2
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_468_fst__def,axiom,
    ( product_fst_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X1: nat,X22: nat] : X1 ) ) ).

% fst_def
thf(fact_469_fst__def,axiom,
    ( produc5563429021163328950ar_nat
    = ( produc2609824543435736768t_char
      @ ^ [X1: list_char,X22: nat] : X1 ) ) ).

% fst_def
thf(fact_470_fst__def,axiom,
    ( produc6204653154488278070t_char
    = ( produc9163108120003956271ar_nat
      @ ^ [X1: nat,X22: list_char] : X1 ) ) ).

% fst_def
thf(fact_471_case__prodD,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char] :
      ( ( produc186087365799766445char_o @ F4 @ ( produc2591414526589101846t_char @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_472_case__prodD,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > $o,A: produc1457211279475724562t_char > prat,B: produc2489117125269924006t_char] :
      ( ( produc8988401804721015065char_o @ F4 @ ( produc2957999048406202538t_char @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_473_case__prodD,axiom,
    ! [F4: ( a > prat ) > produc3375937593958228902prat_a > $o,A: a > prat,B: produc3375937593958228902prat_a] :
      ( ( produc1272685208547636229at_a_o @ F4 @ ( produc2856268368059332798prat_a @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_474_case__prodD,axiom,
    ! [F4: ( a > prat ) > a > $o,A: a > prat,B: a] :
      ( ( produc7462986796979360941at_a_o @ F4 @ ( produc3557287861060643862prat_a @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_475_case__prodD,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat] :
      ( ( produc5069634436926473260prat_o @ F4 @ ( produc1164665353665699927r_prat @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_476_case__prodD,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_477_case__prodD,axiom,
    ! [F4: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F4 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_478_case__prodE,axiom,
    ! [C: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > $o,P7: produc2489117125269924006t_char] :
      ( ( produc186087365799766445char_o @ C @ P7 )
     => ~ ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char] :
            ( ( P7
              = ( produc2591414526589101846t_char @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_479_case__prodE,axiom,
    ! [C: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > $o,P7: produc5803078220529002682t_char] :
      ( ( produc8988401804721015065char_o @ C @ P7 )
     => ~ ! [X2: produc1457211279475724562t_char > prat,Y4: produc2489117125269924006t_char] :
            ( ( P7
              = ( produc2957999048406202538t_char @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_480_case__prodE,axiom,
    ! [C: ( a > prat ) > produc3375937593958228902prat_a > $o,P7: produc4654305426090421966prat_a] :
      ( ( produc1272685208547636229at_a_o @ C @ P7 )
     => ~ ! [X2: a > prat,Y4: produc3375937593958228902prat_a] :
            ( ( P7
              = ( produc2856268368059332798prat_a @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_481_case__prodE,axiom,
    ! [C: ( a > prat ) > a > $o,P7: produc3375937593958228902prat_a] :
      ( ( produc7462986796979360941at_a_o @ C @ P7 )
     => ~ ! [X2: a > prat,Y4: a] :
            ( ( P7
              = ( produc3557287861060643862prat_a @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_482_case__prodE,axiom,
    ! [C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,P7: produc3336119888008106215r_prat] :
      ( ( produc5069634436926473260prat_o @ C @ P7 )
     => ~ ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > prat] :
            ( ( P7
              = ( produc1164665353665699927r_prat @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_483_case__prodE,axiom,
    ! [C: nat > list_char > $o,P7: produc1457211279475724562t_char] :
      ( ( produc8526122873445342073char_o @ C @ P7 )
     => ~ ! [X2: nat,Y4: list_char] :
            ( ( P7
              = ( produc120197956887798346t_char @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_484_case__prodE,axiom,
    ! [C: nat > nat > $o,P7: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P7 )
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P7
              = ( product_Pair_nat_nat @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 ) ) ) ).

% case_prodE
thf(fact_485_case__prodE2,axiom,
    ! [Q: $o > $o,P2: nat > list_char > $o,Z4: produc1457211279475724562t_char] :
      ( ( Q @ ( produc8526122873445342073char_o @ P2 @ Z4 ) )
     => ~ ! [X2: nat,Y4: list_char] :
            ( ( Z4
              = ( produc120197956887798346t_char @ X2 @ Y4 ) )
           => ~ ( Q @ ( P2 @ X2 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_486_case__prodE2,axiom,
    ! [Q: nat > $o,P2: nat > list_char > nat,Z4: produc1457211279475724562t_char] :
      ( ( Q @ ( produc9163108120003956271ar_nat @ P2 @ Z4 ) )
     => ~ ! [X2: nat,Y4: list_char] :
            ( ( Z4
              = ( produc120197956887798346t_char @ X2 @ Y4 ) )
           => ~ ( Q @ ( P2 @ X2 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_487_case__prodE2,axiom,
    ! [Q: product_prod_nat_nat > $o,P2: nat > nat > product_prod_nat_nat,Z4: product_prod_nat_nat] :
      ( ( Q @ ( produc2626176000494625587at_nat @ P2 @ Z4 ) )
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z4
              = ( product_Pair_nat_nat @ X2 @ Y4 ) )
           => ~ ( Q @ ( P2 @ X2 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_488_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P2: nat > nat > product_prod_nat_nat > $o,Z4: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P2 @ Z4 ) )
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z4
              = ( product_Pair_nat_nat @ X2 @ Y4 ) )
           => ~ ( Q @ ( P2 @ X2 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_489_case__prodE2,axiom,
    ! [Q: $o > $o,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Z4: produc3336119888008106215r_prat] :
      ( ( Q @ ( produc5069634436926473260prat_o @ P2 @ Z4 ) )
     => ~ ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > prat] :
            ( ( Z4
              = ( produc1164665353665699927r_prat @ X2 @ Y4 ) )
           => ~ ( Q @ ( P2 @ X2 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_490_case__prodE2,axiom,
    ! [Q: $o > $o,P2: nat > nat > $o,Z4: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P2 @ Z4 ) )
     => ~ ! [X2: nat,Y4: nat] :
            ( ( Z4
              = ( product_Pair_nat_nat @ X2 @ Y4 ) )
           => ~ ( Q @ ( P2 @ X2 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_491_case__prod__eta,axiom,
    ! [F4: produc1457211279475724562t_char > $o] :
      ( ( produc8526122873445342073char_o
        @ ^ [X5: nat,Y2: list_char] : ( F4 @ ( produc120197956887798346t_char @ X5 @ Y2 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_492_case__prod__eta,axiom,
    ! [F4: produc1457211279475724562t_char > nat] :
      ( ( produc9163108120003956271ar_nat
        @ ^ [X5: nat,Y2: list_char] : ( F4 @ ( produc120197956887798346t_char @ X5 @ Y2 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_493_case__prod__eta,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [X5: nat,Y2: nat] : ( F4 @ ( product_Pair_nat_nat @ X5 @ Y2 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_494_case__prod__eta,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X5: nat,Y2: nat] : ( F4 @ ( product_Pair_nat_nat @ X5 @ Y2 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_495_case__prod__eta,axiom,
    ! [F4: produc3336119888008106215r_prat > $o] :
      ( ( produc5069634436926473260prat_o
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( F4 @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_496_case__prod__eta,axiom,
    ! [F4: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X5: nat,Y2: nat] : ( F4 @ ( product_Pair_nat_nat @ X5 @ Y2 ) ) )
      = F4 ) ).

% case_prod_eta
thf(fact_497_mem__case__prodI2,axiom,
    ! [P7: produc1457211279475724562t_char,Z4: $o,C: nat > list_char > set_o] :
      ( ! [A6: nat,B5: list_char] :
          ( ( P7
            = ( produc120197956887798346t_char @ A6 @ B5 ) )
         => ( member_o @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_o @ Z4 @ ( produc3898378359715667545_set_o @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_498_mem__case__prodI2,axiom,
    ! [P7: produc1457211279475724562t_char,Z4: nat,C: nat > list_char > set_nat] :
      ( ! [A6: nat,B5: list_char] :
          ( ( P7
            = ( produc120197956887798346t_char @ A6 @ B5 ) )
         => ( member_nat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_nat @ Z4 @ ( produc2160937464213208549et_nat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_499_mem__case__prodI2,axiom,
    ! [P7: produc1457211279475724562t_char,Z4: set_prat,C: nat > list_char > set_set_prat] :
      ( ! [A6: nat,B5: list_char] :
          ( ( P7
            = ( produc120197956887798346t_char @ A6 @ B5 ) )
         => ( member_set_prat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_set_prat @ Z4 @ ( produc8806316560933437467t_prat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_500_mem__case__prodI2,axiom,
    ! [P7: produc1457211279475724562t_char,Z4: set_nat,C: nat > list_char > set_set_nat] :
      ( ! [A6: nat,B5: list_char] :
          ( ( P7
            = ( produc120197956887798346t_char @ A6 @ B5 ) )
         => ( member_set_nat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_set_nat @ Z4 @ ( produc6675599676517680795et_nat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_501_mem__case__prodI2,axiom,
    ! [P7: produc3375937593958228902prat_a,Z4: $o,C: ( a > prat ) > a > set_o] :
      ( ! [A6: a > prat,B5: a] :
          ( ( P7
            = ( produc3557287861060643862prat_a @ A6 @ B5 ) )
         => ( member_o @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_o @ Z4 @ ( produc7741865399517775757_set_o @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_502_mem__case__prodI2,axiom,
    ! [P7: produc3375937593958228902prat_a,Z4: nat,C: ( a > prat ) > a > set_nat] :
      ( ! [A6: a > prat,B5: a] :
          ( ( P7
            = ( produc3557287861060643862prat_a @ A6 @ B5 ) )
         => ( member_nat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_nat @ Z4 @ ( produc8536393513858091441et_nat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_503_mem__case__prodI2,axiom,
    ! [P7: produc3375937593958228902prat_a,Z4: set_prat,C: ( a > prat ) > a > set_set_prat] :
      ( ! [A6: a > prat,B5: a] :
          ( ( P7
            = ( produc3557287861060643862prat_a @ A6 @ B5 ) )
         => ( member_set_prat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_set_prat @ Z4 @ ( produc4584181444925067239t_prat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_504_mem__case__prodI2,axiom,
    ! [P7: produc3375937593958228902prat_a,Z4: set_nat,C: ( a > prat ) > a > set_set_nat] :
      ( ! [A6: a > prat,B5: a] :
          ( ( P7
            = ( produc3557287861060643862prat_a @ A6 @ B5 ) )
         => ( member_set_nat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_set_nat @ Z4 @ ( produc124007076447527527et_nat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_505_mem__case__prodI2,axiom,
    ! [P7: produc1457211279475724562t_char,Z4: produc1457211279475724562t_char > prat,C: nat > list_char > set_Pr3433046674199179443r_prat] :
      ( ! [A6: nat,B5: list_char] :
          ( ( P7
            = ( produc120197956887798346t_char @ A6 @ B5 ) )
         => ( member2336342582916103932r_prat @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member2336342582916103932r_prat @ Z4 @ ( produc2481503598742377364r_prat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_506_mem__case__prodI2,axiom,
    ! [P7: produc4654305426090421966prat_a,Z4: $o,C: ( a > prat ) > produc3375937593958228902prat_a > set_o] :
      ( ! [A6: a > prat,B5: produc3375937593958228902prat_a] :
          ( ( P7
            = ( produc2856268368059332798prat_a @ A6 @ B5 ) )
         => ( member_o @ Z4 @ ( C @ A6 @ B5 ) ) )
     => ( member_o @ Z4 @ ( produc8802671834256983781_set_o @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_507_mem__case__prodI,axiom,
    ! [Z4: $o,C: nat > list_char > set_o,A: nat,B: list_char] :
      ( ( member_o @ Z4 @ ( C @ A @ B ) )
     => ( member_o @ Z4 @ ( produc3898378359715667545_set_o @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_508_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_509_mem__case__prodI,axiom,
    ! [Z4: set_prat,C: nat > list_char > set_set_prat,A: nat,B: list_char] :
      ( ( member_set_prat @ Z4 @ ( C @ A @ B ) )
     => ( member_set_prat @ Z4 @ ( produc8806316560933437467t_prat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_510_mem__case__prodI,axiom,
    ! [Z4: set_nat,C: nat > list_char > set_set_nat,A: nat,B: list_char] :
      ( ( member_set_nat @ Z4 @ ( C @ A @ B ) )
     => ( member_set_nat @ Z4 @ ( produc6675599676517680795et_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_511_mem__case__prodI,axiom,
    ! [Z4: $o,C: ( a > prat ) > a > set_o,A: a > prat,B: a] :
      ( ( member_o @ Z4 @ ( C @ A @ B ) )
     => ( member_o @ Z4 @ ( produc7741865399517775757_set_o @ C @ ( produc3557287861060643862prat_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_512_mem__case__prodI,axiom,
    ! [Z4: nat,C: ( a > prat ) > a > set_nat,A: a > prat,B: a] :
      ( ( member_nat @ Z4 @ ( C @ A @ B ) )
     => ( member_nat @ Z4 @ ( produc8536393513858091441et_nat @ C @ ( produc3557287861060643862prat_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_513_mem__case__prodI,axiom,
    ! [Z4: set_prat,C: ( a > prat ) > a > set_set_prat,A: a > prat,B: a] :
      ( ( member_set_prat @ Z4 @ ( C @ A @ B ) )
     => ( member_set_prat @ Z4 @ ( produc4584181444925067239t_prat @ C @ ( produc3557287861060643862prat_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_514_mem__case__prodI,axiom,
    ! [Z4: set_nat,C: ( a > prat ) > a > set_set_nat,A: a > prat,B: a] :
      ( ( member_set_nat @ Z4 @ ( C @ A @ B ) )
     => ( member_set_nat @ Z4 @ ( produc124007076447527527et_nat @ C @ ( produc3557287861060643862prat_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_515_mem__case__prodI,axiom,
    ! [Z4: produc1457211279475724562t_char > prat,C: nat > list_char > set_Pr3433046674199179443r_prat,A: nat,B: list_char] :
      ( ( member2336342582916103932r_prat @ Z4 @ ( C @ A @ B ) )
     => ( member2336342582916103932r_prat @ Z4 @ ( produc2481503598742377364r_prat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_516_mem__case__prodI,axiom,
    ! [Z4: $o,C: ( a > prat ) > produc3375937593958228902prat_a > set_o,A: a > prat,B: produc3375937593958228902prat_a] :
      ( ( member_o @ Z4 @ ( C @ A @ B ) )
     => ( member_o @ Z4 @ ( produc8802671834256983781_set_o @ C @ ( produc2856268368059332798prat_a @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_517_case__prodI2_H,axiom,
    ! [P7: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( ( product_Pair_nat_nat @ A6 @ B5 )
            = P7 )
         => ( C @ A6 @ B5 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P7 @ X ) ) ).

% case_prodI2'
thf(fact_518_accp__subset,axiom,
    ! [R1: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,R22: produc5803078220529002682t_char > produc5803078220529002682t_char > $o] :
      ( ( ord_le6348261579570775310char_o @ R1 @ R22 )
     => ( ord_le7815076450745241763char_o @ ( accp_P224314924008452035t_char @ R22 ) @ ( accp_P224314924008452035t_char @ R1 ) ) ) ).

% accp_subset
thf(fact_519_accp__subset,axiom,
    ! [R1: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,R22: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o] :
      ( ( ord_le7575307955986077710at_a_o @ R1 @ R22 )
     => ( ord_le8178139514714933135at_a_o @ ( accp_P8439977412596685783prat_a @ R22 ) @ ( accp_P8439977412596685783prat_a @ R1 ) ) ) ).

% accp_subset
thf(fact_520_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_521_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_522_accp__subset,axiom,
    ! [R1: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,R22: produc1457211279475724562t_char > produc1457211279475724562t_char > $o] :
      ( ( ord_le2827606955515295502char_o @ R1 @ R22 )
     => ( ord_le5722155653058504523char_o @ ( accp_P8892882183480857371t_char @ R22 ) @ ( accp_P8892882183480857371t_char @ R1 ) ) ) ).

% accp_subset
thf(fact_523_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_524_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P7: product_prod_nat_nat,Z4: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P7 @ Z4 )
     => ~ ! [X2: nat,Y4: nat] :
            ( ( P7
              = ( product_Pair_nat_nat @ X2 @ Y4 ) )
           => ~ ( C @ X2 @ Y4 @ Z4 ) ) ) ).

% case_prodE'
thf(fact_525_case__prodD_H,axiom,
    ! [R3: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R3 @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R3 @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_526_accp__subset__induct,axiom,
    ! [D2: produc5803078220529002682t_char > $o,R3: produc5803078220529002682t_char > produc5803078220529002682t_char > $o,X: produc5803078220529002682t_char,P2: produc5803078220529002682t_char > $o] :
      ( ( ord_le7815076450745241763char_o @ D2 @ ( accp_P224314924008452035t_char @ R3 ) )
     => ( ! [X2: produc5803078220529002682t_char,Z2: produc5803078220529002682t_char] :
            ( ( D2 @ X2 )
           => ( ( R3 @ Z2 @ X2 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X2: produc5803078220529002682t_char] :
                ( ( D2 @ X2 )
               => ( ! [Z3: produc5803078220529002682t_char] :
                      ( ( R3 @ Z3 @ X2 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_527_accp__subset__induct,axiom,
    ! [D2: produc4654305426090421966prat_a > $o,R3: produc4654305426090421966prat_a > produc4654305426090421966prat_a > $o,X: produc4654305426090421966prat_a,P2: produc4654305426090421966prat_a > $o] :
      ( ( ord_le8178139514714933135at_a_o @ D2 @ ( accp_P8439977412596685783prat_a @ R3 ) )
     => ( ! [X2: produc4654305426090421966prat_a,Z2: produc4654305426090421966prat_a] :
            ( ( D2 @ X2 )
           => ( ( R3 @ Z2 @ X2 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X2: produc4654305426090421966prat_a] :
                ( ( D2 @ X2 )
               => ( ! [Z3: produc4654305426090421966prat_a] :
                      ( ( R3 @ Z3 @ X2 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_528_accp__subset__induct,axiom,
    ! [D2: product_prod_nat_nat > $o,R3: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D2 @ ( accp_P4275260045618599050at_nat @ R3 ) )
     => ( ! [X2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
            ( ( D2 @ X2 )
           => ( ( R3 @ Z2 @ X2 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( D2 @ X2 )
               => ( ! [Z3: product_prod_nat_nat] :
                      ( ( R3 @ Z3 @ X2 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_529_accp__subset__induct,axiom,
    ! [D2: nat > $o,R3: nat > nat > $o,X: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat_o @ D2 @ ( accp_nat @ R3 ) )
     => ( ! [X2: nat,Z2: nat] :
            ( ( D2 @ X2 )
           => ( ( R3 @ Z2 @ X2 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X2: nat] :
                ( ( D2 @ X2 )
               => ( ! [Z3: nat] :
                      ( ( R3 @ Z3 @ X2 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_530_accp__subset__induct,axiom,
    ! [D2: produc1457211279475724562t_char > $o,R3: produc1457211279475724562t_char > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char,P2: produc1457211279475724562t_char > $o] :
      ( ( ord_le5722155653058504523char_o @ D2 @ ( accp_P8892882183480857371t_char @ R3 ) )
     => ( ! [X2: produc1457211279475724562t_char,Z2: produc1457211279475724562t_char] :
            ( ( D2 @ X2 )
           => ( ( R3 @ Z2 @ X2 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X2: produc1457211279475724562t_char] :
                ( ( D2 @ X2 )
               => ( ! [Z3: produc1457211279475724562t_char] :
                      ( ( R3 @ Z3 @ X2 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_531_accp__subset__induct,axiom,
    ! [D2: ( produc1457211279475724562t_char > prat ) > $o,R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le2830794348966088778prat_o @ D2 @ ( accp_P4765339447491148764r_prat @ R3 ) )
     => ( ! [X2: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > prat] :
            ( ( D2 @ X2 )
           => ( ( R3 @ Z2 @ X2 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X2: produc1457211279475724562t_char > prat] :
                ( ( D2 @ X2 )
               => ( ! [Z3: produc1457211279475724562t_char > prat] :
                      ( ( R3 @ Z3 @ X2 )
                     => ( P2 @ Z3 ) )
                 => ( P2 @ X2 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_532_pred__subset__eq2,axiom,
    ! [R3: set_Pr7632435056502277254t_char,S: set_Pr7632435056502277254t_char] :
      ( ( ord_le1638870297938053199char_o
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ S ) )
      = ( ord_le3803256517986266150t_char @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_533_pred__subset__eq2,axiom,
    ! [R3: set_Pr1935176096852982554t_char,S: set_Pr1935176096852982554t_char] :
      ( ( ord_le6251799596582475195char_o
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ S ) )
      = ( ord_le5742415681470231226t_char @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_534_pred__subset__eq2,axiom,
    ! [R3: set_Pr9117357208957936942prat_a,S: set_Pr9117357208957936942prat_a] :
      ( ( ord_le3742656491906381991at_a_o
        @ ^ [X5: a > prat,Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: a > prat,Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ S ) )
      = ( ord_le7550862147281444046prat_a @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_535_pred__subset__eq2,axiom,
    ! [R3: set_Pr2748872423075033990prat_a,S: set_Pr2748872423075033990prat_a] :
      ( ( ord_le5399397932252467535at_a_o
        @ ^ [X5: a > prat,Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: a > prat,Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ S ) )
      = ( ord_le7405015385136159526prat_a @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_536_pred__subset__eq2,axiom,
    ! [R3: set_Pr1685704261205283783r_prat,S: set_Pr1685704261205283783r_prat] :
      ( ( ord_le6723339807950303054prat_o
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ S ) )
      = ( ord_le295572976866519911r_prat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_537_pred__subset__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( ord_le2712619360209801397char_o
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ S ) )
      = ( ord_le6315511645215477266t_char @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_538_pred__subset__eq,axiom,
    ! [R3: set_o,S: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X5: $o] : ( member_o @ X5 @ R3 )
        @ ^ [X5: $o] : ( member_o @ X5 @ S ) )
      = ( ord_less_eq_set_o @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_539_pred__subset__eq,axiom,
    ! [R3: set_set_prat,S: set_set_prat] :
      ( ( ord_le7917094133057636969prat_o
        @ ^ [X5: set_prat] : ( member_set_prat @ X5 @ R3 )
        @ ^ [X5: set_prat] : ( member_set_prat @ X5 @ S ) )
      = ( ord_le8696906415146163484t_prat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_540_pred__subset__eq,axiom,
    ! [R3: set_set_nat,S: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ R3 )
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ S ) )
      = ( ord_le6893508408891458716et_nat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_541_pred__subset__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X5: nat] : ( member_nat @ X5 @ R3 )
        @ ^ [X5: nat] : ( member_nat @ X5 @ S ) )
      = ( ord_less_eq_set_nat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_542_pred__subset__eq,axiom,
    ! [R3: set_Pr3433046674199179443r_prat,S: set_Pr3433046674199179443r_prat] :
      ( ( ord_le2830794348966088778prat_o
        @ ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ R3 )
        @ ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ S ) )
      = ( ord_le6641467840663156819r_prat @ R3 @ S ) ) ).

% pred_subset_eq
thf(fact_543_less__eq__set__def,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A4: set_o,B6: set_o] :
          ( ord_less_eq_o_o
          @ ^ [X5: $o] : ( member_o @ X5 @ A4 )
          @ ^ [X5: $o] : ( member_o @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_544_less__eq__set__def,axiom,
    ( ord_le8696906415146163484t_prat
    = ( ^ [A4: set_set_prat,B6: set_set_prat] :
          ( ord_le7917094133057636969prat_o
          @ ^ [X5: set_prat] : ( member_set_prat @ X5 @ A4 )
          @ ^ [X5: set_prat] : ( member_set_prat @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_545_less__eq__set__def,axiom,
    ( ord_le6641467840663156819r_prat
    = ( ^ [A4: set_Pr3433046674199179443r_prat,B6: set_Pr3433046674199179443r_prat] :
          ( ord_le2830794348966088778prat_o
          @ ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ A4 )
          @ ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_546_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A4 )
          @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_547_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
          @ ^ [X5: nat] : ( member_nat @ X5 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_548_lex__prod__def,axiom,
    ( lex_pr1681056561245511266t_char
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4480099295215233283t_char] :
          ( collec1162968953386343058t_char
          @ ( produc755220117336157804char_o
            @ ( produc2833417181138109642char_o
              @ ^ [A2: nat,B2: list_char] :
                  ( produc8526122873445342073char_o
                  @ ^ [A7: nat,B7: list_char] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B2 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_549_lex__prod__def,axiom,
    ( lex_pr789199185301838959r_prat
    = ( ^ [Ra: set_Pr1685704261205283783r_prat,Rb: set_Pr1685704261205283783r_prat] :
          ( collec7121748714612160850r_prat
          @ ( produc6083878442619929900prat_o
            @ ( produc4966222556782071362prat_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] :
                  ( produc5069634436926473260prat_o
                  @ ^ [A7: produc1457211279475724562t_char > prat,B7: produc1457211279475724562t_char > prat] :
                      ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ B2 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_550_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
                  @ ^ [A7: nat,B7: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_551_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F4: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6081775807080527818_nat_o @ F4 @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X1: nat,X22: nat] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_552_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F4: nat > list_char > $o,Prod: produc1457211279475724562t_char] :
      ( ( H2 @ ( produc8526122873445342073char_o @ F4 @ Prod ) )
      = ( produc8526122873445342073char_o
        @ ^ [X1: nat,X22: list_char] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_553_prod_Ocase__distrib,axiom,
    ! [H2: $o > nat,F4: nat > list_char > $o,Prod: produc1457211279475724562t_char] :
      ( ( H2 @ ( produc8526122873445342073char_o @ F4 @ Prod ) )
      = ( produc9163108120003956271ar_nat
        @ ^ [X1: nat,X22: list_char] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_554_prod_Ocase__distrib,axiom,
    ! [H2: nat > $o,F4: nat > list_char > nat,Prod: produc1457211279475724562t_char] :
      ( ( H2 @ ( produc9163108120003956271ar_nat @ F4 @ Prod ) )
      = ( produc8526122873445342073char_o
        @ ^ [X1: nat,X22: list_char] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_555_prod_Ocase__distrib,axiom,
    ! [H2: nat > nat,F4: nat > list_char > nat,Prod: produc1457211279475724562t_char] :
      ( ( H2 @ ( produc9163108120003956271ar_nat @ F4 @ Prod ) )
      = ( produc9163108120003956271ar_nat
        @ ^ [X1: nat,X22: list_char] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_556_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_nat_nat,F4: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6081775807080527818_nat_o @ F4 @ Prod ) )
      = ( produc2626176000494625587at_nat
        @ ^ [X1: nat,X22: nat] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_557_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_nat_nat > $o,F4: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc2626176000494625587at_nat @ F4 @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X1: nat,X22: nat] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_558_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_nat_nat > $o,F4: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6081775807080527818_nat_o @ F4 @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X1: nat,X22: nat] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_559_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_nat_nat > product_prod_nat_nat,F4: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc2626176000494625587at_nat @ F4 @ Prod ) )
      = ( produc2626176000494625587at_nat
        @ ^ [X1: nat,X22: nat] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_560_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > $o ) > $o,F4: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc8739625826339149834_nat_o @ F4 @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X1: nat,X22: nat] : ( H2 @ ( F4 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_561_prod_Odisc__eq__case,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( produc8526122873445342073char_o
      @ ^ [Uu: nat,Uv: list_char] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_562_prod_Odisc__eq__case,axiom,
    ! [Prod: produc3336119888008106215r_prat] :
      ( produc5069634436926473260prat_o
      @ ^ [Uu: produc1457211279475724562t_char > prat,Uv: produc1457211279475724562t_char > prat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_563_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_564_cond__case__prod__eta,axiom,
    ! [F4: nat > list_char > $o,G: produc1457211279475724562t_char > $o] :
      ( ! [X2: nat,Y4: list_char] :
          ( ( F4 @ X2 @ Y4 )
          = ( G @ ( produc120197956887798346t_char @ X2 @ Y4 ) ) )
     => ( ( produc8526122873445342073char_o @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_565_cond__case__prod__eta,axiom,
    ! [F4: nat > list_char > nat,G: produc1457211279475724562t_char > nat] :
      ( ! [X2: nat,Y4: list_char] :
          ( ( F4 @ X2 @ Y4 )
          = ( G @ ( produc120197956887798346t_char @ X2 @ Y4 ) ) )
     => ( ( produc9163108120003956271ar_nat @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_566_cond__case__prod__eta,axiom,
    ! [F4: nat > nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X2: nat,Y4: nat] :
          ( ( F4 @ X2 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X2 @ Y4 ) ) )
     => ( ( produc2626176000494625587at_nat @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_567_cond__case__prod__eta,axiom,
    ! [F4: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X2: nat,Y4: nat] :
          ( ( F4 @ X2 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X2 @ Y4 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_568_cond__case__prod__eta,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,G: produc3336119888008106215r_prat > $o] :
      ( ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > prat] :
          ( ( F4 @ X2 @ Y4 )
          = ( G @ ( produc1164665353665699927r_prat @ X2 @ Y4 ) ) )
     => ( ( produc5069634436926473260prat_o @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_569_cond__case__prod__eta,axiom,
    ! [F4: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X2: nat,Y4: nat] :
          ( ( F4 @ X2 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X2 @ Y4 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F4 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_570_relImage__def,axiom,
    ( bNF_Gr5060297752514649919at_nat
    = ( ^ [R4: set_Pr1685704261205283783r_prat,F2: ( produc1457211279475724562t_char > prat ) > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A12 @ A23 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_571_relImage__def,axiom,
    ( bNF_Gr2145404550722348580r_prat
    = ( ^ [R4: set_Pr1685704261205283783r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
          ( collec1156307336577696338r_prat
          @ ^ [Uu: produc3336119888008106215r_prat] :
            ? [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( produc1164665353665699927r_prat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A12 @ A23 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_572_relInvImage__def,axiom,
    ( bNF_Gr7259518749458858748r_prat
    = ( ^ [A4: set_o,R4: set_Pr1685704261205283783r_prat,F2: $o > produc1457211279475724562t_char > prat] :
          ( collec3167064739498627218od_o_o
          @ ^ [Uu: product_prod_o_o] :
            ? [A12: $o,A23: $o] :
              ( ( Uu
                = ( product_Pair_o_o @ A12 @ A23 ) )
              & ( member_o @ A12 @ A4 )
              & ( member_o @ A23 @ A4 )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_573_relInvImage__def,axiom,
    ( bNF_Gr2010402745563153482r_prat
    = ( ^ [A4: set_set_prat,R4: set_Pr1685704261205283783r_prat,F2: set_prat > produc1457211279475724562t_char > prat] :
          ( collec8766186246528183128t_prat
          @ ^ [Uu: produc3531106461118509933t_prat] :
            ? [A12: set_prat,A23: set_prat] :
              ( ( Uu
                = ( produc1046168772652838309t_prat @ A12 @ A23 ) )
              & ( member_set_prat @ A12 @ A4 )
              & ( member_set_prat @ A23 @ A4 )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_574_relInvImage__def,axiom,
    ( bNF_Gr8791779402478003658r_prat
    = ( ^ [A4: set_set_nat,R4: set_Pr1685704261205283783r_prat,F2: set_nat > produc1457211279475724562t_char > prat] :
          ( collec6662362479098859352et_nat
          @ ^ [Uu: produc7819656566062154093et_nat] :
            ? [A12: set_nat,A23: set_nat] :
              ( ( Uu
                = ( produc4532415448927165861et_nat @ A12 @ A23 ) )
              & ( member_set_nat @ A12 @ A4 )
              & ( member_set_nat @ A23 @ A4 )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_575_relInvImage__def,axiom,
    ( bNF_Gr3267885531569300736r_prat
    = ( ^ [A4: set_nat,R4: set_Pr1685704261205283783r_prat,F2: nat > produc1457211279475724562t_char > prat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A12: nat,A23: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A12 @ A23 ) )
              & ( member_nat @ A12 @ A4 )
              & ( member_nat @ A23 @ A4 )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_576_relInvImage__def,axiom,
    ( bNF_Gr7532869937377485441r_prat
    = ( ^ [A4: set_Pr3433046674199179443r_prat,R4: set_Pr1685704261205283783r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
          ( collec1156307336577696338r_prat
          @ ^ [Uu: produc3336119888008106215r_prat] :
            ? [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( produc1164665353665699927r_prat @ A12 @ A23 ) )
              & ( member2336342582916103932r_prat @ A12 @ A4 )
              & ( member2336342582916103932r_prat @ A23 @ A4 )
              & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_577_case__prod__app,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [L: nat,R: nat] : ( F2 @ L @ R @ Y2 )
          @ X5 ) ) ) ).

% case_prod_app
thf(fact_578_conj__subset__def,axiom,
    ! [A3: set_Pr1685704261205283783r_prat,P2: produc3336119888008106215r_prat > $o,Q: produc3336119888008106215r_prat > $o] :
      ( ( ord_le295572976866519911r_prat @ A3
        @ ( collec1156307336577696338r_prat
          @ ^ [X5: produc3336119888008106215r_prat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le295572976866519911r_prat @ A3 @ ( collec1156307336577696338r_prat @ P2 ) )
        & ( ord_le295572976866519911r_prat @ A3 @ ( collec1156307336577696338r_prat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_579_conj__subset__def,axiom,
    ! [A3: set_list_char,P2: list_char > $o,Q: list_char > $o] :
      ( ( ord_le1207710660481778807t_char @ A3
        @ ( collect_list_char
          @ ^ [X5: list_char] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le1207710660481778807t_char @ A3 @ ( collect_list_char @ P2 ) )
        & ( ord_le1207710660481778807t_char @ A3 @ ( collect_list_char @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_580_conj__subset__def,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6641467840663156819r_prat @ A3
        @ ( collec7093475260167521854r_prat
          @ ^ [X5: produc1457211279475724562t_char > prat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le6641467840663156819r_prat @ A3 @ ( collec7093475260167521854r_prat @ P2 ) )
        & ( ord_le6641467840663156819r_prat @ A3 @ ( collec7093475260167521854r_prat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_581_conj__subset__def,axiom,
    ! [A3: set_nat_nat,P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
      ( ( ord_le9059583361652607317at_nat @ A3
        @ ( collect_nat_nat
          @ ^ [X5: nat > nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le9059583361652607317at_nat @ A3 @ ( collect_nat_nat @ P2 ) )
        & ( ord_le9059583361652607317at_nat @ A3 @ ( collect_nat_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_582_conj__subset__def,axiom,
    ! [A3: set_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A3
        @ ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A3 @ ( collect_set_nat @ P2 ) )
        & ( ord_le6893508408891458716et_nat @ A3 @ ( collect_set_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_583_conj__subset__def,axiom,
    ! [A3: set_nat,P2: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P2 ) )
        & ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_584_conj__subset__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ P2 ) )
        & ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_585_prop__restrict,axiom,
    ! [X: $o,Z5: set_o,X8: set_o,P2: $o > $o] :
      ( ( member_o @ X @ Z5 )
     => ( ( ord_less_eq_set_o @ Z5
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_586_prop__restrict,axiom,
    ! [X: set_prat,Z5: set_set_prat,X8: set_set_prat,P2: set_prat > $o] :
      ( ( member_set_prat @ X @ Z5 )
     => ( ( ord_le8696906415146163484t_prat @ Z5
          @ ( collect_set_prat
            @ ^ [X5: set_prat] :
                ( ( member_set_prat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_587_prop__restrict,axiom,
    ! [X: produc3336119888008106215r_prat,Z5: set_Pr1685704261205283783r_prat,X8: set_Pr1685704261205283783r_prat,P2: produc3336119888008106215r_prat > $o] :
      ( ( member1381939458143960592r_prat @ X @ Z5 )
     => ( ( ord_le295572976866519911r_prat @ Z5
          @ ( collec1156307336577696338r_prat
            @ ^ [X5: produc3336119888008106215r_prat] :
                ( ( member1381939458143960592r_prat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_588_prop__restrict,axiom,
    ! [X: list_char,Z5: set_list_char,X8: set_list_char,P2: list_char > $o] :
      ( ( member_list_char @ X @ Z5 )
     => ( ( ord_le1207710660481778807t_char @ Z5
          @ ( collect_list_char
            @ ^ [X5: list_char] :
                ( ( member_list_char @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_589_prop__restrict,axiom,
    ! [X: produc1457211279475724562t_char > prat,Z5: set_Pr3433046674199179443r_prat,X8: set_Pr3433046674199179443r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( member2336342582916103932r_prat @ X @ Z5 )
     => ( ( ord_le6641467840663156819r_prat @ Z5
          @ ( collec7093475260167521854r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat] :
                ( ( member2336342582916103932r_prat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_590_prop__restrict,axiom,
    ! [X: nat > nat,Z5: set_nat_nat,X8: set_nat_nat,P2: ( nat > nat ) > $o] :
      ( ( member_nat_nat @ X @ Z5 )
     => ( ( ord_le9059583361652607317at_nat @ Z5
          @ ( collect_nat_nat
            @ ^ [X5: nat > nat] :
                ( ( member_nat_nat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_591_prop__restrict,axiom,
    ! [X: set_nat,Z5: set_set_nat,X8: set_set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ X @ Z5 )
     => ( ( ord_le6893508408891458716et_nat @ Z5
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_592_prop__restrict,axiom,
    ! [X: nat,Z5: set_nat,X8: set_nat,P2: nat > $o] :
      ( ( member_nat @ X @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_593_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X8: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ Z5 )
     => ( ( ord_le3146513528884898305at_nat @ Z5
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ X8 )
                & ( P2 @ X5 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_594_Collect__restrict,axiom,
    ! [X8: set_o,P2: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_595_Collect__restrict,axiom,
    ! [X8: set_set_prat,P2: set_prat > $o] :
      ( ord_le8696906415146163484t_prat
      @ ( collect_set_prat
        @ ^ [X5: set_prat] :
            ( ( member_set_prat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_596_Collect__restrict,axiom,
    ! [X8: set_Pr1685704261205283783r_prat,P2: produc3336119888008106215r_prat > $o] :
      ( ord_le295572976866519911r_prat
      @ ( collec1156307336577696338r_prat
        @ ^ [X5: produc3336119888008106215r_prat] :
            ( ( member1381939458143960592r_prat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_597_Collect__restrict,axiom,
    ! [X8: set_list_char,P2: list_char > $o] :
      ( ord_le1207710660481778807t_char
      @ ( collect_list_char
        @ ^ [X5: list_char] :
            ( ( member_list_char @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_598_Collect__restrict,axiom,
    ! [X8: set_Pr3433046674199179443r_prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ord_le6641467840663156819r_prat
      @ ( collec7093475260167521854r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_599_Collect__restrict,axiom,
    ! [X8: set_nat_nat,P2: ( nat > nat ) > $o] :
      ( ord_le9059583361652607317at_nat
      @ ( collect_nat_nat
        @ ^ [X5: nat > nat] :
            ( ( member_nat_nat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_600_Collect__restrict,axiom,
    ! [X8: set_set_nat,P2: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X5: set_nat] :
            ( ( member_set_nat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_601_Collect__restrict,axiom,
    ! [X8: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_602_Collect__restrict,axiom,
    ! [X8: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ X8 )
            & ( P2 @ X5 ) ) )
      @ X8 ) ).

% Collect_restrict
thf(fact_603_case__prod__Pair__iden,axiom,
    ! [P7: produc2489117125269924006t_char] :
      ( ( produc289299604985033403t_char @ produc2591414526589101846t_char @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_604_case__prod__Pair__iden,axiom,
    ! [P7: produc5803078220529002682t_char] :
      ( ( produc851857660381163707t_char @ produc2957999048406202538t_char @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_605_case__prod__Pair__iden,axiom,
    ! [P7: produc3336119888008106215r_prat] :
      ( ( produc644870500113131387r_prat @ produc1164665353665699927r_prat @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_606_case__prod__Pair__iden,axiom,
    ! [P7: produc4654305426090421966prat_a] :
      ( ( produc6124589016206478779prat_a @ produc2856268368059332798prat_a @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_607_case__prod__Pair__iden,axiom,
    ! [P7: produc3375937593958228902prat_a] :
      ( ( produc5169377756248968379prat_a @ produc3557287861060643862prat_a @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_608_case__prod__Pair__iden,axiom,
    ! [P7: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_609_case__prod__Pair__iden,axiom,
    ! [P7: produc1457211279475724562t_char] :
      ( ( produc4071494739252518515t_char @ produc120197956887798346t_char @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_610_eq__subset,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ord_le6723339807950303054prat_o
      @ ^ [Y3: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > prat] : ( Y3 = Z )
      @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] :
          ( ( P2 @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ).

% eq_subset
thf(fact_611_same__fst__def,axiom,
    ( same_f652031872680891423t_char
    = ( ^ [P4: nat > $o,R4: nat > set_Pr4480099295215233283t_char] :
          ( collec1162968953386343058t_char
          @ ( produc755220117336157804char_o
            @ ( produc2833417181138109642char_o
              @ ^ [X7: nat,Y6: list_char] :
                  ( produc8526122873445342073char_o
                  @ ^ [X5: nat,Y2: list_char] :
                      ( ( X7 = X5 )
                      & ( P4 @ X5 )
                      & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y6 @ Y2 ) @ ( R4 @ X5 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_612_same__fst__def,axiom,
    ( same_f1825534920405731884r_prat
    = ( ^ [P4: ( produc1457211279475724562t_char > prat ) > $o,R4: ( produc1457211279475724562t_char > prat ) > set_Pr1685704261205283783r_prat] :
          ( collec7121748714612160850r_prat
          @ ( produc6083878442619929900prat_o
            @ ( produc4966222556782071362prat_o
              @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > prat] :
                  ( produc5069634436926473260prat_o
                  @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] :
                      ( ( X7 = X5 )
                      & ( P4 @ X5 )
                      & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y6 @ Y2 ) @ ( R4 @ X5 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_613_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P4: nat > $o,R4: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X7: nat,Y6: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X5: nat,Y2: nat] :
                      ( ( X7 = X5 )
                      & ( P4 @ X5 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y2 ) @ ( R4 @ X5 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_614_inv__image__def,axiom,
    ( inv_im6765648638891743953at_nat
    = ( ^ [R: set_Pr1685704261205283783r_prat,F2: nat > produc1457211279475724562t_char > prat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X5: nat,Y2: nat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ X5 ) @ ( F2 @ Y2 ) ) @ R ) ) ) ) ) ).

% inv_image_def
thf(fact_615_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X5 ) @ ( F2 @ Y2 ) ) @ R ) ) ) ) ) ).

% inv_image_def
thf(fact_616_inv__image__def,axiom,
    ( inv_im3997502450938070034r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ ( F2 @ X5 ) @ ( F2 @ Y2 ) ) @ R ) ) ) ) ) ).

% inv_image_def
thf(fact_617_Collect__case__prod__in__rel__leI,axiom,
    ! [X8: set_Pr2507339680178222706t_char,Y7: set_Pr2507339680178222706t_char] :
      ( ( ord_le6315511645215477266t_char @ X8 @ Y7 )
     => ( ord_le6315511645215477266t_char @ X8 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ ( fun_in175243070162512245t_char @ Y7 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_618_Collect__case__prod__in__rel__leI,axiom,
    ! [X8: set_Pr1685704261205283783r_prat,Y7: set_Pr1685704261205283783r_prat] :
      ( ( ord_le295572976866519911r_prat @ X8 @ Y7 )
     => ( ord_le295572976866519911r_prat @ X8 @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ ( fun_in9115074924180500354r_prat @ Y7 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_619_Collect__case__prod__in__rel__leI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X8 @ Y7 )
     => ( ord_le3146513528884898305at_nat @ X8 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y7 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_620_Collect__case__prod__in__rel__leE,axiom,
    ! [X8: set_Pr2507339680178222706t_char,Y7: set_Pr2507339680178222706t_char] :
      ( ( ord_le6315511645215477266t_char @ X8 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ ( fun_in175243070162512245t_char @ Y7 ) ) ) )
     => ( ord_le6315511645215477266t_char @ X8 @ Y7 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_621_Collect__case__prod__in__rel__leE,axiom,
    ! [X8: set_Pr1685704261205283783r_prat,Y7: set_Pr1685704261205283783r_prat] :
      ( ( ord_le295572976866519911r_prat @ X8 @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ ( fun_in9115074924180500354r_prat @ Y7 ) ) ) )
     => ( ord_le295572976866519911r_prat @ X8 @ Y7 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_622_Collect__case__prod__in__rel__leE,axiom,
    ! [X8: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X8 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y7 ) ) ) )
     => ( ord_le3146513528884898305at_nat @ X8 @ Y7 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_623_Gr__def,axiom,
    ( bNF_Gr4667860031443508641t_char
    = ( ^ [A4: set_Pr3433046674199179443r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char] :
          ( collec8998774337691853201t_char
          @ ^ [Uu: produc2489117125269924006t_char] :
            ? [A2: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( produc2591414526589101846t_char @ A2 @ ( F2 @ A2 ) ) )
              & ( member2336342582916103932r_prat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_624_Gr__def,axiom,
    ( bNF_Gr7494250897068838581t_char
    = ( ^ [A4: set_Pr3433046674199179443r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char] :
          ( collec1912346231567511845t_char
          @ ^ [Uu: produc5803078220529002682t_char] :
            ? [A2: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( produc2957999048406202538t_char @ A2 @ ( F2 @ A2 ) ) )
              & ( member2336342582916103932r_prat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_625_Gr__def,axiom,
    ( bNF_Gr6062854646325018466r_prat
    = ( ^ [A4: set_Pr3433046674199179443r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
          ( collec1156307336577696338r_prat
          @ ^ [Uu: produc3336119888008106215r_prat] :
            ? [A2: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( produc1164665353665699927r_prat @ A2 @ ( F2 @ A2 ) ) )
              & ( member2336342582916103932r_prat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_626_Gr__def,axiom,
    ( bNF_Gr6139830065717436617prat_a
    = ( ^ [A4: set_a_prat,F2: ( a > prat ) > produc3375937593958228902prat_a] :
          ( collec8956631807754062137prat_a
          @ ^ [Uu: produc4654305426090421966prat_a] :
            ? [A2: a > prat] :
              ( ( Uu
                = ( produc2856268368059332798prat_a @ A2 @ ( F2 @ A2 ) ) )
              & ( member_a_prat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_627_Gr__def,axiom,
    ( bNF_Gr_a_prat_a
    = ( ^ [A4: set_a_prat,F2: ( a > prat ) > a] :
          ( collec2652489992548837521prat_a
          @ ^ [Uu: produc3375937593958228902prat_a] :
            ? [A2: a > prat] :
              ( ( Uu
                = ( produc3557287861060643862prat_a @ A2 @ ( F2 @ A2 ) ) )
              & ( member_a_prat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_628_Gr__def,axiom,
    ( bNF_Gr_nat_nat
    = ( ^ [A4: set_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A2: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A2 @ ( F2 @ A2 ) ) )
              & ( member_nat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_629_Gr__def,axiom,
    ( bNF_Gr_nat_list_char
    = ( ^ [A4: set_nat,F2: nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A2: nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ A2 @ ( F2 @ A2 ) ) )
              & ( member_nat @ A2 @ A4 ) ) ) ) ) ).

% Gr_def
thf(fact_630_Eps__case__prod,axiom,
    ! [P2: list_char > nat > $o] :
      ( ( fChoic8626096029427592294ar_nat @ ( produc2492928673396900345_nat_o @ P2 ) )
      = ( fChoic8626096029427592294ar_nat
        @ ^ [Xy: produc7606469099237612818ar_nat] : ( P2 @ ( produc5563429021163328950ar_nat @ Xy ) @ ( produc1259554233722553080ar_nat @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_631_Eps__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( fChoic869448611404120123r_prat @ ( produc5069634436926473260prat_o @ P2 ) )
      = ( fChoic869448611404120123r_prat
        @ ^ [Xy: produc3336119888008106215r_prat] : ( P2 @ ( produc2193917115357690435r_prat @ Xy ) @ ( produc5358294431189718917r_prat @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_632_Eps__case__prod,axiom,
    ! [P2: nat > list_char > $o] :
      ( ( fChoic2476838209665704038t_char @ ( produc8526122873445342073char_o @ P2 ) )
      = ( fChoic2476838209665704038t_char
        @ ^ [Xy: produc1457211279475724562t_char] : ( P2 @ ( produc6204653154488278070t_char @ Xy ) @ ( produc1900778367047502200t_char @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_633_Eps__case__prod,axiom,
    ! [P2: nat > nat > $o] :
      ( ( fChoic6978938873391328853at_nat @ ( produc6081775807080527818_nat_o @ P2 ) )
      = ( fChoic6978938873391328853at_nat
        @ ^ [Xy: product_prod_nat_nat] : ( P2 @ ( product_fst_nat_nat @ Xy ) @ ( product_snd_nat_nat @ Xy ) ) ) ) ).

% Eps_case_prod
thf(fact_634_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec2458894688891239037t_char
            @ ( produc8526122873445342073char_o
              @ ^ [A2: nat,B2: list_char] : P2 ) )
          = top_to1250346684925019074t_char ) )
      & ( ~ P2
       => ( ( collec2458894688891239037t_char
            @ ( produc8526122873445342073char_o
              @ ^ [A2: nat,B2: list_char] : P2 ) )
          = bot_bo1414094081888755934t_char ) ) ) ).

% Collect_const_case_prod
thf(fact_635_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec1156307336577696338r_prat
            @ ( produc5069634436926473260prat_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : P2 ) )
          = top_to6977641080806416663r_prat ) )
      & ( ~ P2
       => ( ( collec1156307336577696338r_prat
            @ ( produc5069634436926473260prat_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : P2 ) )
          = bot_bo6182153949361364531r_prat ) ) ) ).

% Collect_const_case_prod
thf(fact_636_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B2: nat] : P2 ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P2
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B2: nat] : P2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_637_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec1156307336577696338r_prat
            @ ^ [S3: produc3336119888008106215r_prat] : P2 )
          = top_to6977641080806416663r_prat ) )
      & ( ~ P2
       => ( ( collec1156307336577696338r_prat
            @ ^ [S3: produc3336119888008106215r_prat] : P2 )
          = bot_bo6182153949361364531r_prat ) ) ) ).

% Collect_const
thf(fact_638_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collect_list_char
            @ ^ [S3: list_char] : P2 )
          = top_to3190553582279535303t_char ) )
      & ( ~ P2
       => ( ( collect_list_char
            @ ^ [S3: list_char] : P2 )
          = bot_bo8503570744482963627t_char ) ) ) ).

% Collect_const
thf(fact_639_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec7093475260167521854r_prat
            @ ^ [S3: produc1457211279475724562t_char > prat] : P2 )
          = top_to6849109582029272579r_prat ) )
      & ( ~ P2
       => ( ( collec7093475260167521854r_prat
            @ ^ [S3: produc1457211279475724562t_char > prat] : P2 )
          = bot_bo3164680191535668511r_prat ) ) ) ).

% Collect_const
thf(fact_640_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec5514110066124741708at_nat
            @ ^ [S3: set_Pr1261947904930325089at_nat] : P2 )
          = top_to7629004291339433233at_nat ) )
      & ( ~ P2
       => ( ( collec5514110066124741708at_nat
            @ ^ [S3: set_Pr1261947904930325089at_nat] : P2 )
          = bot_bo3083307316010499117at_nat ) ) ) ).

% Collect_const
thf(fact_641_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collect_set_nat
            @ ^ [S3: set_nat] : P2 )
          = top_top_set_set_nat ) )
      & ( ~ P2
       => ( ( collect_set_nat
            @ ^ [S3: set_nat] : P2 )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_const
thf(fact_642_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collect_nat_nat
            @ ^ [S3: nat > nat] : P2 )
          = top_top_set_nat_nat ) )
      & ( ~ P2
       => ( ( collect_nat_nat
            @ ^ [S3: nat > nat] : P2 )
          = bot_bot_set_nat_nat ) ) ) ).

% Collect_const
thf(fact_643_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collect_nat
            @ ^ [S3: nat] : P2 )
          = top_top_set_nat ) )
      & ( ~ P2
       => ( ( collect_nat
            @ ^ [S3: nat] : P2 )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_644_Collect__const,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec3392354462482085612at_nat
            @ ^ [S3: product_prod_nat_nat] : P2 )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P2
       => ( ( collec3392354462482085612at_nat
            @ ^ [S3: product_prod_nat_nat] : P2 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const
thf(fact_645_UNIV__def,axiom,
    ( top_to6977641080806416663r_prat
    = ( collec1156307336577696338r_prat
      @ ^ [X5: produc3336119888008106215r_prat] : $true ) ) ).

% UNIV_def
thf(fact_646_UNIV__def,axiom,
    ( top_to3190553582279535303t_char
    = ( collect_list_char
      @ ^ [X5: list_char] : $true ) ) ).

% UNIV_def
thf(fact_647_UNIV__def,axiom,
    ( top_to6849109582029272579r_prat
    = ( collec7093475260167521854r_prat
      @ ^ [X5: produc1457211279475724562t_char > prat] : $true ) ) ).

% UNIV_def
thf(fact_648_UNIV__def,axiom,
    ( top_to7629004291339433233at_nat
    = ( collec5514110066124741708at_nat
      @ ^ [X5: set_Pr1261947904930325089at_nat] : $true ) ) ).

% UNIV_def
thf(fact_649_UNIV__def,axiom,
    ( top_top_set_set_nat
    = ( collect_set_nat
      @ ^ [X5: set_nat] : $true ) ) ).

% UNIV_def
thf(fact_650_UNIV__def,axiom,
    ( top_top_set_nat_nat
    = ( collect_nat_nat
      @ ^ [X5: nat > nat] : $true ) ) ).

% UNIV_def
thf(fact_651_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X5: nat] : $true ) ) ).

% UNIV_def
thf(fact_652_UNIV__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X5: product_prod_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_653_snd__def,axiom,
    ( product_snd_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X1: nat,X22: nat] : X22 ) ) ).

% snd_def
thf(fact_654_snd__def,axiom,
    ( produc1900778367047502200t_char
    = ( produc7351934036967509312t_char
      @ ^ [X1: nat,X22: list_char] : X22 ) ) ).

% snd_def
thf(fact_655_snd__def,axiom,
    ( produc1259554233722553080ar_nat
    = ( produc7665377738449061807at_nat
      @ ^ [X1: list_char,X22: nat] : X22 ) ) ).

% snd_def
thf(fact_656_pre__get__h_Osimps,axiom,
    pre_get_h_nat_nat = product_snd_nat_nat ).

% pre_get_h.simps
thf(fact_657_pre__get__h_Osimps,axiom,
    pre_ge1487422843495280912t_char = produc1900778367047502200t_char ).

% pre_get_h.simps
thf(fact_658_pre__get__h_Osimps,axiom,
    pre_ge846198710170331792ar_nat = produc1259554233722553080ar_nat ).

% pre_get_h.simps
thf(fact_659_pre__get__h_Oelims,axiom,
    ! [X: product_prod_nat_nat,Y: nat] :
      ( ( ( pre_get_h_nat_nat @ X )
        = Y )
     => ( Y
        = ( product_snd_nat_nat @ X ) ) ) ).

% pre_get_h.elims
thf(fact_660_pre__get__h_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: list_char] :
      ( ( ( pre_ge1487422843495280912t_char @ X )
        = Y )
     => ( Y
        = ( produc1900778367047502200t_char @ X ) ) ) ).

% pre_get_h.elims
thf(fact_661_pre__get__h_Oelims,axiom,
    ! [X: produc7606469099237612818ar_nat,Y: nat] :
      ( ( ( pre_ge846198710170331792ar_nat @ X )
        = Y )
     => ( Y
        = ( produc1259554233722553080ar_nat @ X ) ) ) ).

% pre_get_h.elims
thf(fact_662_GrD1,axiom,
    ! [X: nat,Fx: nat,A3: set_nat,F4: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A3 @ F4 ) )
     => ( member_nat @ X @ A3 ) ) ).

% GrD1
thf(fact_663_GrD1,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc1457211279475724562t_char,A3: set_Pr3433046674199179443r_prat,F4: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char] :
      ( ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X @ Fx ) @ ( bNF_Gr4667860031443508641t_char @ A3 @ F4 ) )
     => ( member2336342582916103932r_prat @ X @ A3 ) ) ).

% GrD1
thf(fact_664_GrD1,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc2489117125269924006t_char,A3: set_Pr3433046674199179443r_prat,F4: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char] :
      ( ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X @ Fx ) @ ( bNF_Gr7494250897068838581t_char @ A3 @ F4 ) )
     => ( member2336342582916103932r_prat @ X @ A3 ) ) ).

% GrD1
thf(fact_665_GrD1,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc1457211279475724562t_char > prat,A3: set_Pr3433046674199179443r_prat,F4: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
      ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X @ Fx ) @ ( bNF_Gr6062854646325018466r_prat @ A3 @ F4 ) )
     => ( member2336342582916103932r_prat @ X @ A3 ) ) ).

% GrD1
thf(fact_666_GrD1,axiom,
    ! [X: a > prat,Fx: produc3375937593958228902prat_a,A3: set_a_prat,F4: ( a > prat ) > produc3375937593958228902prat_a] :
      ( ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X @ Fx ) @ ( bNF_Gr6139830065717436617prat_a @ A3 @ F4 ) )
     => ( member_a_prat @ X @ A3 ) ) ).

% GrD1
thf(fact_667_GrD1,axiom,
    ! [X: a > prat,Fx: a,A3: set_a_prat,F4: ( a > prat ) > a] :
      ( ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X @ Fx ) @ ( bNF_Gr_a_prat_a @ A3 @ F4 ) )
     => ( member_a_prat @ X @ A3 ) ) ).

% GrD1
thf(fact_668_GrD1,axiom,
    ! [X: nat,Fx: list_char,A3: set_nat,F4: nat > list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Fx ) @ ( bNF_Gr_nat_list_char @ A3 @ F4 ) )
     => ( member_nat @ X @ A3 ) ) ).

% GrD1
thf(fact_669_GrD2,axiom,
    ! [X: nat,Fx: nat,A3: set_nat,F4: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_670_GrD2,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc1457211279475724562t_char,A3: set_Pr3433046674199179443r_prat,F4: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char] :
      ( ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X @ Fx ) @ ( bNF_Gr4667860031443508641t_char @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_671_GrD2,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc2489117125269924006t_char,A3: set_Pr3433046674199179443r_prat,F4: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char] :
      ( ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X @ Fx ) @ ( bNF_Gr7494250897068838581t_char @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_672_GrD2,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc1457211279475724562t_char > prat,A3: set_Pr3433046674199179443r_prat,F4: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
      ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X @ Fx ) @ ( bNF_Gr6062854646325018466r_prat @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_673_GrD2,axiom,
    ! [X: a > prat,Fx: produc3375937593958228902prat_a,A3: set_a_prat,F4: ( a > prat ) > produc3375937593958228902prat_a] :
      ( ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X @ Fx ) @ ( bNF_Gr6139830065717436617prat_a @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_674_GrD2,axiom,
    ! [X: a > prat,Fx: a,A3: set_a_prat,F4: ( a > prat ) > a] :
      ( ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X @ Fx ) @ ( bNF_Gr_a_prat_a @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_675_GrD2,axiom,
    ! [X: nat,Fx: list_char,A3: set_nat,F4: nat > list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Fx ) @ ( bNF_Gr_nat_list_char @ A3 @ F4 ) )
     => ( ( F4 @ X )
        = Fx ) ) ).

% GrD2
thf(fact_676_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char,A: produc2489117125269924006t_char] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2591414526589101846t_char @ X @ Y ) )
       => ( P2 @ ( produc5001474941816913666t_char @ A ) @ ( produc5484914016365488452t_char @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_677_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > produc2489117125269924006t_char > $o,X: produc1457211279475724562t_char > prat,Y: produc2489117125269924006t_char,A: produc5803078220529002682t_char] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2957999048406202538t_char @ X @ Y ) )
       => ( P2 @ ( produc583991874214702742t_char @ A ) @ ( produc5882964721085011416t_char @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_678_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > prat,A: produc3336119888008106215r_prat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc1164665353665699927r_prat @ X @ Y ) )
       => ( P2 @ ( produc2193917115357690435r_prat @ A ) @ ( produc5358294431189718917r_prat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_679_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( a > prat ) > produc3375937593958228902prat_a > $o,X: a > prat,Y: produc3375937593958228902prat_a,A: produc4654305426090421966prat_a] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2856268368059332798prat_a @ X @ Y ) )
       => ( P2 @ ( produc725351007466518186prat_a @ A ) @ ( produc7544562691789162988prat_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_680_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( a > prat ) > a > $o,X: a > prat,Y: a,A: produc3375937593958228902prat_a] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc3557287861060643862prat_a @ X @ Y ) )
       => ( P2 @ ( product_fst_a_prat_a @ A ) @ ( product_snd_a_prat_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_681_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( product_Pair_nat_nat @ X @ Y ) )
       => ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_682_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_char > nat > $o,X: list_char,Y: nat,A: produc7606469099237612818ar_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc8702345860417625034ar_nat @ X @ Y ) )
       => ( P2 @ ( produc5563429021163328950ar_nat @ A ) @ ( produc1259554233722553080ar_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_683_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > list_char > $o,X: nat,Y: list_char,A: produc1457211279475724562t_char] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc120197956887798346t_char @ X @ Y ) )
       => ( P2 @ ( produc6204653154488278070t_char @ A ) @ ( produc1900778367047502200t_char @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_684_in__rel__Collect__case__prod__eq,axiom,
    ! [X8: nat > list_char > $o] :
      ( ( fun_in175243070162512245t_char @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ X8 ) ) )
      = X8 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_685_in__rel__Collect__case__prod__eq,axiom,
    ! [X8: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( fun_in9115074924180500354r_prat @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ X8 ) ) )
      = X8 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_686_in__rel__Collect__case__prod__eq,axiom,
    ! [X8: nat > nat > $o] :
      ( ( fun_in_rel_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ X8 ) ) )
      = X8 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_687_case__prod__unfold,axiom,
    ( produc8526122873445342073char_o
    = ( ^ [C2: nat > list_char > $o,P: produc1457211279475724562t_char] : ( C2 @ ( produc6204653154488278070t_char @ P ) @ ( produc1900778367047502200t_char @ P ) ) ) ) ).

% case_prod_unfold
thf(fact_688_case__prod__unfold,axiom,
    ( produc9163108120003956271ar_nat
    = ( ^ [C2: nat > list_char > nat,P: produc1457211279475724562t_char] : ( C2 @ ( produc6204653154488278070t_char @ P ) @ ( produc1900778367047502200t_char @ P ) ) ) ) ).

% case_prod_unfold
thf(fact_689_case__prod__unfold,axiom,
    ( produc2626176000494625587at_nat
    = ( ^ [C2: nat > nat > product_prod_nat_nat,P: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P ) @ ( product_snd_nat_nat @ P ) ) ) ) ).

% case_prod_unfold
thf(fact_690_case__prod__unfold,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [C2: nat > nat > product_prod_nat_nat > $o,P: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P ) @ ( product_snd_nat_nat @ P ) ) ) ) ).

% case_prod_unfold
thf(fact_691_case__prod__unfold,axiom,
    ( produc5069634436926473260prat_o
    = ( ^ [C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,P: produc3336119888008106215r_prat] : ( C2 @ ( produc2193917115357690435r_prat @ P ) @ ( produc5358294431189718917r_prat @ P ) ) ) ) ).

% case_prod_unfold
thf(fact_692_case__prod__unfold,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [C2: nat > nat > $o,P: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P ) @ ( product_snd_nat_nat @ P ) ) ) ) ).

% case_prod_unfold
thf(fact_693_case__prod__beta_H,axiom,
    ( produc8526122873445342073char_o
    = ( ^ [F2: nat > list_char > $o,X5: produc1457211279475724562t_char] : ( F2 @ ( produc6204653154488278070t_char @ X5 ) @ ( produc1900778367047502200t_char @ X5 ) ) ) ) ).

% case_prod_beta'
thf(fact_694_case__prod__beta_H,axiom,
    ( produc9163108120003956271ar_nat
    = ( ^ [F2: nat > list_char > nat,X5: produc1457211279475724562t_char] : ( F2 @ ( produc6204653154488278070t_char @ X5 ) @ ( produc1900778367047502200t_char @ X5 ) ) ) ) ).

% case_prod_beta'
thf(fact_695_case__prod__beta_H,axiom,
    ( produc2626176000494625587at_nat
    = ( ^ [F2: nat > nat > product_prod_nat_nat,X5: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X5 ) @ ( product_snd_nat_nat @ X5 ) ) ) ) ).

% case_prod_beta'
thf(fact_696_case__prod__beta_H,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X5 ) @ ( product_snd_nat_nat @ X5 ) ) ) ) ).

% case_prod_beta'
thf(fact_697_case__prod__beta_H,axiom,
    ( produc5069634436926473260prat_o
    = ( ^ [F2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X5: produc3336119888008106215r_prat] : ( F2 @ ( produc2193917115357690435r_prat @ X5 ) @ ( produc5358294431189718917r_prat @ X5 ) ) ) ) ).

% case_prod_beta'
thf(fact_698_case__prod__beta_H,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,X5: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X5 ) @ ( product_snd_nat_nat @ X5 ) ) ) ) ).

% case_prod_beta'
thf(fact_699_pre__get__h_Opelims,axiom,
    ! [X: produc5803078220529002682t_char,Y: produc2489117125269924006t_char] :
      ( ( ( pre_ge3831893988676007280t_char @ X )
        = Y )
     => ( ( accp_P224314924008452035t_char @ pre_ge5134124999753271535t_char @ X )
       => ~ ( ( Y
              = ( produc5882964721085011416t_char @ X ) )
           => ~ ( accp_P224314924008452035t_char @ pre_ge5134124999753271535t_char @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_700_pre__get__h_Opelims,axiom,
    ! [X: produc4654305426090421966prat_a,Y: produc3375937593958228902prat_a] :
      ( ( ( pre_ge6197302870995220868prat_a @ X )
        = Y )
     => ( ( accp_P8439977412596685783prat_a @ pre_ge4968141634538297091prat_a @ X )
       => ~ ( ( Y
              = ( produc7544562691789162988prat_a @ X ) )
           => ~ ( accp_P8439977412596685783prat_a @ pre_ge4968141634538297091prat_a @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_701_pre__get__h_Opelims,axiom,
    ! [X: product_prod_nat_nat,Y: nat] :
      ( ( ( pre_get_h_nat_nat @ X )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ pre_ge4447934673086891262at_nat @ X )
       => ~ ( ( Y
              = ( product_snd_nat_nat @ X ) )
           => ~ ( accp_P4275260045618599050at_nat @ pre_ge4447934673086891262at_nat @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_702_pre__get__h_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: list_char] :
      ( ( ( pre_ge1487422843495280912t_char @ X )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ pre_ge7016768201484263823t_char @ X )
       => ~ ( ( Y
              = ( produc1900778367047502200t_char @ X ) )
           => ~ ( accp_P8892882183480857371t_char @ pre_ge7016768201484263823t_char @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_703_pre__get__h_Opelims,axiom,
    ! [X: produc7606469099237612818ar_nat,Y: nat] :
      ( ( ( pre_ge846198710170331792ar_nat @ X )
        = Y )
     => ( ( accp_P5818767966387969819ar_nat @ pre_ge6375544068159314703ar_nat @ X )
       => ~ ( ( Y
              = ( produc1259554233722553080ar_nat @ X ) )
           => ~ ( accp_P5818767966387969819ar_nat @ pre_ge6375544068159314703ar_nat @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_704_exE__realizer,axiom,
    ! [P2: list_char > nat > $o,P7: produc1457211279475724562t_char,Q: $o > $o,F4: nat > list_char > $o] :
      ( ( P2 @ ( produc1900778367047502200t_char @ P7 ) @ ( produc6204653154488278070t_char @ P7 ) )
     => ( ! [X2: nat,Y4: list_char] :
            ( ( P2 @ Y4 @ X2 )
           => ( Q @ ( F4 @ X2 @ Y4 ) ) )
       => ( Q @ ( produc8526122873445342073char_o @ F4 @ P7 ) ) ) ) ).

% exE_realizer
thf(fact_705_exE__realizer,axiom,
    ! [P2: list_char > nat > $o,P7: produc1457211279475724562t_char,Q: nat > $o,F4: nat > list_char > nat] :
      ( ( P2 @ ( produc1900778367047502200t_char @ P7 ) @ ( produc6204653154488278070t_char @ P7 ) )
     => ( ! [X2: nat,Y4: list_char] :
            ( ( P2 @ Y4 @ X2 )
           => ( Q @ ( F4 @ X2 @ Y4 ) ) )
       => ( Q @ ( produc9163108120003956271ar_nat @ F4 @ P7 ) ) ) ) ).

% exE_realizer
thf(fact_706_exE__realizer,axiom,
    ! [P2: nat > nat > $o,P7: product_prod_nat_nat,Q: product_prod_nat_nat > $o,F4: nat > nat > product_prod_nat_nat] :
      ( ( P2 @ ( product_snd_nat_nat @ P7 ) @ ( product_fst_nat_nat @ P7 ) )
     => ( ! [X2: nat,Y4: nat] :
            ( ( P2 @ Y4 @ X2 )
           => ( Q @ ( F4 @ X2 @ Y4 ) ) )
       => ( Q @ ( produc2626176000494625587at_nat @ F4 @ P7 ) ) ) ) ).

% exE_realizer
thf(fact_707_exE__realizer,axiom,
    ! [P2: nat > nat > $o,P7: product_prod_nat_nat,Q: ( product_prod_nat_nat > $o ) > $o,F4: nat > nat > product_prod_nat_nat > $o] :
      ( ( P2 @ ( product_snd_nat_nat @ P7 ) @ ( product_fst_nat_nat @ P7 ) )
     => ( ! [X2: nat,Y4: nat] :
            ( ( P2 @ Y4 @ X2 )
           => ( Q @ ( F4 @ X2 @ Y4 ) ) )
       => ( Q @ ( produc8739625826339149834_nat_o @ F4 @ P7 ) ) ) ) ).

% exE_realizer
thf(fact_708_exE__realizer,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,P7: produc3336119888008106215r_prat,Q: $o > $o,F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( P2 @ ( produc5358294431189718917r_prat @ P7 ) @ ( produc2193917115357690435r_prat @ P7 ) )
     => ( ! [X2: produc1457211279475724562t_char > prat,Y4: produc1457211279475724562t_char > prat] :
            ( ( P2 @ Y4 @ X2 )
           => ( Q @ ( F4 @ X2 @ Y4 ) ) )
       => ( Q @ ( produc5069634436926473260prat_o @ F4 @ P7 ) ) ) ) ).

% exE_realizer
thf(fact_709_exE__realizer,axiom,
    ! [P2: nat > nat > $o,P7: product_prod_nat_nat,Q: $o > $o,F4: nat > nat > $o] :
      ( ( P2 @ ( product_snd_nat_nat @ P7 ) @ ( product_fst_nat_nat @ P7 ) )
     => ( ! [X2: nat,Y4: nat] :
            ( ( P2 @ Y4 @ X2 )
           => ( Q @ ( F4 @ X2 @ Y4 ) ) )
       => ( Q @ ( produc6081775807080527818_nat_o @ F4 @ P7 ) ) ) ) ).

% exE_realizer
thf(fact_710_snd__fst__flip,axiom,
    ( produc1660480999306372870r_prat
    = ( comp_P6127525099765932504r_prat @ produc5001474941816913666t_char
      @ ( produc1491052064730943609t_char
        @ ^ [X5: produc1457211279475724562t_char,Y2: produc1457211279475724562t_char > prat] : ( produc2591414526589101846t_char @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_711_snd__fst__flip,axiom,
    ( produc1502536198929214834r_prat
    = ( comp_P5993110389636870232r_prat @ produc583991874214702742t_char
      @ ( produc2751466438118233505t_char
        @ ^ [X5: produc2489117125269924006t_char,Y2: produc1457211279475724562t_char > prat] : ( produc2957999048406202538t_char @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_712_snd__fst__flip,axiom,
    ( produc5358294431189718917r_prat
    = ( comp_P71585005891228568r_prat @ produc2193917115357690435r_prat
      @ ( produc644870500113131387r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_713_snd__fst__flip,axiom,
    ( produc3870545126281465950a_prat
    = ( comp_P4236322220017179244a_prat @ produc725351007466518186prat_a
      @ ( produc786514161592030665prat_a
        @ ^ [X5: produc3375937593958228902prat_a,Y2: a > prat] : ( produc2856268368059332798prat_a @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_714_snd__fst__flip,axiom,
    ( product_snd_a_a_prat
    = ( comp_P3353187207481185516a_prat @ product_fst_a_prat_a
      @ ( produc7867159419813889657prat_a
        @ ^ [X5: a,Y2: a > prat] : ( produc3557287861060643862prat_a @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_715_snd__fst__flip,axiom,
    ( produc1900778367047502200t_char
    = ( comp_P3053329765100910016t_char @ produc5563429021163328950ar_nat
      @ ( produc997380522159630963ar_nat
        @ ^ [X5: nat,Y2: list_char] : ( produc8702345860417625034ar_nat @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_716_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X5: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_717_snd__fst__flip,axiom,
    ( produc1259554233722553080ar_nat
    = ( comp_P3187634502937433873ar_nat @ produc6204653154488278070t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X5: list_char,Y2: nat] : ( produc120197956887798346t_char @ Y2 @ X5 ) ) ) ) ).

% snd_fst_flip
thf(fact_718_fst__snd__flip,axiom,
    ( produc1177041924757798084r_prat
    = ( comp_P1328237187794921817r_prat @ produc5484914016365488452t_char
      @ ( produc1491052064730943609t_char
        @ ^ [X5: produc1457211279475724562t_char,Y2: produc1457211279475724562t_char > prat] : ( produc2591414526589101846t_char @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_719_fst__snd__flip,axiom,
    ( produc5426935388913681968r_prat
    = ( comp_P3658868354089463237r_prat @ produc5882964721085011416t_char
      @ ( produc2751466438118233505t_char
        @ ^ [X5: produc2489117125269924006t_char,Y2: produc1457211279475724562t_char > prat] : ( produc2957999048406202538t_char @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_720_fst__snd__flip,axiom,
    ( produc2193917115357690435r_prat
    = ( comp_P71585005891228568r_prat @ produc5358294431189718917r_prat
      @ ( produc644870500113131387r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_721_fst__snd__flip,axiom,
    ( produc6274705478813596956a_prat
    = ( comp_P4350617202128291781a_prat @ produc7544562691789162988prat_a
      @ ( produc786514161592030665prat_a
        @ ^ [X5: produc3375937593958228902prat_a,Y2: a > prat] : ( produc2856268368059332798prat_a @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_722_fst__snd__flip,axiom,
    ( product_fst_a_a_prat
    = ( comp_P3523072553637217901a_prat @ product_snd_a_prat_a
      @ ( produc7867159419813889657prat_a
        @ ^ [X5: a,Y2: a > prat] : ( produc3557287861060643862prat_a @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_723_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X5: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_724_fst__snd__flip,axiom,
    ( produc5563429021163328950ar_nat
    = ( comp_P26140635296239040ar_nat @ produc1900778367047502200t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X5: list_char,Y2: nat] : ( produc120197956887798346t_char @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_725_fst__snd__flip,axiom,
    ( produc6204653154488278070t_char
    = ( comp_P680174135349460753t_char @ produc1259554233722553080ar_nat
      @ ( produc997380522159630963ar_nat
        @ ^ [X5: nat,Y2: list_char] : ( produc8702345860417625034ar_nat @ Y2 @ X5 ) ) ) ) ).

% fst_snd_flip
thf(fact_726_The__case__prod,axiom,
    ! [P2: list_char > nat > $o] :
      ( ( the_Pr6351503334702488345ar_nat @ ( produc2492928673396900345_nat_o @ P2 ) )
      = ( the_Pr6351503334702488345ar_nat
        @ ^ [Xy: produc7606469099237612818ar_nat] : ( P2 @ ( produc5563429021163328950ar_nat @ Xy ) @ ( produc1259554233722553080ar_nat @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_727_The__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( the_Pr7520103294981519086r_prat @ ( produc5069634436926473260prat_o @ P2 ) )
      = ( the_Pr7520103294981519086r_prat
        @ ^ [Xy: produc3336119888008106215r_prat] : ( P2 @ ( produc2193917115357690435r_prat @ Xy ) @ ( produc5358294431189718917r_prat @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_728_The__case__prod,axiom,
    ! [P2: nat > list_char > $o] :
      ( ( the_Pr202245514940600089t_char @ ( produc8526122873445342073char_o @ P2 ) )
      = ( the_Pr202245514940600089t_char
        @ ^ [Xy: produc1457211279475724562t_char] : ( P2 @ ( produc6204653154488278070t_char @ Xy ) @ ( produc1900778367047502200t_char @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_729_The__case__prod,axiom,
    ! [P2: nat > nat > $o] :
      ( ( the_Pr7557018466319803784at_nat @ ( produc6081775807080527818_nat_o @ P2 ) )
      = ( the_Pr7557018466319803784at_nat
        @ ^ [Xy: product_prod_nat_nat] : ( P2 @ ( product_fst_nat_nat @ Xy ) @ ( product_snd_nat_nat @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_730_Range__Collect__case__prod,axiom,
    ! [P2: nat > list_char > $o] :
      ( ( range_nat_list_char @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P2 ) ) )
      = ( collect_list_char
        @ ^ [Y2: list_char] :
          ? [X5: nat] : ( P2 @ X5 @ Y2 ) ) ) ).

% Range_Collect_case_prod
thf(fact_731_Range__Collect__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( range_652675479909198133r_prat @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ P2 ) ) )
      = ( collec7093475260167521854r_prat
        @ ^ [Y2: produc1457211279475724562t_char > prat] :
          ? [X5: produc1457211279475724562t_char > prat] : ( P2 @ X5 @ Y2 ) ) ) ).

% Range_Collect_case_prod
thf(fact_732_Range__Collect__case__prod,axiom,
    ! [P2: nat > nat > $o] :
      ( ( range_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
      = ( collect_nat
        @ ^ [Y2: nat] :
          ? [X5: nat] : ( P2 @ X5 @ Y2 ) ) ) ).

% Range_Collect_case_prod
thf(fact_733_rp__inv__image__def,axiom,
    ( fun_rp3004530424201022588at_nat
    = ( produc1827612380487965875at_nat
      @ ^ [R4: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat,F2: nat > nat] : ( produc2922128104949294807at_nat @ ( inv_image_nat_nat @ R4 @ F2 ) @ ( inv_image_nat_nat @ S4 @ F2 ) ) ) ) ).

% rp_inv_image_def
thf(fact_734_The__split__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char] :
      ( ( the_Pr6337852224591121453t_char
        @ ( produc186087365799766445char_o
          @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc2591414526589101846t_char @ X @ Y ) ) ).

% The_split_eq
thf(fact_735_The__split__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc2489117125269924006t_char] :
      ( ( the_Pr7226236956677534145t_char
        @ ( produc8988401804721015065char_o
          @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc2489117125269924006t_char] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc2957999048406202538t_char @ X @ Y ) ) ).

% The_split_eq
thf(fact_736_The__split__eq,axiom,
    ! [X: a > prat,Y: produc3375937593958228902prat_a] :
      ( ( the_Pr1956078245877658069prat_a
        @ ( produc1272685208547636229at_a_o
          @ ^ [X7: a > prat,Y6: produc3375937593958228902prat_a] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc2856268368059332798prat_a @ X @ Y ) ) ).

% The_split_eq
thf(fact_737_The__split__eq,axiom,
    ! [X: a > prat,Y: a] :
      ( ( the_Pr690290899479495469prat_a
        @ ( produc7462986796979360941at_a_o
          @ ^ [X7: a > prat,Y6: a] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc3557287861060643862prat_a @ X @ Y ) ) ).

% The_split_eq
thf(fact_738_The__split__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > prat] :
      ( ( the_Pr7520103294981519086r_prat
        @ ( produc5069634436926473260prat_o
          @ ^ [X7: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > prat] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc1164665353665699927r_prat @ X @ Y ) ) ).

% The_split_eq
thf(fact_739_The__split__eq,axiom,
    ! [X: nat,Y: list_char] :
      ( ( the_Pr202245514940600089t_char
        @ ( produc8526122873445342073char_o
          @ ^ [X7: nat,Y6: list_char] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( produc120197956887798346t_char @ X @ Y ) ) ).

% The_split_eq
thf(fact_740_The__split__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X7: nat,Y6: nat] :
              ( ( X = X7 )
              & ( Y = Y6 ) ) ) )
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_741_Nitpick_OThe__psimp,axiom,
    ! [P2: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ( P2
        = ( ^ [Y3: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y3 = Z )
          @ X ) )
     => ( ( the_Pr7557018466319803784at_nat @ P2 )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_742_Nitpick_OThe__psimp,axiom,
    ! [P2: produc1457211279475724562t_char > $o,X: produc1457211279475724562t_char] :
      ( ( P2
        = ( ^ [Y3: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] : ( Y3 = Z )
          @ X ) )
     => ( ( the_Pr202245514940600089t_char @ P2 )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_743_Nitpick_OThe__psimp,axiom,
    ! [P2: nat > $o,X: nat] :
      ( ( P2
        = ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
          @ X ) )
     => ( ( the_nat @ P2 )
        = X ) ) ).

% Nitpick.The_psimp
thf(fact_744_the__sym__eq__trivial,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( ^ [Y3: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y3 = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_745_the__sym__eq__trivial,axiom,
    ! [X: produc1457211279475724562t_char] :
      ( ( the_Pr202245514940600089t_char
        @ ( ^ [Y3: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] : ( Y3 = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_746_the__sym__eq__trivial,axiom,
    ! [X: nat] :
      ( ( the_nat
        @ ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_747_the__eq__trivial,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ^ [X5: product_prod_nat_nat] : ( X5 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_748_the__eq__trivial,axiom,
    ! [A: produc1457211279475724562t_char] :
      ( ( the_Pr202245514940600089t_char
        @ ^ [X5: produc1457211279475724562t_char] : ( X5 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_749_the__eq__trivial,axiom,
    ! [A: nat] :
      ( ( the_nat
        @ ^ [X5: nat] : ( X5 = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_750_the__equality,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( P2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ( the_Pr7557018466319803784at_nat @ P2 )
          = A ) ) ) ).

% the_equality
thf(fact_751_the__equality,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
      ( ( P2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ( the_Pr202245514940600089t_char @ P2 )
          = A ) ) ) ).

% the_equality
thf(fact_752_the__equality,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( P2 @ A )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ( the_nat @ P2 )
          = A ) ) ) ).

% the_equality
thf(fact_753_the1__equality,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( P2 @ X6 )
          & ! [Y4: product_prod_nat_nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P2 @ A )
       => ( ( the_Pr7557018466319803784at_nat @ P2 )
          = A ) ) ) ).

% the1_equality
thf(fact_754_the1__equality,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
      ( ? [X6: produc1457211279475724562t_char] :
          ( ( P2 @ X6 )
          & ! [Y4: produc1457211279475724562t_char] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P2 @ A )
       => ( ( the_Pr202245514940600089t_char @ P2 )
          = A ) ) ) ).

% the1_equality
thf(fact_755_the1__equality,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ? [X6: nat] :
          ( ( P2 @ X6 )
          & ! [Y4: nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P2 @ A )
       => ( ( the_nat @ P2 )
          = A ) ) ) ).

% the1_equality
thf(fact_756_the1I2,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( P2 @ X6 )
          & ! [Y4: product_prod_nat_nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( the_Pr7557018466319803784at_nat @ P2 ) ) ) ) ).

% the1I2
thf(fact_757_the1I2,axiom,
    ! [P2: produc1457211279475724562t_char > $o,Q: produc1457211279475724562t_char > $o] :
      ( ? [X6: produc1457211279475724562t_char] :
          ( ( P2 @ X6 )
          & ! [Y4: produc1457211279475724562t_char] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( the_Pr202245514940600089t_char @ P2 ) ) ) ) ).

% the1I2
thf(fact_758_the1I2,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( P2 @ X6 )
          & ! [Y4: nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) )
       => ( Q @ ( the_nat @ P2 ) ) ) ) ).

% the1I2
thf(fact_759_If__def,axiom,
    ( if_Pro6206227464963214023at_nat
    = ( ^ [P4: $o,X5: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( the_Pr7557018466319803784at_nat
          @ ^ [Z6: product_prod_nat_nat] :
              ( ( P4
               => ( Z6 = X5 ) )
              & ( ~ P4
               => ( Z6 = Y2 ) ) ) ) ) ) ).

% If_def
thf(fact_760_If__def,axiom,
    ( if_Pro2639532401392527832t_char
    = ( ^ [P4: $o,X5: produc1457211279475724562t_char,Y2: produc1457211279475724562t_char] :
          ( the_Pr202245514940600089t_char
          @ ^ [Z6: produc1457211279475724562t_char] :
              ( ( P4
               => ( Z6 = X5 ) )
              & ( ~ P4
               => ( Z6 = Y2 ) ) ) ) ) ) ).

% If_def
thf(fact_761_If__def,axiom,
    ( if_nat
    = ( ^ [P4: $o,X5: nat,Y2: nat] :
          ( the_nat
          @ ^ [Z6: nat] :
              ( ( P4
               => ( Z6 = X5 ) )
              & ( ~ P4
               => ( Z6 = Y2 ) ) ) ) ) ) ).

% If_def
thf(fact_762_theI2,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat,Q: product_prod_nat_nat > $o] :
      ( ( P2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ! [X2: product_prod_nat_nat] :
              ( ( P2 @ X2 )
             => ( Q @ X2 ) )
         => ( Q @ ( the_Pr7557018466319803784at_nat @ P2 ) ) ) ) ) ).

% theI2
thf(fact_763_theI2,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char,Q: produc1457211279475724562t_char > $o] :
      ( ( P2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ! [X2: produc1457211279475724562t_char] :
              ( ( P2 @ X2 )
             => ( Q @ X2 ) )
         => ( Q @ ( the_Pr202245514940600089t_char @ P2 ) ) ) ) ) ).

% theI2
thf(fact_764_theI2,axiom,
    ! [P2: nat > $o,A: nat,Q: nat > $o] :
      ( ( P2 @ A )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( ! [X2: nat] :
              ( ( P2 @ X2 )
             => ( Q @ X2 ) )
         => ( Q @ ( the_nat @ P2 ) ) ) ) ) ).

% theI2
thf(fact_765_theI,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( P2 @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( P2 @ ( the_Pr7557018466319803784at_nat @ P2 ) ) ) ) ).

% theI
thf(fact_766_theI,axiom,
    ! [P2: produc1457211279475724562t_char > $o,A: produc1457211279475724562t_char] :
      ( ( P2 @ A )
     => ( ! [X2: produc1457211279475724562t_char] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( P2 @ ( the_Pr202245514940600089t_char @ P2 ) ) ) ) ).

% theI
thf(fact_767_theI,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( P2 @ A )
     => ( ! [X2: nat] :
            ( ( P2 @ X2 )
           => ( X2 = A ) )
       => ( P2 @ ( the_nat @ P2 ) ) ) ) ).

% theI
thf(fact_768_theI_H,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( P2 @ X6 )
          & ! [Y4: product_prod_nat_nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( P2 @ ( the_Pr7557018466319803784at_nat @ P2 ) ) ) ).

% theI'
thf(fact_769_theI_H,axiom,
    ! [P2: produc1457211279475724562t_char > $o] :
      ( ? [X6: produc1457211279475724562t_char] :
          ( ( P2 @ X6 )
          & ! [Y4: produc1457211279475724562t_char] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( P2 @ ( the_Pr202245514940600089t_char @ P2 ) ) ) ).

% theI'
thf(fact_770_theI_H,axiom,
    ! [P2: nat > $o] :
      ( ? [X6: nat] :
          ( ( P2 @ X6 )
          & ! [Y4: nat] :
              ( ( P2 @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( P2 @ ( the_nat @ P2 ) ) ) ).

% theI'
thf(fact_771_Rangep__Range__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( rangep_nat_nat
        @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: nat] : ( member_nat @ X5 @ ( range_nat_nat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_772_Rangep__Range__eq,axiom,
    ! [R2: set_Pr7632435056502277254t_char] :
      ( ( rangep3692341034030587962t_char
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc1457211279475724562t_char] : ( member484235747325421115t_char @ X5 @ ( range_3772036313548763892t_char @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_773_Rangep__Range__eq,axiom,
    ! [R2: set_Pr1935176096852982554t_char] :
      ( ( rangep7864699256716524494t_char
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc2489117125269924006t_char] : ( member1441224781927977039t_char @ X5 @ ( range_4746785725432321928t_char @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_774_Rangep__Range__eq,axiom,
    ! [R2: set_Pr1685704261205283783r_prat] :
      ( ( rangep8630193229993553787r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc1457211279475724562t_char > prat] : ( member2336342582916103932r_prat @ X5 @ ( range_652675479909198133r_prat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_775_Rangep__Range__eq,axiom,
    ! [R2: set_Pr9117357208957936942prat_a] :
      ( ( rangep7799018733632820194prat_a
        @ ^ [X5: a > prat,Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: produc3375937593958228902prat_a] : ( member1351683188118327119prat_a @ X5 @ ( range_1214164913069337500prat_a @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_776_Rangep__Range__eq,axiom,
    ! [R2: set_Pr2748872423075033990prat_a] :
      ( ( rangep_a_prat_a
        @ ^ [X5: a > prat,Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: a] : ( member_a @ X5 @ ( range_a_prat_a @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_777_Rangep__Range__eq,axiom,
    ! [R2: set_Pr2507339680178222706t_char] :
      ( ( rangep_nat_list_char
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: list_char] : ( member_list_char @ X5 @ ( range_nat_list_char @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_778_Range__def,axiom,
    ( range_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ( rangep_nat_nat
            @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_779_Range__def,axiom,
    ( range_3772036313548763892t_char
    = ( ^ [R: set_Pr7632435056502277254t_char] :
          ( collec2458894688891239037t_char
          @ ( rangep3692341034030587962t_char
            @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_780_Range__def,axiom,
    ( range_4746785725432321928t_char
    = ( ^ [R: set_Pr1935176096852982554t_char] :
          ( collec8998774337691853201t_char
          @ ( rangep7864699256716524494t_char
            @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_781_Range__def,axiom,
    ( range_652675479909198133r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat] :
          ( collec7093475260167521854r_prat
          @ ( rangep8630193229993553787r_prat
            @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_782_Range__def,axiom,
    ( range_1214164913069337500prat_a
    = ( ^ [R: set_Pr9117357208957936942prat_a] :
          ( collec2652489992548837521prat_a
          @ ( rangep7799018733632820194prat_a
            @ ^ [X5: a > prat,Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_783_Range__def,axiom,
    ( range_a_prat_a
    = ( ^ [R: set_Pr2748872423075033990prat_a] :
          ( collect_a
          @ ( rangep_a_prat_a
            @ ^ [X5: a > prat,Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_784_Range__def,axiom,
    ( range_nat_list_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collect_list_char
          @ ( rangep_nat_list_char
            @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_785_K__record__comp,axiom,
    ! [C: list_char,F4: produc7606469099237612818ar_nat > produc1457211279475724562t_char] :
      ( ( comp_P26140635296239040ar_nat
        @ ^ [X5: produc1457211279475724562t_char] : C
        @ F4 )
      = ( ^ [X5: produc7606469099237612818ar_nat] : C ) ) ).

% K_record_comp
thf(fact_786_K__record__comp,axiom,
    ! [C: nat,F4: produc7606469099237612818ar_nat > produc1457211279475724562t_char] :
      ( ( comp_P3187634502937433873ar_nat
        @ ^ [X5: produc1457211279475724562t_char] : C
        @ F4 )
      = ( ^ [X5: produc7606469099237612818ar_nat] : C ) ) ).

% K_record_comp
thf(fact_787_K__record__comp,axiom,
    ! [C: $o,F4: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( comp_P6344650113790022471at_nat
        @ ^ [X5: product_prod_nat_nat] : C
        @ F4 )
      = ( ^ [X5: product_prod_nat_nat] : C ) ) ).

% K_record_comp
thf(fact_788_K__record__comp,axiom,
    ! [C: nat,F4: produc1457211279475724562t_char > product_prod_nat_nat] :
      ( ( comp_P2122579405136797538t_char
        @ ^ [X5: product_prod_nat_nat] : C
        @ F4 )
      = ( ^ [X5: produc1457211279475724562t_char] : C ) ) ).

% K_record_comp
thf(fact_789_K__record__comp,axiom,
    ! [C: product_prod_nat_nat,F4: produc1457211279475724562t_char > nat] :
      ( ( comp_n1645060710405774596t_char
        @ ^ [X5: nat] : C
        @ F4 )
      = ( ^ [X5: produc1457211279475724562t_char] : C ) ) ).

% K_record_comp
thf(fact_790_fst__diag__snd,axiom,
    ( ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( comp_n829915685812334579at_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ product_snd_nat_nat ) )
    = product_snd_nat_nat ) ).

% fst_diag_snd
thf(fact_791_fst__diag__snd,axiom,
    ( ( comp_P4090179523072277615t_char @ produc4841048791867702855t_char
      @ ( comp_l7936247125563682291t_char
        @ ^ [X5: list_char] : ( produc1606739095562851931t_char @ X5 @ X5 )
        @ produc1900778367047502200t_char ) )
    = produc1900778367047502200t_char ) ).

% fst_diag_snd
thf(fact_792_fst__diag__snd,axiom,
    ( ( comp_P8271837224898685794ar_nat @ product_fst_nat_nat
      @ ( comp_n7794318530167662852ar_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ produc1259554233722553080ar_nat ) )
    = produc1259554233722553080ar_nat ) ).

% fst_diag_snd
thf(fact_793_snd__diag__fst,axiom,
    ( ( comp_P1016065305979390063ar_nat @ produc5218843525345763465t_char
      @ ( comp_l4862132908470794739ar_nat
        @ ^ [X5: list_char] : ( produc1606739095562851931t_char @ X5 @ X5 )
        @ produc5563429021163328950ar_nat ) )
    = produc5563429021163328950ar_nat ) ).

% snd_diag_fst
thf(fact_794_snd__diag__fst,axiom,
    ( ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( comp_n829915685812334579at_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ product_fst_nat_nat ) )
    = product_fst_nat_nat ) ).

% snd_diag_fst
thf(fact_795_snd__diag__fst,axiom,
    ( ( comp_P2122579405136797538t_char @ product_snd_nat_nat
      @ ( comp_n1645060710405774596t_char
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ produc6204653154488278070t_char ) )
    = produc6204653154488278070t_char ) ).

% snd_diag_fst
thf(fact_796_old_Orec__unit__def,axiom,
    ( produc4455288259268591162at_nat
    = ( ^ [F1: product_prod_nat_nat,X5: product_unit] : ( the_Pr7557018466319803784at_nat @ ( produc8778658879891000828at_nat @ F1 @ X5 ) ) ) ) ).

% old.rec_unit_def
thf(fact_797_old_Orec__unit__def,axiom,
    ( produc16126686263669963t_char
    = ( ^ [F1: produc1457211279475724562t_char,X5: product_unit] : ( the_Pr202245514940600089t_char @ ( produc7973017197139442061t_char @ F1 @ X5 ) ) ) ) ).

% old.rec_unit_def
thf(fact_798_old_Orec__unit__def,axiom,
    ( product_rec_unit_nat
    = ( ^ [F1: nat,X5: product_unit] : ( the_nat @ ( produc7502264439391792277it_nat @ F1 @ X5 ) ) ) ) ).

% old.rec_unit_def
thf(fact_799_fst__diag__fst,axiom,
    ( ( comp_P1016065305979390063ar_nat @ produc4841048791867702855t_char
      @ ( comp_l4862132908470794739ar_nat
        @ ^ [X5: list_char] : ( produc1606739095562851931t_char @ X5 @ X5 )
        @ produc5563429021163328950ar_nat ) )
    = produc5563429021163328950ar_nat ) ).

% fst_diag_fst
thf(fact_800_fst__diag__fst,axiom,
    ( ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( comp_n829915685812334579at_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ product_fst_nat_nat ) )
    = product_fst_nat_nat ) ).

% fst_diag_fst
thf(fact_801_fst__diag__fst,axiom,
    ( ( comp_P2122579405136797538t_char @ product_fst_nat_nat
      @ ( comp_n1645060710405774596t_char
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ produc6204653154488278070t_char ) )
    = produc6204653154488278070t_char ) ).

% fst_diag_fst
thf(fact_802_snd__diag__snd,axiom,
    ( ( comp_P4090179523072277615t_char @ produc5218843525345763465t_char
      @ ( comp_l7936247125563682291t_char
        @ ^ [X5: list_char] : ( produc1606739095562851931t_char @ X5 @ X5 )
        @ produc1900778367047502200t_char ) )
    = produc1900778367047502200t_char ) ).

% snd_diag_snd
thf(fact_803_snd__diag__snd,axiom,
    ( ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( comp_n829915685812334579at_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ product_snd_nat_nat ) )
    = product_snd_nat_nat ) ).

% snd_diag_snd
thf(fact_804_snd__diag__snd,axiom,
    ( ( comp_P8271837224898685794ar_nat @ product_snd_nat_nat
      @ ( comp_n7794318530167662852ar_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ produc1259554233722553080ar_nat ) )
    = produc1259554233722553080ar_nat ) ).

% snd_diag_snd
thf(fact_805_old_Orec__bool__def,axiom,
    ( produc6024350069564022202at_nat
    = ( ^ [F1: product_prod_nat_nat,F22: product_prod_nat_nat,X5: $o] : ( the_Pr7557018466319803784at_nat @ ( produc865553358233325936at_nat @ F1 @ F22 @ X5 ) ) ) ) ).

% old.rec_bool_def
thf(fact_806_old_Orec__bool__def,axiom,
    ( produc7240177962613183563t_char
    = ( ^ [F1: produc1457211279475724562t_char,F22: produc1457211279475724562t_char,X5: $o] : ( the_Pr202245514940600089t_char @ ( produc7531586246926991105t_char @ F1 @ F22 @ X5 ) ) ) ) ).

% old.rec_bool_def
thf(fact_807_old_Orec__bool__def,axiom,
    ( product_rec_bool_nat
    = ( ^ [F1: nat,F22: nat,X5: $o] : ( the_nat @ ( produc2035648292922657313ol_nat @ F1 @ F22 @ X5 ) ) ) ) ).

% old.rec_bool_def
thf(fact_808_Domain__Collect__case__prod,axiom,
    ! [P2: nat > list_char > $o] :
      ( ( domain_nat_list_char @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P2 ) ) )
      = ( collect_nat
        @ ^ [X5: nat] :
          ? [X3: list_char] : ( P2 @ X5 @ X3 ) ) ) ).

% Domain_Collect_case_prod
thf(fact_809_Domain__Collect__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( domain4089750737140655564r_prat @ ( collec1156307336577696338r_prat @ ( produc5069634436926473260prat_o @ P2 ) ) )
      = ( collec7093475260167521854r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat] :
          ? [X3: produc1457211279475724562t_char > prat] : ( P2 @ X5 @ X3 ) ) ) ).

% Domain_Collect_case_prod
thf(fact_810_Domain__Collect__case__prod,axiom,
    ! [P2: nat > nat > $o] :
      ( ( domain_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
      = ( collect_nat
        @ ^ [X5: nat] :
          ? [X3: nat] : ( P2 @ X5 @ X3 ) ) ) ).

% Domain_Collect_case_prod
thf(fact_811_snd__diag__id,axiom,
    ! [Z4: produc1457211279475724562t_char > prat] :
      ( ( comp_P2783082240043180164r_prat @ produc5358294431189718917r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ X5 @ X5 )
        @ Z4 )
      = ( id_Pro6476562945676181316r_prat @ Z4 ) ) ).

% snd_diag_id
thf(fact_812_snd__diag__id,axiom,
    ! [Z4: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ Z4 )
      = ( id_nat @ Z4 ) ) ).

% snd_diag_id
thf(fact_813_fst__diag__id,axiom,
    ! [Z4: produc1457211279475724562t_char > prat] :
      ( ( comp_P2783082240043180164r_prat @ produc2193917115357690435r_prat
        @ ^ [X5: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ X5 @ X5 )
        @ Z4 )
      = ( id_Pro6476562945676181316r_prat @ Z4 ) ) ).

% fst_diag_id
thf(fact_814_fst__diag__id,axiom,
    ! [Z4: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat
        @ ^ [X5: nat] : ( product_Pair_nat_nat @ X5 @ X5 )
        @ Z4 )
      = ( id_nat @ Z4 ) ) ).

% fst_diag_id
thf(fact_815_case__swap,axiom,
    ! [F4: nat > list_char > $o,P7: produc1457211279475724562t_char] :
      ( ( produc2492928673396900345_nat_o
        @ ^ [Y2: list_char,X5: nat] : ( F4 @ X5 @ Y2 )
        @ ( produc5051419219066781674t_char @ P7 ) )
      = ( produc8526122873445342073char_o @ F4 @ P7 ) ) ).

% case_swap
thf(fact_816_case__swap,axiom,
    ! [F4: nat > list_char > nat,P7: produc1457211279475724562t_char] :
      ( ( produc7665377738449061807at_nat
        @ ^ [Y2: list_char,X5: nat] : ( F4 @ X5 @ Y2 )
        @ ( produc5051419219066781674t_char @ P7 ) )
      = ( produc9163108120003956271ar_nat @ F4 @ P7 ) ) ).

% case_swap
thf(fact_817_case__swap,axiom,
    ! [F4: list_char > nat > $o,P7: produc7606469099237612818ar_nat] :
      ( ( produc8526122873445342073char_o
        @ ^ [Y2: nat,X5: list_char] : ( F4 @ X5 @ Y2 )
        @ ( produc4410195085741832554ar_nat @ P7 ) )
      = ( produc2492928673396900345_nat_o @ F4 @ P7 ) ) ).

% case_swap
thf(fact_818_case__swap,axiom,
    ! [F4: list_char > nat > nat,P7: produc7606469099237612818ar_nat] :
      ( ( produc9163108120003956271ar_nat
        @ ^ [Y2: nat,X5: list_char] : ( F4 @ X5 @ Y2 )
        @ ( produc4410195085741832554ar_nat @ P7 ) )
      = ( produc7665377738449061807at_nat @ F4 @ P7 ) ) ).

% case_swap
thf(fact_819_case__swap,axiom,
    ! [F4: nat > nat > product_prod_nat_nat,P7: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [Y2: nat,X5: nat] : ( F4 @ X5 @ Y2 )
        @ ( product_swap_nat_nat @ P7 ) )
      = ( produc2626176000494625587at_nat @ F4 @ P7 ) ) ).

% case_swap
thf(fact_820_case__swap,axiom,
    ! [F4: nat > nat > product_prod_nat_nat > $o,P7: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [Y2: nat,X5: nat] : ( F4 @ X5 @ Y2 )
        @ ( product_swap_nat_nat @ P7 ) )
      = ( produc8739625826339149834_nat_o @ F4 @ P7 ) ) ).

% case_swap
thf(fact_821_case__swap,axiom,
    ! [F4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,P7: produc3336119888008106215r_prat] :
      ( ( produc5069634436926473260prat_o
        @ ^ [Y2: produc1457211279475724562t_char > prat,X5: produc1457211279475724562t_char > prat] : ( F4 @ X5 @ Y2 )
        @ ( produc82946499796401655r_prat @ P7 ) )
      = ( produc5069634436926473260prat_o @ F4 @ P7 ) ) ).

% case_swap
thf(fact_822_case__swap,axiom,
    ! [F4: nat > nat > $o,P7: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y2: nat,X5: nat] : ( F4 @ X5 @ Y2 )
        @ ( product_swap_nat_nat @ P7 ) )
      = ( produc6081775807080527818_nat_o @ F4 @ P7 ) ) ).

% case_swap
thf(fact_823_relcomp__unfold,axiom,
    ( relcom7222395050640001041t_char
    = ( ^ [R: set_Pr1685704261205283783r_prat,S3: set_Pr7632435056502277254t_char] :
          ( collec8998774337691853201t_char
          @ ( produc186087365799766445char_o
            @ ^ [X5: produc1457211279475724562t_char > prat,Z6: produc1457211279475724562t_char] :
              ? [Y2: produc1457211279475724562t_char > prat] :
                ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R )
                & ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_824_relcomp__unfold,axiom,
    ( relcom2001116082538860837t_char
    = ( ^ [R: set_Pr1685704261205283783r_prat,S3: set_Pr1935176096852982554t_char] :
          ( collec1912346231567511845t_char
          @ ( produc8988401804721015065char_o
            @ ^ [X5: produc1457211279475724562t_char > prat,Z6: produc2489117125269924006t_char] :
              ? [Y2: produc1457211279475724562t_char > prat] :
                ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R )
                & ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_825_relcomp__unfold,axiom,
    ( relcomp_nat_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X5: nat,Z6: nat] :
              ? [Y2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_826_relcomp__unfold,axiom,
    ( relcom6453288877151117700t_char
    = ( ^ [R: set_Pr1261947904930325089at_nat,S3: set_Pr2507339680178222706t_char] :
          ( collec2458894688891239037t_char
          @ ( produc8526122873445342073char_o
            @ ^ [X5: nat,Z6: list_char] :
              ? [Y2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R )
                & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_827_relcomp__unfold,axiom,
    ( relcom5956200836249630357t_char
    = ( ^ [R: set_Pr2507339680178222706t_char,S3: set_Pr4480099295215233283t_char] :
          ( collec2458894688891239037t_char
          @ ( produc8526122873445342073char_o
            @ ^ [X5: nat,Z6: list_char] :
              ? [Y2: list_char] :
                ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R )
                & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_828_relcomp__unfold,axiom,
    ( relcom3397962033580885459r_prat
    = ( ^ [R: set_Pr7632435056502277254t_char,S3: set_Pr4624115533177120456r_prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ^ [X5: produc1457211279475724562t_char > prat,Z6: produc1457211279475724562t_char > prat] :
              ? [Y2: produc1457211279475724562t_char] :
                ( ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R )
                & ( member7861002897371493265r_prat @ ( produc7990353546384762072r_prat @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_829_relcomp__unfold,axiom,
    ( relcom6844059597237840063r_prat
    = ( ^ [R: set_Pr1935176096852982554t_char,S3: set_Pr4683309014602836788r_prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ^ [X5: produc1457211279475724562t_char > prat,Z6: produc1457211279475724562t_char > prat] :
              ? [Y2: produc2489117125269924006t_char] :
                ( ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R )
                & ( member8465322205588144125r_prat @ ( produc7800942563105181764r_prat @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_830_relcomp__unfold,axiom,
    ( relcom6619832327022793170r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat,S3: set_Pr1685704261205283783r_prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ^ [X5: produc1457211279475724562t_char > prat,Z6: produc1457211279475724562t_char > prat] :
              ? [Y2: produc1457211279475724562t_char > prat] :
                ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R )
                & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_831_relcomp__unfold,axiom,
    ( relcom5812064743826168580ar_nat
    = ( ^ [R: set_Pr2507339680178222706t_char,S3: set_Pr8803067691650843250ar_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X5: nat,Z6: nat] :
              ? [Y2: list_char] :
                ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R )
                & ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ Y2 @ Z6 ) @ S3 ) ) ) ) ) ) ).

% relcomp_unfold
thf(fact_832_Domain__unfold,axiom,
    ( domain_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
            ? [Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_833_Domain__unfold,axiom,
    ( domain4991527842029491467t_char
    = ( ^ [R: set_Pr7632435056502277254t_char] :
          ( collec7093475260167521854r_prat
          @ ^ [X5: produc1457211279475724562t_char > prat] :
            ? [Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_834_Domain__unfold,axiom,
    ( domain2005253360264185631t_char
    = ( ^ [R: set_Pr1935176096852982554t_char] :
          ( collec7093475260167521854r_prat
          @ ^ [X5: produc1457211279475724562t_char > prat] :
            ? [Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_835_Domain__unfold,axiom,
    ( domain4089750737140655564r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat] :
          ( collec7093475260167521854r_prat
          @ ^ [X5: produc1457211279475724562t_char > prat] :
            ? [Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_836_Domain__unfold,axiom,
    ( domain2177972146002949427prat_a
    = ( ^ [R: set_Pr9117357208957936942prat_a] :
          ( collect_a_prat
          @ ^ [X5: a > prat] :
            ? [Y2: produc3375937593958228902prat_a] : ( member5370254448528625399prat_a @ ( produc2856268368059332798prat_a @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_837_Domain__unfold,axiom,
    ( domain_a_prat_a
    = ( ^ [R: set_Pr2748872423075033990prat_a] :
          ( collect_a_prat
          @ ^ [X5: a > prat] :
            ? [Y2: a] : ( member1351683188118327119prat_a @ ( produc3557287861060643862prat_a @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_838_Domain__unfold,axiom,
    ( domain_nat_list_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collect_nat
          @ ^ [X5: nat] :
            ? [Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R ) ) ) ) ).

% Domain_unfold
thf(fact_839_image2__Gr,axiom,
    ( bNF_Gr7397543130437154909t_char
    = ( ^ [A4: set_nat,F2: nat > nat,G2: nat > list_char] : ( relcom6453288877151117700t_char @ ( converse_nat_nat @ ( bNF_Gr_nat_nat @ A4 @ F2 ) ) @ ( bNF_Gr_nat_list_char @ A4 @ G2 ) ) ) ) ).

% image2_Gr
thf(fact_840_image2__Gr,axiom,
    ( bNF_Gr7318156773085685964at_nat
    = ( ^ [A4: set_nat,F2: nat > nat,G2: nat > nat] : ( relcomp_nat_nat_nat @ ( converse_nat_nat @ ( bNF_Gr_nat_nat @ A4 @ F2 ) ) @ ( bNF_Gr_nat_nat @ A4 @ G2 ) ) ) ) ).

% image2_Gr
thf(fact_841_image2__Gr,axiom,
    ( bNF_Gr689107047223432174t_char
    = ( ^ [A4: set_nat,F2: nat > list_char,G2: nat > list_char] : ( relcom1214091342717857813t_char @ ( conver1319806385908312430t_char @ ( bNF_Gr_nat_list_char @ A4 @ F2 ) ) @ ( bNF_Gr_nat_list_char @ A4 @ G2 ) ) ) ) ).

% image2_Gr
thf(fact_842_image2__Gr,axiom,
    ( bNF_Gr6756318997112205789ar_nat
    = ( ^ [A4: set_nat,F2: nat > list_char,G2: nat > nat] : ( relcom4314334362271274116at_nat @ ( conver1319806385908312430t_char @ ( bNF_Gr_nat_list_char @ A4 @ F2 ) ) @ ( bNF_Gr_nat_nat @ A4 @ G2 ) ) ) ) ).

% image2_Gr
thf(fact_843_image2__Gr,axiom,
    ( bNF_Gr5258588615557311325at_nat
    = ( ^ [A4: set_list_char,F2: list_char > nat,G2: list_char > nat] : ( relcom5812064743826168580ar_nat @ ( conver678582252583363310ar_nat @ ( bNF_Gr_list_char_nat @ A4 @ F2 ) ) @ ( bNF_Gr_list_char_nat @ A4 @ G2 ) ) ) ) ).

% image2_Gr
thf(fact_844_image2__Gr,axiom,
    ( bNF_Gr5170369590546435438t_char
    = ( ^ [A4: set_list_char,F2: list_char > nat,G2: list_char > list_char] : ( relcom5956200836249630357t_char @ ( conver678582252583363310ar_nat @ ( bNF_Gr_list_char_nat @ A4 @ F2 ) ) @ ( bNF_Gr2197830230412165350t_char @ A4 @ G2 ) ) ) ) ).

% image2_Gr
thf(fact_845_Id__on__Gr,axiom,
    ( id_on_nat
    = ( ^ [A4: set_nat] : ( bNF_Gr_nat_nat @ A4 @ id_nat ) ) ) ).

% Id_on_Gr
thf(fact_846_Id__on__Gr,axiom,
    ( id_on_2554058798563519774at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] : ( bNF_Gr4088865321246080482at_nat @ A4 @ id_Pro7559728917991913458at_nat ) ) ) ).

% Id_on_Gr
thf(fact_847_times__subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C3: set_nat,B3: set_Pr1261947904930325089at_nat,D2: set_nat] :
      ( ( ord_le7094820750414311574at_nat
        @ ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : C3 )
        @ ( produc7672662199629908489at_nat @ B3
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( C3 = bot_bot_set_nat )
        | ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_848_times__subset__iff,axiom,
    ! [A3: set_list_char,C3: set_nat,B3: set_list_char,D2: set_nat] :
      ( ( ord_le3387867619833322002ar_nat
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : C3 )
        @ ( produc5389449503320859771ar_nat @ B3
          @ ^ [Uu: list_char] : D2 ) )
      = ( ( A3 = bot_bo8503570744482963627t_char )
        | ( C3 = bot_bot_set_nat )
        | ( ( ord_le1207710660481778807t_char @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_849_times__subset__iff,axiom,
    ! [A3: set_set_nat,C3: set_set_nat,B3: set_set_nat,D2: set_set_nat] :
      ( ( ord_le4402255868550298733et_nat
        @ ( produc105401717395143638et_nat @ A3
          @ ^ [Uu: set_nat] : C3 )
        @ ( produc105401717395143638et_nat @ B3
          @ ^ [Uu: set_nat] : D2 ) )
      = ( ( A3 = bot_bot_set_set_nat )
        | ( C3 = bot_bot_set_set_nat )
        | ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
          & ( ord_le6893508408891458716et_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_850_times__subset__iff,axiom,
    ! [A3: set_set_nat,C3: set_nat,B3: set_set_nat,D2: set_nat] :
      ( ( ord_le7046853370993658551at_nat
        @ ( produc5318310618559172640at_nat @ A3
          @ ^ [Uu: set_nat] : C3 )
        @ ( produc5318310618559172640at_nat @ B3
          @ ^ [Uu: set_nat] : D2 ) )
      = ( ( A3 = bot_bot_set_set_nat )
        | ( C3 = bot_bot_set_nat )
        | ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_851_times__subset__iff,axiom,
    ! [A3: set_nat,C3: set_Pr1261947904930325089at_nat,B3: set_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3050193496395145148at_nat
        @ ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : C3 )
        @ ( produc1809337555817847783at_nat @ B3
          @ ^ [Uu: nat] : D2 ) )
      = ( ( A3 = bot_bot_set_nat )
        | ( C3 = bot_bo2099793752762293965at_nat )
        | ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_le3146513528884898305at_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_852_times__subset__iff,axiom,
    ! [A3: set_nat,C3: set_list_char,B3: set_nat,D2: set_list_char] :
      ( ( ord_le6315511645215477266t_char
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : C3 )
        @ ( produc6030673636645808891t_char @ B3
          @ ^ [Uu: nat] : D2 ) )
      = ( ( A3 = bot_bot_set_nat )
        | ( C3 = bot_bo8503570744482963627t_char )
        | ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_le1207710660481778807t_char @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_853_times__subset__iff,axiom,
    ! [A3: set_nat,C3: set_set_nat,B3: set_nat,D2: set_set_nat] :
      ( ( ord_le3845944159117341623et_nat
        @ ( produc8883945523214541856et_nat @ A3
          @ ^ [Uu: nat] : C3 )
        @ ( produc8883945523214541856et_nat @ B3
          @ ^ [Uu: nat] : D2 ) )
      = ( ( A3 = bot_bot_set_nat )
        | ( C3 = bot_bot_set_set_nat )
        | ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_le6893508408891458716et_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_854_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_855_relcomp__def,axiom,
    ( relcom7222395050640001041t_char
    = ( ^ [R: set_Pr1685704261205283783r_prat,S3: set_Pr7632435056502277254t_char] :
          ( collec8998774337691853201t_char
          @ ( produc186087365799766445char_o
            @ ( relcom8471531622496164321t_char
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R )
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_856_relcomp__def,axiom,
    ( relcom2001116082538860837t_char
    = ( ^ [R: set_Pr1685704261205283783r_prat,S3: set_Pr1935176096852982554t_char] :
          ( collec1912346231567511845t_char
          @ ( produc8988401804721015065char_o
            @ ( relcom7420010653013420789t_char
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R )
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_857_relcomp__def,axiom,
    ( relcomp_nat_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( relcompp_nat_nat_nat
              @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R )
              @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_858_relcomp__def,axiom,
    ( relcom5956200836249630357t_char
    = ( ^ [R: set_Pr2507339680178222706t_char,S3: set_Pr4480099295215233283t_char] :
          ( collec2458894688891239037t_char
          @ ( produc8526122873445342073char_o
            @ ( relcom7349595004981586117t_char
              @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R )
              @ ^ [X5: list_char,Y2: list_char] : ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_859_relcomp__def,axiom,
    ( relcom6453288877151117700t_char
    = ( ^ [R: set_Pr1261947904930325089at_nat,S3: set_Pr2507339680178222706t_char] :
          ( collec2458894688891239037t_char
          @ ( produc8526122873445342073char_o
            @ ( relcom3288008990786313140t_char
              @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R )
              @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_860_relcomp__def,axiom,
    ( relcom3397962033580885459r_prat
    = ( ^ [R: set_Pr7632435056502277254t_char,S3: set_Pr4624115533177120456r_prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ( relcom4647098605437048739r_prat
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char] : ( member1441224781927977039t_char @ ( produc2591414526589101846t_char @ X5 @ Y2 ) @ R )
              @ ^ [X5: produc1457211279475724562t_char,Y2: produc1457211279475724562t_char > prat] : ( member7861002897371493265r_prat @ ( produc7990353546384762072r_prat @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_861_relcomp__def,axiom,
    ( relcom6844059597237840063r_prat
    = ( ^ [R: set_Pr1935176096852982554t_char,S3: set_Pr4683309014602836788r_prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ( relcom3039582130857624207r_prat
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc2489117125269924006t_char] : ( member5171123587969721059t_char @ ( produc2957999048406202538t_char @ X5 @ Y2 ) @ R )
              @ ^ [X5: produc2489117125269924006t_char,Y2: produc1457211279475724562t_char > prat] : ( member8465322205588144125r_prat @ ( produc7800942563105181764r_prat @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_862_relcomp__def,axiom,
    ( relcom6619832327022793170r_prat
    = ( ^ [R: set_Pr1685704261205283783r_prat,S3: set_Pr1685704261205283783r_prat] :
          ( collec1156307336577696338r_prat
          @ ( produc5069634436926473260prat_o
            @ ( relcom8047291834541808546r_prat
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ R )
              @ ^ [X5: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_863_relcomp__def,axiom,
    ( relcom5812064743826168580ar_nat
    = ( ^ [R: set_Pr2507339680178222706t_char,S3: set_Pr8803067691650843250ar_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( relcom2646784857461364020ar_nat
              @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R )
              @ ^ [X5: list_char,Y2: nat] : ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X5 @ Y2 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_864_in__rel__Gr,axiom,
    ! [A3: set_nat,F4: nat > list_char] :
      ( ( fun_in175243070162512245t_char @ ( bNF_Gr_nat_list_char @ A3 @ F4 ) )
      = ( bNF_Gr2905446856833220225t_char @ A3 @ F4 ) ) ).

% in_rel_Gr
thf(fact_865_in__rel__Gr,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ( fun_in_rel_nat_nat @ ( bNF_Gr_nat_nat @ A3 @ F4 ) )
      = ( bNF_Grp_nat_nat @ A3 @ F4 ) ) ).

% in_rel_Gr
thf(fact_866_fun_Oin__rel,axiom,
    ! [R3: $o > $o > $o,A: nat > $o,B: nat > $o] :
      ( ( bNF_re4705727531993890431at_o_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: nat > product_prod_o_o] :
            ( ( member9175467367257937463od_o_o @ Z6
              @ ( collec5126044608080245625od_o_o
                @ ^ [X5: nat > product_prod_o_o] : ( ord_le192195778054999463od_o_o @ ( image_102748095387869968od_o_o @ X5 @ top_top_set_nat ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ R3 ) ) ) ) )
            & ( ( comp_P7081879951647482084_o_nat @ product_fst_o_o @ Z6 )
              = A )
            & ( ( comp_P7081879951647482084_o_nat @ product_snd_o_o @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_867_fun_Oin__rel,axiom,
    ! [R3: nat > nat > $o,A: nat > nat,B: nat > nat] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: nat > product_prod_nat_nat] :
            ( ( member6743353555990375057at_nat @ Z6
              @ ( collec8718012497556192979at_nat
                @ ^ [X5: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X5 @ 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_868_fun_Oin__rel,axiom,
    ! [R3: list_char > nat > $o,A: nat > list_char,B: nat > nat] :
      ( ( bNF_re1435683047850566962ar_nat
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: nat > produc7606469099237612818ar_nat] :
            ( ( member231257875666289314ar_nat @ Z6
              @ ( collec1892443993989155812ar_nat
                @ ^ [X5: nat > produc7606469099237612818ar_nat] : ( ord_le3387867619833322002ar_nat @ ( image_7152680309924031739ar_nat @ X5 @ top_top_set_nat ) @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P5292914338597962658ar_nat @ produc5563429021163328950ar_nat @ Z6 )
              = A )
            & ( ( comp_P8318769832456242705at_nat @ produc1259554233722553080ar_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_869_fun_Oin__rel,axiom,
    ! [R3: nat > nat > $o,A: set_nat > nat,B: set_nat > nat] :
      ( ( bNF_re2508443332600976141at_nat
        @ ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: set_nat > product_prod_nat_nat] :
            ( ( member7598294160067007323at_nat @ Z6
              @ ( collec8743797477381660573at_nat
                @ ^ [X5: set_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_7073161707855575860at_nat @ X5 @ top_top_set_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P5369868937050299702et_nat @ product_fst_nat_nat @ Z6 )
              = A )
            & ( ( comp_P5369868937050299702et_nat @ product_snd_nat_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_870_fun_Oin__rel,axiom,
    ! [R3: nat > list_char > $o,A: nat > nat,B: nat > list_char] :
      ( ( bNF_re2076907181175516082t_char
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: nat > produc1457211279475724562t_char] :
            ( ( member3158901901048444578t_char @ Z6
              @ ( collec4820088019371311076t_char
                @ ^ [X5: nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_1003422490162143483t_char @ X5 @ 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_871_fun_Oin__rel,axiom,
    ! [R3: list_char > nat > $o,A: set_nat > list_char,B: set_nat > nat] :
      ( ( bNF_re6814798851205064350ar_nat
        @ ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: set_nat > produc7606469099237612818ar_nat] :
            ( ( member8226418253345158252ar_nat @ Z6
              @ ( collec6560429172039693742ar_nat
                @ ^ [X5: set_nat > produc7606469099237612818ar_nat] : ( ord_le3387867619833322002ar_nat @ ( image_6339652515608133317ar_nat @ X5 @ top_top_set_set_nat ) @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P137096244479289176et_nat @ produc5563429021163328950ar_nat @ Z6 )
              = A )
            & ( ( comp_P559749852281925063et_nat @ produc1259554233722553080ar_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_872_fun_Oin__rel,axiom,
    ! [R3: nat > nat > $o,A: ( nat > nat ) > nat,B: ( nat > nat ) > nat] :
      ( ( bNF_re3320998365863425151at_nat
        @ ^ [Y3: nat > nat,Z: nat > nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: ( nat > nat ) > product_prod_nat_nat] :
            ( ( member3666156368143155042at_nat @ Z6
              @ ( collec5327342486466021540at_nat
                @ ^ [X5: ( nat > nat ) > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5687747386542951227at_nat @ X5 @ top_top_set_nat_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P6240816377000802031at_nat @ product_fst_nat_nat @ Z6 )
              = A )
            & ( ( comp_P6240816377000802031at_nat @ product_snd_nat_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_873_fun_Oin__rel,axiom,
    ! [R3: nat > list_char > $o,A: set_nat > nat,B: set_nat > list_char] :
      ( ( bNF_re7456022984530013470t_char
        @ ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: set_nat > produc1457211279475724562t_char] :
            ( ( member1930690241872537708t_char @ Z6
              @ ( collec264701160567073198t_char
                @ ^ [X5: set_nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_190394695846245061t_char @ X5 @ top_top_set_set_nat ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) ) )
            & ( ( comp_P488745826847564231et_nat @ produc6204653154488278070t_char @ Z6 )
              = A )
            & ( ( comp_P7799612233565674328et_nat @ produc1900778367047502200t_char @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_874_fun_Oin__rel,axiom,
    ! [R3: list_char > nat > $o,A: ( nat > nat ) > list_char,B: ( nat > nat ) > nat] :
      ( ( bNF_re6527413442372678928ar_nat
        @ ^ [Y3: nat > nat,Z: nat > nat] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: ( nat > nat ) > produc7606469099237612818ar_nat] :
            ( ( member8116379064319855091ar_nat @ Z6
              @ ( collec4513549283322988597ar_nat
                @ ^ [X5: ( nat > nat ) > produc7606469099237612818ar_nat] : ( ord_le3387867619833322002ar_nat @ ( image_4136341828564351052ar_nat @ X5 @ top_top_set_nat_nat ) @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P1337775633407159825at_nat @ produc5563429021163328950ar_nat @ Z6 )
              = A )
            & ( ( comp_P4430085340299632128at_nat @ produc1259554233722553080ar_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_875_fun_Oin__rel,axiom,
    ! [R3: nat > nat > $o,A: produc1457211279475724562t_char > nat,B: produc1457211279475724562t_char > nat] :
      ( ( bNF_re5269148346131715199at_nat
        @ ^ [Y3: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] : ( Y3 = Z )
        @ R3
        @ A
        @ B )
      = ( ? [Z6: produc1457211279475724562t_char > product_prod_nat_nat] :
            ( ( member5503537209162532075at_nat @ Z6
              @ ( collec3352718687064416553at_nat
                @ ^ [X5: produc1457211279475724562t_char > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_2110972657701675764at_nat @ X5 @ top_to1250346684925019074t_char ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) )
            & ( ( comp_P2122579405136797538t_char @ product_fst_nat_nat @ Z6 )
              = A )
            & ( ( comp_P2122579405136797538t_char @ product_snd_nat_nat @ Z6 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_876_image__ident,axiom,
    ! [Y7: set_nat] :
      ( ( image_nat_nat2
        @ ^ [X5: nat] : X5
        @ Y7 )
      = Y7 ) ).

% image_ident
thf(fact_877_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A2: nat,B2: nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_878_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [A2: nat,B2: product_prod_nat_nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_879_Collect__case__prod,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [A2: product_prod_nat_nat,B2: nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_880_Collect__case__prod,axiom,
    ! [P2: 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] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_881_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q: set_nat > $o] :
      ( ( collec6652498225584317346et_nat
        @ ( produc6720110361610378900_nat_o
          @ ^ [A2: nat,B2: set_nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc8883945523214541856et_nat @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collect_set_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_882_Collect__case__prod,axiom,
    ! [P2: set_nat > $o,Q: nat > $o] :
      ( ( collec2520389976089426594at_nat
        @ ( produc8429754692982665876_nat_o
          @ ^ [A2: set_nat,B2: nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc5318310618559172640at_nat @ ( collect_set_nat @ P2 )
        @ ^ [Uu: set_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_883_Collect__case__prod,axiom,
    ! [P2: list_char > $o,Q: nat > $o] :
      ( ( collec8608152508653127293ar_nat
        @ ( produc2492928673396900345_nat_o
          @ ^ [A2: list_char,B2: nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc5389449503320859771ar_nat @ ( collect_list_char @ P2 )
        @ ^ [Uu: list_char] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_884_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q: list_char > $o] :
      ( ( collec2458894688891239037t_char
        @ ( produc8526122873445342073char_o
          @ ^ [A2: nat,B2: list_char] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc6030673636645808891t_char @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collect_list_char @ Q ) ) ) ).

% Collect_case_prod
thf(fact_885_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q: ( nat > nat ) > $o] :
      ( ( collec1087395353207291739at_nat
        @ ( produc1872543322361064475_nat_o
          @ ^ [A2: nat,B2: nat > nat] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc4526620505022529241at_nat @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collect_nat_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_886_Collect__case__prod,axiom,
    ! [P2: list_char > $o,Q: list_char > $o] :
      ( ( collec7684657524456971406t_char
        @ ( produc5604122169238565672char_o
          @ ^ [A2: list_char,B2: list_char] :
              ( ( P2 @ A2 )
              & ( Q @ B2 ) ) ) )
      = ( produc2134963774413013388t_char @ ( collect_list_char @ P2 )
        @ ^ [Uu: list_char] : ( collect_list_char @ Q ) ) ) ).

% Collect_case_prod
thf(fact_887_Times__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat] :
      ( ( ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 )
        = bot_bo2204296500067978442at_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( B3 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_888_Times__empty,axiom,
    ! [A3: set_list_char,B3: set_nat] :
      ( ( ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 )
        = bot_bo7709822093361376478ar_nat )
      = ( ( A3 = bot_bo8503570744482963627t_char )
        | ( B3 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_889_Times__empty,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ( produc105401717395143638et_nat @ A3
          @ ^ [Uu: set_nat] : B3 )
        = bot_bo4952753636535259449et_nat )
      = ( ( A3 = bot_bot_set_set_nat )
        | ( B3 = bot_bot_set_set_nat ) ) ) ).

% Times_empty
thf(fact_890_Times__empty,axiom,
    ! [A3: set_set_nat,B3: set_nat] :
      ( ( ( produc5318310618559172640at_nat @ A3
          @ ^ [Uu: set_nat] : B3 )
        = bot_bo8052273170156359555at_nat )
      = ( ( A3 = bot_bot_set_set_nat )
        | ( B3 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_891_Times__empty,axiom,
    ! [A3: set_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : B3 )
        = bot_bo7383041282903587824at_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ( B3 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_892_Times__empty,axiom,
    ! [A3: set_nat,B3: set_list_char] :
      ( ( ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 )
        = bot_bo1414094081888755934t_char )
      = ( ( A3 = bot_bot_set_nat )
        | ( B3 = bot_bo8503570744482963627t_char ) ) ) ).

% Times_empty
thf(fact_893_Times__empty,axiom,
    ! [A3: set_nat,B3: set_set_nat] :
      ( ( ( produc8883945523214541856et_nat @ A3
          @ ^ [Uu: nat] : B3 )
        = bot_bo4851363958280042627et_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ( B3 = bot_bot_set_set_nat ) ) ) ).

% Times_empty
thf(fact_894_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_895_Sigma__empty2,axiom,
    ! [A3: set_nat] :
      ( ( produc1809337555817847783at_nat @ A3
        @ ^ [Uu: nat] : bot_bo2099793752762293965at_nat )
      = bot_bo7383041282903587824at_nat ) ).

% Sigma_empty2
thf(fact_896_Sigma__empty2,axiom,
    ! [A3: set_nat] :
      ( ( produc6030673636645808891t_char @ A3
        @ ^ [Uu: nat] : bot_bo8503570744482963627t_char )
      = bot_bo1414094081888755934t_char ) ).

% Sigma_empty2
thf(fact_897_Sigma__empty2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( produc7672662199629908489at_nat @ A3
        @ ^ [Uu: product_prod_nat_nat] : bot_bot_set_nat )
      = bot_bo2204296500067978442at_nat ) ).

% Sigma_empty2
thf(fact_898_Sigma__empty2,axiom,
    ! [A3: set_list_char] :
      ( ( produc5389449503320859771ar_nat @ A3
        @ ^ [Uu: list_char] : bot_bot_set_nat )
      = bot_bo7709822093361376478ar_nat ) ).

% Sigma_empty2
thf(fact_899_Sigma__empty2,axiom,
    ! [A3: set_nat] :
      ( ( produc457027306803732586at_nat @ A3
        @ ^ [Uu: nat] : bot_bot_set_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty2
thf(fact_900_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_901_UNIV__Times__UNIV,axiom,
    ( ( produc5389449503320859771ar_nat @ top_to3190553582279535303t_char
      @ ^ [Uu: list_char] : top_top_set_nat )
    = top_to7546074696397639618ar_nat ) ).

% UNIV_Times_UNIV
thf(fact_902_UNIV__Times__UNIV,axiom,
    ( ( produc5318310618559172640at_nat @ top_top_set_set_nat
      @ ^ [Uu: set_nat] : top_top_set_nat )
    = top_to3374598108630517863at_nat ) ).

% UNIV_Times_UNIV
thf(fact_903_UNIV__Times__UNIV,axiom,
    ( ( produc6030673636645808891t_char @ top_top_set_nat
      @ ^ [Uu: nat] : top_to3190553582279535303t_char )
    = top_to1250346684925019074t_char ) ).

% UNIV_Times_UNIV
thf(fact_904_UNIV__Times__UNIV,axiom,
    ( ( produc8883945523214541856et_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_set_nat )
    = top_to173688896754200935et_nat ) ).

% UNIV_Times_UNIV
thf(fact_905_UNIV__Times__UNIV,axiom,
    ( ( produc7672662199629908489at_nat @ top_to4669805908274784177at_nat
      @ ^ [Uu: product_prod_nat_nat] : top_top_set_nat )
    = top_to2567003362141017830at_nat ) ).

% UNIV_Times_UNIV
thf(fact_906_UNIV__Times__UNIV,axiom,
    ( ( produc105401717395143638et_nat @ top_top_set_set_nat
      @ ^ [Uu: set_nat] : top_top_set_set_nat )
    = top_to3040159065575275037et_nat ) ).

% UNIV_Times_UNIV
thf(fact_907_UNIV__Times__UNIV,axiom,
    ( ( produc1809337555817847783at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_to4669805908274784177at_nat )
    = top_to7745748144976627212at_nat ) ).

% UNIV_Times_UNIV
thf(fact_908_UNIV__Times__UNIV,axiom,
    ( ( produc4526620505022529241at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat_nat )
    = top_to5113444068631394720at_nat ) ).

% UNIV_Times_UNIV
thf(fact_909_UNIV__Times__UNIV,axiom,
    ( ( produc5982696620300550233at_nat @ top_top_set_nat_nat
      @ ^ [Uu: nat > nat] : top_top_set_nat )
    = top_to7836785446628990112at_nat ) ).

% UNIV_Times_UNIV
thf(fact_910_fun_Orel__Grp,axiom,
    ! [A3: set_Pr2507339680178222706t_char,F4: produc1457211279475724562t_char > list_char] :
      ( ( bNF_re7653730848126131926t_char
        @ ^ [Y3: produc7606469099237612818ar_nat,Z: produc7606469099237612818ar_nat] : ( Y3 = Z )
        @ ( bNF_Gr3750019412944930853t_char @ A3 @ F4 ) )
      = ( bNF_Gr7768808254035958895t_char
        @ ( collec1503323772194643258t_char
          @ ^ [X5: produc7606469099237612818ar_nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_688197443715608197t_char @ X5 @ top_to7546074696397639618ar_nat ) @ A3 ) )
        @ ( comp_P26140635296239040ar_nat @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_911_fun_Orel__Grp,axiom,
    ! [A3: set_Pr2507339680178222706t_char,F4: produc1457211279475724562t_char > nat] :
      ( ( bNF_re742088632225134789ar_nat
        @ ^ [Y3: produc7606469099237612818ar_nat,Z: produc7606469099237612818ar_nat] : ( Y3 = Z )
        @ ( bNF_Gr3073014387538510996ar_nat @ A3 @ F4 ) )
      = ( bNF_Gr7503386797330181086at_nat
        @ ( collec1503323772194643258t_char
          @ ^ [X5: produc7606469099237612818ar_nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_688197443715608197t_char @ X5 @ top_to7546074696397639618ar_nat ) @ A3 ) )
        @ ( comp_P3187634502937433873ar_nat @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_912_fun_Orel__Grp,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > $o] :
      ( ( bNF_re6701202036956822900_nat_o
        @ ^ [Y3: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y3 = Z )
        @ ( bNF_Gr8859634586176591525_nat_o @ A3 @ F4 ) )
      = ( bNF_Gr2209344289654512049_nat_o
        @ ( collec248804655031457914at_nat
          @ ^ [X5: product_prod_nat_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ X5 @ top_to4669805908274784177at_nat ) @ A3 ) )
        @ ( comp_P6344650113790022471at_nat @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_913_fun_Orel__Grp,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > nat] :
      ( ( bNF_re7912597853723674548at_nat
        @ ^ [Y3: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] : ( Y3 = Z )
        @ ( bNF_Gr193424670081168899at_nat @ A3 @ F4 ) )
      = ( bNF_Gr298557646745593647ar_nat
        @ ( collec3352718687064416553at_nat
          @ ^ [X5: produc1457211279475724562t_char > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_2110972657701675764at_nat @ X5 @ top_to1250346684925019074t_char ) @ A3 ) )
        @ ( comp_P2122579405136797538t_char @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_914_fun_Orel__Grp,axiom,
    ! [A3: set_o,F4: $o > $o] :
      ( ( bNF_re4705727531993890431at_o_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ ( bNF_Grp_o_o @ A3 @ F4 ) )
      = ( bNF_Grp_nat_o_nat_o
        @ ( collect_nat_o
          @ ^ [X5: nat > $o] : ( ord_less_eq_set_o @ ( image_nat_o @ X5 @ top_top_set_nat ) @ A3 ) )
        @ ( comp_o_o_nat @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_915_fun_Orel__Grp,axiom,
    ! [A3: set_nat_o,F4: ( nat > $o ) > list_char > $o] :
      ( ( bNF_re4495586573714376046char_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ ( bNF_Gr5836264614112020157char_o @ A3 @ F4 ) )
      = ( bNF_Gr1522612872074377213char_o
        @ ( collect_nat_nat_o
          @ ^ [X5: nat > nat > $o] : ( ord_le6029213668185085951_nat_o @ ( image_nat_nat_o @ X5 @ top_top_set_nat ) @ A3 ) )
        @ ( comp_n1757491575669480226_o_nat @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_916_fun_Orel__Grp,axiom,
    ! [A3: set_nat_o,F4: ( nat > $o ) > nat > $o] :
      ( ( bNF_re578469030762574527_nat_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ ( bNF_Grp_nat_o_nat_o @ A3 @ F4 ) )
      = ( bNF_Gr5554451822430810830_nat_o
        @ ( collect_nat_nat_o
          @ ^ [X5: nat > nat > $o] : ( ord_le6029213668185085951_nat_o @ ( image_nat_nat_o @ X5 @ top_top_set_nat ) @ A3 ) )
        @ ( comp_nat_o_nat_o_nat @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_917_fun_Orel__Grp,axiom,
    ! [A3: set_nat,F4: nat > product_prod_nat_nat] :
      ( ( bNF_re2049273209911613842at_nat
        @ ^ [Y3: produc1457211279475724562t_char,Z: produc1457211279475724562t_char] : ( Y3 = Z )
        @ ( bNF_Gr3553472063123884001at_nat @ A3 @ F4 ) )
      = ( bNF_Gr589112650214279513at_nat
        @ ( collec7673535299250625214ar_nat
          @ ^ [X5: produc1457211279475724562t_char > nat] : ( ord_less_eq_set_nat @ ( image_3548405822317538717ar_nat @ X5 @ top_to1250346684925019074t_char ) @ A3 ) )
        @ ( comp_n1645060710405774596t_char @ F4 ) ) ) ).

% fun.rel_Grp
thf(fact_918_product__swap,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat] :
      ( ( image_2402546415023586989at_nat @ produc672552830730091482at_nat
        @ ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 ) )
      = ( produc1809337555817847783at_nat @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% product_swap
thf(fact_919_product__swap,axiom,
    ! [A3: set_list_char,B3: set_nat] :
      ( ( image_688197443715608197t_char @ produc4410195085741832554ar_nat
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 ) )
      = ( produc6030673636645808891t_char @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% product_swap
thf(fact_920_product__swap,axiom,
    ! [A3: set_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( image_8624973904636368301at_nat @ produc4032600223772806584at_nat
        @ ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc7672662199629908489at_nat @ B3
        @ ^ [Uu: product_prod_nat_nat] : A3 ) ) ).

% product_swap
thf(fact_921_product__swap,axiom,
    ! [A3: set_nat,B3: set_list_char] :
      ( ( image_652198164289537669ar_nat @ produc5051419219066781674t_char
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc5389449503320859771ar_nat @ B3
        @ ^ [Uu: list_char] : A3 ) ) ).

% product_swap
thf(fact_922_product__swap,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( image_5168914502847457605at_nat @ product_swap_nat_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% product_swap
thf(fact_923_Id__on__subset__Times,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ord_le3000389064537975527at_nat @ ( id_on_2554058798563519774at_nat @ A3 )
      @ ( produc2761391749766926216at_nat @ A3
        @ ^ [Uu: product_prod_nat_nat] : A3 ) ) ).

% Id_on_subset_Times
thf(fact_924_Id__on__subset__Times,axiom,
    ! [A3: set_nat] :
      ( ord_le3146513528884898305at_nat @ ( id_on_nat @ A3 )
      @ ( produc457027306803732586at_nat @ A3
        @ ^ [Uu: nat] : A3 ) ) ).

% Id_on_subset_Times
thf(fact_925_Sigma__cong,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: product_prod_nat_nat > set_nat,D2: product_prod_nat_nat > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ B3 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( produc7672662199629908489at_nat @ A3 @ C3 )
          = ( produc7672662199629908489at_nat @ B3 @ D2 ) ) ) ) ).

% Sigma_cong
thf(fact_926_Sigma__cong,axiom,
    ! [A3: set_list_char,B3: set_list_char,C3: list_char > set_nat,D2: list_char > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X2: list_char] :
            ( ( member_list_char @ X2 @ B3 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( produc5389449503320859771ar_nat @ A3 @ C3 )
          = ( produc5389449503320859771ar_nat @ B3 @ D2 ) ) ) ) ).

% Sigma_cong
thf(fact_927_Sigma__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C3: nat > set_Pr1261947904930325089at_nat,D2: nat > set_Pr1261947904930325089at_nat] :
      ( ( A3 = B3 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B3 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( produc1809337555817847783at_nat @ A3 @ C3 )
          = ( produc1809337555817847783at_nat @ B3 @ D2 ) ) ) ) ).

% Sigma_cong
thf(fact_928_Sigma__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C3: nat > set_list_char,D2: nat > set_list_char] :
      ( ( A3 = B3 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B3 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( produc6030673636645808891t_char @ A3 @ C3 )
          = ( produc6030673636645808891t_char @ B3 @ D2 ) ) ) ) ).

% Sigma_cong
thf(fact_929_Sigma__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C3: nat > set_nat,D2: nat > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B3 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( produc457027306803732586at_nat @ A3 @ C3 )
          = ( produc457027306803732586at_nat @ B3 @ D2 ) ) ) ) ).

% Sigma_cong
thf(fact_930_Times__eq__cancel2,axiom,
    ! [X: nat,C3: set_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ X @ C3 )
     => ( ( ( produc7672662199629908489at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : C3 )
          = ( produc7672662199629908489at_nat @ B3
            @ ^ [Uu: product_prod_nat_nat] : C3 ) )
        = ( A3 = B3 ) ) ) ).

% Times_eq_cancel2
thf(fact_931_Times__eq__cancel2,axiom,
    ! [X: nat,C3: set_nat,A3: set_list_char,B3: set_list_char] :
      ( ( member_nat @ X @ C3 )
     => ( ( ( produc5389449503320859771ar_nat @ A3
            @ ^ [Uu: list_char] : C3 )
          = ( produc5389449503320859771ar_nat @ B3
            @ ^ [Uu: list_char] : C3 ) )
        = ( A3 = B3 ) ) ) ).

% Times_eq_cancel2
thf(fact_932_Times__eq__cancel2,axiom,
    ! [X: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,A3: set_nat,B3: set_nat] :
      ( ( member8440522571783428010at_nat @ X @ C3 )
     => ( ( ( produc1809337555817847783at_nat @ A3
            @ ^ [Uu: nat] : C3 )
          = ( produc1809337555817847783at_nat @ B3
            @ ^ [Uu: nat] : C3 ) )
        = ( A3 = B3 ) ) ) ).

% Times_eq_cancel2
thf(fact_933_Times__eq__cancel2,axiom,
    ! [X: list_char,C3: set_list_char,A3: set_nat,B3: set_nat] :
      ( ( member_list_char @ X @ C3 )
     => ( ( ( produc6030673636645808891t_char @ A3
            @ ^ [Uu: nat] : C3 )
          = ( produc6030673636645808891t_char @ B3
            @ ^ [Uu: nat] : C3 ) )
        = ( A3 = B3 ) ) ) ).

% Times_eq_cancel2
thf(fact_934_Times__eq__cancel2,axiom,
    ! [X: nat,C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ X @ C3 )
     => ( ( ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : C3 )
          = ( produc457027306803732586at_nat @ B3
            @ ^ [Uu: nat] : C3 ) )
        = ( A3 = B3 ) ) ) ).

% Times_eq_cancel2
thf(fact_935_converse__Times,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat] :
      ( ( conver1442220823525345110at_nat
        @ ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 ) )
      = ( produc1809337555817847783at_nat @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% converse_Times
thf(fact_936_converse__Times,axiom,
    ! [A3: set_list_char,B3: set_nat] :
      ( ( conver678582252583363310ar_nat
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 ) )
      = ( produc6030673636645808891t_char @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% converse_Times
thf(fact_937_converse__Times,axiom,
    ! [A3: set_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( conver4802268216568060212at_nat
        @ ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc7672662199629908489at_nat @ B3
        @ ^ [Uu: product_prod_nat_nat] : A3 ) ) ).

% converse_Times
thf(fact_938_converse__Times,axiom,
    ! [A3: set_nat,B3: set_list_char] :
      ( ( conver1319806385908312430t_char
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc5389449503320859771ar_nat @ B3
        @ ^ [Uu: list_char] : A3 ) ) ).

% converse_Times
thf(fact_939_converse__Times,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( converse_nat_nat
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% converse_Times
thf(fact_940_imageE,axiom,
    ! [B: $o,F4: $o > $o,A3: set_o] :
      ( ( member_o @ B @ ( image_o_o @ F4 @ A3 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_o @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_941_imageE,axiom,
    ! [B: $o,F4: nat > $o,A3: set_nat] :
      ( ( member_o @ B @ ( image_nat_o @ F4 @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_nat @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_942_imageE,axiom,
    ! [B: nat,F4: $o > nat,A3: set_o] :
      ( ( member_nat @ B @ ( image_o_nat @ F4 @ A3 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_o @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_943_imageE,axiom,
    ! [B: nat,F4: nat > nat,A3: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat2 @ F4 @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_nat @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_944_imageE,axiom,
    ! [B: $o,F4: set_prat > $o,A3: set_set_prat] :
      ( ( member_o @ B @ ( image_set_prat_o @ F4 @ A3 ) )
     => ~ ! [X2: set_prat] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_set_prat @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_945_imageE,axiom,
    ! [B: $o,F4: set_nat > $o,A3: set_set_nat] :
      ( ( member_o @ B @ ( image_set_nat_o @ F4 @ A3 ) )
     => ~ ! [X2: set_nat] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_set_nat @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_946_imageE,axiom,
    ! [B: set_prat,F4: $o > set_prat,A3: set_o] :
      ( ( member_set_prat @ B @ ( image_o_set_prat @ F4 @ A3 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_o @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_947_imageE,axiom,
    ! [B: set_prat,F4: nat > set_prat,A3: set_nat] :
      ( ( member_set_prat @ B @ ( image_nat_set_prat @ F4 @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_nat @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_948_imageE,axiom,
    ! [B: set_nat,F4: $o > set_nat,A3: set_o] :
      ( ( member_set_nat @ B @ ( image_o_set_nat @ F4 @ A3 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_o @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_949_imageE,axiom,
    ! [B: set_nat,F4: nat > set_nat,A3: set_nat] :
      ( ( member_set_nat @ B @ ( image_nat_set_nat @ F4 @ A3 ) )
     => ~ ! [X2: nat] :
            ( ( B
              = ( F4 @ X2 ) )
           => ~ ( member_nat @ X2 @ A3 ) ) ) ).

% imageE
thf(fact_950_image__image,axiom,
    ! [F4: nat > nat,G: nat > nat,A3: set_nat] :
      ( ( image_nat_nat2 @ F4 @ ( image_nat_nat2 @ G @ A3 ) )
      = ( image_nat_nat2
        @ ^ [X5: nat] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_951_image__image,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,G: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat,A3: set_Pr5582243495563764594_nat_o] :
      ( ( image_3684629450409544005at_nat @ F4 @ ( image_7124889717316225246at_nat @ G @ A3 ) )
      = ( image_7124889717316225246at_nat
        @ ^ [X5: product_prod_nat_nat > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_952_image__image,axiom,
    ! [F4: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat,G: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,A3: set_Pr5582243495563764594_nat_o] :
      ( ( image_7124889717316225246at_nat @ F4 @ ( image_2964305653812421503_nat_o @ G @ A3 ) )
      = ( image_7124889717316225246at_nat
        @ ^ [X5: product_prod_nat_nat > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_953_image__image,axiom,
    ! [F4: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat,G: ( nat > nat > $o ) > product_prod_nat_nat > $o,A3: set_nat_nat_o] :
      ( ( image_7124889717316225246at_nat @ F4 @ ( image_7429393840292777309_nat_o @ G @ A3 ) )
      = ( image_7010354106137765248at_nat
        @ ^ [X5: nat > nat > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_954_image__image,axiom,
    ! [F4: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,G: ( nat > nat > $o ) > product_prod_nat_nat > $o,A3: set_nat_nat_o] :
      ( ( image_2964305653812421503_nat_o @ F4 @ ( image_7429393840292777309_nat_o @ G @ A3 ) )
      = ( image_7429393840292777309_nat_o
        @ ^ [X5: nat > nat > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_955_image__image,axiom,
    ! [F4: ( nat > nat > $o ) > product_prod_nat_nat > $o,G: ( nat > nat > $o ) > nat > nat > $o,A3: set_nat_nat_o] :
      ( ( image_7429393840292777309_nat_o @ F4 @ ( image_6800380035593188613_nat_o @ G @ A3 ) )
      = ( image_7429393840292777309_nat_o
        @ ^ [X5: nat > nat > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_956_image__image,axiom,
    ! [F4: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char,G: ( produc1457211279475724562t_char > $o ) > set_Pr2507339680178222706t_char,A3: set_Pr2094409768014413473char_o] :
      ( ( image_1164832423002173061t_char @ F4 @ ( image_9058305973422863104t_char @ G @ A3 ) )
      = ( image_9058305973422863104t_char
        @ ^ [X5: produc1457211279475724562t_char > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_957_image__image,axiom,
    ! [F4: ( produc1457211279475724562t_char > $o ) > set_Pr2507339680178222706t_char,G: ( produc1457211279475724562t_char > $o ) > produc1457211279475724562t_char > $o,A3: set_Pr2094409768014413473char_o] :
      ( ( image_9058305973422863104t_char @ F4 @ ( image_7647239082385002461char_o @ G @ A3 ) )
      = ( image_9058305973422863104t_char
        @ ^ [X5: produc1457211279475724562t_char > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_958_image__image,axiom,
    ! [F4: ( produc1457211279475724562t_char > $o ) > set_Pr2507339680178222706t_char,G: ( nat > list_char > $o ) > produc1457211279475724562t_char > $o,A3: set_nat_list_char_o] :
      ( ( image_9058305973422863104t_char @ F4 @ ( image_1531431914690922683char_o @ G @ A3 ) )
      = ( image_2054065556140572834t_char
        @ ^ [X5: nat > list_char > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_959_image__image,axiom,
    ! [F4: ( produc1457211279475724562t_char > $o ) > produc1457211279475724562t_char > $o,G: ( nat > list_char > $o ) > produc1457211279475724562t_char > $o,A3: set_nat_list_char_o] :
      ( ( image_7647239082385002461char_o @ F4 @ ( image_1531431914690922683char_o @ G @ A3 ) )
      = ( image_1531431914690922683char_o
        @ ^ [X5: nat > list_char > $o] : ( F4 @ ( G @ X5 ) )
        @ A3 ) ) ).

% image_image
thf(fact_960_Compr__image__eq,axiom,
    ! [F4: nat > nat,A3: set_nat,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ ( image_nat_nat2 @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_nat_nat2 @ F4
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_961_Compr__image__eq,axiom,
    ! [F4: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ ( image_2486076414777270412at_nat @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_2486076414777270412at_nat @ F4
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_962_Compr__image__eq,axiom,
    ! [F4: nat > product_prod_nat_nat,A3: set_nat,P2: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ ( image_5846123807819985514at_nat @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_5846123807819985514at_nat @ F4
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_963_Compr__image__eq,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ ( image_5168914502847457605at_nat @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_5168914502847457605at_nat @ F4
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_964_Compr__image__eq,axiom,
    ! [F4: $o > $o,A3: set_o,P2: $o > $o] :
      ( ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ ( image_o_o @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_o_o @ F4
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_965_Compr__image__eq,axiom,
    ! [F4: nat > $o,A3: set_nat,P2: $o > $o] :
      ( ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ ( image_nat_o @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_nat_o @ F4
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_966_Compr__image__eq,axiom,
    ! [F4: $o > nat,A3: set_o,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ ( image_o_nat @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_o_nat @ F4
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_967_Compr__image__eq,axiom,
    ! [F4: set_prat > $o,A3: set_set_prat,P2: $o > $o] :
      ( ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ ( image_set_prat_o @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_set_prat_o @ F4
        @ ( collect_set_prat
          @ ^ [X5: set_prat] :
              ( ( member_set_prat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_968_Compr__image__eq,axiom,
    ! [F4: $o > set_prat,A3: set_o,P2: set_prat > $o] :
      ( ( collect_set_prat
        @ ^ [X5: set_prat] :
            ( ( member_set_prat @ X5 @ ( image_o_set_prat @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_o_set_prat @ F4
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_969_Compr__image__eq,axiom,
    ! [F4: nat > set_prat,A3: set_nat,P2: set_prat > $o] :
      ( ( collect_set_prat
        @ ^ [X5: set_prat] :
            ( ( member_set_prat @ X5 @ ( image_nat_set_prat @ F4 @ A3 ) )
            & ( P2 @ X5 ) ) )
      = ( image_nat_set_prat @ F4
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
              & ( P2 @ ( F4 @ X5 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_970_image__paired__Times,axiom,
    ! [F4: nat > nat,G: nat > nat,A3: set_nat,B3: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X5: nat,Y2: nat] : ( product_Pair_nat_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_971_image__paired__Times,axiom,
    ! [F4: list_char > nat,G: nat > nat,A3: set_list_char,B3: set_nat] :
      ( ( image_8239296223871014644at_nat
        @ ( produc7321533897546000994at_nat
          @ ^ [X5: list_char,Y2: nat] : ( product_Pair_nat_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 ) )
      = ( produc457027306803732586at_nat @ ( image_list_char_nat @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_972_image__paired__Times,axiom,
    ! [F4: nat > nat,G: list_char > nat,A3: set_nat,B3: set_list_char] :
      ( ( image_2110972657701675764at_nat
        @ ( produc6775168648774419426at_nat
          @ ^ [X5: nat,Y2: list_char] : ( product_Pair_nat_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_list_char_nat @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_973_image__paired__Times,axiom,
    ! [F4: nat > list_char,G: nat > nat,A3: set_nat,B3: set_nat] :
      ( ( image_6877823656637235030ar_nat
        @ ( produc8523017789120347204ar_nat
          @ ^ [X5: nat,Y2: nat] : ( produc8702345860417625034ar_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc5389449503320859771ar_nat @ ( image_nat_list_char2 @ F4 @ A3 )
        @ ^ [Uu: list_char] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_974_image__paired__Times,axiom,
    ! [F4: nat > nat,G: nat > list_char,A3: set_nat,B3: set_nat] :
      ( ( image_728565836875346774t_char
        @ ( produc2373759969358458948t_char
          @ ^ [X5: nat,Y2: nat] : ( produc120197956887798346t_char @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc6030673636645808891t_char @ ( image_nat_nat2 @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_list_char2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_975_image__paired__Times,axiom,
    ! [F4: product_prod_nat_nat > nat,G: nat > nat,A3: set_Pr1261947904930325089at_nat,B3: set_nat] :
      ( ( image_8369444115148354106at_nat
        @ ( produc373799411880517786at_nat
          @ ^ [X5: product_prod_nat_nat,Y2: nat] : ( product_Pair_nat_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 ) )
      = ( produc457027306803732586at_nat @ ( image_2486076414777270412at_nat @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_976_image__paired__Times,axiom,
    ! [F4: list_char > list_char,G: nat > nat,A3: set_list_char,B3: set_nat] :
      ( ( image_6837455263477496453ar_nat
        @ ( produc3091969462627340531ar_nat
          @ ^ [X5: list_char,Y2: nat] : ( produc8702345860417625034ar_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 ) )
      = ( produc5389449503320859771ar_nat @ ( image_8265130762032581577t_char @ F4 @ A3 )
        @ ^ [Uu: list_char] : ( image_nat_nat2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_977_image__paired__Times,axiom,
    ! [F4: list_char > nat,G: nat > list_char,A3: set_list_char,B3: set_nat] :
      ( ( image_688197443715608197t_char
        @ ( produc6166083679720228083t_char
          @ ^ [X5: list_char,Y2: nat] : ( produc120197956887798346t_char @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B3 ) )
      = ( produc6030673636645808891t_char @ ( image_list_char_nat @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_nat_list_char2 @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_978_image__paired__Times,axiom,
    ! [F4: nat > nat,G: product_prod_nat_nat > nat,A3: set_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( image_6167232457338462356at_nat
        @ ( produc8859641928216934716at_nat
          @ ^ [X5: nat,Y2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat2 @ F4 @ A3 )
        @ ^ [Uu: nat] : ( image_2486076414777270412at_nat @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_979_image__paired__Times,axiom,
    ! [F4: nat > list_char,G: list_char > nat,A3: set_nat,B3: set_list_char] :
      ( ( image_652198164289537669ar_nat
        @ ( produc997380522159630963ar_nat
          @ ^ [X5: nat,Y2: list_char] : ( produc8702345860417625034ar_nat @ ( F4 @ X5 ) @ ( G @ Y2 ) ) )
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc5389449503320859771ar_nat @ ( image_nat_list_char2 @ F4 @ A3 )
        @ ^ [Uu: list_char] : ( image_list_char_nat @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_980_swap__product,axiom,
    ! [A3: set_Pr2507339680178222706t_char,B3: set_Pr3433046674199179443r_prat] :
      ( ( image_4373228992181088323t_char
        @ ( produc1491052064730943609t_char
          @ ^ [I: produc1457211279475724562t_char,J: produc1457211279475724562t_char > prat] : ( produc2591414526589101846t_char @ J @ I ) )
        @ ( produc933333428453274633r_prat @ A3
          @ ^ [Uu: produc1457211279475724562t_char] : B3 ) )
      = ( produc4757766445512390215t_char @ B3
        @ ^ [Uu: produc1457211279475724562t_char > prat] : A3 ) ) ).

% swap_product
thf(fact_981_swap__product,axiom,
    ! [A3: set_Pr7632435056502277254t_char,B3: set_Pr3433046674199179443r_prat] :
      ( ( image_1067201851549782251t_char
        @ ( produc2751466438118233505t_char
          @ ^ [I: produc2489117125269924006t_char,J: produc1457211279475724562t_char > prat] : ( produc2957999048406202538t_char @ J @ I ) )
        @ ( produc6096210268339807733r_prat @ A3
          @ ^ [Uu: produc2489117125269924006t_char] : B3 ) )
      = ( produc1253266753640828507t_char @ B3
        @ ^ [Uu: produc1457211279475724562t_char > prat] : A3 ) ) ).

% swap_product
thf(fact_982_swap__product,axiom,
    ! [A3: set_Pr3433046674199179443r_prat,B3: set_Pr3433046674199179443r_prat] :
      ( ( image_8327928573957436357r_prat
        @ ( produc644870500113131387r_prat
          @ ^ [I: produc1457211279475724562t_char > prat,J: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ J @ I ) )
        @ ( produc2346184296948171016r_prat @ A3
          @ ^ [Uu: produc1457211279475724562t_char > prat] : B3 ) )
      = ( produc2346184296948171016r_prat @ B3
        @ ^ [Uu: produc1457211279475724562t_char > prat] : A3 ) ) ).

% swap_product
thf(fact_983_swap__product,axiom,
    ! [A3: set_Pr2748872423075033990prat_a,B3: set_a_prat] :
      ( ( image_2033801899237171475prat_a
        @ ( produc786514161592030665prat_a
          @ ^ [I: produc3375937593958228902prat_a,J: a > prat] : ( produc2856268368059332798prat_a @ J @ I ) )
        @ ( produc6486629408028388065a_prat @ A3
          @ ^ [Uu: produc3375937593958228902prat_a] : B3 ) )
      = ( produc937274936681309295prat_a @ B3
        @ ^ [Uu: a > prat] : A3 ) ) ).

% swap_product
thf(fact_984_swap__product,axiom,
    ! [A3: set_a,B3: set_a_prat] :
      ( ( image_5844236873215590467prat_a
        @ ( produc7867159419813889657prat_a
          @ ^ [I: a,J: a > prat] : ( produc3557287861060643862prat_a @ J @ I ) )
        @ ( produc5189648166678903049a_prat @ A3
          @ ^ [Uu: a] : B3 ) )
      = ( produc2975647622218653511prat_a @ B3
        @ ^ [Uu: a > prat] : A3 ) ) ).

% swap_product
thf(fact_985_swap__product,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat] :
      ( ( image_2402546415023586989at_nat
        @ ( produc3206169289476954189at_nat
          @ ^ [I: product_prod_nat_nat,J: nat] : ( produc487386426758144856at_nat @ J @ I ) )
        @ ( produc7672662199629908489at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B3 ) )
      = ( produc1809337555817847783at_nat @ B3
        @ ^ [Uu: nat] : A3 ) ) ).

% swap_product
thf(fact_986_swap__product,axiom,
    ! [A3: set_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( image_8624973904636368301at_nat
        @ ( produc2094262906704694021at_nat
          @ ^ [I: nat,J: product_prod_nat_nat] : ( produc6350711070570205562at_nat @ J @ I ) )
        @ ( produc1809337555817847783at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc7672662199629908489at_nat @ B3
        @ ^ [Uu: product_prod_nat_nat] : A3 ) ) ).

% swap_product
thf(fact_987_swap__product,axiom,
    ! [A3: set_nat,B3: set_list_char] :
      ( ( image_652198164289537669ar_nat
        @ ( produc997380522159630963ar_nat
          @ ^ [I: nat,J: list_char] : ( produc8702345860417625034ar_nat @ J @ I ) )
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( produc5389449503320859771ar_nat @ B3
        @ ^ [Uu: list_char] : A3 ) ) ).

% swap_product
thf(fact_988_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_989_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_990_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( nat > $o ) > ( list_char > $o ) > $o,X: nat > nat > $o,G: ( nat > $o ) > list_char > $o,Y: nat > nat > $o] :
      ( ( bNF_re4495586573714376046char_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ Sa
        @ X
        @ ( comp_n1757491575669480226_o_nat @ G @ Y ) )
      = ( bNF_re578469030762574527_nat_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ ^ [X5: nat > $o,Y2: nat > $o] : ( Sa @ X5 @ ( G @ Y2 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_991_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( nat > $o ) > ( nat > $o ) > $o,X: nat > nat > $o,G: ( list_char > $o ) > nat > $o,Y: nat > list_char > $o] :
      ( ( bNF_re578469030762574527_nat_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ Sa
        @ X
        @ ( comp_l2153918996998263428_o_nat @ G @ Y ) )
      = ( bNF_re4495586573714376046char_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ ^ [X5: nat > $o,Y2: list_char > $o] : ( Sa @ X5 @ ( G @ Y2 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_992_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( nat > $o ) > ( nat > $o ) > $o,X: nat > nat > $o,G: ( nat > $o ) > nat > $o,Y: nat > nat > $o] :
      ( ( bNF_re578469030762574527_nat_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ Sa
        @ X
        @ ( comp_nat_o_nat_o_nat @ G @ Y ) )
      = ( bNF_re578469030762574527_nat_o
        @ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
        @ ^ [X5: nat > $o,Y2: nat > $o] : ( Sa @ X5 @ ( G @ Y2 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_993_Collect__case__prod__Sigma,axiom,
    ! [P2: nat > $o,Q: nat > product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [X5: nat,Y2: product_prod_nat_nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 @ Y2 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P2 )
        @ ^ [X5: nat] : ( collec3392354462482085612at_nat @ ( Q @ X5 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_994_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [X5: product_prod_nat_nat,Y2: nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 @ Y2 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X5: product_prod_nat_nat] : ( collect_nat @ ( Q @ X5 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_995_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X5: product_prod_nat_nat,Y2: product_prod_nat_nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 @ Y2 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X5: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q @ X5 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_996_Collect__case__prod__Sigma,axiom,
    ! [P2: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X5: nat,Y2: nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 @ Y2 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
        @ ^ [X5: nat] : ( collect_nat @ ( Q @ X5 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_997_setcompr__eq__image,axiom,
    ! [F4: nat > nat,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X5: nat] :
            ( ( Uu
              = ( F4 @ X5 ) )
            & ( P2 @ X5 ) ) )
      = ( image_nat_nat2 @ F4 @ ( collect_nat @ P2 ) ) ) ).

% setcompr_eq_image
thf(fact_998_setcompr__eq__image,axiom,
    ! [F4: product_prod_nat_nat > nat,P2: product_prod_nat_nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X5: product_prod_nat_nat] :
            ( ( Uu
              = ( F4 @ X5 ) )
            & ( P2 @ X5 ) ) )
      = ( image_2486076414777270412at_nat @ F4 @ ( collec3392354462482085612at_nat @ P2 ) ) ) ).

% setcompr_eq_image
thf(fact_999_setcompr__eq__image,axiom,
    ! [F4: nat > product_prod_nat_nat,P2: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [Uu: product_prod_nat_nat] :
          ? [X5: nat] :
            ( ( Uu
              = ( F4 @ X5 ) )
            & ( P2 @ X5 ) ) )
      = ( image_5846123807819985514at_nat @ F4 @ ( collect_nat @ P2 ) ) ) ).

% setcompr_eq_image
thf(fact_1000_setcompr__eq__image,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [Uu: product_prod_nat_nat] :
          ? [X5: product_prod_nat_nat] :
            ( ( Uu
              = ( F4 @ X5 ) )
            & ( P2 @ X5 ) ) )
      = ( image_5168914502847457605at_nat @ F4 @ ( collec3392354462482085612at_nat @ P2 ) ) ) ).

% setcompr_eq_image
thf(fact_1001_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: nat > list_char > $o,P2: 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
          @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
        @ P2
        @ 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 ) ) )
             => ( ( P2 @ ( produc6204653154488278070t_char @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q @ ( produc1900778367047502200t_char @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_1002_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: nat > nat > $o,P2: 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
          @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
        @ P2
        @ 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 ) ) )
             => ( ( P2 @ ( produc6204653154488278070t_char @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q @ ( produc1900778367047502200t_char @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_1003_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > list_char > $o,P2: 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
          @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
        @ P2
        @ 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 ) ) )
             => ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q @ ( product_snd_nat_nat @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_1004_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,P2: 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
          @ ^ [Y3: $o,Z: $o] : ( Y3 = Z ) )
        @ P2
        @ 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 ) ) )
             => ( ( P2 @ ( 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_1005_Id__on__def_H,axiom,
    ! [A3: product_prod_nat_nat > $o] :
      ( ( id_on_2554058798563519774at_nat @ ( collec3392354462482085612at_nat @ A3 ) )
      = ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X5: product_prod_nat_nat,Y2: product_prod_nat_nat] :
              ( ( X5 = Y2 )
              & ( A3 @ X5 ) ) ) ) ) ).

% Id_on_def'
thf(fact_1006_Id__on__def_H,axiom,
    ! [A3: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A3 ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X5: nat,Y2: nat] :
              ( ( X5 = Y2 )
              & ( A3 @ X5 ) ) ) ) ) ).

% Id_on_def'
thf(fact_1007_relcompp__relcomp__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr2507339680178222706t_char] :
      ( ( relcom3288008990786313140t_char
        @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R2 )
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ S2 ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( relcom6453288877151117700t_char @ R2 @ S2 ) ) ) ) ).

% relcompp_relcomp_eq
thf(fact_1008_relcompp__relcomp__eq,axiom,
    ! [R2: set_Pr2507339680178222706t_char,S2: set_Pr4480099295215233283t_char] :
      ( ( relcom7349595004981586117t_char
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R2 )
        @ ^ [X5: list_char,Y2: list_char] : ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ X5 @ Y2 ) @ S2 ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( relcom5956200836249630357t_char @ R2 @ S2 ) ) ) ) ).

% relcompp_relcomp_eq
thf(fact_1009_mem__Times__iff,axiom,
    ! [X: produc1457211279475724562t_char,A3: set_nat,B3: set_list_char] :
      ( ( member484235747325421115t_char @ X
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B3 ) )
      = ( ( member_nat @ ( produc6204653154488278070t_char @ X ) @ A3 )
        & ( member_list_char @ ( produc1900778367047502200t_char @ X ) @ B3 ) ) ) ).

% mem_Times_iff
thf(fact_1010_converse__unfold,axiom,
    ( conver1319806385908312430t_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collec8608152508653127293ar_nat
          @ ( produc2492928673396900345_nat_o
            @ ^ [Y2: list_char,X5: nat] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% converse_unfold
thf(fact_1011_converse__unfold,axiom,
    ( converse_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [Y2: nat,X5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% converse_unfold
thf(fact_1012_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_1013_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F4: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member_o @ ( F4 @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_1014_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_1015_conversep__converse__eq,axiom,
    ! [R2: set_Pr8803067691650843250ar_nat] :
      ( ( conver3568184245214596648ar_nat
        @ ^ [X5: list_char,Y2: nat] : ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( conver678582252583363310ar_nat @ R2 ) ) ) ) ).

% conversep_converse_eq
thf(fact_1016_conversep__converse__eq,axiom,
    ! [R2: set_Pr2507339680178222706t_char] :
      ( ( conver4209408378539545768t_char
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R2 ) )
      = ( ^ [X5: list_char,Y2: nat] : ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X5 @ Y2 ) @ ( conver1319806385908312430t_char @ R2 ) ) ) ) ).

% conversep_converse_eq
thf(fact_1017_converse__def,axiom,
    ( conver1319806385908312430t_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collec8608152508653127293ar_nat
          @ ( produc2492928673396900345_nat_o
            @ ( conver4209408378539545768t_char
              @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ) ).

% converse_def
thf(fact_1018_converse__def,axiom,
    ( converse_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( conversep_nat_nat
              @ ^ [X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R ) ) ) ) ) ) ).

% converse_def
thf(fact_1019_fst__image__Sigma,axiom,
    ! [A3: set_nat,B3: nat > set_list_char] :
      ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ ( produc6030673636645808891t_char @ A3 @ B3 ) )
      = ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ( ( B3 @ X5 )
             != bot_bo8503570744482963627t_char ) ) ) ) ).

% fst_image_Sigma
thf(fact_1020_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
            @ ^ [X5: list_char,Y2: nat] : ( produc120197956887798346t_char @ Y2 @ X5 ) )
          @ A3 )
        @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_1021_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
            @ ^ [X5: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X5 ) )
          @ A3 )
        @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_1022_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( X5 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1023_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] : ( X5 = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_1024_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y3: nat,Z: nat] : ( Y3 = Z )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1025_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y3: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y3 = Z )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_1026_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A2: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
              ( ( X5 = A2 )
              | ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_1027_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A2: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( X5 = A2 )
              | ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_1028_insert__Collect,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P2 ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A )
           => ( P2 @ U ) ) ) ) ).

% insert_Collect
thf(fact_1029_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U: product_prod_nat_nat] :
            ( ( U != A )
           => ( P2 @ U ) ) ) ) ).

% insert_Collect
thf(fact_1030_Collect__conv__if,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( X5 = A )
                & ( P2 @ X5 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( X5 = A )
                & ( P2 @ X5 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_1031_Collect__conv__if,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( X5 = A )
                & ( P2 @ X5 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( X5 = A )
                & ( P2 @ X5 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_1032_Collect__conv__if2,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ( ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( A = X5 )
                & ( P2 @ X5 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( A = X5 )
                & ( P2 @ X5 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1033_Collect__conv__if2,axiom,
    ! [P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( A = X5 )
                & ( P2 @ X5 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ( ( A = X5 )
                & ( P2 @ X5 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1034_Id__def,axiom,
    ( id_nat2
    = ( collec3392354462482085612at_nat
      @ ^ [P: product_prod_nat_nat] :
        ? [X5: nat] :
          ( P
          = ( product_Pair_nat_nat @ X5 @ X5 ) ) ) ) ).

% Id_def
thf(fact_1035_Id__fstsnd__eq,axiom,
    ( id_nat2
    = ( collec3392354462482085612at_nat
      @ ^ [X5: product_prod_nat_nat] :
          ( ( product_fst_nat_nat @ X5 )
          = ( product_snd_nat_nat @ X5 ) ) ) ) ).

% Id_fstsnd_eq
thf(fact_1036_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_1037_Collect__disj__eq,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ( ( P2 @ X5 )
            | ( Q @ X5 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1038_Collect__disj__eq,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( P2 @ X5 )
            | ( Q @ X5 ) ) )
      = ( sup_su6327502436637775413at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1039_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              | ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1040_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A4 )
              | ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1041_insert__def,axiom,
    ( insert_nat
    = ( ^ [A2: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] : ( X5 = A2 ) ) ) ) ) ).

% insert_def
thf(fact_1042_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A2: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] : ( X5 = A2 ) ) ) ) ) ).

% insert_def
thf(fact_1043_case__prod__map__prod,axiom,
    ! [H2: nat > nat > $o,F4: nat > nat,G: nat > nat,X: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ H2 @ ( produc6977886695330630970at_nat @ F4 @ G @ X ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [L: nat,R: nat] : ( H2 @ ( F4 @ L ) @ ( G @ R ) )
        @ X ) ) ).

% case_prod_map_prod
thf(fact_1044_case__prod__o__map__prod,axiom,
    ! [F4: nat > nat > $o,G1: nat > nat,G22: nat > nat] :
      ( ( comp_P6344650113790022471at_nat @ ( produc6081775807080527818_nat_o @ F4 ) @ ( produc6977886695330630970at_nat @ G1 @ G22 ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [L: nat,R: nat] : ( F4 @ ( G1 @ L ) @ ( G22 @ R ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_1045_Sigma__def,axiom,
    ( produc6030673636645808891t_char
    = ( ^ [A4: set_nat,B6: nat > set_list_char] :
          ( comple9178978156834306527t_char
          @ ( image_7918911184113263195t_char
            @ ^ [X5: nat] :
                ( comple9178978156834306527t_char
                @ ( image_366428816823531018t_char
                  @ ^ [Y2: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ bot_bo1414094081888755934t_char )
                  @ ( B6 @ X5 ) ) )
            @ A4 ) ) ) ) ).

% Sigma_def
thf(fact_1046_inj__on__convol__ident,axiom,
    ! [F4: nat > list_char,X8: set_nat] :
      ( inj_on9177509583204279951t_char
      @ ^ [X5: nat] : ( produc120197956887798346t_char @ X5 @ ( F4 @ X5 ) )
      @ X8 ) ).

% inj_on_convol_ident
thf(fact_1047_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
            @ ^ [X5: nat] : ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1048_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 )
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1049_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_nat
          @ ^ [X5: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X5 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1050_Sup__set__def,axiom,
    ( comple5685304695842803022at_nat
    = ( ^ [A4: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] : ( complete_Sup_Sup_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X5 ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1051_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_1052_bot__empty__eq2,axiom,
    ( bot_bo8065403328713304961char_o
    = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ bot_bo1414094081888755934t_char ) ) ) ).

% bot_empty_eq2
thf(fact_1053_sup__Un__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( sup_su1536542383718553321char_o
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ S ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( sup_su7899656352754981958t_char @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_1054_SUP__Sup__eq2,axiom,
    ! [S: set_se253819897320413138t_char] :
      ( ( comple724453453808405122char_o
        @ ( image_7123945608888165160char_o
          @ ^ [I: set_Pr2507339680178222706t_char,X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( comple9178978156834306527t_char @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_1055_Sup__SUP__eq2,axiom,
    ( comple724453453808405122char_o
    = ( ^ [S4: set_nat_list_char_o,X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( comple9178978156834306527t_char @ ( image_9058305973422863104t_char @ collec2458894688891239037t_char @ ( image_1531431914690922683char_o @ produc8526122873445342073char_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1056_Sup__SUP__eq2,axiom,
    ( comple3592611370556534995_nat_o
    = ( ^ [S4: set_nat_nat_o,X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S4 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_1057_inj__graph,axiom,
    ( inj_on3412903767820855855at_nat
    @ ^ [F2: nat > nat] :
        ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X5: nat,Y2: nat] :
              ( Y2
              = ( F2 @ X5 ) ) ) )
    @ top_top_set_nat_nat ) ).

% inj_graph
thf(fact_1058_vimage__Collect__eq,axiom,
    ! [F4: nat > nat,P2: nat > $o] :
      ( ( vimage_nat_nat @ F4 @ ( collect_nat @ P2 ) )
      = ( collect_nat
        @ ^ [Y2: nat] : ( P2 @ ( F4 @ Y2 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1059_vimage__Collect__eq,axiom,
    ! [F4: product_prod_nat_nat > nat,P2: nat > $o] :
      ( ( vimage4653281326611754070at_nat @ F4 @ ( collect_nat @ P2 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [Y2: product_prod_nat_nat] : ( P2 @ ( F4 @ Y2 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1060_vimage__Collect__eq,axiom,
    ! [F4: nat > product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( vimage8013328719654469172at_nat @ F4 @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( collect_nat
        @ ^ [Y2: nat] : ( P2 @ ( F4 @ Y2 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1061_vimage__Collect__eq,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( vimage2449269961533847803at_nat @ F4 @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [Y2: product_prod_nat_nat] : ( P2 @ ( F4 @ Y2 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_1062_Inf__set__def,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_nat
          @ ^ [X5: nat] : ( complete_Inf_Inf_o @ ( image_set_nat_o @ ( member_nat @ X5 ) @ A4 ) ) ) ) ) ).

% Inf_set_def
thf(fact_1063_Inf__set__def,axiom,
    ( comple1930127330929219688at_nat
    = ( ^ [A4: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] : ( complete_Inf_Inf_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X5 ) @ A4 ) ) ) ) ) ).

% Inf_set_def
thf(fact_1064_INF__Int__eq2,axiom,
    ! [S: set_se253819897320413138t_char] :
      ( ( comple7993840426618028700char_o
        @ ( image_7123945608888165160char_o
          @ ^ [I: set_Pr2507339680178222706t_char,X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( comple4824088763117934073t_char @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_1065_Inf__INT__eq2,axiom,
    ( comple7993840426618028700char_o
    = ( ^ [S4: set_nat_list_char_o,X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( comple4824088763117934073t_char @ ( image_9058305973422863104t_char @ collec2458894688891239037t_char @ ( image_1531431914690922683char_o @ produc8526122873445342073char_o @ S4 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1066_Inf__INT__eq2,axiom,
    ( comple1675732732025788397_nat_o
    = ( ^ [S4: set_nat_nat_o,X5: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ ( comple1930127330929219688at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S4 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_1067_vimage__fst,axiom,
    ! [A3: set_nat] :
      ( ( vimage2352271050577936999ar_nat @ produc6204653154488278070t_char @ A3 )
      = ( produc6030673636645808891t_char @ A3
        @ ^ [Uu: nat] : top_to3190553582279535303t_char ) ) ).

% vimage_fst
thf(fact_1068_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              & ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1069_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A4 )
              & ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_1070_Int__Collect,axiom,
    ! [X: nat,A3: set_nat,P2: nat > $o] :
      ( ( member_nat @ X @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P2 ) ) )
      = ( ( member_nat @ X @ A3 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_1071_Int__Collect,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P2 ) ) )
      = ( ( member8440522571783428010at_nat @ X @ A3 )
        & ( P2 @ X ) ) ) ).

% Int_Collect
thf(fact_1072_Collect__conj__eq,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ( ( P2 @ X5 )
            & ( Q @ X5 ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1073_Collect__conj__eq,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( P2 @ X5 )
            & ( Q @ X5 ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P2 ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1074_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              & ~ ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1075_set__diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A4 )
              & ~ ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1076_top__empty__eq2,axiom,
    ( top_to1537031869075220069char_o
    = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ top_to1250346684925019074t_char ) ) ) ).

% top_empty_eq2
thf(fact_1077_wf__empty,axiom,
    wf_nat @ bot_bo2099793752762293965at_nat ).

% wf_empty
thf(fact_1078_wf__lex__prod,axiom,
    ! [Ra2: set_Pr1261947904930325089at_nat,Rb2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ Ra2 )
     => ( ( wf_nat @ Rb2 )
       => ( wf_Pro7803398752247294826at_nat @ ( lex_prod_nat_nat @ Ra2 @ Rb2 ) ) ) ) ).

% wf_lex_prod
thf(fact_1079_wf__inv__image,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F4: nat > nat] :
      ( ( wf_nat @ R2 )
     => ( wf_nat @ ( inv_image_nat_nat @ R2 @ F4 ) ) ) ).

% wf_inv_image
thf(fact_1080_wf__Int2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( wf_nat @ ( inf_in2572325071724192079at_nat @ R5 @ R2 ) ) ) ).

% wf_Int2
thf(fact_1081_wf__Int1,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,R5: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( wf_nat @ ( inf_in2572325071724192079at_nat @ R2 @ R5 ) ) ) ).

% wf_Int1
thf(fact_1082_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
            @ ^ [X5: nat] : ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1083_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 )
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_1084_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A4: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
            @ ^ [X5: nat] : ( member_nat @ X5 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1085_minus__set__def,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( minus_2270307095948843157_nat_o
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 )
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1086_inf__Int__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ( inf_in8805929356528176899char_o
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R3 )
        @ ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ S ) )
      = ( ^ [X5: nat,Y2: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ ( inf_in3544766959038609504t_char @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1087_wf__subset,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P7: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ P7 @ R2 )
       => ( wf_nat @ P7 ) ) ) ).

% wf_subset
thf(fact_1088_wf__comp__self,axiom,
    ( wf_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] : ( wf_nat @ ( relcomp_nat_nat_nat @ R4 @ R4 ) ) ) ) ).

% wf_comp_self
thf(fact_1089_wf__acc__iff,axiom,
    ( wf_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
        ! [X5: nat] : ( member_nat @ X5 @ ( acc_nat @ R ) ) ) ) ).

% wf_acc_iff
thf(fact_1090_acc__wfI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat] : ( member_nat @ X2 @ ( acc_nat @ R2 ) )
     => ( wf_nat @ R2 ) ) ).

% acc_wfI
thf(fact_1091_acc__wfD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( wf_nat @ R2 )
     => ( member_nat @ X @ ( acc_nat @ R2 ) ) ) ).

% acc_wfD
thf(fact_1092_wf__if__convertible__to__wf,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,F4: nat > nat] :
      ( ( wf_nat @ S2 )
     => ( ! [X2: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ X2 ) @ ( F4 @ Y4 ) ) @ S2 ) )
       => ( wf_nat @ R2 ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_1093_wf__induct__rule,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( wf_nat @ R2 )
     => ( ! [X2: nat] :
            ( ! [Y5: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X2 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_1094_wf__eq__minimal,axiom,
    ( wf_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
        ! [Q2: set_nat] :
          ( ? [X5: nat] : ( member_nat @ X5 @ Q2 )
         => ? [X5: nat] :
              ( ( member_nat @ X5 @ Q2 )
              & ! [Y2: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X5 ) @ R )
                 => ~ ( member_nat @ Y2 @ Q2 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_1095_wf__not__refl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( wf_nat @ R2 )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R2 ) ) ).

% wf_not_refl
thf(fact_1096_wf__not__sym,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,X: nat] :
      ( ( wf_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ R2 ) ) ) ).

% wf_not_sym
thf(fact_1097_wf__irrefl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( wf_nat @ R2 )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R2 ) ) ).

% wf_irrefl
thf(fact_1098_wf__induct,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( wf_nat @ R2 )
     => ( ! [X2: nat] :
            ( ! [Y5: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 )
               => ( P2 @ Y5 ) )
           => ( P2 @ X2 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_1099_wf__asym,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,X: nat] :
      ( ( wf_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ R2 ) ) ) ).

% wf_asym
thf(fact_1100_wfUNIVI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [P8: nat > $o,X2: nat] :
          ( ! [Xa2: nat] :
              ( ! [Y4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Xa2 ) @ R2 )
                 => ( P8 @ Y4 ) )
             => ( P8 @ Xa2 ) )
         => ( P8 @ X2 ) )
     => ( wf_nat @ R2 ) ) ).

% wfUNIVI
thf(fact_1101_wfI__min,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Q3: set_nat] :
          ( ( member_nat @ X2 @ Q3 )
         => ? [Xa2: nat] :
              ( ( member_nat @ Xa2 @ Q3 )
              & ! [Y4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Xa2 ) @ R3 )
                 => ~ ( member_nat @ Y4 @ Q3 ) ) ) )
     => ( wf_nat @ R3 ) ) ).

% wfI_min
thf(fact_1102_wfE__min,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,X: nat,Q: set_nat] :
      ( ( wf_nat @ R3 )
     => ( ( member_nat @ X @ Q )
       => ~ ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Q )
             => ~ ! [Y5: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z2 ) @ R3 )
                   => ~ ( member_nat @ Y5 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_1103_wf__def,axiom,
    ( wf_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
        ! [P4: nat > $o] :
          ( ! [X5: nat] :
              ( ! [Y2: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X5 ) @ R )
                 => ( P4 @ Y2 ) )
             => ( P4 @ X5 ) )
         => ! [X3: nat] : ( P4 @ X3 ) ) ) ) ).

% wf_def
thf(fact_1104_wfE__min_H,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Q: set_nat] :
      ( ( wf_nat @ R3 )
     => ( ( Q != bot_bot_set_nat )
       => ~ ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Q )
             => ~ ! [Y5: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z2 ) @ R3 )
                   => ~ ( member_nat @ Y5 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_1105_wf__relcomp__compatible,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ ( relcomp_nat_nat_nat @ S @ R3 ) )
       => ( wf_nat @ ( relcomp_nat_nat_nat @ S @ R3 ) ) ) ) ).

% wf_relcomp_compatible
thf(fact_1106_wf__union__merge,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ ( sup_su6327502436637775413at_nat @ R3 @ S ) )
      = ( wf_nat @ ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( relcomp_nat_nat_nat @ R3 @ R3 ) @ ( relcomp_nat_nat_nat @ S @ R3 ) ) @ S ) ) ) ).

% wf_union_merge
thf(fact_1107_wf__union__compatible,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R3 )
     => ( ( wf_nat @ S )
       => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R3 @ S ) @ R3 )
         => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R3 @ S ) ) ) ) ) ).

% wf_union_compatible
thf(fact_1108_wfI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R2
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B3 ) )
     => ( ! [X2: nat,P8: nat > $o] :
            ( ! [Xa2: nat] :
                ( ! [Y4: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Xa2 ) @ R2 )
                   => ( P8 @ Y4 ) )
               => ( P8 @ Xa2 ) )
           => ( ( member_nat @ X2 @ A3 )
             => ( ( member_nat @ X2 @ B3 )
               => ( P8 @ X2 ) ) ) )
       => ( wf_nat @ R2 ) ) ) ).

% wfI
thf(fact_1109_wf__map__prod__image,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F4: nat > nat] :
      ( ( wf_nat @ R2 )
     => ( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
       => ( wf_nat @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F4 @ F4 ) @ R2 ) ) ) ) ).

% wf_map_prod_image
thf(fact_1110_wf__map__prod__image__Dom__Ran,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F4: nat > nat] :
      ( ( wf_nat @ R2 )
     => ( ! [A6: nat,A8: nat] :
            ( ( member_nat @ A6 @ ( domain_nat_nat @ R2 ) )
           => ( ( member_nat @ A8 @ ( range_nat_nat @ R2 ) )
             => ( ( ( F4 @ A6 )
                  = ( F4 @ A8 ) )
               => ( A6 = A8 ) ) ) )
       => ( wf_nat @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F4 @ F4 ) @ R2 ) ) ) ) ).

% wf_map_prod_image_Dom_Ran
thf(fact_1111_wf__Un,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( wf_nat @ S2 )
       => ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R2 ) @ ( range_nat_nat @ S2 ) )
            = bot_bot_set_nat )
         => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S2 ) ) ) ) ) ).

% wf_Un
thf(fact_1112_wf__Union,axiom,
    ! [R3: set_se7855581050983116737at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ R3 )
         => ( wf_nat @ X2 ) )
     => ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ R3 )
           => ! [Xa3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa3 @ R3 )
               => ( ( X2 != Xa3 )
                 => ( ( inf_inf_set_nat @ ( domain_nat_nat @ X2 ) @ ( range_nat_nat @ Xa3 ) )
                    = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ R3 ) ) ) ) ).

% wf_Union
thf(fact_1113_wf__measure,axiom,
    ! [F4: nat > nat] : ( wf_nat @ ( measure_nat @ F4 ) ) ).

% wf_measure
thf(fact_1114_bij__image__Collect__eq,axiom,
    ! [F4: nat > nat,P2: nat > $o] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ( ( image_nat_nat2 @ F4 @ ( collect_nat @ P2 ) )
        = ( collect_nat
          @ ^ [Y2: nat] : ( P2 @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F4 @ Y2 ) ) ) ) ) ).

% bij_image_Collect_eq
thf(fact_1115_bij__image__Collect__eq,axiom,
    ! [F4: nat > product_prod_nat_nat,P2: nat > $o] :
      ( ( bij_be8693218025023041337at_nat @ F4 @ top_top_set_nat @ top_to4669805908274784177at_nat )
     => ( ( image_5846123807819985514at_nat @ F4 @ ( collect_nat @ P2 ) )
        = ( collec3392354462482085612at_nat
          @ ^ [Y2: product_prod_nat_nat] : ( P2 @ ( hilber4375958665455852643at_nat @ top_top_set_nat @ F4 @ Y2 ) ) ) ) ) ).

% bij_image_Collect_eq
thf(fact_1116_bij__image__Collect__eq,axiom,
    ! [F4: product_prod_nat_nat > nat,P2: product_prod_nat_nat > $o] :
      ( ( bij_be5333170631980326235at_nat @ F4 @ top_to4669805908274784177at_nat @ top_top_set_nat )
     => ( ( image_2486076414777270412at_nat @ F4 @ ( collec3392354462482085612at_nat @ P2 ) )
        = ( collect_nat
          @ ^ [Y2: nat] : ( P2 @ ( hilber1015911272413137541at_nat @ top_to4669805908274784177at_nat @ F4 @ Y2 ) ) ) ) ) ).

% bij_image_Collect_eq
thf(fact_1117_bij__image__Collect__eq,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( bij_be1923491781212474806at_nat @ F4 @ top_to4669805908274784177at_nat @ top_to4669805908274784177at_nat )
     => ( ( image_5168914502847457605at_nat @ F4 @ ( collec3392354462482085612at_nat @ P2 ) )
        = ( collec3392354462482085612at_nat
          @ ^ [Y2: product_prod_nat_nat] : ( P2 @ ( hilber1766663248461054988at_nat @ top_to4669805908274784177at_nat @ F4 @ Y2 ) ) ) ) ) ).

% bij_image_Collect_eq
thf(fact_1118_mono__cSup,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F4 @ A3 ) ) @ ( F4 @ ( complete_Sup_Sup_nat @ A3 ) ) ) ) ) ) ).

% mono_cSup
thf(fact_1119_Collect__imp__eq,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ( ( P2 @ X5 )
           => ( Q @ X5 ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_1120_Collect__imp__eq,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ( ( P2 @ X5 )
           => ( Q @ X5 ) ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P2 ) ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_1121_antimonoD,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X5: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X5 )
        @ F4 )
     => ( ( ord_less_eq_nat @ X @ Y )
       => ( ord_less_eq_nat @ ( F4 @ Y ) @ ( F4 @ X ) ) ) ) ).

% antimonoD
thf(fact_1122_antimonoE,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X5: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X5 )
        @ F4 )
     => ( ( ord_less_eq_nat @ X @ Y )
       => ( ord_less_eq_nat @ ( F4 @ Y ) @ ( F4 @ X ) ) ) ) ).

% antimonoE
thf(fact_1123_antimonoI,axiom,
    ! [F4: nat > nat] :
      ( ! [X2: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y4 )
         => ( ord_less_eq_nat @ ( F4 @ Y4 ) @ ( F4 @ X2 ) ) )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X5: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X5 )
        @ F4 ) ) ).

% antimonoI
thf(fact_1124_mono__inv,axiom,
    ! [F4: nat > nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
       => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F4 ) ) ) ) ).

% mono_inv
thf(fact_1125_mono__cInf,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ord_less_eq_nat @ ( F4 @ ( complete_Inf_Inf_nat @ A3 ) ) @ ( complete_Inf_Inf_nat @ ( image_nat_nat2 @ F4 @ A3 ) ) ) ) ) ) ).

% mono_cInf
thf(fact_1126_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A4: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
              ~ ( member_nat @ X5 @ A4 ) ) ) ) ).

% Compl_eq
thf(fact_1127_Compl__eq,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ~ ( member8440522571783428010at_nat @ X5 @ A4 ) ) ) ) ).

% Compl_eq
thf(fact_1128_Collect__neg__eq,axiom,
    ! [P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X5: nat] :
            ~ ( P2 @ X5 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_1129_Collect__neg__eq,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] :
            ~ ( P2 @ X5 ) )
      = ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P2 ) ) ) ).

% Collect_neg_eq
thf(fact_1130_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A4: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X5: nat] : ( member_nat @ X5 @ A4 ) ) ) ) ) ).

% uminus_set_def
thf(fact_1131_uminus__set__def,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( uminus8676089048583255045_nat_o
            @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 ) ) ) ) ) ).

% uminus_set_def
thf(fact_1132_bdd__below__image__antimono,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X5: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X5 )
        @ F4 )
     => ( ( condit2214826472909112428ve_nat @ A3 )
       => ( condit1738341127787009408ow_nat @ ( image_nat_nat2 @ F4 @ A3 ) ) ) ) ).

% bdd_below_image_antimono
thf(fact_1133_bdd__above__image__antimono,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X5: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X5 )
        @ F4 )
     => ( ( condit1738341127787009408ow_nat @ A3 )
       => ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F4 @ A3 ) ) ) ) ).

% bdd_above_image_antimono
thf(fact_1134_Least__mono,axiom,
    ! [F4: nat > nat,S: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ? [X6: nat] :
            ( ( member_nat @ X6 @ S )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ S )
               => ( ord_less_eq_nat @ X6 @ Xa3 ) ) )
       => ( ( ord_Least_nat
            @ ^ [Y2: nat] : ( member_nat @ Y2 @ ( image_nat_nat2 @ F4 @ S ) ) )
          = ( F4
            @ ( ord_Least_nat
              @ ^ [X5: nat] : ( member_nat @ X5 @ S ) ) ) ) ) ) ).

% Least_mono
thf(fact_1135_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
              @ ^ [X5: nat] :
                ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ S )
                 => ( ord_less_eq_nat @ Y2 @ X5 ) ) ) ) ) ) ) ).

% cSup_cInf
thf(fact_1136_Inter__eq,axiom,
    ( comple7806235888213564991et_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
            ! [Y2: set_nat] :
              ( ( member_set_nat @ Y2 @ A4 )
             => ( member_nat @ X5 @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1137_Inter__eq,axiom,
    ( comple1930127330929219688at_nat
    = ( ^ [A4: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
            ! [Y2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y2 @ A4 )
             => ( member8440522571783428010at_nat @ X5 @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1138_refl__on__def_H,axiom,
    ( refl_on_nat
    = ( ^ [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
          ( ! [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ R )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y2: nat,Z6: nat] :
                    ( ( member_nat @ Y2 @ A4 )
                    & ( member_nat @ Z6 @ A4 ) )
                @ X5 ) )
          & ! [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1139_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
              @ ^ [X5: nat] :
                ! [Y2: nat] :
                  ( ( member_nat @ Y2 @ S )
                 => ( ord_less_eq_nat @ X5 @ Y2 ) ) ) ) ) ) ) ).

% cInf_cSup
thf(fact_1140_Least__le,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ K ) ) ).

% Least_le
thf(fact_1141_chain__compr,axiom,
    ! [Ord: nat > nat > $o,A3: set_nat,P2: nat > $o] :
      ( ( comple7016393980872852640in_nat @ Ord @ A3 )
     => ( comple7016393980872852640in_nat @ Ord
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
              & ( P2 @ X5 ) ) ) ) ) ).

% chain_compr
thf(fact_1142_chain__compr,axiom,
    ! [Ord: product_prod_nat_nat > product_prod_nat_nat > $o,A3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( comple3588573705683770417at_nat @ Ord @ A3 )
     => ( comple3588573705683770417at_nat @ Ord
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A3 )
              & ( P2 @ X5 ) ) ) ) ) ).

% chain_compr
thf(fact_1143_lfp__induct2,axiom,
    ! [A: nat,B: list_char,F4: set_Pr2507339680178222706t_char > set_Pr2507339680178222706t_char,P2: nat > list_char > $o] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( comple3227850406620335403t_char @ F4 ) )
     => ( ( monoto7809364549836873959t_char @ top_to6005184720552305954t_char @ ord_le6315511645215477266t_char @ ord_le6315511645215477266t_char @ F4 )
       => ( ! [A6: nat,B5: list_char] :
              ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A6 @ B5 ) @ ( F4 @ ( inf_in3544766959038609504t_char @ ( comple3227850406620335403t_char @ F4 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P2 ) ) ) ) )
             => ( P2 @ A6 @ B5 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% lfp_induct2
thf(fact_1144_lfp__induct2,axiom,
    ! [A: nat,B: nat,F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( comple1738246912304005786at_nat @ F4 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F4 )
       => ( ! [A6: nat,B5: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ ( F4 @ ( inf_in2572325071724192079at_nat @ ( comple1738246912304005786at_nat @ F4 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) ) ) )
             => ( P2 @ A6 @ B5 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% lfp_induct2
thf(fact_1145_acyclicP__converse,axiom,
    ! [R2: nat > nat > $o] :
      ( ( transi3139058337892597825ic_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( conversep_nat_nat @ R2 ) ) ) )
      = ( transi3139058337892597825ic_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ) ).

% acyclicP_converse
thf(fact_1146_wf__acyclic,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( transi3139058337892597825ic_nat @ R2 ) ) ).

% wf_acyclic
thf(fact_1147_lfp__induct__set,axiom,
    ! [A: nat,F4: set_nat > set_nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( comple7975543026063415949et_nat @ F4 ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F4 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ ( F4 @ ( inf_inf_set_nat @ ( comple7975543026063415949et_nat @ F4 ) @ ( collect_nat @ P2 ) ) ) )
             => ( P2 @ X2 ) )
         => ( P2 @ A ) ) ) ) ).

% lfp_induct_set
thf(fact_1148_lfp__induct__set,axiom,
    ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( comple1738246912304005786at_nat @ F4 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F4 )
       => ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ ( F4 @ ( inf_in2572325071724192079at_nat @ ( comple1738246912304005786at_nat @ F4 ) @ ( collec3392354462482085612at_nat @ P2 ) ) ) )
             => ( P2 @ X2 ) )
         => ( P2 @ A ) ) ) ) ).

% lfp_induct_set
thf(fact_1149_def__lfp__induct__set,axiom,
    ! [A3: set_nat,F4: set_nat > set_nat,A: nat,P2: nat > $o] :
      ( ( A3
        = ( comple7975543026063415949et_nat @ F4 ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat @ F4 )
       => ( ( member_nat @ A @ A3 )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ ( F4 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P2 ) ) ) )
               => ( P2 @ X2 ) )
           => ( P2 @ A ) ) ) ) ) ).

% def_lfp_induct_set
thf(fact_1150_def__lfp__induct__set,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( A3
        = ( comple1738246912304005786at_nat @ F4 ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat @ F4 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ ( F4 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P2 ) ) ) )
               => ( P2 @ X2 ) )
           => ( P2 @ A ) ) ) ) ) ).

% def_lfp_induct_set
thf(fact_1151_def__Collect__coinduct,axiom,
    ! [A3: set_nat,P2: set_nat > nat > $o,A: nat,X8: set_nat] :
      ( ( A3
        = ( comple1596078789208929544et_nat
          @ ^ [W: set_nat] : ( collect_nat @ ( P2 @ W ) ) ) )
     => ( ( monoto1748750089227133045et_nat @ top_top_set_set_nat @ ord_less_eq_set_nat @ ord_less_eq_set_nat
          @ ^ [W: set_nat] : ( collect_nat @ ( P2 @ W ) ) )
       => ( ( member_nat @ A @ X8 )
         => ( ! [Z2: nat] :
                ( ( member_nat @ Z2 @ X8 )
               => ( P2 @ ( sup_sup_set_nat @ X8 @ A3 ) @ Z2 ) )
           => ( member_nat @ A @ A3 ) ) ) ) ) ).

% def_Collect_coinduct
thf(fact_1152_def__Collect__coinduct,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,X8: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( comple8166066674555273823at_nat
          @ ^ [W: set_Pr1261947904930325089at_nat] : ( collec3392354462482085612at_nat @ ( P2 @ W ) ) ) )
     => ( ( monoto3897784414697140647at_nat @ top_to7629004291339433233at_nat @ ord_le3146513528884898305at_nat @ ord_le3146513528884898305at_nat
          @ ^ [W: set_Pr1261947904930325089at_nat] : ( collec3392354462482085612at_nat @ ( P2 @ W ) ) )
       => ( ( member8440522571783428010at_nat @ A @ X8 )
         => ( ! [Z2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Z2 @ X8 )
               => ( P2 @ ( sup_su6327502436637775413at_nat @ X8 @ A3 ) @ Z2 ) )
           => ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% def_Collect_coinduct
thf(fact_1153_finite__def,axiom,
    ( finite_finite_nat
    = ( comple7308811355306212098_nat_o
      @ ^ [P: set_nat > $o,X5: set_nat] :
          ( ( X5 = bot_bot_set_nat )
          | ? [A4: set_nat,A2: nat] :
              ( ( X5
                = ( insert_nat @ A2 @ A4 ) )
              & ( P @ A4 ) ) ) ) ) ).

% finite_def
thf(fact_1154_finite__Collect__conjI,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1155_finite__Collect__conjI,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( P2 @ X5 )
              & ( Q @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_1156_finite__Collect__disjI,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( P2 @ X5 )
              | ( Q @ X5 ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1157_finite__Collect__disjI,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( P2 @ X5 )
              | ( Q @ X5 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_1158_finite__Collect__not,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
                ~ ( P2 @ X5 ) ) )
        = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Collect_not
thf(fact_1159_finite__Collect__not,axiom,
    ! [P2: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ~ ( P2 @ X5 ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_1160_finite__Collect__bounded__ex,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
              ? [Y2: product_prod_nat_nat] :
                ( ( P2 @ Y2 )
                & ( Q @ X5 @ Y2 ) ) ) )
        = ( ! [Y2: product_prod_nat_nat] :
              ( ( P2 @ Y2 )
             => ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [X5: product_prod_nat_nat] : ( Q @ X5 @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1161_finite__Collect__bounded__ex,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
              ? [Y2: product_prod_nat_nat] :
                ( ( P2 @ Y2 )
                & ( Q @ X5 @ Y2 ) ) ) )
        = ( ! [Y2: product_prod_nat_nat] :
              ( ( P2 @ Y2 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X5: nat] : ( Q @ X5 @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1162_finite__Collect__bounded__ex,axiom,
    ! [P2: nat > $o,Q: product_prod_nat_nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
              ? [Y2: nat] :
                ( ( P2 @ Y2 )
                & ( Q @ X5 @ Y2 ) ) ) )
        = ( ! [Y2: nat] :
              ( ( P2 @ Y2 )
             => ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [X5: product_prod_nat_nat] : ( Q @ X5 @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1163_finite__Collect__bounded__ex,axiom,
    ! [P2: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
              ? [Y2: nat] :
                ( ( P2 @ Y2 )
                & ( Q @ X5 @ Y2 ) ) ) )
        = ( ! [Y2: nat] :
              ( ( P2 @ Y2 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X5: nat] : ( Q @ X5 @ Y2 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_1164_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1165_finite__SigmaI,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A6 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_1166_finite__UN__I,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A6 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1167_infinite__iff__countable__subset,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ? [F2: nat > nat] :
            ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
            & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).

% infinite_iff_countable_subset
thf(fact_1168_infinite__countable__subset,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ? [F: nat > nat] :
          ( ( inj_on_nat_nat @ F @ top_top_set_nat )
          & ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) @ S ) ) ) ).

% infinite_countable_subset
thf(fact_1169_pigeonhole__infinite,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > nat] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F4 @ A3 ) )
       => ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
            & ~ ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [A2: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ A2 @ A3 )
                      & ( ( F4 @ A2 )
                        = ( F4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1170_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat2 @ F4 @ A3 ) )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A2: nat] :
                      ( ( member_nat @ A2 @ A3 )
                      & ( ( F4 @ A2 )
                        = ( F4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1171_finite__image__set2,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F4: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X5: product_prod_nat_nat,Y2: product_prod_nat_nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1172_finite__image__set2,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o,F4: product_prod_nat_nat > product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X5: product_prod_nat_nat,Y2: product_prod_nat_nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1173_finite__image__set2,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: nat > $o,F4: product_prod_nat_nat > nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X5: product_prod_nat_nat,Y2: nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1174_finite__image__set2,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: nat > $o,F4: product_prod_nat_nat > nat > nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X5: product_prod_nat_nat,Y2: nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1175_finite__image__set2,axiom,
    ! [P2: nat > $o,Q: product_prod_nat_nat > $o,F4: nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X5: nat,Y2: product_prod_nat_nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1176_finite__image__set2,axiom,
    ! [P2: nat > $o,Q: product_prod_nat_nat > $o,F4: nat > product_prod_nat_nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X5: nat,Y2: product_prod_nat_nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1177_finite__image__set2,axiom,
    ! [P2: nat > $o,Q: nat > $o,F4: nat > nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [Uu: product_prod_nat_nat] :
              ? [X5: nat,Y2: nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1178_finite__image__set2,axiom,
    ! [P2: nat > $o,Q: nat > $o,F4: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X5: nat,Y2: nat] :
                ( ( Uu
                  = ( F4 @ X5 @ Y2 ) )
                & ( P2 @ X5 )
                & ( Q @ Y2 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_1179_finite__image__set,axiom,
    ! [P2: product_prod_nat_nat > $o,F4: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [X5: product_prod_nat_nat] :
              ( ( Uu
                = ( F4 @ X5 ) )
              & ( P2 @ X5 ) ) ) ) ) ).

% finite_image_set
thf(fact_1180_finite__image__set,axiom,
    ! [P2: product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X5: product_prod_nat_nat] :
              ( ( Uu
                = ( F4 @ X5 ) )
              & ( P2 @ X5 ) ) ) ) ) ).

% finite_image_set
thf(fact_1181_finite__image__set,axiom,
    ! [P2: nat > $o,F4: nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [X5: nat] :
              ( ( Uu
                = ( F4 @ X5 ) )
              & ( P2 @ X5 ) ) ) ) ) ).

% finite_image_set
thf(fact_1182_finite__image__set,axiom,
    ! [P2: nat > $o,F4: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X5: nat] :
              ( ( Uu
                = ( F4 @ X5 ) )
              & ( P2 @ X5 ) ) ) ) ) ).

% finite_image_set
thf(fact_1183_finite__subset__Union,axiom,
    ! [A3: set_nat,B8: set_set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( comple7399068483239264473et_nat @ B8 ) )
       => ~ ! [F5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F5 )
             => ( ( ord_le6893508408891458716et_nat @ F5 @ B8 )
               => ~ ( ord_less_eq_set_nat @ A3 @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1184_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_1185_finite__SigmaI2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A3 )
              & ( ( B3 @ X5 )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A6: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A6 ) ) )
       => ( finite8785817246233100311at_nat @ ( produc7672662199629908489at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_1186_finite__SigmaI2,axiom,
    ! [A3: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
              & ( ( B3 @ X5 )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A6: nat] :
            ( ( member_nat @ A6 @ A3 )
           => ( finite_finite_nat @ ( B3 @ A6 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A3 @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_1187_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_1188_not__finite__existsD,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P2 ) )
     => ? [X_12: product_prod_nat_nat] : ( P2 @ X_12 ) ) ).

% not_finite_existsD
thf(fact_1189_not__finite__existsD,axiom,
    ! [P2: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ? [X_12: nat] : ( P2 @ X_12 ) ) ).

% not_finite_existsD
thf(fact_1190_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 )
       => ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B3 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A2 @ A3 )
                        & ( R3 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1191_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X2 @ Xa2 ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat @ A2 @ A3 )
                        & ( R3 @ A2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_1192_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F4: product_prod_nat_nat > nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on2178005380612969504at_nat @ F4 @ D2 )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ J @ D2 )
                & ( member_nat @ ( F4 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_1193_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F4: nat > nat,D2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F4 @ D2 )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D2 )
                & ( member_nat @ ( F4 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_1194_wf__iff__acyclic__if__finite,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R2 )
     => ( ( wf_nat @ R2 )
        = ( transi3139058337892597825ic_nat @ R2 ) ) ) ).

% wf_iff_acyclic_if_finite
thf(fact_1195_finite__acyclic__wf,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R2 )
     => ( ( transi3139058337892597825ic_nat @ R2 )
       => ( wf_nat @ R2 ) ) ) ).

% finite_acyclic_wf
thf(fact_1196_finite__inverse__image,axiom,
    ! [A3: set_nat,F4: product_prod_nat_nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on2178005380612969504at_nat @ F4 @ top_to4669805908274784177at_nat )
       => ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [J: product_prod_nat_nat] : ( member_nat @ ( F4 @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1197_finite__inverse__image,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( F4 @ J ) @ A3 ) ) ) ) ) ).

% finite_inverse_image
thf(fact_1198_finite__acyclic__wf__converse,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R2 )
     => ( ( transi3139058337892597825ic_nat @ R2 )
       => ( wf_nat @ ( converse_nat_nat @ R2 ) ) ) ) ).

% finite_acyclic_wf_converse
thf(fact_1199_infinite__imp__bij__betw2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A3 @ ( sup_sup_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_imp_bij_betw2
thf(fact_1200_infinite__imp__bij__betw,axiom,
    ! [A3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A3 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_imp_bij_betw
thf(fact_1201_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_1202_insert__relcomp__union__fold,axiom,
    ! [S: set_Pr4480099295215233283t_char,X: produc1457211279475724562t_char,X8: set_Pr2507339680178222706t_char] :
      ( ( finite3801884896073088588t_char @ S )
     => ( ( sup_su7899656352754981958t_char @ ( relcom5956200836249630357t_char @ ( insert3372640069787401954t_char @ X @ bot_bo1414094081888755934t_char ) @ S ) @ X8 )
        = ( finite859346576728541399t_char
          @ ( produc6181602235716527061t_char
            @ ^ [W: list_char,Z6: list_char,A9: set_Pr2507339680178222706t_char] :
                ( if_set1248378584462554936t_char
                @ ( ( produc1900778367047502200t_char @ X )
                  = W )
                @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ X ) @ Z6 ) @ A9 )
                @ A9 ) )
          @ X8
          @ S ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1203_fold__union__pair,axiom,
    ! [B3: set_list_char,X: nat,A3: set_Pr2507339680178222706t_char] :
      ( ( finite3619747586194699384t_char @ B3 )
     => ( ( sup_su7899656352754981958t_char
          @ ( comple9178978156834306527t_char
            @ ( image_366428816823531018t_char
              @ ^ [Y2: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X @ Y2 ) @ bot_bo1414094081888755934t_char )
              @ B3 ) )
          @ A3 )
        = ( finite3391364926417624327t_char
          @ ^ [Y2: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X @ Y2 ) )
          @ A3
          @ B3 ) ) ) ).

% fold_union_pair
thf(fact_1204_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
            @ ^ [X5: nat,Z6: set_Pr1261947904930325089at_nat] :
                ( finite3745491028973389255at_nat
                @ ^ [Y2: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) )
                @ Z6
                @ B3 )
            @ bot_bo2099793752762293965at_nat
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_1205_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
            @ ^ [X5: nat,Z6: set_Pr2507339680178222706t_char] :
                ( finite3391364926417624327t_char
                @ ^ [Y2: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) )
                @ Z6
                @ B3 )
            @ bot_bo1414094081888755934t_char
            @ A3 ) ) ) ) ).

% product_fold
thf(fact_1206_Id__on__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( id_on_nat @ A3 )
        = ( finite3745491028973389255at_nat
          @ ^ [X5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) )
          @ bot_bo2099793752762293965at_nat
          @ A3 ) ) ) ).

% Id_on_fold
thf(fact_1207_insert__relcomp__fold,axiom,
    ! [S: set_Pr4480099295215233283t_char,X: produc1457211279475724562t_char,R3: set_Pr2507339680178222706t_char] :
      ( ( finite3801884896073088588t_char @ S )
     => ( ( relcom5956200836249630357t_char @ ( insert3372640069787401954t_char @ X @ R3 ) @ S )
        = ( finite859346576728541399t_char
          @ ( produc6181602235716527061t_char
            @ ^ [W: list_char,Z6: list_char,A9: set_Pr2507339680178222706t_char] :
                ( if_set1248378584462554936t_char
                @ ( ( produc1900778367047502200t_char @ X )
                  = W )
                @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ X ) @ Z6 ) @ A9 )
                @ A9 ) )
          @ ( relcom5956200836249630357t_char @ R3 @ S )
          @ S ) ) ) ).

% insert_relcomp_fold
thf(fact_1208_Set__filter__fold,axiom,
    ! [A3: set_nat,P2: nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( filter_nat2 @ P2 @ A3 )
        = ( finite5529483035118572448et_nat
          @ ^ [X5: nat,A9: set_nat] : ( if_set_nat @ ( P2 @ X5 ) @ ( insert_nat @ X5 @ A9 ) @ A9 )
          @ bot_bot_set_nat
          @ A3 ) ) ) ).

% Set_filter_fold
thf(fact_1209_Set_Ofilter__def,axiom,
    ( filter_nat2
    = ( ^ [P4: nat > $o,A4: set_nat] :
          ( collect_nat
          @ ^ [A2: nat] :
              ( ( member_nat @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1210_Set_Ofilter__def,axiom,
    ( filter5640266504077782706at_nat
    = ( ^ [P4: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [A2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A2 @ A4 )
              & ( P4 @ A2 ) ) ) ) ) ).

% Set.filter_def
thf(fact_1211_inter__Set__filter,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = ( filter_nat2
          @ ^ [X5: nat] : ( member_nat @ X5 @ A3 )
          @ B3 ) ) ) ).

% inter_Set_filter
thf(fact_1212_comp__fun__commute__product__fold,axiom,
    ! [B3: set_list_char] :
      ( ( finite3619747586194699384t_char @ B3 )
     => ( finite1002329793627884815t_char
        @ ^ [X5: nat,Z6: set_Pr2507339680178222706t_char] :
            ( finite3391364926417624327t_char
            @ ^ [Y2: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) )
            @ Z6
            @ B3 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1213_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 )
             => ~ ! [X6: nat] :
                    ( ( member_nat @ X6 @ A13 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A24 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1214_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 )
        & ! [X5: nat] :
            ( ( member_nat @ X5 @ A13 )
           => ? [Y2: nat] :
                ( ( member_nat @ Y2 @ A24 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1215_max__ext__wf,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( wf_set_nat @ ( max_ext_nat @ R2 ) ) ) ).

% max_ext_wf
thf(fact_1216_max__ext_Omax__extI,axiom,
    ! [X8: set_nat,Y7: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X8 )
     => ( ( finite_finite_nat @ Y7 )
       => ( ( Y7 != bot_bot_set_nat )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ X8 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ Y7 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Xa2 ) @ R3 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X8 @ Y7 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1217_Pow__fold,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( pow_nat @ A3 )
        = ( finite4178521680790401110et_nat
          @ ^ [X5: nat,A4: set_set_nat] : ( sup_sup_set_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X5 ) @ A4 ) )
          @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
          @ A3 ) ) ) ).

% Pow_fold
thf(fact_1218_max__extp__eq,axiom,
    ( max_extp_nat
    = ( ^ [R: nat > nat > $o,X5: set_nat,Y2: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y2 ) @ ( max_ext_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_1219_min__ext__wf,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( wf_set_nat @ ( min_ext_nat @ R2 ) ) ) ).

% min_ext_wf
thf(fact_1220_wfE__pf,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( wf_nat @ R3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( image_nat_nat @ R3 @ A3 ) )
       => ( A3 = bot_bot_set_nat ) ) ) ).

% wfE_pf
thf(fact_1221_wfI__pf,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ! [A10: set_nat] :
          ( ( ord_less_eq_set_nat @ A10 @ ( image_nat_nat @ R3 @ A10 ) )
         => ( A10 = bot_bot_set_nat ) )
     => ( wf_nat @ R3 ) ) ).

% wfI_pf
thf(fact_1222_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_1223_finite__equiv__class,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X8: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ R2
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : A3 ) )
       => ( ( member_set_nat @ X8 @ ( equiv_quotient_nat @ A3 @ R2 ) )
         => ( finite_finite_nat @ X8 ) ) ) ) ).

% finite_equiv_class
thf(fact_1224_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_1225_trans__wf__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( wf_nat @ R2 )
        = ( ! [A2: nat] :
              ( wf_nat
              @ ( inf_in2572325071724192079at_nat @ R2
                @ ( produc457027306803732586at_nat @ ( image_nat_nat @ ( converse_nat_nat @ R2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
                  @ ^ [Uu: nat] : ( image_nat_nat @ ( converse_nat_nat @ R2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% trans_wf_iff
thf(fact_1226_finite__Collect__bex,axiom,
    ! [A3: set_nat,Q: product_prod_nat_nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X5: product_prod_nat_nat] :
              ? [Y2: nat] :
                ( ( member_nat @ Y2 @ A3 )
                & ( Q @ X5 @ Y2 ) ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [Y2: product_prod_nat_nat] : ( Q @ Y2 @ X5 ) ) ) ) ) ) ) ).

% finite_Collect_bex
thf(fact_1227_finite__Collect__bex,axiom,
    ! [A3: set_nat,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
              ? [Y2: nat] :
                ( ( member_nat @ Y2 @ A3 )
                & ( Q @ X5 @ Y2 ) ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [Y2: nat] : ( Q @ Y2 @ X5 ) ) ) ) ) ) ) ).

% finite_Collect_bex
thf(fact_1228_Image__Collect__case__prod,axiom,
    ! [P2: nat > nat > $o,A3: set_nat] :
      ( ( image_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) @ A3 )
      = ( collect_nat
        @ ^ [Y2: nat] :
          ? [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
            & ( P2 @ X5 @ Y2 ) ) ) ) ).

% Image_Collect_case_prod
thf(fact_1229_max__extp_Ocases,axiom,
    ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,A13: set_Pr1261947904930325089at_nat,A24: set_Pr1261947904930325089at_nat] :
      ( ( max_ex4864111882549613972at_nat @ R3 @ A13 @ A24 )
     => ~ ( ( finite6177210948735845034at_nat @ A13 )
         => ( ( finite6177210948735845034at_nat @ A24 )
           => ( ( A24
               != ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) )
             => ~ ! [X6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X6 @ A13 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A24 )
                        & ( R3 @ X6 @ Xa3 ) ) ) ) ) ) ) ).

% max_extp.cases
thf(fact_1230_max__extp_Ocases,axiom,
    ! [R3: nat > nat > $o,A13: set_nat,A24: set_nat] :
      ( ( max_extp_nat @ R3 @ A13 @ A24 )
     => ~ ( ( finite_finite_nat @ A13 )
         => ( ( finite_finite_nat @ A24 )
           => ( ( A24
               != ( collect_nat @ bot_bot_nat_o ) )
             => ~ ! [X6: nat] :
                    ( ( member_nat @ X6 @ A13 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A24 )
                        & ( R3 @ X6 @ Xa3 ) ) ) ) ) ) ) ).

% max_extp.cases
thf(fact_1231_max__extp_Osimps,axiom,
    ( max_ex4864111882549613972at_nat
    = ( ^ [R4: product_prod_nat_nat > product_prod_nat_nat > $o,A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat] :
          ( ( finite6177210948735845034at_nat @ A12 )
          & ( finite6177210948735845034at_nat @ A23 )
          & ( A23
           != ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) )
          & ! [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ A12 )
             => ? [Y2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y2 @ A23 )
                  & ( R4 @ X5 @ Y2 ) ) ) ) ) ) ).

% max_extp.simps
thf(fact_1232_max__extp_Osimps,axiom,
    ( max_extp_nat
    = ( ^ [R4: nat > nat > $o,A12: set_nat,A23: set_nat] :
          ( ( finite_finite_nat @ A12 )
          & ( finite_finite_nat @ A23 )
          & ( A23
           != ( collect_nat @ bot_bot_nat_o ) )
          & ! [X5: nat] :
              ( ( member_nat @ X5 @ A12 )
             => ? [Y2: nat] :
                  ( ( member_nat @ Y2 @ A23 )
                  & ( R4 @ X5 @ Y2 ) ) ) ) ) ) ).

% max_extp.simps
thf(fact_1233_max__extp_Omax__extI,axiom,
    ! [X8: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ X8 )
     => ( ( finite6177210948735845034at_nat @ Y7 )
       => ( ( Y7
           != ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ X8 )
               => ? [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ Y7 )
                    & ( R3 @ X2 @ Xa2 ) ) )
           => ( max_ex4864111882549613972at_nat @ R3 @ X8 @ Y7 ) ) ) ) ) ).

% max_extp.max_extI
thf(fact_1234_max__extp_Omax__extI,axiom,
    ! [X8: set_nat,Y7: set_nat,R3: nat > nat > $o] :
      ( ( finite_finite_nat @ X8 )
     => ( ( finite_finite_nat @ Y7 )
       => ( ( Y7
           != ( collect_nat @ bot_bot_nat_o ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ X8 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ Y7 )
                    & ( R3 @ X2 @ Xa2 ) ) )
           => ( max_extp_nat @ R3 @ X8 @ Y7 ) ) ) ) ) ).

% max_extp.max_extI
thf(fact_1235_Image__def,axiom,
    ( image_nat_list_char
    = ( ^ [R: set_Pr2507339680178222706t_char,S3: set_nat] :
          ( collect_list_char
          @ ^ [Y2: list_char] :
            ? [X5: nat] :
              ( ( member_nat @ X5 @ S3 )
              & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X5 @ Y2 ) @ R ) ) ) ) ) ).

% Image_def
thf(fact_1236_Union__eq,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
            ? [Y2: set_nat] :
              ( ( member_set_nat @ Y2 @ A4 )
              & ( member_nat @ X5 @ Y2 ) ) ) ) ) ).

% Union_eq
thf(fact_1237_Union__eq,axiom,
    ( comple5685304695842803022at_nat
    = ( ^ [A4: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X5: product_prod_nat_nat] :
            ? [Y2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y2 @ A4 )
              & ( member8440522571783428010at_nat @ X5 @ Y2 ) ) ) ) ) ).

% Union_eq
thf(fact_1238_trans__on__join,axiom,
    ( trans_on_nat
    = ( ^ [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ R )
         => ( produc6081775807080527818_nat_o
            @ ^ [Y2: nat,Y1: nat] :
                ( ( member_nat @ Y2 @ A4 )
               => ( ( member_nat @ Y1 @ A4 )
                 => ! [Z6: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ Z6 @ R )
                     => ( produc6081775807080527818_nat_o
                        @ ^ [Y22: nat,Aa2: nat] :
                            ( ( Y1 = Y22 )
                           => ( ( member_nat @ Aa2 @ A4 )
                             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Aa2 ) @ R ) ) )
                        @ Z6 ) ) ) )
            @ X5 ) ) ) ) ).

% trans_on_join
thf(fact_1239_trans__join,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ R2 )
           => ( produc6081775807080527818_nat_o
              @ ^ [Y2: nat,Y1: nat] :
                ! [Z6: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z6 @ R2 )
                 => ( produc6081775807080527818_nat_o
                    @ ^ [Y22: nat,Aa2: nat] :
                        ( ( Y1 = Y22 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Aa2 ) @ R2 ) )
                    @ Z6 ) )
              @ X5 ) ) ) ) ).

% trans_join
thf(fact_1240_wf__finite__segments,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
       => ( ! [X2: product_prod_nat_nat] :
              ( finite6177210948735845034at_nat
              @ ( collec3392354462482085612at_nat
                @ ^ [Y2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y2 @ X2 ) @ R2 ) ) )
         => ( wf_Pro7803398752247294826at_nat @ R2 ) ) ) ) ).

% wf_finite_segments
thf(fact_1241_wf__finite__segments,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R2 )
     => ( ( trans_on_nat @ top_top_set_nat @ R2 )
       => ( ! [X2: nat] :
              ( finite_finite_nat
              @ ( collect_nat
                @ ^ [Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ X2 ) @ R2 ) ) )
         => ( wf_nat @ R2 ) ) ) ) ).

% wf_finite_segments
thf(fact_1242_wf__converse,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R2 )
     => ( ( trans_on_nat @ top_top_set_nat @ R2 )
       => ( ( finite6177210948735845034at_nat @ R2 )
         => ( wf_nat @ ( converse_nat_nat @ R2 ) ) ) ) ) ).

% wf_converse
thf(fact_1243_wf__imp__irrefl,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( irrefl_on_nat @ top_top_set_nat @ R2 ) ) ).

% wf_imp_irrefl
thf(fact_1244_irrefl__on__distinct,axiom,
    ( irrefl_on_nat
    = ( ^ [A4: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ R )
         => ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] :
                ( ( member_nat @ A2 @ A4 )
               => ( ( member_nat @ B2 @ A4 )
                 => ( A2 != B2 ) ) )
            @ X5 ) ) ) ) ).

% irrefl_on_distinct
thf(fact_1245_max__ext__eq,axiom,
    ( max_ext_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ^ [X3: set_nat,Y8: set_nat] :
                ( ( finite_finite_nat @ X3 )
                & ( finite_finite_nat @ Y8 )
                & ( Y8 != bot_bot_set_nat )
                & ! [X5: nat] :
                    ( ( member_nat @ X5 @ X3 )
                   => ? [Y2: nat] :
                        ( ( member_nat @ Y2 @ Y8 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y2 ) @ R4 ) ) ) ) ) ) ) ) ).

% max_ext_eq
thf(fact_1246_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P4: nat > nat > $o,A4: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 )
                  @ ( produc457027306803732586at_nat @ A4
                    @ ^ [Uu: nat] : A4 ) )
                & ( P4 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_1247_finite__set__of__finite__funs,axiom,
    ! [A3: set_nat,B3: set_nat,D: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite2115694454571419734at_nat
          @ ( collect_nat_nat
            @ ^ [F2: nat > nat] :
              ! [X5: nat] :
                ( ( ( member_nat @ X5 @ A3 )
                 => ( member_nat @ ( F2 @ X5 ) @ B3 ) )
                & ( ~ ( member_nat @ X5 @ A3 )
                 => ( ( F2 @ X5 )
                    = D ) ) ) ) ) ) ) ).

% finite_set_of_finite_funs
thf(fact_1248_Least__def,axiom,
    ( ord_Least_nat
    = ( ^ [P4: nat > $o] :
          ( the_nat
          @ ^ [X5: nat] :
              ( ( P4 @ X5 )
              & ! [Y2: nat] :
                  ( ( P4 @ Y2 )
                 => ( ord_less_eq_nat @ X5 @ Y2 ) ) ) ) ) ) ).

% Least_def
thf(fact_1249_Greatest__def,axiom,
    ( order_Greatest_nat
    = ( ^ [P4: nat > $o] :
          ( the_nat
          @ ^ [X5: nat] :
              ( ( P4 @ X5 )
              & ! [Y2: nat] :
                  ( ( P4 @ Y2 )
                 => ( ord_less_eq_nat @ Y2 @ X5 ) ) ) ) ) ) ).

% Greatest_def
thf(fact_1250_equiv__Eps__preserves,axiom,
    ! [A3: set_prat,R2: set_Pr3298805672376727137t_prat,X8: set_prat] :
      ( ( equiv_equiv_prat @ A3 @ R2 )
     => ( ( member_set_prat @ X8 @ ( equiv_quotient_prat @ A3 @ R2 ) )
       => ( member_prat
          @ ( fChoice_prat
            @ ^ [X5: prat] : ( member_prat @ X5 @ X8 ) )
          @ A3 ) ) ) ).

% equiv_Eps_preserves
thf(fact_1251_equiv__Eps__in,axiom,
    ! [A3: set_prat,R2: set_Pr3298805672376727137t_prat,X8: set_prat] :
      ( ( equiv_equiv_prat @ A3 @ R2 )
     => ( ( member_set_prat @ X8 @ ( equiv_quotient_prat @ A3 @ R2 ) )
       => ( member_prat
          @ ( fChoice_prat
            @ ^ [X5: prat] : ( member_prat @ X5 @ X8 ) )
          @ X8 ) ) ) ).

% equiv_Eps_in
thf(fact_1252_proj__Eps,axiom,
    ! [A3: set_prat,R2: set_Pr3298805672376727137t_prat,X8: set_prat] :
      ( ( equiv_equiv_prat @ A3 @ R2 )
     => ( ( member_set_prat @ X8 @ ( equiv_quotient_prat @ A3 @ R2 ) )
       => ( ( equiv_proj_prat_prat @ R2
            @ ( fChoice_prat
              @ ^ [X5: prat] : ( member_prat @ X5 @ X8 ) ) )
          = X8 ) ) ) ).

% proj_Eps
thf(fact_1253_finite__subset__wf,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( wf_set_nat
        @ ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ^ [X3: set_nat,Y8: set_nat] :
                ( ( ord_less_set_nat @ X3 @ Y8 )
                & ( ord_less_eq_set_nat @ Y8 @ A3 ) ) ) ) ) ) ).

% finite_subset_wf
thf(fact_1254_wf,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf
thf(fact_1255_not__less__Least,axiom,
    ! [K: nat,P2: nat > $o] :
      ( ( ord_less_nat @ K @ ( ord_Least_nat @ P2 ) )
     => ~ ( P2 @ K ) ) ).

% not_less_Least
thf(fact_1256_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1257_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
      = ( ord_less_nat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1258_wf__bounded__set,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > set_nat,F4: nat > set_nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A6 ) @ R2 )
         => ( ( finite_finite_nat @ ( Ub @ A6 ) )
            & ( ord_less_eq_set_nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_eq_set_nat @ ( F4 @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_set_nat @ ( F4 @ A6 ) @ ( F4 @ B5 ) ) ) )
     => ( wf_nat @ R2 ) ) ).

% wf_bounded_set
thf(fact_1259_strict__mono__inv__on__range,axiom,
    ! [F4: nat > nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( monotone_on_nat_nat @ ( image_nat_nat2 @ F4 @ top_top_set_nat ) @ ord_less_nat @ ord_less_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F4 ) ) ) ).

% strict_mono_inv_on_range
thf(fact_1260_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_1261_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A4: set_nat,B6: set_nat] :
            ( ( ord_less_set_nat @ A4 @ B6 )
            & ( finite_finite_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_1262_wf__if__measure,axiom,
    ! [P2: nat > $o,F4: nat > nat,G: nat > nat] :
      ( ! [X2: nat] :
          ( ( P2 @ X2 )
         => ( ord_less_nat @ ( F4 @ ( G @ X2 ) ) @ ( F4 @ X2 ) ) )
     => ( wf_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [Y2: nat,X5: nat] :
                ( ( P2 @ X5 )
                & ( Y2
                  = ( G @ X5 ) ) ) ) ) ) ) ).

% wf_if_measure
thf(fact_1263_wf__bounded__measure,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Ub: nat > nat,F4: nat > nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A6 ) @ R2 )
         => ( ( ord_less_eq_nat @ ( Ub @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_eq_nat @ ( F4 @ B5 ) @ ( Ub @ A6 ) )
            & ( ord_less_nat @ ( F4 @ A6 ) @ ( F4 @ B5 ) ) ) )
     => ( wf_nat @ R2 ) ) ).

% wf_bounded_measure
thf(fact_1264_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
        ? [N: nat,F2: nat > nat] :
          ( A4
          = ( image_nat_nat2 @ F2
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1265_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F4: nat > nat,N2: nat] :
      ( ( A3
        = ( image_nat_nat2 @ F4
          @ ( collect_nat
            @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
     => ( finite_finite_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_1266_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [F: nat > nat,N3: nat] :
          ( ( ( image_nat_nat2 @ F @ A3 )
            = ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) )
          & ( inj_on_nat_nat @ F @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_1267_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [N3: nat,F: nat > nat] :
          ( ( A3
            = ( image_nat_nat2 @ F
              @ ( collect_nat
                @ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) ) )
          & ( inj_on_nat_nat @ F
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_1268_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1269_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1270_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_1271_eventually__le__at__bot,axiom,
    ! [C: nat] :
      ( eventually_nat
      @ ^ [X5: nat] : ( ord_less_eq_nat @ X5 @ C )
      @ at_bot_nat ) ).

% eventually_le_at_bot

% Helper facts (27)
thf(help_fChoice_1_1_fChoice_001_Eo_T,axiom,
    ! [P2: $o > $o] :
      ( ( P2 @ ( fChoice_o @ P2 ) )
      = ( ? [X3: $o] : ( P2 @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X: prat,Y: prat] :
      ( ( if_prat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X: prat,Y: prat] :
      ( ( if_prat @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ ( fChoice_nat @ P2 ) )
      = ( ? [X3: nat] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__PosRat__Oprat_T,axiom,
    ! [P2: prat > $o] :
      ( ( P2 @ ( fChoice_prat @ P2 ) )
      = ( ? [X3: prat] : ( P2 @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [P2: set_nat > $o] :
      ( ( P2 @ ( fChoice_set_nat @ P2 ) )
      = ( ? [X3: set_nat] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Set__Oset_It__PosRat__Oprat_J_T,axiom,
    ! [P2: set_prat > $o] :
      ( ( P2 @ ( fChoice_set_prat @ P2 ) )
      = ( ? [X3: set_prat] : ( P2 @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( P2 @ ( fChoic6978938873391328853at_nat @ P2 ) )
      = ( ? [X3: product_prod_nat_nat] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J_T,axiom,
    ! [P2: produc3375937593958228902prat_a > $o] :
      ( ( P2 @ ( fChoic952938374269289722prat_a @ P2 ) )
      = ( ? [X3: produc3375937593958228902prat_a] : ( P2 @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_T,axiom,
    ! [X: produc1457211279475724562t_char,Y: produc1457211279475724562t_char] :
      ( ( if_Pro2639532401392527832t_char @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_T,axiom,
    ! [X: produc1457211279475724562t_char,Y: produc1457211279475724562t_char] :
      ( ( if_Pro2639532401392527832t_char @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_T,axiom,
    ! [P2: produc7606469099237612818ar_nat > $o] :
      ( ( P2 @ ( fChoic8626096029427592294ar_nat @ P2 ) )
      = ( ? [X3: produc7606469099237612818ar_nat] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_T,axiom,
    ! [P2: produc1457211279475724562t_char > $o] :
      ( ( P2 @ ( fChoic2476838209665704038t_char @ P2 ) )
      = ( ? [X3: produc1457211279475724562t_char] : ( P2 @ X3 ) ) ) ).

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,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $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,
    ! [X: set_Pr2507339680178222706t_char,Y: set_Pr2507339680178222706t_char] :
      ( ( if_set1248378584462554936t_char @ $false @ X @ 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,
    ! [X: set_Pr2507339680178222706t_char,Y: set_Pr2507339680178222706t_char] :
      ( ( if_set1248378584462554936t_char @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_T,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( P2 @ ( fChoic3782516149722485159r_prat @ P2 ) )
      = ( ? [X3: produc1457211279475724562t_char > prat] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_I_062_Itf__a_Mt__PosRat__Oprat_J_Mtf__a_J_J_T,axiom,
    ! [P2: produc4654305426090421966prat_a > $o] :
      ( ( P2 @ ( fChoic9085910172109011490prat_a @ P2 ) )
      = ( ? [X3: produc4654305426090421966prat_a] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_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_T,axiom,
    ! [P2: produc2489117125269924006t_char > $o] :
      ( ( P2 @ ( fChoic1284671975056783866t_char @ P2 ) )
      = ( ? [X3: produc2489117125269924006t_char] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_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_T,axiom,
    ! [P2: produc3336119888008106215r_prat > $o] :
      ( ( P2 @ ( fChoic869448611404120123r_prat @ P2 ) )
      = ( ? [X3: produc3336119888008106215r_prat] : ( P2 @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_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_T,axiom,
    ! [P2: produc5803078220529002682t_char > $o] :
      ( ( P2 @ ( fChoic3450237333247147022t_char @ P2 ) )
      = ( ? [X3: produc5803078220529002682t_char] : ( P2 @ X3 ) ) ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( pi2 @ hl )
    = ( add_masks_a @ pi
      @ ^ [Hl2: a] :
          ( fChoice_prat
          @ ^ [P: prat] :
              ( ( pi2 @ Hl2 )
              = ( padd @ ( pi @ Hl2 ) @ P ) ) )
      @ hl ) ) ).

%------------------------------------------------------------------------------