TPTP Problem File: SLH0185^1.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/0002_PartialHeapSA/prob_00213_006443__7251614_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2128 ( 657 unt; 924 typ;   0 def)
%            Number of atoms       : 3559 (1219 equ;   0 cnn)
%            Maximal formula atoms :   10 (   2 avg)
%            Number of connectives : 12076 ( 244   ~;  51   |; 296   &;10274   @)
%                                         (   0 <=>;1211  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   6 avg)
%            Number of types       :  105 ( 104 usr)
%            Number of type conns  : 4392 (4392   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  822 ( 820 usr;  42 con; 0-5 aty)
%            Number of variables   : 4304 (1039   ^;3162   !; 103   ?;4304   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:08:33.071
%------------------------------------------------------------------------------
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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__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_Gr7520394057480975179on_val: set_Pr3433046674199179443r_prat > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val ) > set_Pr7649655576376376530on_val ).

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_OGr_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Gr1527221351080490466at_nat: set_se7855581050983116737at_nat > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Def_OGr_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bNF_Gr7888059597227825506at_nat: set_se5479201105039950375at_nat > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ) > set_Pr7459493094073627847at_nat ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_Gr2620532623635037156on_val: set_na3731733139948239937on_val > ( ( nat > produc9003781588162623004on_val ) > nat > produc1457211279475724562t_char > option_val ) > ( nat > produc9003781588162623004on_val ) > ( nat > produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    bNF_Gr5444735849185432863r_prat: set_na3731733139948239937on_val > ( ( nat > produc9003781588162623004on_val ) > nat > produc1457211279475724562t_char > prat ) > ( nat > produc9003781588162623004on_val ) > ( nat > produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr3388551082464497497at_nat: set_na8845969913329135560at_nat > ( ( nat > product_prod_nat_nat ) > nat > nat ) > ( nat > product_prod_nat_nat ) > ( 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__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_Gr4383009045007556383on_val: set_Pr3433046674199179443r_prat > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_Gr5270771597446195660on_val: set_Pr4636196140548262638on_val > ( ( product_prod_nat_nat > produc9003781588162623004on_val ) > product_prod_nat_nat > produc1457211279475724562t_char > option_val ) > ( product_prod_nat_nat > produc9003781588162623004on_val ) > ( product_prod_nat_nat > produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_OGrp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    bNF_Gr2950598962552920375r_prat: set_Pr4636196140548262638on_val > ( ( product_prod_nat_nat > produc9003781588162623004on_val ) > product_prod_nat_nat > produc1457211279475724562t_char > prat ) > ( product_prod_nat_nat > produc9003781588162623004on_val ) > ( product_prod_nat_nat > produc1457211279475724562t_char > prat ) > $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_Mt__Nat__Onat_J,type,
    bNF_Gr7802473308256045197at_nat: set_Pr2458342521480944603at_nat > ( ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > nat ) > $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,
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thf(sy_c_BNF__Def_OGrp_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__Option__Ooption_It__Mask__Oval_J_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_Gr1324552783004614022on_val: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_OGrp_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__Option__Ooption_It__Mask__Oval_J_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    bNF_Gr2025013579966528743r_prat: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr7275288873855648526at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Gr6677838525845560078at_nat: set_se7855581050983116737at_nat > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_BNF__Def_Oconvol_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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_co5850895267955675587on_val: ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ) > ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val ).

thf(sy_c_BNF__Def_Oconvol_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_co805650143699787099at_nat: ( nat > nat ) > ( nat > nat ) > nat > product_prod_nat_nat ).

thf(sy_c_BNF__Def_Oconvol_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__Option__Ooption_It__Mask__Oval_J_J_J_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__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_co3837107326993794344on_val: ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_BNF__Def_Oconvol_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__Option__Ooption_It__Mask__Oval_J_J_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_co6248701753126501031at_nat: ( produc9003781588162623004on_val > nat ) > ( produc9003781588162623004on_val > nat ) > produc9003781588162623004on_val > product_prod_nat_nat ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_co3675872600281376917on_val: ( product_prod_nat_nat > produc1457211279475724562t_char > prat ) > ( product_prod_nat_nat > produc1457211279475724562t_char > option_val ) > product_prod_nat_nat > produc9003781588162623004on_val ).

thf(sy_c_BNF__Def_Oconvol_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_co8276296344278409722at_nat: ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_BNF__Def_Oconvol_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,type,
    bNF_co3897469944465340696at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_BNF__Def_Oconvol_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_co147202357524637304at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).

thf(sy_c_BNF__Def_Ocsquare_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__Option__Ooption_It__Mask__Oval_J_J_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_cs5223968648578849398on_val: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > ( produc1757238238866670357on_val > produc1457211279475724562t_char > option_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > ( produc9003781588162623004on_val > produc1757238238866670357on_val ) > $o ).

thf(sy_c_BNF__Def_Ocsquare_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__Option__Ooption_It__Mask__Oval_J_J_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J_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_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    bNF_cs2794013118909459695on_val: set_Pr7649655576376376530on_val > ( produc3336119888008106215r_prat > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc3336119888008106215r_prat ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > $o ).

thf(sy_c_BNF__Def_OfstOp_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__Option__Ooption_It__Mask__Oval_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_fs7489134050367444823on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_BNF__Def_OfstOp_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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_fs7282908041383899532on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > produc3336119888008106215r_prat ).

thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_fs6799373675401686564at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_BNF__Def_OfstOp_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,type,
    bNF_fs8257191039976258191at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc859450856879609959at_nat > produc859450856879609959at_nat ).

thf(sy_c_BNF__Def_OfstOp_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_fs1946732770515719151at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > produc3843707927480180839at_nat > produc3843707927480180839at_nat ).

thf(sy_c_BNF__Def_Opick__middlep_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__Option__Ooption_It__Mask__Oval_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_pi6941616487637503653on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val ).

thf(sy_c_BNF__Def_Opick__middlep_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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_pi5559183403598038974on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > prat ).

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    bNF_re753112871316139152_nat_o: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > nat > $o ) > ( ( produc1457211279475724562t_char > option_val ) > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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__Option__Ooption_It__Mask__Oval_J_J_001_Eo_001_Eo,type,
    bNF_re6067813816088466256al_o_o: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( $o > $o > $o ) > ( ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > option_val ) > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_Eo_J_001_062_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_M_Eo_J,type,
    bNF_re4892053619688935818_val_o: ( nat > nat > $o ) > ( ( ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > option_val ) > $o ) > $o ) > ( nat > ( produc1457211279475724562t_char > prat ) > $o ) > ( nat > ( produc1457211279475724562t_char > option_val ) > $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_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__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_re2896540935372247726on_val: ( nat > nat > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( nat > produc1457211279475724562t_char > prat ) > ( nat > produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re3272885123108610425_nat_o: ( nat > nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( nat > product_prod_nat_nat > $o ) > ( nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    bNF_re7533008290153783993_nat_o: ( nat > nat > $o ) > ( ( set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > $o ) > $o ) > ( nat > set_Pr1261947904930325089at_nat > $o ) > ( nat > set_Pr1261947904930325089at_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__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__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re1392313733760654173_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( product_prod_nat_nat > nat > $o ) > ( product_prod_nat_nat > nat > $o ) > $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_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__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_re8250582710452816528on_val: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( product_prod_nat_nat > produc1457211279475724562t_char > prat ) > ( product_prod_nat_nat > produc1457211279475724562t_char > option_val ) > $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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $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_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
    bNF_re6631849262158050903_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > $o ) > $o ) > ( product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ) > ( product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ) > $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_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $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__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re3326851373202864989_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( set_Pr1261947904930325089at_nat > nat > $o ) > ( set_Pr1261947904930325089at_nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re1007418676577076503_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o ) > ( set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo_001_Eo,type,
    bNF_re2156183356565917981at_o_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( $o > $o > $o ) > ( set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > $o ) > $o ).

thf(sy_c_BNF__Def_OsndOp_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__Option__Ooption_It__Mask__Oval_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_sn3802693149471413141on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > produc1757238238866670357on_val ).

thf(sy_c_BNF__Def_OsndOp_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_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    bNF_sn6607638108379713230on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_BNF__Def_OsndOp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_sn8916957246138178918at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_BNF__Def_OsndOp_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,type,
    bNF_sn5923954573255187405at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc859450856879609959at_nat > produc859450856879609959at_nat ).

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    bNF_sn1857951156667551021at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > produc3843707927480180839at_nat > produc3843707927480180839at_nat ).

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    bNF_vi2254905643487785511_val_o: ( ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ) > ( ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

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    bNF_vi2516110938607903234_nat_o: ( ( produc1457211279475724562t_char > prat ) > nat ) > ( ( produc1457211279475724562t_char > option_val ) > nat ) > ( nat > nat > $o ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

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    bNF_vi6495413567062537508_val_o: ( nat > produc1457211279475724562t_char > prat ) > ( nat > produc1457211279475724562t_char > option_val ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > nat > nat > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    bNF_vi6667456707345531737_nat_o: ( nat > nat ) > ( nat > nat ) > ( nat > nat > $o ) > nat > nat > $o ).

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    bNF_Gr3695791879626193176at_nat: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > nat ) > ( produc9003781588162623004on_val > nat ) > set_Pr1261947904930325089at_nat ).

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thf(sy_c_PartialHeapSA_Ounit,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4971006070143632144at_nat: produc5602960702702913767at_nat > set_Pr1386498806471529671at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
    member3480088626692287306at_nat: produc2087150632810049441at_nat > set_Pr5070339090747523585at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6233145392104960784at_nat: produc6865100024664242407at_nat > set_Pr5975235057897686215at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member1953685042357896106at_nat: produc560747048475658241at_nat > set_Pr1868028757359454817at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6009668390874684176at_nat: produc1906917888121415911at_nat > set_Pr5564308138774400199at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member8215962448328497936at_nat: produc1429535001352425191at_nat > set_Pr1160274906116719815at_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6273874620674182256at_nat: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat > $o ).

thf(sy_v_x,type,
    x: state ).

% Relevant facts (1203)
thf(fact_0_valid__state_Ocases,axiom,
    ! [X: produc9003781588162623004on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( X
         != ( produc74620467203698198on_val @ Pi @ H ) ) ).

% valid_state.cases
thf(fact_1_Rep__state__inject,axiom,
    ! [X: state,Y: state] :
      ( ( ( rep_state @ X )
        = ( rep_state @ Y ) )
      = ( X = Y ) ) ).

% Rep_state_inject
thf(fact_2_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ X1 @ X2 )
        = ( produc9060074326276436823at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X2 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_4_prod_Oinject,axiom,
    ! [X1: nat,X2: list_char,Y1: nat,Y2: list_char] :
      ( ( ( produc120197956887798346t_char @ X1 @ X2 )
        = ( produc120197956887798346t_char @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_5_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_6_prod_Oinject,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X2: produc1457211279475724562t_char > option_val,Y1: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ X1 @ X2 )
        = ( produc74620467203698198on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_7_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_9_old_Oprod_Oinject,axiom,
    ! [A: nat,B: list_char,A2: nat,B2: list_char] :
      ( ( ( produc120197956887798346t_char @ A @ B )
        = ( produc120197956887798346t_char @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_11_old_Oprod_Oinject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_12_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1319942482725812455at_nat] :
      ~ ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
          ( Y
         != ( produc9060074326276436823at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_13_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_14_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1457211279475724562t_char] :
      ~ ! [A3: nat,B3: list_char] :
          ( Y
         != ( produc120197956887798346t_char @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_15_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_16_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ~ ! [A3: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc74620467203698198on_val @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_17_surj__pair,axiom,
    ! [P: produc1319942482725812455at_nat] :
    ? [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P
      = ( produc9060074326276436823at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_18_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_19_surj__pair,axiom,
    ! [P: produc1457211279475724562t_char] :
    ? [X3: nat,Y3: list_char] :
      ( P
      = ( produc120197956887798346t_char @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_20_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_21_surj__pair,axiom,
    ! [P: produc9003781588162623004on_val] :
    ? [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
      ( P
      = ( produc74620467203698198on_val @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_22_prod__cases,axiom,
    ! [P2: produc1319942482725812455at_nat > $o,P: produc1319942482725812455at_nat] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( P2 @ ( produc9060074326276436823at_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_23_prod__cases,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_24_prod__cases,axiom,
    ! [P2: produc1457211279475724562t_char > $o,P: produc1457211279475724562t_char] :
      ( ! [A3: nat,B3: list_char] : ( P2 @ ( produc120197956887798346t_char @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_25_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_26_prod__cases,axiom,
    ! [P2: produc9003781588162623004on_val > $o,P: produc9003781588162623004on_val] :
      ( ! [A3: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] : ( P2 @ ( produc74620467203698198on_val @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_27_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_28_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_29_Pair__inject,axiom,
    ! [A: nat,B: list_char,A2: nat,B2: list_char] :
      ( ( ( produc120197956887798346t_char @ A @ B )
        = ( produc120197956887798346t_char @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_30_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_31_Pair__inject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_32_add__states_Ocases,axiom,
    ! [X: produc1317171644805857433on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
          ( X
         != ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi2 @ H2 ) ) ) ).

% add_states.cases
thf(fact_33_ssubst__Pair__rhs,axiom,
    ! [R: product_prod_nat_nat,S: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_34_ssubst__Pair__rhs,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_35_ssubst__Pair__rhs,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_36_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: list_char,R2: set_Pr2507339680178222706t_char,S2: list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_37_ssubst__Pair__rhs,axiom,
    ! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_38_ssubst__Pair__rhs,axiom,
    ! [R: produc1457211279475724562t_char > prat,S: produc1457211279475724562t_char > option_val,R2: set_Pr7649655576376376530on_val,S2: produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_39_get__m_Osimps,axiom,
    ( get_m
    = ( ^ [A4: state] : ( produc5239619271231976746on_val @ ( rep_state @ A4 ) ) ) ) ).

% get_m.simps
thf(fact_40_get__m_Oelims,axiom,
    ! [X: state,Y: produc1457211279475724562t_char > prat] :
      ( ( ( get_m @ X )
        = Y )
     => ( Y
        = ( produc5239619271231976746on_val @ ( rep_state @ X ) ) ) ) ).

% get_m.elims
thf(fact_41_curryI,axiom,
    ! [F: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( F @ ( produc9060074326276436823at_nat @ A @ B ) )
     => ( produc447603880154497357_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_42_curryI,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( produc7820828192191229133_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_43_curryI,axiom,
    ! [F: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( F @ ( produc120197956887798346t_char @ A @ B ) )
     => ( produc80901228256969370char_o @ F @ A @ B ) ) ).

% curryI
thf(fact_44_curryI,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( produc1310100445399344235_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_45_curryI,axiom,
    ! [F: produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( F @ ( produc74620467203698198on_val @ A @ B ) )
     => ( produc9184331568460336180_val_o @ F @ A @ B ) ) ).

% curryI
thf(fact_46_get__h_Osimps,axiom,
    ( get_h
    = ( ^ [A4: state] : ( produc2319605628138516840on_val @ ( rep_state @ A4 ) ) ) ) ).

% get_h.simps
thf(fact_47_get__h_Oelims,axiom,
    ! [X: state,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( get_h @ X )
        = Y )
     => ( Y
        = ( produc2319605628138516840on_val @ ( rep_state @ X ) ) ) ) ).

% get_h.elims
thf(fact_48_Rep__state__inverse,axiom,
    ! [X: state] :
      ( ( abs_state @ ( rep_state @ X ) )
      = X ) ).

% Rep_state_inverse
thf(fact_49_swap__simp,axiom,
    ! [X: list_char,Y: nat] :
      ( ( produc4410195085741832554ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) )
      = ( produc120197956887798346t_char @ Y @ X ) ) ).

% swap_simp
thf(fact_50_swap__simp,axiom,
    ! [X: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > prat] :
      ( ( produc5819413633627552064r_prat @ ( produc1897429902272487136r_prat @ X @ Y ) )
      = ( produc74620467203698198on_val @ Y @ X ) ) ).

% swap_simp
thf(fact_51_swap__simp,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc5396952528824381687at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_52_swap__simp,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc3946833401584108663at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_53_swap__simp,axiom,
    ! [X: nat,Y: list_char] :
      ( ( produc5051419219066781674t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc8702345860417625034ar_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_54_swap__simp,axiom,
    ! [X: nat,Y: nat] :
      ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_55_swap__simp,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( produc3996604198558763126on_val @ ( produc74620467203698198on_val @ X @ Y ) )
      = ( produc1897429902272487136r_prat @ Y @ X ) ) ).

% swap_simp
thf(fact_56_core__def,axiom,
    ( core
    = ( ^ [Phi: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi ) ) ) ) ) ).

% core_def
thf(fact_57_prod_Ocollapse,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_58_prod_Ocollapse,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_59_prod_Ocollapse,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_60_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_61_prod_Ocollapse,axiom,
    ! [Prod: produc9003781588162623004on_val] :
      ( ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_62_fst__swap,axiom,
    ! [X: produc7776263825685906734r_prat] :
      ( ( produc5239619271231976746on_val @ ( produc5819413633627552064r_prat @ X ) )
      = ( produc4142415063207305778r_prat @ X ) ) ).

% fst_swap
thf(fact_63_fst__swap,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ( produc7062428706300765684r_prat @ ( produc3996604198558763126on_val @ X ) )
      = ( produc2319605628138516840on_val @ X ) ) ).

% fst_swap
thf(fact_64_snd__swap,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ( produc4142415063207305778r_prat @ ( produc3996604198558763126on_val @ X ) )
      = ( produc5239619271231976746on_val @ X ) ) ).

% snd_swap
thf(fact_65_snd__swap,axiom,
    ! [X: produc7776263825685906734r_prat] :
      ( ( produc2319605628138516840on_val @ ( produc5819413633627552064r_prat @ X ) )
      = ( produc7062428706300765684r_prat @ X ) ) ).

% snd_swap
thf(fact_66_prod_Oexpand,axiom,
    ! [Prod: produc9003781588162623004on_val,Prod2: produc9003781588162623004on_val] :
      ( ( ( ( produc5239619271231976746on_val @ Prod )
          = ( produc5239619271231976746on_val @ Prod2 ) )
        & ( ( produc2319605628138516840on_val @ Prod )
          = ( produc2319605628138516840on_val @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_67_prod_Oswap__def,axiom,
    ( produc5396952528824381687at_nat
    = ( ^ [P3: produc1319942482725812455at_nat] : ( produc9060074326276436823at_nat @ ( produc826222791318859909at_nat @ P3 ) @ ( produc4486455499028916547at_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_68_prod_Oswap__def,axiom,
    ( produc3946833401584108663at_nat
    = ( ^ [P3: produc3843707927480180839at_nat] : ( produc2922128104949294807at_nat @ ( produc2937545259423399429at_nat @ P3 ) @ ( produc6011863565407846595at_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_69_prod_Oswap__def,axiom,
    ( produc4410195085741832554ar_nat
    = ( ^ [P3: produc7606469099237612818ar_nat] : ( produc120197956887798346t_char @ ( produc1259554233722553080ar_nat @ P3 ) @ ( produc5563429021163328950ar_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_70_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P3: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P3 ) @ ( product_fst_nat_nat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_71_prod_Oswap__def,axiom,
    ( produc5819413633627552064r_prat
    = ( ^ [P3: produc7776263825685906734r_prat] : ( produc74620467203698198on_val @ ( produc4142415063207305778r_prat @ P3 ) @ ( produc7062428706300765684r_prat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_72_prod_Oswap__def,axiom,
    ( produc3996604198558763126on_val
    = ( ^ [P3: produc9003781588162623004on_val] : ( produc1897429902272487136r_prat @ ( produc2319605628138516840on_val @ P3 ) @ ( produc5239619271231976746on_val @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_73_prod__eqI,axiom,
    ! [P: produc9003781588162623004on_val,Q: produc9003781588162623004on_val] :
      ( ( ( produc5239619271231976746on_val @ P )
        = ( produc5239619271231976746on_val @ Q ) )
     => ( ( ( produc2319605628138516840on_val @ P )
          = ( produc2319605628138516840on_val @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_74_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] : ( Y4 = Z ) )
    = ( ^ [S3: produc9003781588162623004on_val,T: produc9003781588162623004on_val] :
          ( ( ( produc5239619271231976746on_val @ S3 )
            = ( produc5239619271231976746on_val @ T ) )
          & ( ( produc2319605628138516840on_val @ S3 )
            = ( produc2319605628138516840on_val @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_75_surjective__pairing,axiom,
    ! [T2: produc1319942482725812455at_nat] :
      ( T2
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ T2 ) @ ( produc826222791318859909at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_76_surjective__pairing,axiom,
    ! [T2: produc3843707927480180839at_nat] :
      ( T2
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ T2 ) @ ( produc2937545259423399429at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_77_surjective__pairing,axiom,
    ! [T2: produc1457211279475724562t_char] :
      ( T2
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ T2 ) @ ( produc1900778367047502200t_char @ T2 ) ) ) ).

% surjective_pairing
thf(fact_78_surjective__pairing,axiom,
    ! [T2: product_prod_nat_nat] :
      ( T2
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_79_surjective__pairing,axiom,
    ! [T2: produc9003781588162623004on_val] :
      ( T2
      = ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ T2 ) @ ( produc2319605628138516840on_val @ T2 ) ) ) ).

% surjective_pairing
thf(fact_80_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( Prod
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_81_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( Prod
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_82_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( Prod
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_83_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_84_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9003781588162623004on_val] :
      ( Prod
      = ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_85_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
    ! [A: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ A @ ( collec7475294502966882545on_val @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_88_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_89_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_90_Collect__mem__eq,axiom,
    ! [A5: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
    ! [A5: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [X4: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_93_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_94_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_95_Collect__cong,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: produc9003781588162623004on_val > $o] :
      ( ! [X3: produc9003781588162623004on_val] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec7475294502966882545on_val @ P2 )
        = ( collec7475294502966882545on_val @ Q2 ) ) ) ).

% Collect_cong
thf(fact_96_Collect__cong,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P2 )
        = ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_97_Collect__cong,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_98_snd__eqD,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_99_snd__eqD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_100_snd__eqD,axiom,
    ! [X: nat,Y: list_char,A: list_char] :
      ( ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_101_snd__eqD,axiom,
    ! [X: nat,Y: nat,A: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_102_snd__eqD,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > option_val] :
      ( ( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_103_snd__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_104_snd__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_105_snd__conv,axiom,
    ! [X1: nat,X2: list_char] :
      ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_106_snd__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_107_snd__conv,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X2: produc1457211279475724562t_char > option_val] :
      ( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_108_fst__eqD,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_109_fst__eqD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_110_fst__eqD,axiom,
    ! [X: nat,Y: list_char,A: nat] :
      ( ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_111_fst__eqD,axiom,
    ! [X: nat,Y: nat,A: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_112_fst__eqD,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > prat] :
      ( ( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_113_fst__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
      ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_114_fst__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
      ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_115_fst__conv,axiom,
    ! [X1: nat,X2: list_char] :
      ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_116_fst__conv,axiom,
    ! [X1: nat,X2: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_117_fst__conv,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X2: produc1457211279475724562t_char > option_val] :
      ( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_118_curryD,axiom,
    ! [F: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc447603880154497357_nat_o @ F @ A @ B )
     => ( F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_119_curryD,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F @ A @ B )
     => ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_120_curryD,axiom,
    ! [F: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( produc80901228256969370char_o @ F @ A @ B )
     => ( F @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% curryD
thf(fact_121_curryD,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryD
thf(fact_122_curryD,axiom,
    ! [F: produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc9184331568460336180_val_o @ F @ A @ B )
     => ( F @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% curryD
thf(fact_123_curryE,axiom,
    ! [F: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc447603880154497357_nat_o @ F @ A @ B )
     => ( F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_124_curryE,axiom,
    ! [F: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F @ A @ B )
     => ( F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_125_curryE,axiom,
    ! [F: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( produc80901228256969370char_o @ F @ A @ B )
     => ( F @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% curryE
thf(fact_126_curryE,axiom,
    ! [F: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A @ B )
     => ( F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryE
thf(fact_127_curryE,axiom,
    ! [F: produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc9184331568460336180_val_o @ F @ A @ B )
     => ( F @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% curryE
thf(fact_128_exI__realizer,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_129_exI__realizer,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_130_exI__realizer,axiom,
    ! [P2: list_char > nat > $o,Y: list_char,X: nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_131_exI__realizer,axiom,
    ! [P2: nat > nat > $o,Y: nat,X: nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_132_exI__realizer,axiom,
    ! [P2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > $o,Y: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char > prat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X @ Y ) ) @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_133_conjI__realizer,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > $o,P: set_Pr4329608150637261639at_nat,Q2: set_Pr4329608150637261639at_nat > $o,Q: set_Pr4329608150637261639at_nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_134_conjI__realizer,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_135_conjI__realizer,axiom,
    ! [P2: nat > $o,P: nat,Q2: list_char > $o,Q: list_char] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ P @ Q ) ) )
          & ( Q2 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_136_conjI__realizer,axiom,
    ! [P2: nat > $o,P: nat,Q2: nat > $o,Q: nat] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_137_conjI__realizer,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,P: produc1457211279475724562t_char > prat,Q2: ( produc1457211279475724562t_char > option_val ) > $o,Q: produc1457211279475724562t_char > option_val] :
      ( ( P2 @ P )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ P @ Q ) ) )
          & ( Q2 @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_138_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc9060074326276436823at_nat @ X @ Y ) )
       => ( P2 @ ( produc4486455499028916547at_nat @ A ) @ ( produc826222791318859909at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_139_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2922128104949294807at_nat @ X @ Y ) )
       => ( P2 @ ( produc6011863565407846595at_nat @ A ) @ ( produc2937545259423399429at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_140_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_141_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_142_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val,A: produc9003781588162623004on_val] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc74620467203698198on_val @ X @ Y ) )
       => ( P2 @ ( produc5239619271231976746on_val @ A ) @ ( produc2319605628138516840on_val @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_143_unit__def,axiom,
    ( unit
    = ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).

% unit_def
thf(fact_144_exE__realizer_H,axiom,
    ! [P2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > $o,P: produc9003781588162623004on_val] :
      ( ( P2 @ ( produc2319605628138516840on_val @ P ) @ ( produc5239619271231976746on_val @ P ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ~ ( P2 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_145_sndI,axiom,
    ! [X: produc1319942482725812455at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( X
        = ( produc9060074326276436823at_nat @ Y @ Z2 ) )
     => ( ( produc826222791318859909at_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_146_sndI,axiom,
    ! [X: produc3843707927480180839at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc2922128104949294807at_nat @ Y @ Z2 ) )
     => ( ( produc2937545259423399429at_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_147_sndI,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat,Z2: list_char] :
      ( ( X
        = ( produc120197956887798346t_char @ Y @ Z2 ) )
     => ( ( produc1900778367047502200t_char @ X )
        = Z2 ) ) ).

% sndI
thf(fact_148_sndI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z2: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z2 ) )
     => ( ( product_snd_nat_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_149_sndI,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > option_val] :
      ( ( X
        = ( produc74620467203698198on_val @ Y @ Z2 ) )
     => ( ( produc2319605628138516840on_val @ X )
        = Z2 ) ) ).

% sndI
thf(fact_150_eq__snd__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
      ( ( B
        = ( produc826222791318859909at_nat @ P ) )
      = ( ? [A4: set_Pr4329608150637261639at_nat] :
            ( P
            = ( produc9060074326276436823at_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_151_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( B
        = ( produc2937545259423399429at_nat @ P ) )
      = ( ? [A4: set_Pr1261947904930325089at_nat] :
            ( P
            = ( produc2922128104949294807at_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_152_eq__snd__iff,axiom,
    ! [B: list_char,P: produc1457211279475724562t_char] :
      ( ( B
        = ( produc1900778367047502200t_char @ P ) )
      = ( ? [A4: nat] :
            ( P
            = ( produc120197956887798346t_char @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_153_eq__snd__iff,axiom,
    ! [B: nat,P: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P ) )
      = ( ? [A4: nat] :
            ( P
            = ( product_Pair_nat_nat @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_154_eq__snd__iff,axiom,
    ! [B: produc1457211279475724562t_char > option_val,P: produc9003781588162623004on_val] :
      ( ( B
        = ( produc2319605628138516840on_val @ P ) )
      = ( ? [A4: produc1457211279475724562t_char > prat] :
            ( P
            = ( produc74620467203698198on_val @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_155_fstI,axiom,
    ! [X: produc1319942482725812455at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( X
        = ( produc9060074326276436823at_nat @ Y @ Z2 ) )
     => ( ( produc4486455499028916547at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_156_fstI,axiom,
    ! [X: produc3843707927480180839at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc2922128104949294807at_nat @ Y @ Z2 ) )
     => ( ( produc6011863565407846595at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_157_fstI,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat,Z2: list_char] :
      ( ( X
        = ( produc120197956887798346t_char @ Y @ Z2 ) )
     => ( ( produc6204653154488278070t_char @ X )
        = Y ) ) ).

% fstI
thf(fact_158_fstI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z2: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z2 ) )
     => ( ( product_fst_nat_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_159_fstI,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > option_val] :
      ( ( X
        = ( produc74620467203698198on_val @ Y @ Z2 ) )
     => ( ( produc5239619271231976746on_val @ X )
        = Y ) ) ).

% fstI
thf(fact_160_eq__fst__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
      ( ( A
        = ( produc4486455499028916547at_nat @ P ) )
      = ( ? [B4: set_Pr4329608150637261639at_nat] :
            ( P
            = ( produc9060074326276436823at_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_161_eq__fst__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( A
        = ( produc6011863565407846595at_nat @ P ) )
      = ( ? [B4: set_Pr1261947904930325089at_nat] :
            ( P
            = ( produc2922128104949294807at_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_162_eq__fst__iff,axiom,
    ! [A: nat,P: produc1457211279475724562t_char] :
      ( ( A
        = ( produc6204653154488278070t_char @ P ) )
      = ( ? [B4: list_char] :
            ( P
            = ( produc120197956887798346t_char @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_163_eq__fst__iff,axiom,
    ! [A: nat,P: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P ) )
      = ( ? [B4: nat] :
            ( P
            = ( product_Pair_nat_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_164_eq__fst__iff,axiom,
    ! [A: produc1457211279475724562t_char > prat,P: produc9003781588162623004on_val] :
      ( ( A
        = ( produc5239619271231976746on_val @ P ) )
      = ( ? [B4: produc1457211279475724562t_char > option_val] :
            ( P
            = ( produc74620467203698198on_val @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_165_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y5: list_char,Y: list_char,R2: nat > set_Pr4480099295215233283t_char] :
      ( ( P2 @ X )
     => ( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y5 @ Y ) @ ( R2 @ X ) )
       => ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ X @ Y5 ) @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( same_f652031872680891423t_char @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_166_same__fstI,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val,R2: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
      ( ( P2 @ X )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y5 @ Y ) @ ( R2 @ X ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X @ Y5 ) @ ( produc74620467203698198on_val @ X @ Y ) ) @ ( same_f8138205353876899841on_val @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_167_same__fstI,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat > set_Pr7459493094073627847at_nat] :
      ( ( P2 @ X )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ Y ) @ ( R2 @ X ) )
       => ( member8215962448328497936at_nat @ ( produc2220582066590712663at_nat @ ( produc9060074326276436823at_nat @ X @ Y5 ) @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( same_f4504843259822650924at_nat @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_168_same__fstI,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Y ) @ ( R2 @ X ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X @ Y5 ) @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( same_f1460074399991292076at_nat @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_169_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y5: nat,Y: nat,R2: nat > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Y ) @ ( R2 @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_170_pre__get__h_Oelims,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( pre_ge1486375753531218896on_val @ X )
        = Y )
     => ( Y
        = ( produc2319605628138516840on_val @ X ) ) ) ).

% pre_get_h.elims
thf(fact_171_pre__get__h_Osimps,axiom,
    pre_ge1486375753531218896on_val = produc2319605628138516840on_val ).

% pre_get_h.simps
thf(fact_172_valid__pre__unit,axiom,
    valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).

% valid_pre_unit
thf(fact_173_pre__get__m_Osimps,axiom,
    pre_ge8481574585653477131on_val = produc5239619271231976746on_val ).

% pre_get_m.simps
thf(fact_174_pre__get__m_Oelims,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat] :
      ( ( ( pre_ge8481574585653477131on_val @ X )
        = Y )
     => ( Y
        = ( produc5239619271231976746on_val @ X ) ) ) ).

% pre_get_m.elims
thf(fact_175_get__m_Opelims,axiom,
    ! [X: state,Y: produc1457211279475724562t_char > prat] :
      ( ( ( get_m @ X )
        = Y )
     => ( ( accp_state @ get_m_rel @ X )
       => ~ ( ( Y
              = ( produc5239619271231976746on_val @ ( rep_state @ X ) ) )
           => ~ ( accp_state @ get_m_rel @ X ) ) ) ) ).

% get_m.pelims
thf(fact_176_sndOp__def,axiom,
    ( bNF_sn6607638108379713230on_val
    = ( ^ [P4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Ac: produc9003781588162623004on_val] : ( produc74620467203698198on_val @ ( bNF_pi5559183403598038974on_val @ P4 @ Q3 @ ( produc5239619271231976746on_val @ Ac ) @ ( produc2319605628138516840on_val @ Ac ) ) @ ( produc2319605628138516840on_val @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_177_fstOp__def,axiom,
    ( bNF_fs7489134050367444823on_val
    = ( ^ [P4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q3: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,Ac: produc9003781588162623004on_val] : ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Ac ) @ ( bNF_pi6941616487637503653on_val @ P4 @ Q3 @ ( produc5239619271231976746on_val @ Ac ) @ ( produc2319605628138516840on_val @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_178_get__h_Opelims,axiom,
    ! [X: state,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( get_h @ X )
        = Y )
     => ( ( accp_state @ get_h_rel @ X )
       => ~ ( ( Y
              = ( produc2319605628138516840on_val @ ( rep_state @ X ) ) )
           => ~ ( accp_state @ get_h_rel @ X ) ) ) ) ).

% get_h.pelims
thf(fact_179_image2__eqI,axiom,
    ! [B: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,C: nat,G: product_prod_nat_nat > nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr8954129895687480649at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_180_image2__eqI,axiom,
    ! [B: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,C: list_char,G: product_prod_nat_nat > list_char,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr3738443398878377306t_char @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_181_image2__eqI,axiom,
    ! [B: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,C: product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ ( bNF_Gr8808630244184418791at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_182_image2__eqI,axiom,
    ! [B: nat,F: produc859450856879609959at_nat > nat,X: produc859450856879609959at_nat,C: nat,G: produc859450856879609959at_nat > nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr97278709799963567at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_183_image2__eqI,axiom,
    ! [B: nat,F: produc859450856879609959at_nat > nat,X: produc859450856879609959at_nat,C: list_char,G: produc859450856879609959at_nat > list_char,A5: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A5 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr197193474242742976t_char @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_184_image2__eqI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( bNF_Gr2618164000252033511at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_185_image2__eqI,axiom,
    ! [B: nat,F: produc3843707927480180839at_nat > nat,X: produc3843707927480180839at_nat,C: nat,G: produc3843707927480180839at_nat > nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8757157785044589968at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr1074675703407240111at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_186_image2__eqI,axiom,
    ! [B: nat,F: produc3843707927480180839at_nat > nat,X: produc3843707927480180839at_nat,C: list_char,G: produc3843707927480180839at_nat > list_char,A5: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8757157785044589968at_nat @ X @ A5 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr3647166678560580288t_char @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_187_image2__eqI,axiom,
    ! [B: product_prod_nat_nat,F: produc859450856879609959at_nat > product_prod_nat_nat,X: produc859450856879609959at_nat,C: product_prod_nat_nat,G: produc859450856879609959at_nat > product_prod_nat_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ ( bNF_Gr978357984662150989at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_188_image2__eqI,axiom,
    ! [B: product_prod_nat_nat,F: produc3843707927480180839at_nat > product_prod_nat_nat,X: produc3843707927480180839at_nat,C: product_prod_nat_nat,G: produc3843707927480180839at_nat > product_prod_nat_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8757157785044589968at_nat @ X @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ ( bNF_Gr253540273889441613at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_189_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_190_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_191_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_192_curry__case__prod,axiom,
    ! [F: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_193_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc8197505143624133779at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_194_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat,P: produc1457211279475724562t_char] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc3665563422017311042at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_195_mem__case__prodE,axiom,
    ! [Z2: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,P: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( produc2423807293333670521at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_196_mem__case__prodE,axiom,
    ! [Z2: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat,P: produc1457211279475724562t_char] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( produc3763337624047832232at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_197_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc1433696742876942453at_nat @ C @ P ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_198_mem__case__prodE,axiom,
    ! [Z2: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,P: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( produc3705604114177432185at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_199_mem__case__prodE,axiom,
    ! [Z2: produc3843707927480180839at_nat,C: nat > list_char > set_Pr4329608150637261639at_nat,P: produc1457211279475724562t_char] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( produc7856237077994235048at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_200_mem__case__prodE,axiom,
    ! [Z2: produc859450856879609959at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat,P: produc3843707927480180839at_nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( produc275446641266004827at_nat @ C @ P ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_201_mem__case__prodE,axiom,
    ! [Z2: produc9003781588162623004on_val,C: nat > nat > set_Pr7649655576376376530on_val,P: product_prod_nat_nat] :
      ( ( member2879030624002546611on_val @ Z2 @ ( produc9171785124108960544on_val @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member2879030624002546611on_val @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_202_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat,P: produc9003781588162623004on_val] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc3420870819319363678at_nat @ C @ P ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P
              = ( produc74620467203698198on_val @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_203_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X2: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_204_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc859450856879609959at_nat,A5: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ X @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ A5 ) ) )
     => ( A5 @ ( produc3213797794245857475at_nat @ X ) @ ( produc6408287024330202629at_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_205_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc3843707927480180839at_nat,A5: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ A5 ) ) )
     => ( A5 @ ( produc6011863565407846595at_nat @ X ) @ ( produc2937545259423399429at_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_206_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc9003781588162623004on_val,A5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member2879030624002546611on_val @ X @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ A5 ) ) )
     => ( A5 @ ( produc5239619271231976746on_val @ X ) @ ( produc2319605628138516840on_val @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_207_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_nat_nat,A5: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A5 ) ) )
     => ( A5 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_208_split__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,Prod3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_209_case__prod__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,P3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_210_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_211_split__cong,axiom,
    ! [Q: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q )
       => ( ( produc6081775807080527818_nat_o @ F @ P )
          = ( produc6081775807080527818_nat_o @ G @ Q ) ) ) ) ).

% split_cong
thf(fact_212_csquare__fstOp__sndOp,axiom,
    ! [F: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > $o,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_cs2794013118909459695on_val @ ( collec7475294502966882545on_val @ ( F @ ( relcom8684624306960484107on_val @ P2 @ Q2 ) ) ) @ produc5358294431189718917r_prat @ produc5239619271231976746on_val @ ( bNF_fs7282908041383899532on_val @ P2 @ Q2 ) @ ( bNF_sn6607638108379713230on_val @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_213_csquare__fstOp__sndOp,axiom,
    ! [F: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > $o,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_cs5223968648578849398on_val @ ( collec7475294502966882545on_val @ ( F @ ( relcom7279225663578213656on_val @ P2 @ Q2 ) ) ) @ produc2319605628138516840on_val @ produc6116476718443979321on_val @ ( bNF_fs7489134050367444823on_val @ P2 @ Q2 ) @ ( bNF_sn3802693149471413141on_val @ P2 @ Q2 ) ) ).

% csquare_fstOp_sndOp
thf(fact_214_accp__induct__rule,axiom,
    ! [R: state > state > $o,A: state,P2: state > $o] :
      ( ( accp_state @ R @ A )
     => ( ! [X3: state] :
            ( ( accp_state @ R @ X3 )
           => ( ! [Y6: state] :
                  ( ( R @ Y6 @ X3 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_215_not__accp__down,axiom,
    ! [R2: state > state > $o,X: state] :
      ( ~ ( accp_state @ R2 @ X )
     => ~ ! [Z3: state] :
            ( ( R2 @ Z3 @ X )
           => ( accp_state @ R2 @ Z3 ) ) ) ).

% not_accp_down
thf(fact_216_accp__downward,axiom,
    ! [R: state > state > $o,B: state,A: state] :
      ( ( accp_state @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_state @ R @ A ) ) ) ).

% accp_downward
thf(fact_217_accp__induct,axiom,
    ! [R: state > state > $o,A: state,P2: state > $o] :
      ( ( accp_state @ R @ A )
     => ( ! [X3: state] :
            ( ( accp_state @ R @ X3 )
           => ( ! [Y6: state] :
                  ( ( R @ Y6 @ X3 )
                 => ( P2 @ Y6 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_218_accpI,axiom,
    ! [R: state > state > $o,X: state] :
      ( ! [Y3: state] :
          ( ( R @ Y3 @ X )
         => ( accp_state @ R @ Y3 ) )
     => ( accp_state @ R @ X ) ) ).

% accpI
thf(fact_219_accp_Osimps,axiom,
    ( accp_state
    = ( ^ [R3: state > state > $o,A4: state] :
        ? [X4: state] :
          ( ( A4 = X4 )
          & ! [Y7: state] :
              ( ( R3 @ Y7 @ X4 )
             => ( accp_state @ R3 @ Y7 ) ) ) ) ) ).

% accp.simps
thf(fact_220_accp_Ocases,axiom,
    ! [R: state > state > $o,A: state] :
      ( ( accp_state @ R @ A )
     => ! [Y6: state] :
          ( ( R @ Y6 @ A )
         => ( accp_state @ R @ Y6 ) ) ) ).

% accp.cases
thf(fact_221_sndOp__in,axiom,
    ! [Ac2: produc859450856879609959at_nat,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ Ac2 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( relcom282571730187242960at_nat @ P2 @ Q2 ) ) ) )
     => ( member8206827879206165904at_nat @ ( bNF_sn5923954573255187405at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_222_sndOp__in,axiom,
    ! [Ac2: produc3843707927480180839at_nat,P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Q2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ Ac2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( relcom8363394528208558896at_nat @ P2 @ Q2 ) ) ) )
     => ( member8757157785044589968at_nat @ ( bNF_sn1857951156667551021at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_223_sndOp__in,axiom,
    ! [Ac2: produc9003781588162623004on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member2879030624002546611on_val @ Ac2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( relcom8684624306960484107on_val @ P2 @ Q2 ) ) ) )
     => ( member2879030624002546611on_val @ ( bNF_sn6607638108379713230on_val @ P2 @ Q2 @ Ac2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_224_sndOp__in,axiom,
    ! [Ac2: product_prod_nat_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q2 ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_sn8916957246138178918at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q2 ) ) ) ) ).

% sndOp_in
thf(fact_225_fstOp__in,axiom,
    ! [Ac2: produc859450856879609959at_nat,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ Ac2 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( relcom282571730187242960at_nat @ P2 @ Q2 ) ) ) )
     => ( member8206827879206165904at_nat @ ( bNF_fs8257191039976258191at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_226_fstOp__in,axiom,
    ! [Ac2: produc3843707927480180839at_nat,P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Q2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ Ac2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( relcom8363394528208558896at_nat @ P2 @ Q2 ) ) ) )
     => ( member8757157785044589968at_nat @ ( bNF_fs1946732770515719151at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_227_fstOp__in,axiom,
    ! [Ac2: produc9003781588162623004on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member2879030624002546611on_val @ Ac2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( relcom7279225663578213656on_val @ P2 @ Q2 ) ) ) )
     => ( member2879030624002546611on_val @ ( bNF_fs7489134050367444823on_val @ P2 @ Q2 @ Ac2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_228_fstOp__in,axiom,
    ! [Ac2: product_prod_nat_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q2 ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_fs6799373675401686564at_nat @ P2 @ Q2 @ Ac2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_229_pre__get__h_Opelims,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > option_val] :
      ( ( ( pre_ge1486375753531218896on_val @ X )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ pre_ge6140436789258014353on_val @ X )
       => ~ ( ( Y
              = ( produc2319605628138516840on_val @ X ) )
           => ~ ( accp_P4405556307444930259on_val @ pre_ge6140436789258014353on_val @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_230_pre__get__m_Opelims,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat] :
      ( ( ( pre_ge8481574585653477131on_val @ X )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ pre_ge4462515927930180886on_val @ X )
       => ~ ( ( Y
              = ( produc5239619271231976746on_val @ X ) )
           => ~ ( accp_P4405556307444930259on_val @ pre_ge4462515927930180886on_val @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_231_in__lex__prod,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_232_in__lex__prod,axiom,
    ! [A: nat,B: list_char,A2: nat,B2: list_char,R: set_Pr1261947904930325089at_nat,S: set_Pr4480099295215233283t_char] :
      ( ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc120197956887798346t_char @ A2 @ B2 ) ) @ ( lex_pr1681056561245511266t_char @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_233_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: nat,A2: product_prod_nat_nat,B2: nat,R: set_Pr8693737435421807431at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A2 @ B2 ) ) @ ( lex_pr4669217892513370978at_nat @ R @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_234_in__lex__prod,axiom,
    ! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A2 @ B2 ) ) @ ( lex_pr8029265285556086080at_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_235_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: nat,A2: set_Pr1261947904930325089at_nat,B2: nat,R: set_Pr4329608150637261639at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member3480088626692287306at_nat @ ( produc3585720979125124569at_nat @ ( produc3149970401713881818at_nat @ A @ B ) @ ( produc3149970401713881818at_nat @ A2 @ B2 ) ) @ ( lex_pr1951919194703979714at_nat @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_236_in__lex__prod,axiom,
    ! [A: nat,B: set_Pr1261947904930325089at_nat,A2: nat,B2: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4107301959146914710at_nat @ ( produc8824474337961625893at_nat @ ( produc5463602008962177208at_nat @ A @ B ) @ ( produc5463602008962177208at_nat @ A2 @ B2 ) ) @ ( lex_pr4265550801952275104at_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_237_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A2 @ B2 ) ) @ ( lex_pr8801849515957261039at_nat @ R @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_238_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4971006070143632144at_nat @ ( produc3335200116822430551at_nat @ ( produc5966227314700062775at_nat @ A @ B ) @ ( produc5966227314700062775at_nat @ A2 @ B2 ) ) @ ( lex_pr864846108736269391at_nat @ R @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_239_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member6233145392104960784at_nat @ ( produc9001802813318374231at_nat @ ( produc4320700058201173367at_nat @ A @ B ) @ ( produc4320700058201173367at_nat @ A2 @ B2 ) ) @ ( lex_pr8442690889092155791at_nat @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_240_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ A2 @ B2 ) ) @ ( lex_pr8000699672184251119at_nat @ R @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A2 ) @ R )
        | ( ( A = A2 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_241_valid__state__decompose,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A @ B ) @ H3 ) )
     => ( valid_state @ ( produc74620467203698198on_val @ A @ H3 ) ) ) ).

% valid_state_decompose
thf(fact_242_pair__in__swap__image,axiom,
    ! [Y: product_prod_nat_nat,X: product_prod_nat_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( image_6321802852832981957at_nat @ produc7225117575323628663at_nat @ A5 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_243_pair__in__swap__image,axiom,
    ! [Y: list_char,X: nat,A5: set_Pr2507339680178222706t_char] :
      ( ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ Y @ X ) @ ( image_652198164289537669ar_nat @ produc5051419219066781674t_char @ A5 ) )
      = ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_244_pair__in__swap__image,axiom,
    ! [Y: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char > prat,A5: set_Pr7649655576376376530on_val] :
      ( ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ Y @ X ) @ ( image_2943372182239103057r_prat @ produc3996604198558763126on_val @ A5 ) )
      = ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_245_pair__in__swap__image,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,A5: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ ( image_3313167189156628421at_nat @ produc5396952528824381687at_nat @ A5 ) )
      = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_246_pair__in__swap__image,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ ( image_960689587819713477at_nat @ produc3946833401584108663at_nat @ A5 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_247_pair__in__swap__image,axiom,
    ! [Y: nat,X: list_char,A5: set_Pr8803067691650843250ar_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y @ X ) @ ( image_688197443715608197t_char @ produc4410195085741832554ar_nat @ A5 ) )
      = ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_248_pair__in__swap__image,axiom,
    ! [Y: nat,X: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A5 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_249_pair__in__swap__image,axiom,
    ! [Y: produc1457211279475724562t_char > prat,X: produc1457211279475724562t_char > option_val,A5: set_Pr2871993341207995108r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Y @ X ) @ ( image_1698057272686278225on_val @ produc5819413633627552064r_prat @ A5 ) )
      = ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_250_in__inv__image,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_251_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im2857804880773004637at_nat @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_252_in__inv__image,axiom,
    ! [X: nat,Y: nat,R: set_Pr8693737435421807431at_nat,F: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im8721129524585065343at_nat @ R @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_253_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im6750873321330482877at_nat @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_254_in__inv__image,axiom,
    ! [X: nat,Y: nat,R: set_Pr4329608150637261639at_nat,F: nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im4437241714082187487at_nat @ R @ F ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_255_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im5896302226278515346at_nat @ R @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_256_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr4329608150637261639at_nat,F: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im8562125120543730994at_nat @ R @ F ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_257_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,F: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im984280340187844594at_nat @ R @ F ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_258_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im7023619800666061458at_nat @ R @ F ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_259_in__inv__image,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R: set_Pr1261947904930325089at_nat,F: set_Pr4329608150637261639at_nat > nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( inv_im2354225701593503139at_nat @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_260_snd__sndOp,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( produc2319605628138516840on_val
      = ( comp_P291491979669821239on_val @ produc2319605628138516840on_val @ ( bNF_sn6607638108379713230on_val @ P2 @ Q2 ) ) ) ).

% snd_sndOp
thf(fact_261_fst__fstOp,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( produc5239619271231976746on_val
      = ( comp_P6028404540003424944on_val @ produc5239619271231976746on_val @ ( bNF_fs7489134050367444823on_val @ P2 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_262_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_263_empty__mask_Opelims,axiom,
    ! [X: state,Y: prat] :
      ( ( ( empty_mask_state @ X )
        = Y )
     => ( ( accp_state @ empty_mask_rel_state @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_state @ empty_mask_rel_state @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_264_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A5 @ F ) ) ) )
     => ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ Z2 ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_265_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc3843707927480180839at_nat,A5: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A5 @ F ) ) ) )
     => ( member2643936169264416010at_nat @ ( produc6011863565407846595at_nat @ Z2 ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_266_Collect__case__prod__Grp__in,axiom,
    ! [Z2: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ Z2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_Gr4383009045007556383on_val @ A5 @ F ) ) ) )
     => ( member2336342582916103932r_prat @ ( produc5239619271231976746on_val @ Z2 ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_267_Collect__case__prod__Grp__in,axiom,
    ! [Z2: product_prod_nat_nat,A5: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A5 @ F ) ) ) )
     => ( member_nat @ ( product_fst_nat_nat @ Z2 ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_268_add__masks__cancellative,axiom,
    ! [B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B @ C )
        = ( add_ma6586698345353345811t_char @ B @ D ) )
     => ( C = D ) ) ).

% add_masks_cancellative
thf(fact_269_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A4: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B4 @ A4 ) ) ) ).

% add_masks_comm
thf(fact_270_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_271_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A5 @ F ) ) ) )
     => ( ( comp_P3189659738270478212at_nat @ F @ produc3213797794245857475at_nat @ Z2 )
        = ( produc6408287024330202629at_nat @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_272_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc3843707927480180839at_nat,A5: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A5 @ F ) ) ) )
     => ( ( comp_s7355055891894327172at_nat @ F @ produc6011863565407846595at_nat @ Z2 )
        = ( produc2937545259423399429at_nat @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_273_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ Z2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_Gr4383009045007556383on_val @ A5 @ F ) ) ) )
     => ( ( comp_P123150374798284520on_val @ F @ produc5239619271231976746on_val @ Z2 )
        = ( produc2319605628138516840on_val @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_274_Collect__case__prod__Grp__eqD,axiom,
    ! [Z2: product_prod_nat_nat,A5: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A5 @ F ) ) ) )
     => ( ( comp_n2711240574518091068at_nat @ F @ product_fst_nat_nat @ Z2 )
        = ( product_snd_nat_nat @ Z2 ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_275_minus__empty,axiom,
    ! [Pi3: produc1457211279475724562t_char > prat] :
      ( Pi3
      = ( add_ma6586698345353345811t_char @ Pi3 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_276_empty__mask_Osimps,axiom,
    ( empty_3446695950879338768t_char
    = ( ^ [Hl: produc1457211279475724562t_char] : pnone ) ) ).

% empty_mask.simps
thf(fact_277_empty__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_278_snd__comp__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val] :
      ( ( comp_P291491979669821239on_val @ produc2319605628138516840on_val @ ( produc1735415053012026015r_prat @ F ) )
      = ( comp_P4718904269643193781on_val @ F @ produc2319605628138516840on_val ) ) ).

% snd_comp_apsnd
thf(fact_279_fst__comp__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
      ( ( comp_P6028404540003424944on_val @ produc5239619271231976746on_val @ ( produc8604563337118726038on_val @ F ) )
      = ( comp_P4687317905303122495on_val @ F @ produc5239619271231976746on_val ) ) ).

% fst_comp_apfst
thf(fact_280_Collect__split__mono__strong,axiom,
    ! [X5: set_Pr3433046674199179443r_prat,A5: set_Pr7649655576376376530on_val,Y8: set_Pr1004501353197068112on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( X5
        = ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ A5 ) )
     => ( ( Y8
          = ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ A5 ) )
       => ( ! [X3: produc1457211279475724562t_char > prat] :
              ( ( member2336342582916103932r_prat @ X3 @ X5 )
             => ! [Xa: produc1457211279475724562t_char > option_val] :
                  ( ( member1577473829074448177on_val @ Xa @ Y8 )
                 => ( ( P2 @ X3 @ Xa )
                   => ( Q2 @ X3 @ Xa ) ) ) )
         => ( ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P2 ) ) )
           => ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_281_Collect__split__mono__strong,axiom,
    ! [X5: set_nat,A5: set_Pr1261947904930325089at_nat,Y8: set_nat,P2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( X5
        = ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A5 ) )
     => ( ( Y8
          = ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A5 ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ X5 )
             => ! [Xa: nat] :
                  ( ( member_nat @ Xa @ Y8 )
                 => ( ( P2 @ X3 @ Xa )
                   => ( Q2 @ X3 @ Xa ) ) ) )
         => ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
           => ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_282_snd__comp__map__prod,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val] :
      ( ( comp_P291491979669821239on_val @ produc2319605628138516840on_val @ ( produc8229809728907007980on_val @ F @ G ) )
      = ( comp_P4718904269643193781on_val @ G @ produc2319605628138516840on_val ) ) ).

% snd_comp_map_prod
thf(fact_283_fst__comp__map__prod,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val] :
      ( ( comp_P6028404540003424944on_val @ produc5239619271231976746on_val @ ( produc8229809728907007980on_val @ F @ G ) )
      = ( comp_P4687317905303122495on_val @ F @ produc5239619271231976746on_val ) ) ).

% fst_comp_map_prod
thf(fact_284_fst__comp__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val] :
      ( ( comp_P6028404540003424944on_val @ produc5239619271231976746on_val @ ( produc1735415053012026015r_prat @ F ) )
      = produc5239619271231976746on_val ) ).

% fst_comp_apsnd
thf(fact_285_snd__comp__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat] :
      ( ( comp_P291491979669821239on_val @ produc2319605628138516840on_val @ ( produc8604563337118726038on_val @ F ) )
      = produc2319605628138516840on_val ) ).

% snd_comp_apfst
thf(fact_286_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_287_surj__swap,axiom,
    ( ( image_5168914502847457605at_nat @ product_swap_nat_nat @ top_to4669805908274784177at_nat )
    = top_to4669805908274784177at_nat ) ).

% surj_swap
thf(fact_288_map__prod__simp,axiom,
    ! [F: nat > nat,G: nat > nat,A: nat,B: nat] :
      ( ( produc6977886695330630970at_nat @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_289_map__prod__simp,axiom,
    ! [F: nat > nat,G: list_char > nat,A: nat,B: list_char] :
      ( ( produc3271292527955122507ar_nat @ F @ G @ ( produc120197956887798346t_char @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_290_map__prod__simp,axiom,
    ! [F: nat > nat,G: nat > list_char,A: nat,B: nat] :
      ( ( produc3912516661280071627t_char @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_291_map__prod__simp,axiom,
    ! [F: nat > nat,G: list_char > list_char,A: nat,B: list_char] :
      ( ( produc5890707981334396508t_char @ F @ G @ ( produc120197956887798346t_char @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_292_map__prod__simp,axiom,
    ! [F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7374836315209622838at_nat @ F @ G @ ( produc2922128104949294807at_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_293_map__prod__simp,axiom,
    ! [F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( produc7508474501381341110at_nat @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc2922128104949294807at_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_294_map__prod__simp,axiom,
    ! [F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > list_char,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc4151949830376967623t_char @ F @ G @ ( produc2922128104949294807at_nat @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_295_map__prod__simp,axiom,
    ! [F: nat > set_Pr1261947904930325089at_nat,G: list_char > set_Pr1261947904930325089at_nat,A: nat,B: list_char] :
      ( ( produc4168075969930867685at_nat @ F @ G @ ( produc120197956887798346t_char @ A @ B ) )
      = ( produc2922128104949294807at_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_296_map__prod__simp,axiom,
    ! [F: nat > produc1457211279475724562t_char > prat,G: nat > produc1457211279475724562t_char > option_val,A: nat,B: nat] :
      ( ( produc7620138996006934647on_val @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc74620467203698198on_val @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_297_map__prod__simp,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > nat,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc43609526583453465al_nat @ F @ G @ ( produc74620467203698198on_val @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_298_fst__map__prod,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
      ( ( produc5239619271231976746on_val @ ( produc8229809728907007980on_val @ F @ G @ X ) )
      = ( F @ ( produc5239619271231976746on_val @ X ) ) ) ).

% fst_map_prod
thf(fact_299_snd__map__prod,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
      ( ( produc2319605628138516840on_val @ ( produc8229809728907007980on_val @ F @ G @ X ) )
      = ( G @ ( produc2319605628138516840on_val @ X ) ) ) ).

% snd_map_prod
thf(fact_300_apfst__conv,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc6529295065807390923at_nat @ F @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_301_apfst__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc5380969780118646373at_nat @ F @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_302_apfst__conv,axiom,
    ! [F: nat > nat,X: nat,Y: list_char] :
      ( ( produc2835005367637779135t_char @ F @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc120197956887798346t_char @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_303_apfst__conv,axiom,
    ! [F: nat > nat,X: nat,Y: nat] :
      ( ( produc986720760941809198at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_304_apfst__conv,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( produc8604563337118726038on_val @ F @ ( produc74620467203698198on_val @ X @ Y ) )
      = ( produc74620467203698198on_val @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_305_apsnd__conv,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc8057050257526108681at_nat @ F @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_306_apsnd__conv,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc6468969725218530979at_nat @ F @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_307_apsnd__conv,axiom,
    ! [F: list_char > list_char,X: nat,Y: list_char] :
      ( ( produc5571979602231644306ar_nat @ F @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc120197956887798346t_char @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_308_apsnd__conv,axiom,
    ! [F: list_char > nat,X: nat,Y: list_char] :
      ( ( produc8286888221270194945at_nat @ F @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( product_Pair_nat_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_309_apsnd__conv,axiom,
    ! [F: nat > list_char,X: nat,Y: nat] :
      ( ( produc561246565970313601ar_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( produc120197956887798346t_char @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_310_apsnd__conv,axiom,
    ! [F: nat > nat,X: nat,Y: nat] :
      ( ( produc3094765310956027504at_nat @ F @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_311_apsnd__conv,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( produc1735415053012026015r_prat @ F @ ( produc74620467203698198on_val @ X @ Y ) )
      = ( produc74620467203698198on_val @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_312_fst__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,X: produc9003781588162623004on_val] :
      ( ( produc5239619271231976746on_val @ ( produc8604563337118726038on_val @ F @ X ) )
      = ( F @ ( produc5239619271231976746on_val @ X ) ) ) ).

% fst_apfst
thf(fact_313_snd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,X: produc9003781588162623004on_val] :
      ( ( produc2319605628138516840on_val @ ( produc8604563337118726038on_val @ F @ X ) )
      = ( produc2319605628138516840on_val @ X ) ) ).

% snd_apfst
thf(fact_314_fst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
      ( ( produc5239619271231976746on_val @ ( produc1735415053012026015r_prat @ F @ X ) )
      = ( produc5239619271231976746on_val @ X ) ) ).

% fst_apsnd
thf(fact_315_snd__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
      ( ( produc2319605628138516840on_val @ ( produc1735415053012026015r_prat @ F @ X ) )
      = ( F @ ( produc2319605628138516840on_val @ X ) ) ) ).

% snd_apsnd
thf(fact_316_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_317_map__prod__imageI,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char,F: nat > nat,G: list_char > nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_2110972657701675764at_nat @ ( produc3271292527955122507ar_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_318_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat,G: nat > list_char] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F @ A ) @ ( G @ B ) ) @ ( image_728565836875346774t_char @ ( produc3912516661280071627t_char @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_319_map__prod__imageI,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char,F: nat > nat,G: list_char > list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
     => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F @ A ) @ ( G @ B ) ) @ ( image_3726312381382425221t_char @ ( produc5890707981334396508t_char @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_320_map__prod__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_321_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_322_map__prod__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > nat,G: product_prod_nat_nat > list_char] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F @ A ) @ ( G @ B ) ) @ ( image_4209661836242193520t_char @ ( produc8778638755205509575t_char @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_323_map__prod__imageI,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char,F: nat > product_prod_nat_nat,G: list_char > product_prod_nat_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_5263967778105348570at_nat @ ( produc6494513283846573029at_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_324_map__prod__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_7325330684212377951at_nat @ ( produc7374836315209622838at_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_325_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_1672857615394513579at_nat @ ( produc7508474501381341110at_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_326_range__fst,axiom,
    ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ top_to1561438921946393474on_val )
    = top_to6849109582029272579r_prat ) ).

% range_fst
thf(fact_327_range__fst,axiom,
    ( ( image_2486076414777270412at_nat @ product_fst_nat_nat @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% range_fst
thf(fact_328_range__snd,axiom,
    ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ top_to1561438921946393474on_val )
    = top_to3326603054294128128on_val ) ).

% range_snd
thf(fact_329_range__snd,axiom,
    ( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% range_snd
thf(fact_330_apfst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > set_Pr4329608150637261639at_nat,G: ( produc1457211279475724562t_char > option_val ) > set_Pr4329608150637261639at_nat,X: produc9003781588162623004on_val] :
      ( ( produc3036412456634812887at_nat @ F @ ( produc8076712183867557860r_prat @ G @ X ) )
      = ( produc9060074326276436823at_nat @ ( F @ ( produc5239619271231976746on_val @ X ) ) @ ( G @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_331_apfst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > set_Pr1261947904930325089at_nat,G: ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat,X: produc9003781588162623004on_val] :
      ( ( produc1982046010511310167at_nat @ F @ ( produc818678705018253770r_prat @ G @ X ) )
      = ( produc2922128104949294807at_nat @ ( F @ ( produc5239619271231976746on_val @ X ) ) @ ( G @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_332_apfst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > list_char,X: produc9003781588162623004on_val] :
      ( ( produc2791019445495265482t_char @ F @ ( produc4576295017903145368r_prat @ G @ X ) )
      = ( produc120197956887798346t_char @ ( F @ ( produc5239619271231976746on_val @ X ) ) @ ( G @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_333_apfst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > nat,X: produc9003781588162623004on_val] :
      ( ( produc7691703001529815225at_nat @ F @ ( produc6779024961348319977r_prat @ G @ X ) )
      = ( product_Pair_nat_nat @ ( F @ ( produc5239619271231976746on_val @ X ) ) @ ( G @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_334_apfst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,X: produc9003781588162623004on_val] :
      ( ( produc8604563337118726038on_val @ F @ ( produc1735415053012026015r_prat @ G @ X ) )
      = ( produc74620467203698198on_val @ ( F @ ( produc5239619271231976746on_val @ X ) ) @ ( G @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_335_apsnd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > set_Pr4329608150637261639at_nat,G: ( produc1457211279475724562t_char > prat ) > set_Pr4329608150637261639at_nat,X: produc9003781588162623004on_val] :
      ( ( produc8540158291859151192at_nat @ F @ ( produc2235273608163267722on_val @ G @ X ) )
      = ( produc9060074326276436823at_nat @ ( G @ ( produc5239619271231976746on_val @ X ) ) @ ( F @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_336_apsnd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat,G: ( produc1457211279475724562t_char > prat ) > set_Pr1261947904930325089at_nat,X: produc9003781588162623004on_val] :
      ( ( produc1458986165287348184at_nat @ F @ ( produc3730451593991314596on_val @ G @ X ) )
      = ( produc2922128104949294807at_nat @ ( G @ ( produc5239619271231976746on_val @ X ) ) @ ( F @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_337_apsnd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > list_char,G: ( produc1457211279475724562t_char > prat ) > nat,X: produc9003781588162623004on_val] :
      ( ( produc1904874682975057099ar_nat @ F @ ( produc7822372560581557379on_val @ G @ X ) )
      = ( produc120197956887798346t_char @ ( G @ ( produc5239619271231976746on_val @ X ) ) @ ( F @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_338_apsnd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > nat,G: ( produc1457211279475724562t_char > prat ) > nat,X: produc9003781588162623004on_val] :
      ( ( produc3479886823388342458at_nat @ F @ ( produc7822372560581557379on_val @ G @ X ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc5239619271231976746on_val @ X ) ) @ ( F @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_339_apsnd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,G: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,X: produc9003781588162623004on_val] :
      ( ( produc1735415053012026015r_prat @ F @ ( produc8604563337118726038on_val @ G @ X ) )
      = ( produc74620467203698198on_val @ ( G @ ( produc5239619271231976746on_val @ X ) ) @ ( F @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_340_map__prod__surj,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ top_to4669805908274784177at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_341_map__prod__surj,axiom,
    ! [F: nat > nat,G: nat > product_prod_nat_nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_5846123807819985514at_nat @ G @ top_top_set_nat )
          = top_to4669805908274784177at_nat )
       => ( ( image_1862905170067107298at_nat @ ( produc9062365837733128983at_nat @ F @ G ) @ top_to4669805908274784177at_nat )
          = top_to7745748144976627212at_nat ) ) ) ).

% map_prod_surj
thf(fact_342_map__prod__surj,axiom,
    ! [F: nat > nat,G: product_prod_nat_nat > nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_2486076414777270412at_nat @ G @ top_to4669805908274784177at_nat )
          = top_top_set_nat )
       => ( ( image_6167232457338462356at_nat @ ( produc5702318444690413881at_nat @ F @ G ) @ top_to7745748144976627212at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_343_map__prod__surj,axiom,
    ! [F: nat > product_prod_nat_nat,G: nat > nat] :
      ( ( ( image_5846123807819985514at_nat @ F @ top_top_set_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_2988392154034408380at_nat @ ( produc8306735474602065755at_nat @ F @ G ) @ top_to4669805908274784177at_nat )
          = top_to2567003362141017830at_nat ) ) ) ).

% map_prod_surj
thf(fact_344_map__prod__surj,axiom,
    ! [F: product_prod_nat_nat > nat,G: nat > nat] :
      ( ( ( image_2486076414777270412at_nat @ F @ top_to4669805908274784177at_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_8369444115148354106at_nat @ ( produc3053662887905975677at_nat @ F @ G ) @ top_to2567003362141017830at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_345_map__prod__surj,axiom,
    ! [F: nat > nat,G: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_5168914502847457605at_nat @ G @ top_to4669805908274784177at_nat )
          = top_to4669805908274784177at_nat )
       => ( ( image_7499486920669067219at_nat @ ( produc415658797863765080at_nat @ F @ G ) @ top_to7745748144976627212at_nat )
          = top_to7745748144976627212at_nat ) ) ) ).

% map_prod_surj
thf(fact_346_map__prod__surj,axiom,
    ! [F: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat] :
      ( ( ( image_5846123807819985514at_nat @ F @ top_top_set_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( image_5846123807819985514at_nat @ G @ top_top_set_nat )
          = top_to4669805908274784177at_nat )
       => ( ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F @ G ) @ top_to4669805908274784177at_nat )
          = top_to8454300755403694231at_nat ) ) ) ).

% map_prod_surj
thf(fact_347_map__prod__surj,axiom,
    ! [F: nat > product_prod_nat_nat,G: product_prod_nat_nat > nat] :
      ( ( ( image_5846123807819985514at_nat @ F @ top_top_set_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( image_2486076414777270412at_nat @ G @ top_to4669805908274784177at_nat )
          = top_top_set_nat )
       => ( ( image_8624973904636368301at_nat @ ( produc7016512962194184664at_nat @ F @ G ) @ top_to7745748144976627212at_nat )
          = top_to2567003362141017830at_nat ) ) ) ).

% map_prod_surj
thf(fact_348_map__prod__surj,axiom,
    ! [F: product_prod_nat_nat > nat,G: nat > product_prod_nat_nat] :
      ( ( ( image_2486076414777270412at_nat @ F @ top_to4669805908274784177at_nat )
        = top_top_set_nat )
     => ( ( ( image_5846123807819985514at_nat @ G @ top_top_set_nat )
          = top_to4669805908274784177at_nat )
       => ( ( image_2402546415023586989at_nat @ ( produc8516801393723169556at_nat @ F @ G ) @ top_to2567003362141017830at_nat )
          = top_to7745748144976627212at_nat ) ) ) ).

% map_prod_surj
thf(fact_349_map__prod__surj,axiom,
    ! [F: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
      ( ( ( image_2486076414777270412at_nat @ F @ top_to4669805908274784177at_nat )
        = top_top_set_nat )
     => ( ( ( image_2486076414777270412at_nat @ G @ top_to4669805908274784177at_nat )
          = top_top_set_nat )
       => ( ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F @ G ) @ top_to8454300755403694231at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_350_subrelI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% subrelI
thf(fact_351_subrelI,axiom,
    ! [R: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le5997549366648089703at_nat @ R @ S ) ) ).

% subrelI
thf(fact_352_subrelI,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R @ S ) ) ).

% subrelI
thf(fact_353_subrelI,axiom,
    ! [R: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ! [X3: nat,Y3: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ S ) )
     => ( ord_le6315511645215477266t_char @ R @ S ) ) ).

% subrelI
thf(fact_354_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% subrelI
thf(fact_355_subrelI,axiom,
    ! [R: set_Pr7649655576376376530on_val,S: set_Pr7649655576376376530on_val] :
      ( ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ R )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ S ) )
     => ( ord_le6900622908513426738on_val @ R @ S ) ) ).

% subrelI
thf(fact_356_ssubsetI,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [Pi: product_prod_nat_nat,H: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Pi @ H ) @ A5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Pi @ H ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_357_ssubsetI,axiom,
    ! [A5: set_Pr7459493094073627847at_nat,B5: set_Pr7459493094073627847at_nat] :
      ( ! [Pi: set_Pr4329608150637261639at_nat,H: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Pi @ H ) @ A5 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Pi @ H ) @ B5 ) )
     => ( ord_le5997549366648089703at_nat @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_358_ssubsetI,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [Pi: set_Pr1261947904930325089at_nat,H: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi @ H ) @ A5 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi @ H ) @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_359_ssubsetI,axiom,
    ! [A5: set_Pr2507339680178222706t_char,B5: set_Pr2507339680178222706t_char] :
      ( ! [Pi: nat,H: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi @ H ) @ A5 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi @ H ) @ B5 ) )
     => ( ord_le6315511645215477266t_char @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_360_ssubsetI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [Pi: nat,H: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi @ H ) @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi @ H ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_361_ssubsetI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
          ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ A5 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ B5 ) )
     => ( ord_le6900622908513426738on_val @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_362_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: nat > nat,G: nat > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_363_prod__fun__imageE,axiom,
    ! [C: produc1457211279475724562t_char,F: nat > nat,G: nat > list_char,R2: set_Pr1261947904930325089at_nat] :
      ( ( member484235747325421115t_char @ C @ ( image_728565836875346774t_char @ ( produc3912516661280071627t_char @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc120197956887798346t_char @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_364_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: nat > nat,G: list_char > nat,R2: set_Pr2507339680178222706t_char] :
      ( ( member8440522571783428010at_nat @ C @ ( image_2110972657701675764at_nat @ ( produc3271292527955122507ar_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_365_prod__fun__imageE,axiom,
    ! [C: produc1457211279475724562t_char,F: nat > nat,G: list_char > list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member484235747325421115t_char @ C @ ( image_3726312381382425221t_char @ ( produc5890707981334396508t_char @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( C
              = ( produc120197956887798346t_char @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_366_prod__fun__imageE,axiom,
    ! [C: produc859450856879609959at_nat,F: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_367_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_368_prod__fun__imageE,axiom,
    ! [C: produc859450856879609959at_nat,F: nat > product_prod_nat_nat,G: list_char > product_prod_nat_nat,R2: set_Pr2507339680178222706t_char] :
      ( ( member8206827879206165904at_nat @ C @ ( image_5263967778105348570at_nat @ ( produc6494513283846573029at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( C
              = ( produc6161850002892822231at_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_369_prod__fun__imageE,axiom,
    ! [C: produc1457211279475724562t_char,F: product_prod_nat_nat > nat,G: product_prod_nat_nat > list_char,R2: set_Pr8693737435421807431at_nat] :
      ( ( member484235747325421115t_char @ C @ ( image_4209661836242193520t_char @ ( produc8778638755205509575t_char @ F @ G ) @ R2 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( C
              = ( produc120197956887798346t_char @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_370_prod__fun__imageE,axiom,
    ! [C: produc3843707927480180839at_nat,F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( image_1672857615394513579at_nat @ ( produc7508474501381341110at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc2922128104949294807at_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_371_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( image_7325330684212377951at_nat @ ( produc7374836315209622838at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_372_valid__null__def,axiom,
    ( valid_null
    = ( ^ [Pi4: produc1457211279475724562t_char > prat] :
        ! [F2: list_char] :
          ( ( Pi4 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
          = pnone ) ) ) ).

% valid_null_def
thf(fact_373_convol__image__vimage2p,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ ( bNF_co3837107326993794344on_val @ ( comp_P4687317905303122495on_val @ F @ produc5239619271231976746on_val ) @ ( comp_P4718904269643193781on_val @ G @ produc2319605628138516840on_val ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_vi2254905643487785511_val_o @ F @ G @ R2 ) ) ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ).

% convol_image_vimage2p
thf(fact_374_convol__image__vimage2p,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > nat,R2: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_3154293121060339486at_nat @ ( bNF_co6248701753126501031at_nat @ ( comp_P5781055251786105118on_val @ F @ produc5239619271231976746on_val ) @ ( comp_P4527276930459016555on_val @ G @ produc2319605628138516840on_val ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_vi2516110938607903234_nat_o @ F @ G @ R2 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ).

% convol_image_vimage2p
thf(fact_375_convol__image__vimage2p,axiom,
    ! [F: nat > produc1457211279475724562t_char > prat,G: nat > produc1457211279475724562t_char > option_val,R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( ord_le6900622908513426738on_val @ ( image_1778330534009140952on_val @ ( bNF_co3675872600281376917on_val @ ( comp_n4993841364546111009at_nat @ F @ product_fst_nat_nat ) @ ( comp_n3217000323701810034at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi6495413567062537508_val_o @ F @ G @ R2 ) ) ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ).

% convol_image_vimage2p
thf(fact_376_convol__image__vimage2p,axiom,
    ! [F: nat > nat,G: nat > nat,R2: nat > nat > $o] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ ( bNF_co8276296344278409722at_nat @ ( comp_n2711240574518091068at_nat @ F @ product_fst_nat_nat ) @ ( comp_n2711240574518091068at_nat @ G @ product_snd_nat_nat ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_vi6667456707345531737_nat_o @ F @ G @ R2 ) ) ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ).

% convol_image_vimage2p
thf(fact_377_relChain__def,axiom,
    ( bNF_Ca8308629720386654381at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > nat] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_378_relChain__def,axiom,
    ( bNF_Ca8906692637563564659at_nat
    = ( ^ [R3: set_Pr7459493094073627847at_nat,As: set_Pr4329608150637261639at_nat > nat] :
        ! [I: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_379_relChain__def,axiom,
    ( bNF_Ca4949091444211332621at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat,As: set_Pr1261947904930325089at_nat > nat] :
        ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_380_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_381_Collect__case__prod__in__rel__leI,axiom,
    ! [X5: set_Pr7649655576376376530on_val,Y8: set_Pr7649655576376376530on_val] :
      ( ( ord_le6900622908513426738on_val @ X5 @ Y8 )
     => ( ord_le6900622908513426738on_val @ X5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( fun_in3588502033776859947on_val @ Y8 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_382_Collect__case__prod__in__rel__leI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X5 @ Y8 )
     => ( ord_le3146513528884898305at_nat @ X5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y8 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_383_Collect__case__prod__in__rel__leE,axiom,
    ! [X5: set_Pr7649655576376376530on_val,Y8: set_Pr7649655576376376530on_val] :
      ( ( ord_le6900622908513426738on_val @ X5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( fun_in3588502033776859947on_val @ Y8 ) ) ) )
     => ( ord_le6900622908513426738on_val @ X5 @ Y8 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_384_Collect__case__prod__in__rel__leE,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y8 ) ) ) )
     => ( ord_le3146513528884898305at_nat @ X5 @ Y8 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_385_add__masks_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa2: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa2 @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X @ Xb ) @ ( Xa2 @ Xb ) ) )
           => ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa2 @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_386_asymI,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R )
         => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X3 ) @ R ) )
     => ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R ) ) ).

% asymI
thf(fact_387_asymI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
         => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R ) )
     => ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% asymI
thf(fact_388_asymI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) )
     => ( asym_on_nat @ top_top_set_nat @ R ) ) ).

% asymI
thf(fact_389_asymI,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
         => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R ) )
     => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% asymI
thf(fact_390_asym__onI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X3 @ A5 )
         => ( ( member2879030624002546611on_val @ Y3 @ A5 )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_o6325306393319210090on_val @ A5 @ R ) ) ).

% asym_onI
thf(fact_391_asym__onI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A5 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A5 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_o8546584678760569651at_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_392_asym__onI,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A5 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A5 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_o9200482492015546265at_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_393_asym__onI,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A5 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A5 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
             => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_o9101232109321078681at_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_394_asym__onI,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A5 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A5 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R )
             => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_o8654330542403097209at_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_395_asym__onI,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A5 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A5 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_o5033339280856310931at_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_396_asym__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( ( member_nat @ Y3 @ A5 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) )
     => ( asym_on_nat @ A5 @ R ) ) ).

% asym_onI
thf(fact_397_accp__subset,axiom,
    ! [R1: state > state > $o,R22: state > state > $o] :
      ( ( ord_le4344933087531911346tate_o @ R1 @ R22 )
     => ( ord_less_eq_state_o @ ( accp_state @ R22 ) @ ( accp_state @ R1 ) ) ) ).

% accp_subset
thf(fact_398_accp__subset__induct,axiom,
    ! [D2: state > $o,R2: state > state > $o,X: state,P2: state > $o] :
      ( ( ord_less_eq_state_o @ D2 @ ( accp_state @ R2 ) )
     => ( ! [X3: state,Z3: state] :
            ( ( D2 @ X3 )
           => ( ( R2 @ Z3 @ X3 )
             => ( D2 @ Z3 ) ) )
       => ( ( D2 @ X )
         => ( ! [X3: state] :
                ( ( D2 @ X3 )
               => ( ! [Z4: state] :
                      ( ( R2 @ Z4 @ X3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_399_asym__on__def,axiom,
    ( asym_o8546584678760569651at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A6 )
         => ! [Y7: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y7 @ A6 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R3 )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_400_asym__on__def,axiom,
    ( asym_o8654330542403097209at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
        ! [X4: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X4 @ A6 )
         => ! [Y7: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y7 @ A6 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R3 )
               => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_401_asym__on__def,axiom,
    ( asym_o5033339280856310931at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ A6 )
         => ! [Y7: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y7 @ A6 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R3 )
               => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_402_asym__on__def,axiom,
    ( asym_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ! [Y7: nat] :
              ( ( member_nat @ Y7 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R3 )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X4 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_403_asym__onD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( asym_o6325306393319210090on_val @ A5 @ R )
     => ( ( member2879030624002546611on_val @ X @ A5 )
       => ( ( member2879030624002546611on_val @ Y @ A5 )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_404_asym__onD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( ( member8440522571783428010at_nat @ Y @ A5 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_405_asym__onD,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( asym_o9200482492015546265at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ X @ A5 )
       => ( ( member8206827879206165904at_nat @ Y @ A5 )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R )
           => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_406_asym__onD,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( asym_o9101232109321078681at_nat @ A5 @ R )
     => ( ( member8757157785044589968at_nat @ X @ A5 )
       => ( ( member8757157785044589968at_nat @ Y @ A5 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
           => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_407_asym__onD,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( asym_o8654330542403097209at_nat @ A5 @ R )
     => ( ( member6273874620674182256at_nat @ X @ A5 )
       => ( ( member6273874620674182256at_nat @ Y @ A5 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
           => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_408_asym__onD,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( asym_o5033339280856310931at_nat @ A5 @ R )
     => ( ( member2643936169264416010at_nat @ X @ A5 )
       => ( ( member2643936169264416010at_nat @ Y @ A5 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_409_asym__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ A5 @ R )
     => ( ( member_nat @ X @ A5 )
       => ( ( member_nat @ Y @ A5 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ) ) ).

% asym_onD
thf(fact_410_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi5: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( padd @ ( Pi5 @ Hl ) @ ( Pi4 @ Hl ) ) ) ) ).

% add_masks.simps
thf(fact_411_add__masks_Oelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa2: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X @ Xa2 @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X @ Xb ) @ ( Xa2 @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_412_in__rel__def,axiom,
    ( fun_in3916733394112622082at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_413_in__rel__def,axiom,
    ( fun_in6812141508599261570at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat,X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_414_in__rel__def,axiom,
    ( fun_in8056734985859589634at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_415_in__rel__def,axiom,
    ( fun_in175243070162512245t_char
    = ( ^ [R4: set_Pr2507339680178222706t_char,X4: nat,Y7: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X4 @ Y7 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_416_in__rel__def,axiom,
    ( fun_in_rel_nat_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_417_in__rel__def,axiom,
    ( fun_in3588502033776859947on_val
    = ( ^ [R4: set_Pr7649655576376376530on_val,X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y7 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_418_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_419_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( asym_o3144787024281459316at_nat @ top_to7745748144976627212at_nat @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_420_asym__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o4270274008248760398at_nat @ top_to2567003362141017830at_nat @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_421_asym__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( asym_o9200482492015546265at_nat @ top_to8454300755403694231at_nat @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_422_in__rel__Collect__case__prod__eq,axiom,
    ! [X5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( fun_in3588502033776859947on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ X5 ) ) )
      = X5 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_423_in__rel__Collect__case__prod__eq,axiom,
    ! [X5: nat > nat > $o] :
      ( ( fun_in_rel_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ X5 ) ) )
      = X5 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_424_Collect__case__prod__mono,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,B5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( ord_le3852884270828607079_val_o @ A5 @ B5 )
     => ( ord_le6900622908513426738on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ A5 ) ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_425_Collect__case__prod__mono,axiom,
    ! [A5: nat > nat > $o,B5: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ A5 @ B5 )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A5 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_426_asym__iff,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R )
      = ( ! [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] :
            ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R )
           => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_427_asym__iff,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R )
      = ( ! [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_428_asym__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
      = ( ! [X4: nat,Y7: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_429_asym__iff,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
      = ( ! [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ X4 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_430_asymD,axiom,
    ! [R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
       => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_431_asymD,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_432_asymD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_433_asymD,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_434_GrD1,axiom,
    ! [X: product_prod_nat_nat,Fx: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Fx ) @ ( bNF_Gr4088865321246080482at_nat @ A5 @ F ) )
     => ( member8440522571783428010at_nat @ X @ A5 ) ) ).

% GrD1
thf(fact_435_GrD1,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Fx: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat,F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Fx ) @ ( bNF_Gr7888059597227825506at_nat @ A5 @ F ) )
     => ( member6273874620674182256at_nat @ X @ A5 ) ) ).

% GrD1
thf(fact_436_GrD1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Fx: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Fx ) @ ( bNF_Gr1527221351080490466at_nat @ A5 @ F ) )
     => ( member2643936169264416010at_nat @ X @ A5 ) ) ).

% GrD1
thf(fact_437_GrD1,axiom,
    ! [X: nat,Fx: list_char,A5: set_nat,F: nat > list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Fx ) @ ( bNF_Gr_nat_list_char @ A5 @ F ) )
     => ( member_nat @ X @ A5 ) ) ).

% GrD1
thf(fact_438_GrD1,axiom,
    ! [X: nat,Fx: nat,A5: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A5 @ F ) )
     => ( member_nat @ X @ A5 ) ) ).

% GrD1
thf(fact_439_GrD1,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc1457211279475724562t_char > option_val,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Fx ) @ ( bNF_Gr7520394057480975179on_val @ A5 @ F ) )
     => ( member2336342582916103932r_prat @ X @ A5 ) ) ).

% GrD1
thf(fact_440_GrD2,axiom,
    ! [X: product_prod_nat_nat,Fx: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Fx ) @ ( bNF_Gr4088865321246080482at_nat @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_441_GrD2,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Fx: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat,F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Fx ) @ ( bNF_Gr7888059597227825506at_nat @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_442_GrD2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Fx: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Fx ) @ ( bNF_Gr1527221351080490466at_nat @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_443_GrD2,axiom,
    ! [X: nat,Fx: list_char,A5: set_nat,F: nat > list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Fx ) @ ( bNF_Gr_nat_list_char @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_444_GrD2,axiom,
    ! [X: nat,Fx: nat,A5: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Fx ) @ ( bNF_Gr_nat_nat @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_445_GrD2,axiom,
    ! [X: produc1457211279475724562t_char > prat,Fx: produc1457211279475724562t_char > option_val,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X @ Fx ) @ ( bNF_Gr7520394057480975179on_val @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_446_mult__add__states,axiom,
    ! [A: prat,B: prat,Pi3: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ ( padd @ A @ B ) @ Pi3 )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ Pi3 ) @ ( multiply_mask @ B @ Pi3 ) ) ) ).

% mult_add_states
thf(fact_447_fun_Oin__rel,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: nat > produc1457211279475724562t_char > prat,B: nat > produc1457211279475724562t_char > option_val] :
      ( ( bNF_re2896540935372247726on_val
        @ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
        @ R2
        @ A
        @ B )
      = ( ? [Z5: nat > produc9003781588162623004on_val] :
            ( ( member798016022416105634on_val @ Z5
              @ ( collec2477365322772234464on_val
                @ ^ [X4: nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ X4 @ top_top_set_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ) )
            & ( ( comp_P4447563945128311908at_nat @ produc5239619271231976746on_val @ Z5 )
              = A )
            & ( ( comp_P6187037784939234283al_nat @ produc2319605628138516840on_val @ Z5 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_448_fun_Oin__rel,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: product_prod_nat_nat > produc1457211279475724562t_char > prat,B: product_prod_nat_nat > produc1457211279475724562t_char > option_val] :
      ( ( bNF_re8250582710452816528on_val
        @ ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z )
        @ R2
        @ A
        @ B )
      = ( ? [Z5: product_prod_nat_nat > produc9003781588162623004on_val] :
            ( ( member3223413302433764535on_val @ Z5
              @ ( collec5030907905214230265on_val
                @ ^ [X4: product_prod_nat_nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_1778330534009140952on_val @ X4 @ top_to4669805908274784177at_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ) )
            & ( ( comp_P7748810038695468909at_nat @ produc5239619271231976746on_val @ Z5 )
              = A )
            & ( ( comp_P157993088335196966at_nat @ produc2319605628138516840on_val @ Z5 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_449_fun_Oin__rel,axiom,
    ! [R2: nat > nat > $o,A: nat > nat,B: nat > nat] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
        @ R2
        @ A
        @ B )
      = ( ? [Z5: nat > product_prod_nat_nat] :
            ( ( member6743353555990375057at_nat @ Z5
              @ ( collec8718012497556192979at_nat
                @ ^ [X4: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X4 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ) )
            & ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat @ Z5 )
              = A )
            & ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat @ Z5 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_450_fun_Oin__rel,axiom,
    ! [R2: nat > nat > $o,A: product_prod_nat_nat > nat,B: product_prod_nat_nat > nat] :
      ( ( bNF_re8246922863344978751at_nat
        @ ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z )
        @ R2
        @ A
        @ B )
      = ( ? [Z5: product_prod_nat_nat > product_prod_nat_nat] :
            ( ( member8885076297122219836at_nat @ Z5
              @ ( collec248804655031457914at_nat
                @ ^ [X4: product_prod_nat_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ X4 @ top_to4669805908274784177at_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ) )
            & ( ( comp_P1567445206330693457at_nat @ product_fst_nat_nat @ Z5 )
              = A )
            & ( ( comp_P1567445206330693457at_nat @ product_snd_nat_nat @ Z5 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_451_convol__mem__GrpI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,G: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ X @ A5 )
     => ( member8757157785044589968at_nat @ ( bNF_co147202357524637304at_nat @ id_set2334057505560298834at_nat @ G @ X ) @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A5 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_452_convol__mem__GrpI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( member8206827879206165904at_nat @ ( bNF_co3897469944465340696at_nat @ id_Pro7559728917991913458at_nat @ G @ X ) @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A5 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_453_convol__mem__GrpI,axiom,
    ! [X: produc1457211279475724562t_char > prat,A5: set_Pr3433046674199179443r_prat,G: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2336342582916103932r_prat @ X @ A5 )
     => ( member2879030624002546611on_val @ ( bNF_co5850895267955675587on_val @ id_Pro6476562945676181316r_prat @ G @ X ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_Gr4383009045007556383on_val @ A5 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_454_convol__mem__GrpI,axiom,
    ! [X: nat,A5: set_nat,G: nat > nat] :
      ( ( member_nat @ X @ A5 )
     => ( member8440522571783428010at_nat @ ( bNF_co805650143699787099at_nat @ id_nat @ G @ X ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A5 @ G ) ) ) ) ) ).

% convol_mem_GrpI
thf(fact_455_transp__relcompp,axiom,
    ! [R: nat > nat > $o] :
      ( ( transp_on_nat @ top_top_set_nat @ R )
      = ( ord_le2646555220125990790_nat_o @ ( relcompp_nat_nat_nat @ R @ R ) @ R ) ) ).

% transp_relcompp
thf(fact_456_transp__relcompp,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( transp6142674538225245873at_nat @ top_to4669805908274784177at_nat @ R )
      = ( ord_le5604493270027003598_nat_o @ ( relcom282571730187242960at_nat @ R @ R ) @ R ) ) ).

% transp_relcompp
thf(fact_457_transp__relcompp__less__eq,axiom,
    ! [R: nat > nat > $o] :
      ( ( transp_on_nat @ top_top_set_nat @ R )
     => ( ord_le2646555220125990790_nat_o @ ( relcompp_nat_nat_nat @ R @ R ) @ R ) ) ).

% transp_relcompp_less_eq
thf(fact_458_transp__relcompp__less__eq,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( transp6142674538225245873at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ord_le5604493270027003598_nat_o @ ( relcom282571730187242960at_nat @ R @ R ) @ R ) ) ).

% transp_relcompp_less_eq
thf(fact_459_split__part,axiom,
    ! [P2: $o,Q2: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A4: nat,B4: nat] :
            ( P2
            & ( Q2 @ A4 @ B4 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P2
            & ( produc6081775807080527818_nat_o @ Q2 @ Ab ) ) ) ) ).

% split_part
thf(fact_460_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_461_case__prodI,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( F @ A @ B )
     => ( produc9020218426428501292_nat_o @ F @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_462_case__prodI,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ A @ B )
     => ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_463_case__prodI,axiom,
    ! [F: nat > list_char > $o,A: nat,B: list_char] :
      ( ( F @ A @ B )
     => ( produc8526122873445342073char_o @ F @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_464_case__prodI,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( F @ A @ B )
     => ( produc5248837549992124053_val_o @ F @ ( produc74620467203698198on_val @ A @ B ) ) ) ).

% case_prodI
thf(fact_465_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_466_case__prodI2,axiom,
    ! [P: produc1319942482725812455at_nat,C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ! [A3: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
          ( ( P
            = ( produc9060074326276436823at_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc9020218426428501292_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_467_case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc2922128104949294807at_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc410239310623530412_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_468_case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,C: nat > list_char > $o] :
      ( ! [A3: nat,B3: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc8526122873445342073char_o @ C @ P ) ) ).

% case_prodI2
thf(fact_469_case__prodI2,axiom,
    ! [P: produc9003781588162623004on_val,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ! [A3: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( ( P
            = ( produc74620467203698198on_val @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5248837549992124053_val_o @ C @ P ) ) ).

% case_prodI2
thf(fact_470_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_471_case__prod__Pair,axiom,
    ( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat )
    = id_Pro6043886709289754456at_nat ) ).

% case_prod_Pair
thf(fact_472_case__prod__Pair,axiom,
    ( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat )
    = id_Pro8722402698821300696at_nat ) ).

% case_prod_Pair
thf(fact_473_case__prod__Pair,axiom,
    ( ( produc4071494739252518515t_char @ produc120197956887798346t_char )
    = id_Pro6264179822262827651t_char ) ).

% case_prod_Pair
thf(fact_474_case__prod__Pair,axiom,
    ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat )
    = id_Pro7559728917991913458at_nat ) ).

% case_prod_Pair
thf(fact_475_case__prod__Pair,axiom,
    ( ( produc4192255817509754207on_val @ produc74620467203698198on_val )
    = id_Pro1402184692240358763on_val ) ).

% case_prod_Pair
thf(fact_476_case__swap,axiom,
    ! [F: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y7: nat,X4: nat] : ( F @ X4 @ Y7 )
        @ ( product_swap_nat_nat @ P ) )
      = ( produc6081775807080527818_nat_o @ F @ P ) ) ).

% case_swap
thf(fact_477_mem__case__prodI,axiom,
    ! [Z2: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_478_mem__case__prodI,axiom,
    ! [Z2: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat,A: nat,B: list_char] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc3665563422017311042at_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_479_mem__case__prodI,axiom,
    ! [Z2: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,A: nat,B: nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z2 @ ( produc2423807293333670521at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_480_mem__case__prodI,axiom,
    ! [Z2: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat,A: nat,B: list_char] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z2 @ ( produc3763337624047832232at_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_481_mem__case__prodI,axiom,
    ! [Z2: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc1433696742876942453at_nat @ C @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_482_mem__case__prodI,axiom,
    ! [Z2: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,A: nat,B: nat] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8757157785044589968at_nat @ Z2 @ ( produc3705604114177432185at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_483_mem__case__prodI,axiom,
    ! [Z2: produc3843707927480180839at_nat,C: nat > list_char > set_Pr4329608150637261639at_nat,A: nat,B: list_char] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8757157785044589968at_nat @ Z2 @ ( produc7856237077994235048at_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_484_mem__case__prodI,axiom,
    ! [Z2: produc859450856879609959at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z2 @ ( produc275446641266004827at_nat @ C @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_485_mem__case__prodI,axiom,
    ! [Z2: produc9003781588162623004on_val,C: nat > nat > set_Pr7649655576376376530on_val,A: nat,B: nat] :
      ( ( member2879030624002546611on_val @ Z2 @ ( C @ A @ B ) )
     => ( member2879030624002546611on_val @ Z2 @ ( produc9171785124108960544on_val @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_486_mem__case__prodI,axiom,
    ! [Z2: product_prod_nat_nat,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc3420870819319363678at_nat @ C @ ( produc74620467203698198on_val @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_487_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc8197505143624133779at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_488_mem__case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,Z2: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat] :
      ( ! [A3: nat,B3: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc3665563422017311042at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_489_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z2 @ ( produc2423807293333670521at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_490_mem__case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,Z2: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat] :
      ( ! [A3: nat,B3: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z2 @ ( produc3763337624047832232at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_491_mem__case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,Z2: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc2922128104949294807at_nat @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc1433696742876942453at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_492_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member8757157785044589968at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8757157785044589968at_nat @ Z2 @ ( produc3705604114177432185at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_493_mem__case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,Z2: produc3843707927480180839at_nat,C: nat > list_char > set_Pr4329608150637261639at_nat] :
      ( ! [A3: nat,B3: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A3 @ B3 ) )
         => ( member8757157785044589968at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8757157785044589968at_nat @ Z2 @ ( produc7856237077994235048at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_494_mem__case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,Z2: produc859450856879609959at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc2922128104949294807at_nat @ A3 @ B3 ) )
         => ( member8206827879206165904at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8206827879206165904at_nat @ Z2 @ ( produc275446641266004827at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_495_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: produc9003781588162623004on_val,C: nat > nat > set_Pr7649655576376376530on_val] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member2879030624002546611on_val @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member2879030624002546611on_val @ Z2 @ ( produc9171785124108960544on_val @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_496_mem__case__prodI2,axiom,
    ! [P: produc9003781588162623004on_val,Z2: product_prod_nat_nat,C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat] :
      ( ! [A3: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( ( P
            = ( produc74620467203698198on_val @ A3 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc3420870819319363678at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_497_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member_o @ ( F @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_498_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_499_pair__imageI,axiom,
    ! [A: nat,B: list_char,A5: set_Pr2507339680178222706t_char,F: nat > list_char > product_prod_nat_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_2110972657701675764at_nat @ ( produc6775168648774419426at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_500_pair__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_6706873702294942047at_nat @ ( produc2982569246487150357at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_501_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member8206827879206165904at_nat @ ( F @ A @ B ) @ ( image_1119892238107900587at_nat @ ( produc2382474379627796377at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_502_pair__imageI,axiom,
    ! [A: nat,B: list_char,A5: set_Pr2507339680178222706t_char,F: nat > list_char > produc859450856879609959at_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A5 )
     => ( member8206827879206165904at_nat @ ( F @ A @ B ) @ ( image_5263967778105348570at_nat @ ( produc849435112462063048at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_503_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_7325330684212377951at_nat @ ( produc2376066862417776405at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_504_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member8757157785044589968at_nat @ ( F @ A @ B ) @ ( image_1672857615394513579at_nat @ ( produc153460504799203225at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_505_pair__imageI,axiom,
    ! [A: nat,B: list_char,A5: set_Pr2507339680178222706t_char,F: nat > list_char > produc3843707927480180839at_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A5 )
     => ( member8757157785044589968at_nat @ ( F @ A @ B ) @ ( image_5870379653388829146at_nat @ ( produc5612128348879181256at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_506_pair__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ A5 )
     => ( member8206827879206165904at_nat @ ( F @ A @ B ) @ ( image_6321802852832981957at_nat @ ( produc6744312701629110395at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_507_subset__CollectI,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat,Q2: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A5 )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_508_subset__CollectI,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat,Q2: produc3843707927480180839at_nat > $o,P2: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B5 @ A5 )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le1268244103169919719at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_509_subset__CollectI,axiom,
    ! [B5: set_Pr7649655576376376530on_val,A5: set_Pr7649655576376376530on_val,Q2: produc9003781588162623004on_val > $o,P2: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ B5 @ A5 )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le6900622908513426738on_val
          @ ( collec7475294502966882545on_val
            @ ^ [X4: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collec7475294502966882545on_val
            @ ^ [X4: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_510_subset__CollectI,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,Q2: product_prod_nat_nat > $o,P2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A5 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_511_subset__CollectI,axiom,
    ! [B5: set_nat,A5: set_nat,Q2: nat > $o,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B5 @ A5 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B5 )
           => ( ( Q2 @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ B5 )
                & ( Q2 @ X4 ) ) )
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_512_subset__Collect__iff,axiom,
    ! [B5: set_Pr8693737435421807431at_nat,A5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B5 @ A5 )
     => ( ( ord_le3000389064537975527at_nat @ B5
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_513_subset__Collect__iff,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ B5 @ A5 )
     => ( ( ord_le1268244103169919719at_nat @ B5
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_514_subset__Collect__iff,axiom,
    ! [B5: set_Pr7649655576376376530on_val,A5: set_Pr7649655576376376530on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ B5 @ A5 )
     => ( ( ord_le6900622908513426738on_val @ B5
          @ ( collec7475294502966882545on_val
            @ ^ [X4: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_515_subset__Collect__iff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A5 )
     => ( ( ord_le3146513528884898305at_nat @ B5
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_516_subset__Collect__iff,axiom,
    ! [B5: set_nat,A5: set_nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ B5 @ A5 )
     => ( ( ord_less_eq_set_nat @ B5
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A5 )
                & ( P2 @ X4 ) ) ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ B5 )
             => ( P2 @ X4 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_517_prop__restrict,axiom,
    ! [X: produc859450856879609959at_nat,Z6: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X @ Z6 )
     => ( ( ord_le3000389064537975527at_nat @ Z6
          @ ( collec7088162979684241874at_nat
            @ ^ [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_518_prop__restrict,axiom,
    ! [X: produc3843707927480180839at_nat,Z6: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X @ Z6 )
     => ( ( ord_le1268244103169919719at_nat @ Z6
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_519_prop__restrict,axiom,
    ! [X: produc9003781588162623004on_val,Z6: set_Pr7649655576376376530on_val,X5: set_Pr7649655576376376530on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ X @ Z6 )
     => ( ( ord_le6900622908513426738on_val @ Z6
          @ ( collec7475294502966882545on_val
            @ ^ [X4: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_520_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z6: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ Z6 )
     => ( ( ord_le3146513528884898305at_nat @ Z6
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_521_prop__restrict,axiom,
    ! [X: nat,Z6: set_nat,X5: set_nat,P2: nat > $o] :
      ( ( member_nat @ X @ Z6 )
     => ( ( ord_less_eq_set_nat @ Z6
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ X5 )
                & ( P2 @ X4 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_522_Collect__restrict,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_523_Collect__restrict,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ord_le1268244103169919719at_nat
      @ ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_524_Collect__restrict,axiom,
    ! [X5: set_Pr7649655576376376530on_val,P2: produc9003781588162623004on_val > $o] :
      ( ord_le6900622908513426738on_val
      @ ( collec7475294502966882545on_val
        @ ^ [X4: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_525_Collect__restrict,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_526_Collect__restrict,axiom,
    ! [X5: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
            & ( P2 @ X4 ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_527_case__prod__Pair__iden,axiom,
    ! [P: produc1319942482725812455at_nat] :
      ( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_528_case__prod__Pair__iden,axiom,
    ! [P: produc3843707927480180839at_nat] :
      ( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_529_case__prod__Pair__iden,axiom,
    ! [P: produc1457211279475724562t_char] :
      ( ( produc4071494739252518515t_char @ produc120197956887798346t_char @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_530_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_531_case__prod__Pair__iden,axiom,
    ! [P: produc9003781588162623004on_val] :
      ( ( produc4192255817509754207on_val @ produc74620467203698198on_val @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_532_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_533_prod_Ocase__distrib,axiom,
    ! [H3: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X12: nat,X22: nat] : ( H3 @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_534_pred__equals__eq2,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R2 ) )
        = ( ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_535_pred__equals__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R2 ) )
        = ( ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_536_pred__equals__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R2 ) )
        = ( ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_537_pred__equals__eq2,axiom,
    ! [R2: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
      ( ( ( ^ [X4: nat,Y7: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X4 @ Y7 ) @ R2 ) )
        = ( ^ [X4: nat,Y7: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X4 @ Y7 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_538_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R2 ) )
        = ( ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_539_pred__equals__eq2,axiom,
    ! [R2: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
      ( ( ( ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y7 ) @ R2 ) )
        = ( ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y7 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_540_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member8440522571783428010at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_541_image__Collect__subsetI,axiom,
    ! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
         => ( member8440522571783428010at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_542_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member8206827879206165904at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ ( image_3276603626957510736at_nat @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_543_image__Collect__subsetI,axiom,
    ! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
         => ( member8206827879206165904at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ ( image_1119892238107900587at_nat @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_544_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member8757157785044589968at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ ( image_2499013659892586576at_nat @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_545_image__Collect__subsetI,axiom,
    ! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
         => ( member8757157785044589968at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ ( image_1672857615394513579at_nat @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_546_image__Collect__subsetI,axiom,
    ! [P2: nat > $o,F: nat > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( member2879030624002546611on_val @ ( F @ X3 ) @ B5 ) )
     => ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ F @ ( collect_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_547_image__Collect__subsetI,axiom,
    ! [P2: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: produc9003781588162623004on_val] :
          ( ( P2 @ X3 )
         => ( member8440522571783428010at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_3154293121060339486at_nat @ F @ ( collec7475294502966882545on_val @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_548_image__Collect__subsetI,axiom,
    ! [P2: product_prod_nat_nat > $o,F: product_prod_nat_nat > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
         => ( member2879030624002546611on_val @ ( F @ X3 ) @ B5 ) )
     => ( ord_le6900622908513426738on_val @ ( image_1778330534009140952on_val @ F @ ( collec3392354462482085612at_nat @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_549_image__Collect__subsetI,axiom,
    ! [P2: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc9003781588162623004on_val] :
          ( ( P2 @ X3 )
         => ( member8206827879206165904at_nat @ ( F @ X3 ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ ( image_4697052935620052740at_nat @ F @ ( collec7475294502966882545on_val @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_550_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_551_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X4: nat,Y7: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y7 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_552_case__prodE2,axiom,
    ! [Q2: $o > $o,P2: nat > nat > $o,Z2: product_prod_nat_nat] :
      ( ( Q2 @ ( produc6081775807080527818_nat_o @ P2 @ Z2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q2 @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_553_fst__diag__id,axiom,
    ! [Z2: set_Pr4329608150637261639at_nat] :
      ( ( comp_P8235817900121468292at_nat @ produc4486455499028916547at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X4 @ X4 )
        @ Z2 )
      = ( id_set7849613212452405432at_nat @ Z2 ) ) ).

% fst_diag_id
thf(fact_554_fst__diag__id,axiom,
    ! [Z2: set_Pr1261947904930325089at_nat] :
      ( ( comp_P4528037747288561284at_nat @ produc6011863565407846595at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X4 @ X4 )
        @ Z2 )
      = ( id_set2334057505560298834at_nat @ Z2 ) ) ).

% fst_diag_id
thf(fact_555_fst__diag__id,axiom,
    ! [Z2: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat
        @ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ X4 )
        @ Z2 )
      = ( id_nat @ Z2 ) ) ).

% fst_diag_id
thf(fact_556_snd__diag__id,axiom,
    ! [Z2: set_Pr4329608150637261639at_nat] :
      ( ( comp_P8235817900121468292at_nat @ produc826222791318859909at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X4 @ X4 )
        @ Z2 )
      = ( id_set7849613212452405432at_nat @ Z2 ) ) ).

% snd_diag_id
thf(fact_557_snd__diag__id,axiom,
    ! [Z2: set_Pr1261947904930325089at_nat] :
      ( ( comp_P4528037747288561284at_nat @ produc2937545259423399429at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X4 @ X4 )
        @ Z2 )
      = ( id_set2334057505560298834at_nat @ Z2 ) ) ).

% snd_diag_id
thf(fact_558_snd__diag__id,axiom,
    ! [Z2: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat
        @ ^ [X4: nat] : ( product_Pair_nat_nat @ X4 @ X4 )
        @ Z2 )
      = ( id_nat @ Z2 ) ) ).

% snd_diag_id
thf(fact_559_case__prodD,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc9020218426428501292_nat_o @ F @ ( produc9060074326276436823at_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_560_case__prodD,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_561_case__prodD,axiom,
    ! [F: nat > list_char > $o,A: nat,B: list_char] :
      ( ( produc8526122873445342073char_o @ F @ ( produc120197956887798346t_char @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_562_case__prodD,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc5248837549992124053_val_o @ F @ ( produc74620467203698198on_val @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_563_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_564_case__prodE,axiom,
    ! [C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,P: produc1319942482725812455at_nat] :
      ( ( produc9020218426428501292_nat_o @ C @ P )
     => ~ ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
            ( ( P
              = ( produc9060074326276436823at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_565_case__prodE,axiom,
    ! [C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ( produc410239310623530412_nat_o @ C @ P )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_566_case__prodE,axiom,
    ! [C: nat > list_char > $o,P: produc1457211279475724562t_char] :
      ( ( produc8526122873445342073char_o @ C @ P )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_567_case__prodE,axiom,
    ! [C: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P: produc9003781588162623004on_val] :
      ( ( produc5248837549992124053_val_o @ C @ P )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P
              = ( produc74620467203698198on_val @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_568_case__prodE,axiom,
    ! [C: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_569_fst__def,axiom,
    ( produc5239619271231976746on_val
    = ( produc5787256515117077072r_prat
      @ ^ [X12: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] : X12 ) ) ).

% fst_def
thf(fact_570_snd__def,axiom,
    ( produc2319605628138516840on_val
    = ( produc985966260317727453on_val
      @ ^ [X12: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] : X22 ) ) ).

% snd_def
thf(fact_571_case__prod__map__prod,axiom,
    ! [H3: nat > nat > $o,F: nat > nat,G: nat > nat,X: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ H3 @ ( produc6977886695330630970at_nat @ F @ G @ X ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [L: nat,R3: nat] : ( H3 @ ( F @ L ) @ ( G @ R3 ) )
        @ X ) ) ).

% case_prod_map_prod
thf(fact_572_flip__pred,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R2: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( conver4900003399366114050r_prat @ R2 ) ) ) )
     => ( ord_le2122960673345045316r_prat
        @ ( image_2943372182239103057r_prat
          @ ( produc2964738055033037937r_prat
            @ ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y7 @ X4 ) )
          @ A5 )
        @ ( collec6247776740490166275r_prat @ ( produc5226666187743293259prat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_573_flip__pred,axiom,
    ! [A5: set_Pr7459493094073627847at_nat,R2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ( ord_le5997549366648089703at_nat @ A5 @ ( collec7872792499061297490at_nat @ ( produc9020218426428501292_nat_o @ ( conver9095369822871743413at_nat @ R2 ) ) ) )
     => ( ord_le5997549366648089703at_nat
        @ ( image_3313167189156628421at_nat
          @ ( produc5456455677105495163at_nat
            @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y7 @ X4 ) )
          @ A5 )
        @ ( collec7872792499061297490at_nat @ ( produc9020218426428501292_nat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_574_flip__pred,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ A5 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( conver5243090411617913141at_nat @ R2 ) ) ) )
     => ( ord_le1268244103169919719at_nat
        @ ( image_960689587819713477at_nat
          @ ( produc7548845685434208379at_nat
            @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y7 @ X4 ) )
          @ A5 )
        @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_575_flip__pred,axiom,
    ! [A5: set_Pr8803067691650843250ar_nat,R2: nat > list_char > $o] :
      ( ( ord_le3387867619833322002ar_nat @ A5 @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ ( conver4209408378539545768t_char @ R2 ) ) ) )
     => ( ord_le6315511645215477266t_char
        @ ( image_688197443715608197t_char
          @ ( produc6166083679720228083t_char
            @ ^ [X4: list_char,Y7: nat] : ( produc120197956887798346t_char @ Y7 @ X4 ) )
          @ A5 )
        @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_576_flip__pred,axiom,
    ! [A5: set_Pr2871993341207995108r_prat,R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( ord_le2122960673345045316r_prat @ A5 @ ( collec6247776740490166275r_prat @ ( produc5226666187743293259prat_o @ ( conver3077193964297325112on_val @ R2 ) ) ) )
     => ( ord_le6900622908513426738on_val
        @ ( image_1698057272686278225on_val
          @ ( produc4110721796449090601on_val
            @ ^ [X4: produc1457211279475724562t_char > option_val,Y7: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y7 @ X4 ) )
          @ A5 )
        @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_577_flip__pred,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R2: nat > nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( conversep_nat_nat @ R2 ) ) ) )
     => ( ord_le3146513528884898305at_nat
        @ ( image_5168914502847457605at_nat
          @ ( produc2626176000494625587at_nat
            @ ^ [X4: nat,Y7: nat] : ( product_Pair_nat_nat @ Y7 @ X4 ) )
          @ A5 )
        @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_578_exE__realizer,axiom,
    ! [P2: nat > nat > $o,P: product_prod_nat_nat,Q2: $o > $o,F: nat > nat > $o] :
      ( ( P2 @ ( product_snd_nat_nat @ P ) @ ( product_fst_nat_nat @ P ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P2 @ Y3 @ X3 )
           => ( Q2 @ ( F @ X3 @ Y3 ) ) )
       => ( Q2 @ ( produc6081775807080527818_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_579_case__prod__unfold,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [C2: nat > nat > $o,P3: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_580_case__prod__beta_H,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,X4: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X4 ) @ ( product_snd_nat_nat @ X4 ) ) ) ) ).

% case_prod_beta'
thf(fact_581_pred__subset__eq2,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R2 )
        @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_582_pred__subset__eq2,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R2 )
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ S4 ) )
      = ( ord_le5997549366648089703at_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_583_pred__subset__eq2,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R2 )
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ S4 ) )
      = ( ord_le1268244103169919719at_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_584_pred__subset__eq2,axiom,
    ! [R2: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
      ( ( ord_le2712619360209801397char_o
        @ ^ [X4: nat,Y7: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X4 @ Y7 ) @ R2 )
        @ ^ [X4: nat,Y7: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X4 @ Y7 ) @ S4 ) )
      = ( ord_le6315511645215477266t_char @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_585_pred__subset__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R2 )
        @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_586_pred__subset__eq2,axiom,
    ! [R2: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
      ( ( ord_le3852884270828607079_val_o
        @ ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y7 ) @ R2 )
        @ ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X4 @ Y7 ) @ S4 ) )
      = ( ord_le6900622908513426738on_val @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_587_Grp__UNIV__id,axiom,
    ! [F: nat > nat] :
      ( ( F = id_nat )
     => ( ( relcompp_nat_nat_nat @ ( conversep_nat_nat @ ( bNF_Grp_nat_nat @ top_top_set_nat @ F ) ) @ ( bNF_Grp_nat_nat @ top_top_set_nat @ F ) )
        = ( bNF_Grp_nat_nat @ top_top_set_nat @ F ) ) ) ).

% Grp_UNIV_id
thf(fact_588_Grp__UNIV__id,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( F = id_Pro7559728917991913458at_nat )
     => ( ( relcom282571730187242960at_nat @ ( conver6965824823119931701at_nat @ ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ F ) ) @ ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ F ) )
        = ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ F ) ) ) ).

% Grp_UNIV_id
thf(fact_589_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A4: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_590_lex__prod__def,axiom,
    ( lex_pr4669217892513370978at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2200730763347676620at_nat
          @ ( produc3600351907215576810_nat_o
            @ ( produc6198819664464039784_nat_o
              @ ^ [A4: product_prod_nat_nat,B4: nat] :
                  ( produc8758269395934548017_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_591_lex__prod__def,axiom,
    ( lex_pr8029265285556086080at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec3253044014973990424at_nat
          @ ( produc2869316064735525022_nat_o
            @ ( produc7480869177107369060_nat_o
              @ ^ [A4: nat,B4: product_prod_nat_nat] :
                  ( produc5864757623865647827_nat_o
                  @ ^ [A7: nat,B6: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_592_lex__prod__def,axiom,
    ( lex_pr1951919194703979714at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec1461227866750739596at_nat
          @ ( produc8701595406162815018_nat_o
            @ ( produc7110675705651593384_nat_o
              @ ^ [A4: set_Pr1261947904930325089at_nat,B4: nat] :
                  ( produc3595121349294487249_nat_o
                  @ ^ [A7: set_Pr1261947904930325089at_nat,B6: nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_593_lex__prod__def,axiom,
    ( lex_pr4265550801952275104at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec2088441199205367000at_nat
          @ ( produc7830147906768200158_nat_o
            @ ( produc1908958340173245860_nat_o
              @ ^ [A4: nat,B4: set_Pr1261947904930325089at_nat] :
                  ( produc7997411485876045171_nat_o
                  @ ^ [A7: nat,B6: set_Pr1261947904930325089at_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_594_lex__prod__def,axiom,
    ( lex_pr8801849515957261039at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7334067512558549330at_nat
          @ ( produc3603361296207914284_nat_o
            @ ( produc5077327844045112642_nat_o
              @ ^ [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
                  ( produc6590410687421337004_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_595_lex__prod__def,axiom,
    ( lex_pr864846108736269391at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec1712228713741422930at_nat
          @ ( produc3676778756977593644_nat_o
            @ ( produc8821334275656303938_nat_o
              @ ^ [A4: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
                  ( produc7695416289158663244_nat_o
                  @ ^ [A7: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_596_lex__prod__def,axiom,
    ( lex_pr8442690889092155791at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec2974368035702751570at_nat
          @ ( produc4641403137511177516_nat_o
            @ ( produc3500598050016679234_nat_o
              @ ^ [A4: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
                  ( produc743548188471107340_nat_o
                  @ ^ [A7: set_Pr1261947904930325089at_nat,B6: product_prod_nat_nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_597_lex__prod__def,axiom,
    ( lex_pr8000699672184251119at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec223960383693904210at_nat
          @ ( produc1561160019221259564_nat_o
            @ ( produc1971064617671185730_nat_o
              @ ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
                  ( produc410239310623530412_nat_o
                  @ ^ [A7: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_598_lex__prod__def,axiom,
    ( lex_pr1916850892004341160at_nat
    = ( ^ [Ra: set_Pr7459493094073627847at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec6386531722299773144at_nat
          @ ( produc6304834068385745886_nat_o
            @ ( produc3582348002201993308_nat_o
              @ ^ [A4: set_Pr4329608150637261639at_nat,B4: nat] :
                  ( produc6668815691043536491_nat_o
                  @ ^ [A7: set_Pr4329608150637261639at_nat,B6: nat] :
                      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A4 @ A7 ) @ Ra )
                      | ( ( A4 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_599_inv__image__def,axiom,
    ( inv_im8721129524585065343at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y7: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X4 ) @ ( F2 @ Y7 ) ) @ R3 ) ) ) ) ) ).

% inv_image_def
thf(fact_600_inv__image__def,axiom,
    ( inv_im2903009620893676101at_nat
    = ( ^ [R3: set_Pr7459493094073627847at_nat,F2: nat > set_Pr4329608150637261639at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y7: nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( F2 @ X4 ) @ ( F2 @ Y7 ) ) @ R3 ) ) ) ) ) ).

% inv_image_def
thf(fact_601_inv__image__def,axiom,
    ( inv_im4437241714082187487at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y7: nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ X4 ) @ ( F2 @ Y7 ) ) @ R3 ) ) ) ) ) ).

% inv_image_def
thf(fact_602_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X4 ) @ ( F2 @ Y7 ) ) @ R3 ) ) ) ) ) ).

% inv_image_def
thf(fact_603_case__prod__o__map__prod,axiom,
    ! [F: nat > nat > $o,G1: nat > nat,G2: nat > nat] :
      ( ( comp_P6344650113790022471at_nat @ ( produc6081775807080527818_nat_o @ F ) @ ( produc6977886695330630970at_nat @ G1 @ G2 ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [L: nat,R3: nat] : ( F @ ( G1 @ L ) @ ( G2 @ R3 ) ) ) ) ).

% case_prod_o_map_prod
thf(fact_604_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P4: nat > $o,R4: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X6: nat,Y9: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X4: nat,Y7: nat] :
                      ( ( X6 = X4 )
                      & ( P4 @ X4 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y9 @ Y7 ) @ ( R4 @ X4 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_605_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,P2: nat > nat > $o,Q2: nat > nat > $o,A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re578469030762574527_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_606_predicate2__transferD,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: nat > nat > $o,P2: product_prod_nat_nat > nat > $o,Q2: product_prod_nat_nat > nat > $o,A: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re1392313733760654173_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8206827879206165904at_nat @ A @ A5 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le3000389064537975527at_nat @ A5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P2 @ ( produc3213797794245857475at_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc6408287024330202629at_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_607_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o,P2: nat > product_prod_nat_nat > $o,Q2: nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( bNF_re3272885123108610425_nat_o @ R1
        @ ( bNF_re3666534408544137501at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ( ( member8206827879206165904at_nat @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le3000389064537975527at_nat @ B5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R22 ) ) )
             => ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( produc3213797794245857475at_nat @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( produc6408287024330202629at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_608_predicate2__transferD,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,R22: nat > nat > $o,P2: set_Pr1261947904930325089at_nat > nat > $o,Q2: set_Pr1261947904930325089at_nat > nat > $o,A: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re3326851373202864989_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8757157785044589968at_nat @ A @ A5 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le1268244103169919719at_nat @ A5 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P2 @ ( produc6011863565407846595at_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc2937545259423399429at_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_609_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P2: nat > set_Pr1261947904930325089at_nat > $o,Q2: nat > set_Pr1261947904930325089at_nat > $o,A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( bNF_re7533008290153783993_nat_o @ R1
        @ ( bNF_re2156183356565917981at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ( ( member8757157785044589968at_nat @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le1268244103169919719at_nat @ B5 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R22 ) ) )
             => ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( produc6011863565407846595at_nat @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( produc2937545259423399429at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_610_predicate2__transferD,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o,P2: product_prod_nat_nat > product_prod_nat_nat > $o,Q2: product_prod_nat_nat > product_prod_nat_nat > $o,A: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( bNF_re4202695980764964119_nat_o @ R1
        @ ( bNF_re3666534408544137501at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8206827879206165904at_nat @ A @ A5 )
       => ( ( member8206827879206165904at_nat @ B @ B5 )
         => ( ( ord_le3000389064537975527at_nat @ A5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R1 ) ) )
           => ( ( ord_le3000389064537975527at_nat @ B5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R22 ) ) )
             => ( ( P2 @ ( produc3213797794245857475at_nat @ A ) @ ( produc3213797794245857475at_nat @ B ) )
                = ( Q2 @ ( produc6408287024330202629at_nat @ A ) @ ( produc6408287024330202629at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_611_predicate2__transferD,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P2: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o,Q2: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o,A: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( bNF_re6631849262158050903_nat_o @ R1
        @ ( bNF_re2156183356565917981at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8206827879206165904at_nat @ A @ A5 )
       => ( ( member8757157785044589968at_nat @ B @ B5 )
         => ( ( ord_le3000389064537975527at_nat @ A5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R1 ) ) )
           => ( ( ord_le1268244103169919719at_nat @ B5 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R22 ) ) )
             => ( ( P2 @ ( produc3213797794245857475at_nat @ A ) @ ( produc6011863565407846595at_nat @ B ) )
                = ( Q2 @ ( produc6408287024330202629at_nat @ A ) @ ( produc2937545259423399429at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_612_predicate2__transferD,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o,P2: set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o,Q2: set_Pr1261947904930325089at_nat > product_prod_nat_nat > $o,A: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat,B: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( bNF_re1007418676577076503_nat_o @ R1
        @ ( bNF_re3666534408544137501at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8757157785044589968at_nat @ A @ A5 )
       => ( ( member8206827879206165904at_nat @ B @ B5 )
         => ( ( ord_le1268244103169919719at_nat @ A5 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R1 ) ) )
           => ( ( ord_le3000389064537975527at_nat @ B5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R22 ) ) )
             => ( ( P2 @ ( produc6011863565407846595at_nat @ A ) @ ( produc3213797794245857475at_nat @ B ) )
                = ( Q2 @ ( produc2937545259423399429at_nat @ A ) @ ( produc6408287024330202629at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_613_predicate2__transferD,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R22: nat > nat > $o,P2: ( produc1457211279475724562t_char > prat ) > nat > $o,Q2: ( produc1457211279475724562t_char > option_val ) > nat > $o,A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re753112871316139152_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member2879030624002546611on_val @ A @ A5 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P2 @ ( produc5239619271231976746on_val @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc2319605628138516840on_val @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_614_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P2: nat > ( produc1457211279475724562t_char > prat ) > $o,Q2: nat > ( produc1457211279475724562t_char > option_val ) > $o,A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( bNF_re4892053619688935818_val_o @ R1
        @ ( bNF_re6067813816088466256al_o_o @ R22
          @ ^ [Y4: $o,Z: $o] : ( Y4 = Z ) )
        @ P2
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ( ( member2879030624002546611on_val @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le6900622908513426738on_val @ B5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R22 ) ) )
             => ( ( P2 @ ( product_fst_nat_nat @ A ) @ ( produc5239619271231976746on_val @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( produc2319605628138516840on_val @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_615_snd__fst__flip,axiom,
    ( produc826222791318859909at_nat
    = ( comp_P3728553842126764068at_nat @ produc4486455499028916547at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y7 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_616_snd__fst__flip,axiom,
    ( produc2937545259423399429at_nat
    = ( comp_P4974406650942564362at_nat @ produc6011863565407846595at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y7 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_617_snd__fst__flip,axiom,
    ( produc1259554233722553080ar_nat
    = ( comp_P3187634502937433873ar_nat @ produc6204653154488278070t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X4: list_char,Y7: nat] : ( produc120197956887798346t_char @ Y7 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_618_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y7: nat] : ( product_Pair_nat_nat @ Y7 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_619_snd__fst__flip,axiom,
    ( produc4142415063207305778r_prat
    = ( comp_P4800886777526708674r_prat @ produc5239619271231976746on_val
      @ ( produc4110721796449090601on_val
        @ ^ [X4: produc1457211279475724562t_char > option_val,Y7: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y7 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_620_snd__fst__flip,axiom,
    ( produc2319605628138516840on_val
    = ( comp_P1319385742190247241on_val @ produc7062428706300765684r_prat
      @ ( produc2964738055033037937r_prat
        @ ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y7 @ X4 ) ) ) ) ).

% snd_fst_flip
thf(fact_621_fst__snd__flip,axiom,
    ( produc4486455499028916547at_nat
    = ( comp_P3728553842126764068at_nat @ produc826222791318859909at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y7 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_622_fst__snd__flip,axiom,
    ( produc6011863565407846595at_nat
    = ( comp_P4974406650942564362at_nat @ produc2937545259423399429at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y7 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_623_fst__snd__flip,axiom,
    ( produc5563429021163328950ar_nat
    = ( comp_P26140635296239040ar_nat @ produc1900778367047502200t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X4: list_char,Y7: nat] : ( produc120197956887798346t_char @ Y7 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_624_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y7: nat] : ( product_Pair_nat_nat @ Y7 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_625_fst__snd__flip,axiom,
    ( produc5239619271231976746on_val
    = ( comp_P1591131706424975134on_val @ produc4142415063207305778r_prat
      @ ( produc2964738055033037937r_prat
        @ ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y7 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_626_fst__snd__flip,axiom,
    ( produc7062428706300765684r_prat
    = ( comp_P8287346254047880777r_prat @ produc2319605628138516840on_val
      @ ( produc4110721796449090601on_val
        @ ^ [X4: produc1457211279475724562t_char > option_val,Y7: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y7 @ X4 ) ) ) ) ).

% fst_snd_flip
thf(fact_627_eq__alt,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat ) ) ).

% eq_alt
thf(fact_628_eq__alt,axiom,
    ( ( ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z ) )
    = ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ id_Pro7559728917991913458at_nat ) ) ).

% eq_alt
thf(fact_629_Grp__UNIV__idI,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat @ X @ Y ) ) ).

% Grp_UNIV_idI
thf(fact_630_Grp__UNIV__idI,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( X = Y )
     => ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ id_Pro7559728917991913458at_nat @ X @ Y ) ) ).

% Grp_UNIV_idI
thf(fact_631_fst__diag__fst,axiom,
    ( ( comp_P5489260687716092971on_val @ produc2193917115357690435r_prat
      @ ( comp_P4016055770202289235on_val
        @ ^ [X4: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ X4 @ X4 )
        @ produc5239619271231976746on_val ) )
    = produc5239619271231976746on_val ) ).

% fst_diag_fst
thf(fact_632_snd__diag__snd,axiom,
    ( ( comp_P3343515995226764714on_val @ produc1305954679496617083on_val
      @ ( comp_P3324790395486970996on_val
        @ ^ [X4: produc1457211279475724562t_char > option_val] : ( produc8008623296994167373on_val @ X4 @ X4 )
        @ produc2319605628138516840on_val ) )
    = produc2319605628138516840on_val ) ).

% snd_diag_snd
thf(fact_633_mult__distr__masks,axiom,
    ! [A: prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B ) @ ( multiply_mask @ A @ C ) ) ) ).

% mult_distr_masks
thf(fact_634_fun_Orel__compp__Grp,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( bNF_re2896540935372247726on_val
        @ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
        @ R2 )
      = ( relcom7349770684181336927on_val
        @ ( conver8203838583239980984r_prat
          @ ( bNF_Gr5444735849185432863r_prat
            @ ( collec2477365322772234464on_val
              @ ^ [X4: nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ X4 @ top_top_set_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) )
            @ ( comp_P4447563945128311908at_nat @ produc5239619271231976746on_val ) ) )
        @ ( bNF_Gr2620532623635037156on_val
          @ ( collec2477365322772234464on_val
            @ ^ [X4: nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ X4 @ top_top_set_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) )
          @ ( comp_P6187037784939234283al_nat @ produc2319605628138516840on_val ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_635_fun_Orel__compp__Grp,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( bNF_re8250582710452816528on_val
        @ ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z )
        @ R2 )
      = ( relcom756628661608191142on_val
        @ ( conver3034012162531316304r_prat
          @ ( bNF_Gr2950598962552920375r_prat
            @ ( collec5030907905214230265on_val
              @ ^ [X4: product_prod_nat_nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_1778330534009140952on_val @ X4 @ top_to4669805908274784177at_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) )
            @ ( comp_P7748810038695468909at_nat @ produc5239619271231976746on_val ) ) )
        @ ( bNF_Gr5270771597446195660on_val
          @ ( collec5030907905214230265on_val
            @ ^ [X4: product_prod_nat_nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_1778330534009140952on_val @ X4 @ top_to4669805908274784177at_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) )
          @ ( comp_P157993088335196966at_nat @ produc2319605628138516840on_val ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_636_fun_Orel__compp__Grp,axiom,
    ! [R2: nat > nat > $o] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y4: nat,Z: nat] : ( Y4 = Z )
        @ R2 )
      = ( relcom8186637903969514807at_nat
        @ ( conver3498139987538047730at_nat
          @ ( bNF_Gr3388551082464497497at_nat
            @ ( collec8718012497556192979at_nat
              @ ^ [X4: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X4 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) )
            @ ( comp_P1955610211387027840at_nat @ product_fst_nat_nat ) ) )
        @ ( bNF_Gr3388551082464497497at_nat
          @ ( collec8718012497556192979at_nat
            @ ^ [X4: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X4 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) )
          @ ( comp_P1955610211387027840at_nat @ product_snd_nat_nat ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_637_fun_Orel__compp__Grp,axiom,
    ! [R2: nat > nat > $o] :
      ( ( bNF_re8246922863344978751at_nat
        @ ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z )
        @ R2 )
      = ( relcom4236271064425356490at_nat
        @ ( conver4470774000191062694at_nat
          @ ( bNF_Gr7802473308256045197at_nat
            @ ( collec248804655031457914at_nat
              @ ^ [X4: product_prod_nat_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ X4 @ top_to4669805908274784177at_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) )
            @ ( comp_P1567445206330693457at_nat @ product_fst_nat_nat ) ) )
        @ ( bNF_Gr7802473308256045197at_nat
          @ ( collec248804655031457914at_nat
            @ ^ [X4: product_prod_nat_nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5168914502847457605at_nat @ X4 @ top_to4669805908274784177at_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) )
          @ ( comp_P1567445206330693457at_nat @ product_snd_nat_nat ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_638_snd__diag__fst,axiom,
    ( ( comp_P5489260687716092971on_val @ produc5358294431189718917r_prat
      @ ( comp_P4016055770202289235on_val
        @ ^ [X4: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ X4 @ X4 )
        @ produc5239619271231976746on_val ) )
    = produc5239619271231976746on_val ) ).

% snd_diag_fst
thf(fact_639_fst__diag__snd,axiom,
    ( ( comp_P3343515995226764714on_val @ produc6116476718443979321on_val
      @ ( comp_P3324790395486970996on_val
        @ ^ [X4: produc1457211279475724562t_char > option_val] : ( produc8008623296994167373on_val @ X4 @ X4 )
        @ produc2319605628138516840on_val ) )
    = produc2319605628138516840on_val ) ).

% fst_diag_snd
thf(fact_640_Grp__fst__snd,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( relcom4197558024698484890on_val @ ( conver1816161830495851136r_prat @ ( bNF_Gr2025013579966528743r_prat @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) @ produc5239619271231976746on_val ) ) @ ( bNF_Gr1324552783004614022on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) @ produc2319605628138516840on_val ) )
      = R2 ) ).

% Grp_fst_snd
thf(fact_641_Grp__fst__snd,axiom,
    ! [R2: nat > nat > $o] :
      ( ( relcom8572034174491624592at_nat @ ( conver4691796276509297820at_nat @ ( bNF_Gr193424670081168899at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) @ product_fst_nat_nat ) ) @ ( bNF_Gr193424670081168899at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) @ product_snd_nat_nat ) )
      = R2 ) ).

% Grp_fst_snd
thf(fact_642_Quotient__alt__def5,axiom,
    ( quotient_nat_nat
    = ( ^ [R4: nat > nat > $o,Abs: nat > nat,Rep: nat > nat,T3: nat > nat > $o] :
          ( ( ord_le2646555220125990790_nat_o @ T3 @ ( bNF_Grp_nat_nat @ top_top_set_nat @ Abs ) )
          & ( ord_le2646555220125990790_nat_o @ ( bNF_Grp_nat_nat @ top_top_set_nat @ Rep ) @ ( conversep_nat_nat @ T3 ) )
          & ( R4
            = ( relcompp_nat_nat_nat @ T3 @ ( conversep_nat_nat @ T3 ) ) ) ) ) ) ).

% Quotient_alt_def5
thf(fact_643_Quotient__alt__def5,axiom,
    ( quotie4557386371875466009at_nat
    = ( ^ [R4: nat > nat > $o,Abs: nat > product_prod_nat_nat,Rep: product_prod_nat_nat > nat,T3: nat > product_prod_nat_nat > $o] :
          ( ( ord_le4212822800974190475_nat_o @ T3 @ ( bNF_Gr3553472063123884001at_nat @ top_top_set_nat @ Abs ) )
          & ( ord_le2374372785170350083_nat_o @ ( bNF_Gr193424670081168899at_nat @ top_to4669805908274784177at_nat @ Rep ) @ ( conver8051843669552012922at_nat @ T3 ) )
          & ( R4
            = ( relcom8572034174491624592at_nat @ T3 @ ( conver8051843669552012922at_nat @ T3 ) ) ) ) ) ) ).

% Quotient_alt_def5
thf(fact_644_Quotient__alt__def5,axiom,
    ( quotie1197338978832750907at_nat
    = ( ^ [R4: product_prod_nat_nat > product_prod_nat_nat > $o,Abs: product_prod_nat_nat > nat,Rep: nat > product_prod_nat_nat,T3: product_prod_nat_nat > nat > $o] :
          ( ( ord_le2374372785170350083_nat_o @ T3 @ ( bNF_Gr193424670081168899at_nat @ top_to4669805908274784177at_nat @ Abs ) )
          & ( ord_le4212822800974190475_nat_o @ ( bNF_Gr3553472063123884001at_nat @ top_top_set_nat @ Rep ) @ ( conver4691796276509297820at_nat @ T3 ) )
          & ( R4
            = ( relcom3050870033064515487at_nat @ T3 @ ( conver4691796276509297820at_nat @ T3 ) ) ) ) ) ) ).

% Quotient_alt_def5
thf(fact_645_Quotient__alt__def5,axiom,
    ( quotie4166942682936818134at_nat
    = ( ^ [R4: product_prod_nat_nat > product_prod_nat_nat > $o,Abs: product_prod_nat_nat > product_prod_nat_nat,Rep: product_prod_nat_nat > product_prod_nat_nat,T3: product_prod_nat_nat > product_prod_nat_nat > $o] :
          ( ( ord_le5604493270027003598_nat_o @ T3 @ ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ Abs ) )
          & ( ord_le5604493270027003598_nat_o @ ( bNF_Gr7275288873855648526at_nat @ top_to4669805908274784177at_nat @ Rep ) @ ( conver6965824823119931701at_nat @ T3 ) )
          & ( R4
            = ( relcom282571730187242960at_nat @ T3 @ ( conver6965824823119931701at_nat @ T3 ) ) ) ) ) ) ).

% Quotient_alt_def5
thf(fact_646_asymp__asym__eq,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ( asymp_7147290840880511973at_nat @ top_to1754375979021635447at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R ) ) ).

% asymp_asym_eq
thf(fact_647_asymp__asym__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( asymp_8721585889738326015at_nat @ top_to7629004291339433233at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% asymp_asym_eq
thf(fact_648_asymp__asym__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ top_top_set_nat
        @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_on_nat @ top_top_set_nat @ R ) ) ).

% asymp_asym_eq
thf(fact_649_asymp__asym__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( asymp_3201933403969103519at_nat @ top_to4669805908274784177at_nat
        @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% asymp_asym_eq
thf(fact_650_asym__on__lex__prod,axiom,
    ! [A5: set_nat,R_A: set_Pr1261947904930325089at_nat,B5: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ A5 @ R_A )
     => ( ( asym_on_nat @ B5 @ R_B )
       => ( asym_o8546584678760569651at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : B5 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% asym_on_lex_prod
thf(fact_651_Gr__incl,axiom,
    ! [A5: set_nat,F: nat > nat,B5: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( bNF_Gr_nat_nat @ A5 @ F )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 ) ) ).

% Gr_incl
thf(fact_652_SigmaI,axiom,
    ! [A: nat,A5: set_nat,B: nat,B5: nat > set_nat] :
      ( ( member_nat @ A @ A5 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_653_SigmaI,axiom,
    ! [A: nat,A5: set_nat,B: list_char,B5: nat > set_list_char] :
      ( ( member_nat @ A @ A5 )
     => ( ( member_list_char @ B @ ( B5 @ A ) )
       => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_654_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A5 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_655_SigmaI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A5 )
     => ( ( member2643936169264416010at_nat @ B @ ( B5 @ A ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_656_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc859450856879609959at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A5 )
     => ( ( member8206827879206165904at_nat @ B @ ( B5 @ A ) )
       => ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_657_SigmaI,axiom,
    ! [A: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A5 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_658_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat,B5: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A5 )
     => ( ( member8757157785044589968at_nat @ B @ ( B5 @ A ) )
       => ( member600271025443689462at_nat @ ( produc8594291983450332477at_nat @ A @ B ) @ ( produc3090783324798197870at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_659_SigmaI,axiom,
    ! [A: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat,B: product_prod_nat_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A5 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member1953685042357896106at_nat @ ( produc5023393015413421041at_nat @ A @ B ) @ ( produc8743256393616062242at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_660_SigmaI,axiom,
    ! [A: produc1457211279475724562t_char > prat,A5: set_Pr3433046674199179443r_prat,B: produc1457211279475724562t_char > option_val,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2336342582916103932r_prat @ A @ A5 )
     => ( ( member1577473829074448177on_val @ B @ ( B5 @ A ) )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_661_SigmaI,axiom,
    ! [A: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B: produc859450856879609959at_nat,B5: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A5 )
     => ( ( member8206827879206165904at_nat @ B @ ( B5 @ A ) )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( produc18557253517588744at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_662_mem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A5: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
      = ( ( member_nat @ A @ A5 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_663_mem__Sigma__iff,axiom,
    ! [A: nat,B: list_char,A5: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A5 @ B5 ) )
      = ( ( member_nat @ A @ A5 )
        & ( member_list_char @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_664_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A5 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_665_mem__Sigma__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A5 @ B5 ) )
      = ( ( member2643936169264416010at_nat @ A @ A5 )
        & ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_666_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A5 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A5 )
        & ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_667_mem__Sigma__iff,axiom,
    ! [A: produc859450856879609959at_nat,B: product_prod_nat_nat,A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A5 @ B5 ) )
      = ( ( member8206827879206165904at_nat @ A @ A5 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_668_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: produc3843707927480180839at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( member600271025443689462at_nat @ ( produc8594291983450332477at_nat @ A @ B ) @ ( produc3090783324798197870at_nat @ A5 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A5 )
        & ( member8757157785044589968at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_669_mem__Sigma__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: product_prod_nat_nat,A5: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member1953685042357896106at_nat @ ( produc5023393015413421041at_nat @ A @ B ) @ ( produc8743256393616062242at_nat @ A5 @ B5 ) )
      = ( ( member8757157785044589968at_nat @ A @ A5 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_670_mem__Sigma__iff,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
      = ( ( member2336342582916103932r_prat @ A @ A5 )
        & ( member1577473829074448177on_val @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_671_mem__Sigma__iff,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( produc18557253517588744at_nat @ A5 @ B5 ) )
      = ( ( member8206827879206165904at_nat @ A @ A5 )
        & ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_672_Collect__case__prod,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: produc9003781588162623004on_val > $o] :
      ( ( collec5166231387257825412on_val
        @ ( produc1845727980445322802_val_o
          @ ^ [A4: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc4067394217145614850on_val @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [Uu: produc9003781588162623004on_val] : ( collec7475294502966882545on_val @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_673_Collect__case__prod,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: product_prod_nat_nat > $o] :
      ( ( collec927009928073930287at_nat
        @ ( produc8420946574280188703_nat_o
          @ ^ [A4: produc9003781588162623004on_val,B4: product_prod_nat_nat] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc8411625936386031515at_nat @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [Uu: produc9003781588162623004on_val] : ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_674_Collect__case__prod,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: nat > $o] :
      ( ( collec8317772394953758008al_nat
        @ ( produc5057885034201482878_nat_o
          @ ^ [A4: produc9003781588162623004on_val,B4: nat] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc7812486985998511542al_nat @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [Uu: produc9003781588162623004on_val] : ( collect_nat @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_675_Collect__case__prod,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: produc9003781588162623004on_val > $o] :
      ( ( collec7293267056801676193on_val
        @ ( produc3458391012123660517_val_o
          @ ^ [A4: product_prod_nat_nat,B4: produc9003781588162623004on_val] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc7035663349334832981on_val @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu: product_prod_nat_nat] : ( collec7475294502966882545on_val @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_676_Collect__case__prod,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_677_Collect__case__prod,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [A4: product_prod_nat_nat,B4: nat] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_678_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q2: produc9003781588162623004on_val > $o] :
      ( ( collec5872124358691874360on_val
        @ ( produc1906844705236591486_val_o
          @ ^ [A4: nat,B4: produc9003781588162623004on_val] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc2110153610967699638on_val @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collec7475294502966882545on_val @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_679_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [A4: nat,B4: product_prod_nat_nat] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_680_Collect__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,Q2: ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( collec7475294502966882545on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [A4: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc1366239611153933861on_val @ ( collec7093475260167521854r_prat @ P2 )
        @ ^ [Uu: produc1457211279475724562t_char > prat] : ( collec8455114619259388271on_val @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_681_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A4: nat,B4: nat] :
              ( ( P2 @ A4 )
              & ( Q2 @ B4 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
        @ ^ [Uu: nat] : ( collect_nat @ Q2 ) ) ) ).

% Collect_case_prod
thf(fact_682_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_683_UNIV__Times__UNIV,axiom,
    ( ( produc1809337555817847783at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_to4669805908274784177at_nat )
    = top_to7745748144976627212at_nat ) ).

% UNIV_Times_UNIV
thf(fact_684_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_685_UNIV__Times__UNIV,axiom,
    ( ( produc2761391749766926216at_nat @ top_to4669805908274784177at_nat
      @ ^ [Uu: product_prod_nat_nat] : top_to4669805908274784177at_nat )
    = top_to8454300755403694231at_nat ) ).

% UNIV_Times_UNIV
thf(fact_686_Times__eq__cancel2,axiom,
    ! [X: nat,C3: set_nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ X @ C3 )
     => ( ( ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : C3 )
          = ( produc457027306803732586at_nat @ B5
            @ ^ [Uu: nat] : C3 ) )
        = ( A5 = B5 ) ) ) ).

% Times_eq_cancel2
thf(fact_687_Sigma__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C3: nat > set_nat,D2: nat > set_nat] :
      ( ( A5 = B5 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D2 @ X3 ) ) )
       => ( ( produc457027306803732586at_nat @ A5 @ C3 )
          = ( produc457027306803732586at_nat @ B5 @ D2 ) ) ) ) ).

% Sigma_cong
thf(fact_688_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A5: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_689_SigmaE,axiom,
    ! [C: produc1457211279475724562t_char,A5: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ C @ ( produc6030673636645808891t_char @ A5 @ B5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ! [Y3: list_char] :
                ( ( member_list_char @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc120197956887798346t_char @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_690_SigmaE,axiom,
    ! [C: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_691_SigmaE,axiom,
    ! [C: produc3843707927480180839at_nat,A5: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( produc7726050920695093128at_nat @ A5 @ B5 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A5 )
           => ! [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_692_SigmaE,axiom,
    ! [C: produc39072733474941005at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ C @ ( produc7465995907834253422at_nat @ A5 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc5496787636967163197at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_693_SigmaE,axiom,
    ! [C: produc7148259590854449153at_nat,A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ C @ ( produc3829605335166519074at_nat @ A5 @ B5 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A5 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc1860397064299428849at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_694_SigmaE,axiom,
    ! [C: produc8430705068416227405at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( member600271025443689462at_nat @ C @ ( produc3090783324798197870at_nat @ A5 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ! [Y3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc8594291983450332477at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_695_SigmaE,axiom,
    ! [C: produc560747048475658241at_nat,A5: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member1953685042357896106at_nat @ C @ ( produc8743256393616062242at_nat @ A5 @ B5 ) )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A5 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc5023393015413421041at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_696_SigmaE,axiom,
    ! [C: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ C @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X3 @ A5 )
           => ! [Y3: produc1457211279475724562t_char > option_val] :
                ( ( member1577473829074448177on_val @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc74620467203698198on_val @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_697_SigmaE,axiom,
    ! [C: produc349518998152878311at_nat,A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C @ ( produc18557253517588744at_nat @ A5 @ B5 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A5 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc4662710985925991255at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_698_SigmaD1,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
     => ( member8440522571783428010at_nat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_699_SigmaD1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat,B5: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc6258361011560675592at_nat @ A5 @ B5 ) )
     => ( member6273874620674182256at_nat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_700_SigmaD1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A5 @ B5 ) )
     => ( member2643936169264416010at_nat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_701_SigmaD1,axiom,
    ! [A: nat,B: list_char,A5: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A5 @ B5 ) )
     => ( member_nat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_702_SigmaD1,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
     => ( member2336342582916103932r_prat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_703_SigmaD1,axiom,
    ! [A: nat,B: nat,A5: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
     => ( member_nat @ A @ A5 ) ) ).

% SigmaD1
thf(fact_704_SigmaD2,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
     => ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_705_SigmaD2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat,B5: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc6258361011560675592at_nat @ A5 @ B5 ) )
     => ( member6273874620674182256at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_706_SigmaD2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A5 @ B5 ) )
     => ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_707_SigmaD2,axiom,
    ! [A: nat,B: list_char,A5: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A5 @ B5 ) )
     => ( member_list_char @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_708_SigmaD2,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
     => ( member1577473829074448177on_val @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_709_SigmaD2,axiom,
    ! [A: nat,B: nat,A5: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
     => ( member_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_710_SigmaE2,axiom,
    ! [A: nat,B: nat,A5: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
     => ~ ( ( member_nat @ A @ A5 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_711_SigmaE2,axiom,
    ! [A: nat,B: list_char,A5: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A5 @ B5 ) )
     => ~ ( ( member_nat @ A @ A5 )
         => ~ ( member_list_char @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_712_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A5 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_713_SigmaE2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A5 @ B5 ) )
     => ~ ( ( member2643936169264416010at_nat @ A @ A5 )
         => ~ ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_714_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A5 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A5 )
         => ~ ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_715_SigmaE2,axiom,
    ! [A: produc859450856879609959at_nat,B: product_prod_nat_nat,A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A5 @ B5 ) )
     => ~ ( ( member8206827879206165904at_nat @ A @ A5 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_716_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: produc3843707927480180839at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( member600271025443689462at_nat @ ( produc8594291983450332477at_nat @ A @ B ) @ ( produc3090783324798197870at_nat @ A5 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A5 )
         => ~ ( member8757157785044589968at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_717_SigmaE2,axiom,
    ! [A: produc3843707927480180839at_nat,B: product_prod_nat_nat,A5: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member1953685042357896106at_nat @ ( produc5023393015413421041at_nat @ A @ B ) @ ( produc8743256393616062242at_nat @ A5 @ B5 ) )
     => ~ ( ( member8757157785044589968at_nat @ A @ A5 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_718_SigmaE2,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
     => ~ ( ( member2336342582916103932r_prat @ A @ A5 )
         => ~ ( member1577473829074448177on_val @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_719_SigmaE2,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( produc18557253517588744at_nat @ A5 @ B5 ) )
     => ~ ( ( member8206827879206165904at_nat @ A @ A5 )
         => ~ ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_720_Collect__case__prod__Sigma,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( collec5166231387257825412on_val
        @ ( produc1845727980445322802_val_o
          @ ^ [X4: produc9003781588162623004on_val,Y7: produc9003781588162623004on_val] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc4067394217145614850on_val @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [X4: produc9003781588162623004on_val] : ( collec7475294502966882545on_val @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_721_Collect__case__prod__Sigma,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: produc9003781588162623004on_val > product_prod_nat_nat > $o] :
      ( ( collec927009928073930287at_nat
        @ ( produc8420946574280188703_nat_o
          @ ^ [X4: produc9003781588162623004on_val,Y7: product_prod_nat_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc8411625936386031515at_nat @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [X4: produc9003781588162623004on_val] : ( collec3392354462482085612at_nat @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_722_Collect__case__prod__Sigma,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q2: produc9003781588162623004on_val > nat > $o] :
      ( ( collec8317772394953758008al_nat
        @ ( produc5057885034201482878_nat_o
          @ ^ [X4: produc9003781588162623004on_val,Y7: nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc7812486985998511542al_nat @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [X4: produc9003781588162623004on_val] : ( collect_nat @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_723_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > produc9003781588162623004on_val > $o] :
      ( ( collec7293267056801676193on_val
        @ ( produc3458391012123660517_val_o
          @ ^ [X4: product_prod_nat_nat,Y7: produc9003781588162623004on_val] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc7035663349334832981on_val @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X4: product_prod_nat_nat] : ( collec7475294502966882545on_val @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_724_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X4: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_725_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [X4: product_prod_nat_nat,Y7: nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X4: product_prod_nat_nat] : ( collect_nat @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_726_Collect__case__prod__Sigma,axiom,
    ! [P2: nat > $o,Q2: nat > produc9003781588162623004on_val > $o] :
      ( ( collec5872124358691874360on_val
        @ ( produc1906844705236591486_val_o
          @ ^ [X4: nat,Y7: produc9003781588162623004on_val] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc2110153610967699638on_val @ ( collect_nat @ P2 )
        @ ^ [X4: nat] : ( collec7475294502966882545on_val @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_727_Collect__case__prod__Sigma,axiom,
    ! [P2: nat > $o,Q2: nat > product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [X4: nat,Y7: product_prod_nat_nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P2 )
        @ ^ [X4: nat] : ( collec3392354462482085612at_nat @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_728_Collect__case__prod__Sigma,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,Q2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( collec7475294502966882545on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [X4: produc1457211279475724562t_char > prat,Y7: produc1457211279475724562t_char > option_val] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc1366239611153933861on_val @ ( collec7093475260167521854r_prat @ P2 )
        @ ^ [X4: produc1457211279475724562t_char > prat] : ( collec8455114619259388271on_val @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_729_Collect__case__prod__Sigma,axiom,
    ! [P2: nat > $o,Q2: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( P2 @ X4 )
              & ( Q2 @ X4 @ Y7 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
        @ ^ [X4: nat] : ( collect_nat @ ( Q2 @ X4 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_730_Sigma__mono,axiom,
    ! [A5: set_nat,C3: set_nat,B5: nat > set_nat,D2: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A5 @ C3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ( ord_less_eq_set_nat @ ( B5 @ X3 ) @ ( D2 @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A5 @ B5 ) @ ( produc457027306803732586at_nat @ C3 @ D2 ) ) ) ) ).

% Sigma_mono
thf(fact_731_Times__subset__cancel2,axiom,
    ! [X: nat,C3: set_nat,A5: set_nat,B5: set_nat] :
      ( ( member_nat @ X @ C3 )
     => ( ( ord_le3146513528884898305at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : C3 )
          @ ( produc457027306803732586at_nat @ B5
            @ ^ [Uu: nat] : C3 ) )
        = ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ).

% Times_subset_cancel2
thf(fact_732_mem__Times__iff,axiom,
    ! [X: product_prod_nat_nat,A5: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A5 )
        & ( member_nat @ ( product_snd_nat_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_733_mem__Times__iff,axiom,
    ! [X: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ X ) @ A5 )
        & ( member8440522571783428010at_nat @ ( produc6408287024330202629at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_734_mem__Times__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A5: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ X
        @ ( produc7726050920695093128at_nat @ A5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) )
      = ( ( member2643936169264416010at_nat @ ( produc6011863565407846595at_nat @ X ) @ A5 )
        & ( member2643936169264416010at_nat @ ( produc2937545259423399429at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_735_mem__Times__iff,axiom,
    ! [X: produc39072733474941005at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ X
        @ ( produc7465995907834253422at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc7998413079968509225at_nat @ X ) @ A5 )
        & ( member8206827879206165904at_nat @ ( produc4828242618666536811at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_736_mem__Times__iff,axiom,
    ! [X: produc7148259590854449153at_nat,A5: set_Pr8693737435421807431at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ X
        @ ( produc3829605335166519074at_nat @ A5
          @ ^ [Uu: produc859450856879609959at_nat] : B5 ) )
      = ( ( member8206827879206165904at_nat @ ( produc4362022507300774877at_nat @ X ) @ A5 )
        & ( member8440522571783428010at_nat @ ( produc1191852045998802463at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_737_mem__Times__iff,axiom,
    ! [X: produc8430705068416227405at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member600271025443689462at_nat @ X
        @ ( produc3090783324798197870at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc3408962115674613033at_nat @ X ) @ A5 )
        & ( member8757157785044589968at_nat @ ( produc9109627233917564779at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_738_mem__Times__iff,axiom,
    ! [X: produc560747048475658241at_nat,A5: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member1953685042357896106at_nat @ X
        @ ( produc8743256393616062242at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
      = ( ( member8757157785044589968at_nat @ ( produc9061435184492477405at_nat @ X ) @ A5 )
        & ( member8440522571783428010at_nat @ ( produc5538728265880653343at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_739_mem__Times__iff,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,B5: set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ X
        @ ( produc1366239611153933861on_val @ A5
          @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) )
      = ( ( member2336342582916103932r_prat @ ( produc5239619271231976746on_val @ X ) @ A5 )
        & ( member1577473829074448177on_val @ ( produc2319605628138516840on_val @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_740_mem__Times__iff,axiom,
    ! [X: produc349518998152878311at_nat,A5: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ X
        @ ( produc18557253517588744at_nat @ A5
          @ ^ [Uu: produc859450856879609959at_nat] : B5 ) )
      = ( ( member8206827879206165904at_nat @ ( produc7253857937018302787at_nat @ X ) @ A5 )
        & ( member8206827879206165904at_nat @ ( produc458091870023641221at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_741_mem__Times__iff,axiom,
    ! [X: produc1034191653824608487at_nat,A5: set_Pr8693737435421807431at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member402237021265326864at_nat @ X
        @ ( produc8155347593902061832at_nat @ A5
          @ ^ [Uu: produc859450856879609959at_nat] : B5 ) )
      = ( ( member8206827879206165904at_nat @ ( produc7486072714475936067at_nat @ X ) @ A5 )
        & ( member8757157785044589968at_nat @ ( produc3561673524491468933at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_742_swap__product,axiom,
    ! [A5: set_se5479201105039950375at_nat,B5: set_se5479201105039950375at_nat] :
      ( ( image_3313167189156628421at_nat
        @ ( produc5456455677105495163at_nat
          @ ^ [I: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ J @ I ) )
        @ ( produc6258361011560675592at_nat @ A5
          @ ^ [Uu: set_Pr4329608150637261639at_nat] : B5 ) )
      = ( produc6258361011560675592at_nat @ B5
        @ ^ [Uu: set_Pr4329608150637261639at_nat] : A5 ) ) ).

% swap_product
thf(fact_743_swap__product,axiom,
    ! [A5: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( image_960689587819713477at_nat
        @ ( produc7548845685434208379at_nat
          @ ^ [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ J @ I ) )
        @ ( produc7726050920695093128at_nat @ A5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) )
      = ( produc7726050920695093128at_nat @ B5
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : A5 ) ) ).

% swap_product
thf(fact_744_swap__product,axiom,
    ! [A5: set_list_char,B5: set_nat] :
      ( ( image_688197443715608197t_char
        @ ( produc6166083679720228083t_char
          @ ^ [I: list_char,J: nat] : ( produc120197956887798346t_char @ J @ I ) )
        @ ( produc5389449503320859771ar_nat @ A5
          @ ^ [Uu: list_char] : B5 ) )
      = ( produc6030673636645808891t_char @ B5
        @ ^ [Uu: nat] : A5 ) ) ).

% swap_product
thf(fact_745_swap__product,axiom,
    ! [A5: set_Pr1004501353197068112on_val,B5: set_Pr3433046674199179443r_prat] :
      ( ( image_1698057272686278225on_val
        @ ( produc4110721796449090601on_val
          @ ^ [I: produc1457211279475724562t_char > option_val,J: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ J @ I ) )
        @ ( produc3189049046222722799r_prat @ A5
          @ ^ [Uu: produc1457211279475724562t_char > option_val] : B5 ) )
      = ( produc1366239611153933861on_val @ B5
        @ ^ [Uu: produc1457211279475724562t_char > prat] : A5 ) ) ).

% swap_product
thf(fact_746_swap__product,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ B5
        @ ^ [Uu: nat] : A5 ) ) ).

% swap_product
thf(fact_747_product__swap,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( image_5168914502847457605at_nat @ product_swap_nat_nat
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ B5
        @ ^ [Uu: nat] : A5 ) ) ).

% product_swap
thf(fact_748_image__paired__Times,axiom,
    ! [F: nat > set_Pr4329608150637261639at_nat,G: nat > set_Pr4329608150637261639at_nat,A5: set_nat,B5: set_nat] :
      ( ( image_1048906336043994923at_nat
        @ ( produc5613386679804371481at_nat
          @ ^ [X4: nat,Y7: nat] : ( produc9060074326276436823at_nat @ ( F @ X4 ) @ ( G @ Y7 ) ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc6258361011560675592at_nat @ ( image_1087905878993176368at_nat @ F @ A5 )
        @ ^ [Uu: set_Pr4329608150637261639at_nat] : ( image_1087905878993176368at_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_749_image__paired__Times,axiom,
    ! [F: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat,A5: set_nat,B5: set_nat] :
      ( ( image_1672857615394513579at_nat
        @ ( produc153460504799203225at_nat
          @ ^ [X4: nat,Y7: nat] : ( produc2922128104949294807at_nat @ ( F @ X4 ) @ ( G @ Y7 ) ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc7726050920695093128at_nat @ ( image_7178329752028323786at_nat @ F @ A5 )
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : ( image_7178329752028323786at_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_750_image__paired__Times,axiom,
    ! [F: nat > nat,G: nat > list_char,A5: set_nat,B5: set_nat] :
      ( ( image_728565836875346774t_char
        @ ( produc2373759969358458948t_char
          @ ^ [X4: nat,Y7: nat] : ( produc120197956887798346t_char @ ( F @ X4 ) @ ( G @ Y7 ) ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc6030673636645808891t_char @ ( image_nat_nat @ F @ A5 )
        @ ^ [Uu: nat] : ( image_nat_list_char @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_751_image__paired__Times,axiom,
    ! [F: nat > produc1457211279475724562t_char > prat,G: nat > produc1457211279475724562t_char > option_val,A5: set_nat,B5: set_nat] :
      ( ( image_1778330534009140952on_val
        @ ( produc5050726480593351786on_val
          @ ^ [X4: nat,Y7: nat] : ( produc74620467203698198on_val @ ( F @ X4 ) @ ( G @ Y7 ) ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc1366239611153933861on_val @ ( image_1918150703725107644r_prat @ F @ A5 )
        @ ^ [Uu: produc1457211279475724562t_char > prat] : ( image_954018951882482801on_val @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_752_image__paired__Times,axiom,
    ! [F: nat > nat,G: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X4: nat,Y7: nat] : ( product_Pair_nat_nat @ ( F @ X4 ) @ ( G @ Y7 ) ) )
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat @ F @ A5 )
        @ ^ [Uu: nat] : ( image_nat_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_753_map__prod__surj__on,axiom,
    ! [F: nat > nat,A5: set_nat,A8: set_nat,G: nat > nat,B5: set_nat,B7: set_nat] :
      ( ( ( image_nat_nat @ F @ A5 )
        = A8 )
     => ( ( ( image_nat_nat @ G @ B5 )
          = B7 )
       => ( ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G )
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : B5 ) )
          = ( produc457027306803732586at_nat @ A8
            @ ^ [Uu: nat] : B7 ) ) ) ) ).

% map_prod_surj_on
thf(fact_754_asymp__on__asym__on__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( asymp_3201933403969103519at_nat @ A5
        @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_o8546584678760569651at_nat @ A5 @ R ) ) ).

% asymp_on_asym_on_eq
thf(fact_755_asymp__on__asym__on__eq,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( asymp_7147290840880511973at_nat @ A5
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_o8654330542403097209at_nat @ A5 @ R ) ) ).

% asymp_on_asym_on_eq
thf(fact_756_asymp__on__asym__on__eq,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( asymp_8721585889738326015at_nat @ A5
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_o5033339280856310931at_nat @ A5 @ R ) ) ).

% asymp_on_asym_on_eq
thf(fact_757_asymp__on__asym__on__eq,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ A5
        @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R ) )
      = ( asym_on_nat @ A5 @ R ) ) ).

% asymp_on_asym_on_eq
thf(fact_758_subset__fst__snd,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat @ A5
      @ ( produc457027306803732586at_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A5 )
        @ ^ [Uu: nat] : ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A5 ) ) ) ).

% subset_fst_snd
thf(fact_759_subset__fst__snd,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ord_le6900622908513426738on_val @ A5
      @ ( produc1366239611153933861on_val @ ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ A5 )
        @ ^ [Uu: produc1457211279475724562t_char > prat] : ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ A5 ) ) ) ).

% subset_fst_snd
thf(fact_760_rp__inv__image__rp,axiom,
    ! [P2: produc1319942482725812455at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( fun_re2478310338295953701at_nat @ P2 )
     => ( fun_re2478310338295953701at_nat @ ( fun_rp8445993449185644698at_nat @ P2 @ F ) ) ) ).

% rp_inv_image_rp
thf(fact_761_member__product,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,B5: set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ X @ ( produc747609993485717659on_val @ A5 @ B5 ) )
      = ( member2879030624002546611on_val @ X
        @ ( produc1366239611153933861on_val @ A5
          @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) ) ) ).

% member_product
thf(fact_762_member__product,axiom,
    ! [X: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( produc6773009516019950738at_nat @ A5 @ B5 ) )
      = ( member8206827879206165904at_nat @ X
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ).

% member_product
thf(fact_763_member__product,axiom,
    ! [X: produc3843707927480180839at_nat,A5: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( produc3241716407206042770at_nat @ A5 @ B5 ) )
      = ( member8757157785044589968at_nat @ X
        @ ( produc7726050920695093128at_nat @ A5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) ) ) ).

% member_product
thf(fact_764_member__product,axiom,
    ! [X: product_prod_nat_nat,A5: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( produc929334515565554804at_nat @ A5 @ B5 ) )
      = ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 ) ) ) ).

% member_product
thf(fact_765_Product__Type_Oproduct__def,axiom,
    ( produc929334515565554804at_nat
    = ( ^ [A6: set_nat,B8: set_nat] :
          ( produc457027306803732586at_nat @ A6
          @ ^ [Uu: nat] : B8 ) ) ) ).

% Product_Type.product_def
thf(fact_766_snd__image__times,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( ( A5 = bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( A5 != bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_767_snd__image__times,axiom,
    ! [A5: set_Pr3433046674199179443r_prat,B5: set_Pr1004501353197068112on_val] :
      ( ( ( A5 = bot_bo3164680191535668511r_prat )
       => ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) )
          = bot_bo5671083314777306084on_val ) )
      & ( ( A5 != bot_bo3164680191535668511r_prat )
       => ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_768_snd__image__times,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( A5 = bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc6408287024330202629at_nat
            @ ( produc2761391749766926216at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A5 != bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc6408287024330202629at_nat
            @ ( produc2761391749766926216at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_769_snd__image__times,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( A5 = bot_bo2099793752762293965at_nat )
       => ( ( image_2020710134525389919at_nat @ produc9109627233917564779at_nat
            @ ( produc3090783324798197870at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A5 != bot_bo2099793752762293965at_nat )
       => ( ( image_2020710134525389919at_nat @ produc9109627233917564779at_nat
            @ ( produc3090783324798197870at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_770_snd__image__times,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( A5 = bot_bo228742789529271731at_nat )
       => ( ( image_1780646082475352133at_nat @ produc5538728265880653343at_nat
            @ ( produc8743256393616062242at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A5 != bot_bo228742789529271731at_nat )
       => ( ( image_1780646082475352133at_nat @ produc5538728265880653343at_nat
            @ ( produc8743256393616062242at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_771_snd__image__times,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( A5 = bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6345472114899077253at_nat
            @ ( produc4441300235185154312at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A5 != bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6345472114899077253at_nat
            @ ( produc4441300235185154312at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_772_fst__image__times,axiom,
    ! [B5: set_nat,A5: set_nat] :
      ( ( ( B5 = bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( B5 != bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
            @ ( produc457027306803732586at_nat @ A5
              @ ^ [Uu: nat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_773_fst__image__times,axiom,
    ! [B5: set_Pr1004501353197068112on_val,A5: set_Pr3433046674199179443r_prat] :
      ( ( ( B5 = bot_bo5671083314777306084on_val )
       => ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) )
          = bot_bo3164680191535668511r_prat ) )
      & ( ( B5 != bot_bo5671083314777306084on_val )
       => ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_774_fst__image__times,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( ( B5 = bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc3213797794245857475at_nat
            @ ( produc2761391749766926216at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( B5 != bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc3213797794245857475at_nat
            @ ( produc2761391749766926216at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_775_fst__image__times,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( ( B5 = bot_bo2099793752762293965at_nat )
       => ( ( image_8065348968207030699at_nat @ produc9061435184492477405at_nat
            @ ( produc8743256393616062242at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( B5 != bot_bo2099793752762293965at_nat )
       => ( ( image_8065348968207030699at_nat @ produc9061435184492477405at_nat
            @ ( produc8743256393616062242at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_776_fst__image__times,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( ( B5 = bot_bo228742789529271731at_nat )
       => ( ( image_7444983900720083193at_nat @ produc3408962115674613033at_nat
            @ ( produc3090783324798197870at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( B5 != bot_bo228742789529271731at_nat )
       => ( ( image_7444983900720083193at_nat @ produc3408962115674613033at_nat
            @ ( produc3090783324798197870at_nat @ A5
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_777_fst__image__times,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( ( B5 = bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6021103657482432835at_nat
            @ ( produc4441300235185154312at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( B5 != bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6021103657482432835at_nat
            @ ( produc4441300235185154312at_nat @ A5
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_778_The__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( the_Pr9161796738952339797on_val @ ( produc5248837549992124053_val_o @ P2 ) )
      = ( the_Pr9161796738952339797on_val
        @ ^ [Xy: produc9003781588162623004on_val] : ( P2 @ ( produc5239619271231976746on_val @ Xy ) @ ( produc2319605628138516840on_val @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_779_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_780_Sigma__empty1,axiom,
    ! [B5: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B5 )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty1
thf(fact_781_Sigma__empty1,axiom,
    ! [B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ bot_bo3083307316010499117at_nat @ B5 )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty1
thf(fact_782_Sigma__empty2,axiom,
    ! [A5: set_nat] :
      ( ( produc457027306803732586at_nat @ A5
        @ ^ [Uu: nat] : bot_bot_set_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty2
thf(fact_783_Sigma__empty2,axiom,
    ! [A5: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ A5
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : bot_bo3083307316010499117at_nat )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty2
thf(fact_784_Times__empty,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = bot_bo5327735625951526323at_nat )
      = ( ( A5 = bot_bo2099793752762293965at_nat )
        | ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_785_Times__empty,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( produc3090783324798197870at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = bot_bo3433114129128967449at_nat )
      = ( ( A5 = bot_bo2099793752762293965at_nat )
        | ( B5 = bot_bo228742789529271731at_nat ) ) ) ).

% Times_empty
thf(fact_786_Times__empty,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8743256393616062242at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = bot_bo7908653319349560013at_nat )
      = ( ( A5 = bot_bo228742789529271731at_nat )
        | ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_787_Times__empty,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4441300235185154312at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = bot_bo7193250159860175667at_nat )
      = ( ( A5 = bot_bo228742789529271731at_nat )
        | ( B5 = bot_bo228742789529271731at_nat ) ) ) ).

% Times_empty
thf(fact_788_Times__empty,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A5 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_789_Times__empty,axiom,
    ! [A5: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ A5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 )
        = bot_bo228742789529271731at_nat )
      = ( ( A5 = bot_bo3083307316010499117at_nat )
        | ( B5 = bot_bo3083307316010499117at_nat ) ) ) ).

% Times_empty
thf(fact_790_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec7475294502966882545on_val
            @ ( produc5248837549992124053_val_o
              @ ^ [A4: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] : P2 ) )
          = top_to1561438921946393474on_val ) )
      & ( ~ P2
       => ( ( collec7475294502966882545on_val
            @ ( produc5248837549992124053_val_o
              @ ^ [A4: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] : P2 ) )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Collect_const_case_prod
thf(fact_791_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec6321179662152712658at_nat
            @ ( produc410239310623530412_nat_o
              @ ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : P2 ) )
          = top_to6833984726390702231at_nat ) )
      & ( ~ P2
       => ( ( collec6321179662152712658at_nat
            @ ( produc410239310623530412_nat_o
              @ ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : P2 ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_792_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A4: nat,B4: nat] : P2 ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P2
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A4: nat,B4: nat] : P2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_793_The__split__eq,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( the_Pr4877703158196064750at_nat
        @ ( produc9020218426428501292_nat_o
          @ ^ [X6: set_Pr4329608150637261639at_nat,Y9: set_Pr4329608150637261639at_nat] :
              ( ( X = X6 )
              & ( Y = Y9 ) ) ) )
      = ( produc9060074326276436823at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_794_The__split__eq,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( the_Pr4604535484834969198at_nat
        @ ( produc410239310623530412_nat_o
          @ ^ [X6: set_Pr1261947904930325089at_nat,Y9: set_Pr1261947904930325089at_nat] :
              ( ( X = X6 )
              & ( Y = Y9 ) ) ) )
      = ( produc2922128104949294807at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_795_The__split__eq,axiom,
    ! [X: nat,Y: list_char] :
      ( ( the_Pr202245514940600089t_char
        @ ( produc8526122873445342073char_o
          @ ^ [X6: nat,Y9: list_char] :
              ( ( X = X6 )
              & ( Y = Y9 ) ) ) )
      = ( produc120197956887798346t_char @ X @ Y ) ) ).

% The_split_eq
thf(fact_796_The__split__eq,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( the_Pr9161796738952339797on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [X6: produc1457211279475724562t_char > prat,Y9: produc1457211279475724562t_char > option_val] :
              ( ( X = X6 )
              & ( Y = Y9 ) ) ) )
      = ( produc74620467203698198on_val @ X @ Y ) ) ).

% The_split_eq
thf(fact_797_The__split__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X6: nat,Y9: nat] :
              ( ( X = X6 )
              & ( Y = Y9 ) ) ) )
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_798_Sigma__empty__iff,axiom,
    ! [I2: set_nat,X5: nat > set_nat] :
      ( ( ( produc457027306803732586at_nat @ I2 @ X5 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ I2 )
           => ( ( X5 @ X4 )
              = bot_bot_set_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_799_Sigma__empty__iff,axiom,
    ! [I2: set_se7855581050983116737at_nat,X5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ I2 @ X5 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ I2 )
           => ( ( X5 @ X4 )
              = bot_bo3083307316010499117at_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_800_subset__emptyI,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ! [X3: produc9003781588162623004on_val] :
          ~ ( member2879030624002546611on_val @ X3 @ A5 )
     => ( ord_le6900622908513426738on_val @ A5 @ bot_bo6722054719108133734on_val ) ) ).

% subset_emptyI
thf(fact_801_subset__emptyI,axiom,
    ! [A5: set_Pr8693737435421807431at_nat] :
      ( ! [X3: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ X3 @ A5 )
     => ( ord_le3000389064537975527at_nat @ A5 @ bot_bo5327735625951526323at_nat ) ) ).

% subset_emptyI
thf(fact_802_subset__emptyI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X3 @ A5 )
     => ( ord_le3146513528884898305at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_803_subset__emptyI,axiom,
    ! [A5: set_Pr4329608150637261639at_nat] :
      ( ! [X3: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ X3 @ A5 )
     => ( ord_le1268244103169919719at_nat @ A5 @ bot_bo228742789529271731at_nat ) ) ).

% subset_emptyI
thf(fact_804_less__by__empty,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( A5 = bot_bo2099793752762293965at_nat )
     => ( ord_le3146513528884898305at_nat @ A5 @ B5 ) ) ).

% less_by_empty
thf(fact_805_less__by__empty,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( A5 = bot_bo228742789529271731at_nat )
     => ( ord_le1268244103169919719at_nat @ A5 @ B5 ) ) ).

% less_by_empty
thf(fact_806_times__eq__iff,axiom,
    ! [A5: set_nat,B5: set_nat,C3: set_nat,D2: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : B5 )
        = ( produc457027306803732586at_nat @ C3
          @ ^ [Uu: nat] : D2 ) )
      = ( ( ( A5 = C3 )
          & ( B5 = D2 ) )
        | ( ( ( A5 = bot_bot_set_nat )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C3 = bot_bot_set_nat )
            | ( D2 = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_807_times__eq__iff,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = ( produc2761391749766926216at_nat @ C3
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( ( A5 = C3 )
          & ( B5 = D2 ) )
        | ( ( ( A5 = bot_bo2099793752762293965at_nat )
            | ( B5 = bot_bo2099793752762293965at_nat ) )
          & ( ( C3 = bot_bo2099793752762293965at_nat )
            | ( D2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_808_times__eq__iff,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat,C3: set_Pr1261947904930325089at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc3090783324798197870at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = ( produc3090783324798197870at_nat @ C3
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( ( A5 = C3 )
          & ( B5 = D2 ) )
        | ( ( ( A5 = bot_bo2099793752762293965at_nat )
            | ( B5 = bot_bo228742789529271731at_nat ) )
          & ( ( C3 = bot_bo2099793752762293965at_nat )
            | ( D2 = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_809_times__eq__iff,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr4329608150637261639at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8743256393616062242at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = ( produc8743256393616062242at_nat @ C3
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( ( A5 = C3 )
          & ( B5 = D2 ) )
        | ( ( ( A5 = bot_bo228742789529271731at_nat )
            | ( B5 = bot_bo2099793752762293965at_nat ) )
          & ( ( C3 = bot_bo228742789529271731at_nat )
            | ( D2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_810_times__eq__iff,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4441300235185154312at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = ( produc4441300235185154312at_nat @ C3
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( ( A5 = C3 )
          & ( B5 = D2 ) )
        | ( ( ( A5 = bot_bo228742789529271731at_nat )
            | ( B5 = bot_bo228742789529271731at_nat ) )
          & ( ( C3 = bot_bo228742789529271731at_nat )
            | ( D2 = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_811_times__subset__iff,axiom,
    ! [A5: set_nat,C3: set_nat,B5: set_nat,D2: set_nat] :
      ( ( ord_le3146513528884898305at_nat
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : C3 )
        @ ( produc457027306803732586at_nat @ B5
          @ ^ [Uu: nat] : D2 ) )
      = ( ( A5 = bot_bot_set_nat )
        | ( C3 = bot_bot_set_nat )
        | ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( ord_less_eq_set_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_812_times__subset__iff,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : C3 )
        @ ( produc2761391749766926216at_nat @ B5
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( A5 = bot_bo2099793752762293965at_nat )
        | ( C3 = bot_bo2099793752762293965at_nat )
        | ( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
          & ( ord_le3146513528884898305at_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_813_times__subset__iff,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,C3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le7005211468645301325at_nat
        @ ( produc3090783324798197870at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : C3 )
        @ ( produc3090783324798197870at_nat @ B5
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( A5 = bot_bo2099793752762293965at_nat )
        | ( C3 = bot_bo228742789529271731at_nat )
        | ( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
          & ( ord_le1268244103169919719at_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_814_times__subset__iff,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,C3: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2257378622011118081at_nat
        @ ( produc8743256393616062242at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : C3 )
        @ ( produc8743256393616062242at_nat @ B5
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( A5 = bot_bo228742789529271731at_nat )
        | ( C3 = bot_bo2099793752762293965at_nat )
        | ( ( ord_le1268244103169919719at_nat @ A5 @ B5 )
          & ( ord_le3146513528884898305at_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_815_times__subset__iff,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,C3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat
        @ ( produc4441300235185154312at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : C3 )
        @ ( produc4441300235185154312at_nat @ B5
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( A5 = bot_bo228742789529271731at_nat )
        | ( C3 = bot_bo228742789529271731at_nat )
        | ( ( ord_le1268244103169919719at_nat @ A5 @ B5 )
          & ( ord_le1268244103169919719at_nat @ C3 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_816_fst__image__Sigma,axiom,
    ! [A5: set_nat,B5: nat > set_nat] :
      ( ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
      = ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bot_set_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_817_fst__image__Sigma,axiom,
    ! [A5: set_nat,B5: nat > set_Pr1261947904930325089at_nat] :
      ( ( image_2960450939771910141at_nat @ produc7982242761980330092at_nat @ ( produc1809337555817847783at_nat @ A5 @ B5 ) )
      = ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_818_fst__image__Sigma,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_6706873702294942047at_nat @ produc3213797794245857475at_nat @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_819_fst__image__Sigma,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_229310333994078635at_nat @ produc4362022507300774877at_nat @ ( produc3829605335166519074at_nat @ A5 @ B5 ) )
      = ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_820_fst__image__Sigma,axiom,
    ! [A5: set_nat,B5: nat > set_Pr4329608150637261639at_nat] :
      ( ( image_8126111929026432099at_nat @ produc5794669274478512978at_nat @ ( produc396409126344608333at_nat @ A5 @ B5 ) )
      = ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_821_fst__image__Sigma,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_8065348968207030699at_nat @ produc9061435184492477405at_nat @ ( produc8743256393616062242at_nat @ A5 @ B5 ) )
      = ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_822_fst__image__Sigma,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( image_7444983900720083193at_nat @ produc3408962115674613033at_nat @ ( produc3090783324798197870at_nat @ A5 @ B5 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_823_fst__image__Sigma,axiom,
    ! [A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
      = ( collec7093475260167521854r_prat
        @ ^ [X4: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo5671083314777306084on_val ) ) ) ) ).

% fst_image_Sigma
thf(fact_824_fst__image__Sigma,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr4329608150637261639at_nat] :
      ( ( image_2925403533633446213at_nat @ produc7486072714475936067at_nat @ ( produc8155347593902061832at_nat @ A5 @ B5 ) )
      = ( collec7088162979684241874at_nat
        @ ^ [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_825_fst__image__Sigma,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ( image_3229096548743918333on_val @ produc2323562575164511008at_nat @ ( produc8411625936386031515at_nat @ A5 @ B5 ) )
      = ( collec7475294502966882545on_val
        @ ^ [X4: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X4 @ A5 )
            & ( ( B5 @ X4 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_826_Abs__state__inverse,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu: produc9003781588162623004on_val] :
            ? [Phi: produc9003781588162623004on_val] :
              ( ( Uu = Phi )
              & ( valid_state @ Phi ) ) ) )
     => ( ( rep_state @ ( abs_state @ Y ) )
        = Y ) ) ).

% Abs_state_inverse
thf(fact_827_type__definition__state,axiom,
    ( type_d7385716227342696708on_val @ rep_state @ abs_state
    @ ( collec7475294502966882545on_val
      @ ^ [Uu: produc9003781588162623004on_val] :
        ? [Phi: produc9003781588162623004on_val] :
          ( ( Uu = Phi )
          & ( valid_state @ Phi ) ) ) ) ).

% type_definition_state
thf(fact_828_refl__on__def,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
          ( ( ord_le3000389064537975527at_nat @ R3
            @ ( produc2761391749766926216at_nat @ A6
              @ ^ [Uu: product_prod_nat_nat] : A6 ) )
          & ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def
thf(fact_829_refl__on__def,axiom,
    ( refl_o6272981002401706822at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
          ( ( ord_le5997549366648089703at_nat @ R3
            @ ( produc6258361011560675592at_nat @ A6
              @ ^ [Uu: set_Pr4329608150637261639at_nat] : A6 ) )
          & ! [X4: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X4 @ A6 )
             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def
thf(fact_830_refl__on__def,axiom,
    ( refl_o6955724340487615712at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
          ( ( ord_le1268244103169919719at_nat @ R3
            @ ( produc7726050920695093128at_nat @ A6
              @ ^ [Uu: set_Pr1261947904930325089at_nat] : A6 ) )
          & ! [X4: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X4 @ A6 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def
thf(fact_831_refl__on__def,axiom,
    ( refl_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ R3
            @ ( produc457027306803732586at_nat @ A6
              @ ^ [Uu: nat] : A6 ) )
          & ! [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def
thf(fact_832_refl__onD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A5 @ R )
     => ( ( member2879030624002546611on_val @ A @ A5 )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_833_refl__onD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_834_refl__onD,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ A @ A5 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_835_refl__onD,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A5 @ R )
     => ( ( member8757157785044589968at_nat @ A @ A5 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_836_refl__onD,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A5 @ R )
     => ( ( member6273874620674182256at_nat @ A @ A5 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_837_refl__onD,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A5 @ R )
     => ( ( member2643936169264416010at_nat @ A @ A5 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_838_refl__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member_nat @ A @ A5 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_839_refl__onD1,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A5 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
       => ( member2879030624002546611on_val @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_840_refl__onD1,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A5 @ R )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R )
       => ( member8206827879206165904at_nat @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_841_refl__onD1,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A5 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
       => ( member8757157785044589968at_nat @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_842_refl__onD1,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
       => ( member8440522571783428010at_nat @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_843_refl__onD1,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A5 @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
       => ( member6273874620674182256at_nat @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_844_refl__onD1,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A5 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( member2643936169264416010at_nat @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_845_refl__onD1,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member_nat @ X @ A5 ) ) ) ).

% refl_onD1
thf(fact_846_refl__onD2,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A5 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
       => ( member2879030624002546611on_val @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_847_refl__onD2,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A5 @ R )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R )
       => ( member8206827879206165904at_nat @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_848_refl__onD2,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A5 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
       => ( member8757157785044589968at_nat @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_849_refl__onD2,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
       => ( member8440522571783428010at_nat @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_850_refl__onD2,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A5 @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
       => ( member6273874620674182256at_nat @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_851_refl__onD2,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A5 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( member2643936169264416010at_nat @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_852_refl__onD2,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member_nat @ Y @ A5 ) ) ) ).

% refl_onD2
thf(fact_853_refl__lex__prod,axiom,
    ! [R_B: set_Pr1261947904930325089at_nat,R_A: set_Pr1261947904930325089at_nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R_B )
     => ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ).

% refl_lex_prod
thf(fact_854_reflI,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ R )
     => ( refl_o6272981002401706822at_nat @ top_to1754375979021635447at_nat @ R ) ) ).

% reflI
thf(fact_855_reflI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ R )
     => ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% reflI
thf(fact_856_reflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( refl_on_nat @ top_top_set_nat @ R ) ) ).

% reflI
thf(fact_857_reflI,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R )
     => ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% reflI
thf(fact_858_reflD,axiom,
    ! [R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ top_to1754375979021635447at_nat @ R )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_859_reflD,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_860_reflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_861_reflD,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ R )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_862_Gr__def,axiom,
    ( bNF_Gr7888059597227825506at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,F2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat] :
          ( collec7872792499061297490at_nat
          @ ^ [Uu: produc1319942482725812455at_nat] :
            ? [A4: set_Pr4329608150637261639at_nat] :
              ( ( Uu
                = ( produc9060074326276436823at_nat @ A4 @ ( F2 @ A4 ) ) )
              & ( member6273874620674182256at_nat @ A4 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_863_Gr__def,axiom,
    ( bNF_Gr1527221351080490466at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A4: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ A4 @ ( F2 @ A4 ) ) )
              & ( member2643936169264416010at_nat @ A4 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_864_Gr__def,axiom,
    ( bNF_Gr_nat_list_char
    = ( ^ [A6: set_nat,F2: nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A4: nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ A4 @ ( F2 @ A4 ) ) )
              & ( member_nat @ A4 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_865_Gr__def,axiom,
    ( bNF_Gr_nat_nat
    = ( ^ [A6: set_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A4: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A4 @ ( F2 @ A4 ) ) )
              & ( member_nat @ A4 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_866_Gr__def,axiom,
    ( bNF_Gr7520394057480975179on_val
    = ( ^ [A6: set_Pr3433046674199179443r_prat,F2: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
          ( collec7475294502966882545on_val
          @ ^ [Uu: produc9003781588162623004on_val] :
            ? [A4: produc1457211279475724562t_char > prat] :
              ( ( Uu
                = ( produc74620467203698198on_val @ A4 @ ( F2 @ A4 ) ) )
              & ( member2336342582916103932r_prat @ A4 @ A6 ) ) ) ) ) ).

% Gr_def
thf(fact_867_Rep__state,axiom,
    ! [X: state] :
      ( member2879030624002546611on_val @ ( rep_state @ X )
      @ ( collec7475294502966882545on_val
        @ ^ [Uu: produc9003781588162623004on_val] :
          ? [Phi: produc9003781588162623004on_val] :
            ( ( Uu = Phi )
            & ( valid_state @ Phi ) ) ) ) ).

% Rep_state
thf(fact_868_Rep__state__cases,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y @ ( collec7475294502966882545on_val @ valid_state ) )
     => ~ ! [X3: state] :
            ( Y
           != ( rep_state @ X3 ) ) ) ).

% Rep_state_cases
thf(fact_869_Rep__state__induct,axiom,
    ! [Y: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu: produc9003781588162623004on_val] :
            ? [Phi: produc9003781588162623004on_val] :
              ( ( Uu = Phi )
              & ( valid_state @ Phi ) ) ) )
     => ( ! [X3: state] : ( P2 @ ( rep_state @ X3 ) )
       => ( P2 @ Y ) ) ) ).

% Rep_state_induct
thf(fact_870_Abs__state__inject,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ X
        @ ( collec7475294502966882545on_val
          @ ^ [Uu: produc9003781588162623004on_val] :
            ? [Phi: produc9003781588162623004on_val] :
              ( ( Uu = Phi )
              & ( valid_state @ Phi ) ) ) )
     => ( ( member2879030624002546611on_val @ Y
          @ ( collec7475294502966882545on_val
            @ ^ [Uu: produc9003781588162623004on_val] :
              ? [Phi: produc9003781588162623004on_val] :
                ( ( Uu = Phi )
                & ( valid_state @ Phi ) ) ) )
       => ( ( ( abs_state @ X )
            = ( abs_state @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_state_inject
thf(fact_871_Abs__state__induct,axiom,
    ! [P2: state > $o,X: state] :
      ( ! [Y3: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ Y3
            @ ( collec7475294502966882545on_val
              @ ^ [Uu: produc9003781588162623004on_val] :
                ? [Phi: produc9003781588162623004on_val] :
                  ( ( Uu = Phi )
                  & ( valid_state @ Phi ) ) ) )
         => ( P2 @ ( abs_state @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% Abs_state_induct
thf(fact_872_Abs__state__cases,axiom,
    ! [X: state] :
      ~ ! [Y3: produc9003781588162623004on_val] :
          ( ( X
            = ( abs_state @ Y3 ) )
         => ~ ( member2879030624002546611on_val @ Y3 @ ( collec7475294502966882545on_val @ valid_state ) ) ) ).

% Abs_state_cases
thf(fact_873_image2__def,axiom,
    ( bNF_Gr8954129895687480649at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat,G3: product_prod_nat_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A4: product_prod_nat_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8440522571783428010at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_874_image2__def,axiom,
    ( bNF_Gr3738443398878377306t_char
    = ( ^ [A6: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat,G3: product_prod_nat_nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A4: product_prod_nat_nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8440522571783428010at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_875_image2__def,axiom,
    ( bNF_Gr97278709799963567at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > nat,G3: produc859450856879609959at_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A4: produc859450856879609959at_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8206827879206165904at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_876_image2__def,axiom,
    ( bNF_Gr197193474242742976t_char
    = ( ^ [A6: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > nat,G3: produc859450856879609959at_nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A4: produc859450856879609959at_nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8206827879206165904at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_877_image2__def,axiom,
    ( bNF_Gr2618164000252033511at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat,G3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A4: product_prod_nat_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8440522571783428010at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_878_image2__def,axiom,
    ( bNF_Gr1074675703407240111at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > nat,G3: produc3843707927480180839at_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A4: produc3843707927480180839at_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8757157785044589968at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_879_image2__def,axiom,
    ( bNF_Gr3647166678560580288t_char
    = ( ^ [A6: set_Pr4329608150637261639at_nat,F2: produc3843707927480180839at_nat > nat,G3: produc3843707927480180839at_nat > list_char] :
          ( collec2458894688891239037t_char
          @ ^ [Uu: produc1457211279475724562t_char] :
            ? [A4: produc3843707927480180839at_nat] :
              ( ( Uu
                = ( produc120197956887798346t_char @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8757157785044589968at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_880_image2__def,axiom,
    ( bNF_Gr5887040420617142093at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,F2: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat,G3: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A4: produc859450856879609959at_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8206827879206165904at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_881_image2__def,axiom,
    ( bNF_Gr3695791879626193176at_nat
    = ( ^ [A6: set_Pr7649655576376376530on_val,F2: produc9003781588162623004on_val > nat,G3: produc9003781588162623004on_val > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A4: produc9003781588162623004on_val] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member2879030624002546611on_val @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_882_image2__def,axiom,
    ( bNF_Gr1122962726781069062on_val
    = ( ^ [A6: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > produc1457211279475724562t_char > prat,G3: product_prod_nat_nat > produc1457211279475724562t_char > option_val] :
          ( collec7475294502966882545on_val
          @ ^ [Uu: produc9003781588162623004on_val] :
            ? [A4: product_prod_nat_nat] :
              ( ( Uu
                = ( produc74620467203698198on_val @ ( F2 @ A4 ) @ ( G3 @ A4 ) ) )
              & ( member8440522571783428010at_nat @ A4 @ A6 ) ) ) ) ) ).

% image2_def
thf(fact_883_refl__onI,axiom,
    ! [R: set_Pr4305388979797279737on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( ord_le7224449410216889241on_val @ R
        @ ( produc4067394217145614850on_val @ A5
          @ ^ [Uu: produc9003781588162623004on_val] : A5 ) )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A5 )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ X3 ) @ R ) )
       => ( refl_o6679066329051764061on_val @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_884_refl__onI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : A5 ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R ) )
       => ( refl_o7122697186039111552at_nat @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_885_refl__onI,axiom,
    ! [R: set_Pr553994874890374343at_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( ord_le2286497330737841255at_nat @ R
        @ ( produc18557253517588744at_nat @ A5
          @ ^ [Uu: produc859450856879609959at_nat] : A5 ) )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A5 )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ X3 ) @ R ) )
       => ( refl_o6679620997798956134at_nat @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_886_refl__onI,axiom,
    ! [R: set_Pr5564308138774400199at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat @ R
        @ ( produc4441300235185154312at_nat @ A5
          @ ^ [Uu: produc3843707927480180839at_nat] : A5 ) )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A5 )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ X3 ) @ R ) )
       => ( refl_o7446464703481863270at_nat @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_887_refl__onI,axiom,
    ! [R: set_Pr7459493094073627847at_nat,A5: set_se5479201105039950375at_nat] :
      ( ( ord_le5997549366648089703at_nat @ R
        @ ( produc6258361011560675592at_nat @ A5
          @ ^ [Uu: set_Pr4329608150637261639at_nat] : A5 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A5 )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ R ) )
       => ( refl_o6272981002401706822at_nat @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_888_refl__onI,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A5: set_se7855581050983116737at_nat] :
      ( ( ord_le1268244103169919719at_nat @ R
        @ ( produc7726050920695093128at_nat @ A5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : A5 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A5 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ R ) )
       => ( refl_o6955724340487615712at_nat @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_889_refl__onI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : A5 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R ) )
       => ( refl_on_nat @ A5 @ R ) ) ) ).

% refl_onI
thf(fact_890_refl__on__domain,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ A5 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ( member2879030624002546611on_val @ A @ A5 )
          & ( member2879030624002546611on_val @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_891_refl__on__domain,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A5 @ R )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R )
       => ( ( member8206827879206165904at_nat @ A @ A5 )
          & ( member8206827879206165904at_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_892_refl__on__domain,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A5 @ R )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R )
       => ( ( member8757157785044589968at_nat @ A @ A5 )
          & ( member8757157785044589968at_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_893_refl__on__domain,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
       => ( ( member8440522571783428010at_nat @ A @ A5 )
          & ( member8440522571783428010at_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_894_refl__on__domain,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A5 @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R )
       => ( ( member6273874620674182256at_nat @ A @ A5 )
          & ( member6273874620674182256at_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_895_refl__on__domain,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A5 @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ( member2643936169264416010at_nat @ A @ A5 )
          & ( member2643936169264416010at_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_896_refl__on__domain,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( refl_on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( ( member_nat @ A @ A5 )
          & ( member_nat @ B @ A5 ) ) ) ) ).

% refl_on_domain
thf(fact_897_relImage__def,axiom,
    ( bNF_Gr3178947973406375814at_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: nat,A22: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_898_relImage__def,axiom,
    ( bNF_Gr6341192048662747117at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: product_prod_nat_nat,A22: product_prod_nat_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_899_relImage__def,axiom,
    ( bNF_Gr4029507825166881579at_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A1: nat,A22: nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_900_relImage__def,axiom,
    ( bNF_Gr1715876217918586189at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_901_relImage__def,axiom,
    ( bNF_Gr5304801575501797892at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A1: product_prod_nat_nat,A22: product_prod_nat_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_902_relImage__def,axiom,
    ( bNF_Gr4979461995297504420at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_903_relImage__def,axiom,
    ( bNF_Gr3232541621826403601at_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,F2: nat > set_Pr4329608150637261639at_nat] :
          ( collec7872792499061297490at_nat
          @ ^ [Uu: produc1319942482725812455at_nat] :
            ? [A1: nat,A22: nat] :
              ( ( Uu
                = ( produc9060074326276436823at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_904_relImage__def,axiom,
    ( bNF_Gr3781325541126576563at_nat
    = ( ^ [R4: set_Pr7459493094073627847at_nat,F2: set_Pr4329608150637261639at_nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_905_relImage__def,axiom,
    ( bNF_Gr909174496254210666at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
          ( collec7872792499061297490at_nat
          @ ^ [Uu: produc1319942482725812455at_nat] :
            ? [A1: product_prod_nat_nat,A22: product_prod_nat_nat] :
              ( ( Uu
                = ( produc9060074326276436823at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_906_relImage__def,axiom,
    ( bNF_Gr7449055099601069322at_nat
    = ( ^ [R4: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
          ( collec7872792499061297490at_nat
          @ ^ [Uu: produc1319942482725812455at_nat] :
            ? [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( produc9060074326276436823at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ R4 ) ) ) ) ) ).

% relImage_def
thf(fact_907_relInvImage__def,axiom,
    ( bNF_Gr8020361688147858275at_nat
    = ( ^ [A6: set_nat,R4: set_Pr1261947904930325089at_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: nat,A22: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A1 @ A22 ) )
              & ( member_nat @ A1 @ A6 )
              & ( member_nat @ A22 @ A6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_908_relInvImage__def,axiom,
    ( bNF_Gr4292214848008494032at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
          ( collec7088162979684241874at_nat
          @ ^ [Uu: produc859450856879609959at_nat] :
            ? [A1: product_prod_nat_nat,A22: product_prod_nat_nat] :
              ( ( Uu
                = ( produc6161850002892822231at_nat @ A1 @ A22 ) )
              & ( member8440522571783428010at_nat @ A1 @ A6 )
              & ( member8440522571783428010at_nat @ A22 @ A6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_909_relInvImage__def,axiom,
    ( bNF_Gr7652262241051209134at_nat
    = ( ^ [A6: set_nat,R4: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: nat,A22: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A1 @ A22 ) )
              & ( member_nat @ A1 @ A6 )
              & ( member_nat @ A22 @ A6 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_910_relInvImage__def,axiom,
    ( bNF_Gr8117433302616110384at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R4: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ A1 @ A22 ) )
              & ( member2643936169264416010at_nat @ A1 @ A6 )
              & ( member2643936169264416010at_nat @ A22 @ A6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_911_relInvImage__def,axiom,
    ( bNF_Gr1207692873009629966at_nat
    = ( ^ [A6: set_nat,R4: set_Pr4329608150637261639at_nat,F2: nat > set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [Uu: product_prod_nat_nat] :
            ? [A1: nat,A22: nat] :
              ( ( Uu
                = ( product_Pair_nat_nat @ A1 @ A22 ) )
              & ( member_nat @ A1 @ A6 )
              & ( member_nat @ A22 @ A6 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_912_relInvImage__def,axiom,
    ( bNF_Gr6489406079561386241at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [Uu: produc859450856879609959at_nat] :
            ? [A1: product_prod_nat_nat,A22: product_prod_nat_nat] :
              ( ( Uu
                = ( produc6161850002892822231at_nat @ A1 @ A22 ) )
              & ( member8440522571783428010at_nat @ A1 @ A6 )
              & ( member8440522571783428010at_nat @ A22 @ A6 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_913_relInvImage__def,axiom,
    ( bNF_Gr2050058807657765473at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat,F2: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [Uu: produc859450856879609959at_nat] :
            ? [A1: product_prod_nat_nat,A22: product_prod_nat_nat] :
              ( ( Uu
                = ( produc6161850002892822231at_nat @ A1 @ A22 ) )
              & ( member8440522571783428010at_nat @ A1 @ A6 )
              & ( member8440522571783428010at_nat @ A22 @ A6 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_914_relInvImage__def,axiom,
    ( bNF_Gr404531551158876065at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R4: set_Pr8693737435421807431at_nat,F2: set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ A1 @ A22 ) )
              & ( member2643936169264416010at_nat @ A1 @ A6 )
              & ( member2643936169264416010at_nat @ A22 @ A6 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_915_relInvImage__def,axiom,
    ( bNF_Gr7352552370965628470at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,R4: set_Pr1261947904930325089at_nat,F2: produc859450856879609959at_nat > nat] :
          ( collec7334067512558549330at_nat
          @ ^ [Uu: produc349518998152878311at_nat] :
            ? [A1: produc859450856879609959at_nat,A22: produc859450856879609959at_nat] :
              ( ( Uu
                = ( produc4662710985925991255at_nat @ A1 @ A22 ) )
              & ( member8206827879206165904at_nat @ A1 @ A6 )
              & ( member8206827879206165904at_nat @ A22 @ A6 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_916_relInvImage__def,axiom,
    ( bNF_Gr3754292640012470529at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat,F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [Uu: produc3843707927480180839at_nat] :
            ? [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat] :
              ( ( Uu
                = ( produc2922128104949294807at_nat @ A1 @ A22 ) )
              & ( member2643936169264416010at_nat @ A1 @ A6 )
              & ( member2643936169264416010at_nat @ A22 @ A6 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F2 @ A1 ) @ ( F2 @ A22 ) ) @ R4 ) ) ) ) ) ).

% relInvImage_def
thf(fact_917_relation__of__def,axiom,
    ( order_1381308103030626383at_nat
    = ( ^ [P4: product_prod_nat_nat > product_prod_nat_nat > $o,A6: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B4 )
                  @ ( produc2761391749766926216at_nat @ A6
                    @ ^ [Uu: product_prod_nat_nat] : A6 ) )
                & ( P4 @ A4 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_918_relation__of__def,axiom,
    ( order_8308734813930759573at_nat
    = ( ^ [P4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A6: set_se5479201105039950375at_nat] :
          ( collec7872792499061297490at_nat
          @ ( produc9020218426428501292_nat_o
            @ ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
                ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A4 @ B4 )
                  @ ( produc6258361011560675592at_nat @ A6
                    @ ^ [Uu: set_Pr4329608150637261639at_nat] : A6 ) )
                & ( P4 @ A4 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_919_relation__of__def,axiom,
    ( order_8809246210994205103at_nat
    = ( ^ [P4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A6: set_se7855581050983116737at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B4 )
                  @ ( produc7726050920695093128at_nat @ A6
                    @ ^ [Uu: set_Pr1261947904930325089at_nat] : A6 ) )
                & ( P4 @ A4 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_920_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P4: nat > nat > $o,A6: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A4: nat,B4: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 )
                  @ ( produc457027306803732586at_nat @ A6
                    @ ^ [Uu: nat] : A6 ) )
                & ( P4 @ A4 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_921_aboveS__def,axiom,
    ( order_86566288019506545on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A4: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] :
              ( ( B4 != A4 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_922_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A4: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] :
              ( ( B4 != A4 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_923_aboveS__def,axiom,
    ( order_2442354982230848562at_nat
    = ( ^ [R3: set_Pr7459493094073627847at_nat,A4: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B4: set_Pr4329608150637261639at_nat] :
              ( ( B4 != A4 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A4 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_924_aboveS__def,axiom,
    ( order_4440095836999870412at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat,A4: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B4: set_Pr1261947904930325089at_nat] :
              ( ( B4 != A4 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_925_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A4: nat] :
          ( collect_nat
          @ ^ [B4: nat] :
              ( ( B4 != A4 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_926_above__def,axiom,
    ( order_646696875765705500on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A4: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_927_above__def,axiom,
    ( order_793026671032835073at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A4: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_928_above__def,axiom,
    ( order_3445360974473395783at_nat
    = ( ^ [R3: set_Pr7459493094073627847at_nat,A4: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A4 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_929_above__def,axiom,
    ( order_4986753419052871009at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat,A4: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_930_above__def,axiom,
    ( order_above_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A4: nat] :
          ( collect_nat
          @ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_931_snd__image__Sigma,axiom,
    ! [A5: set_nat,B5: nat > set_nat] :
      ( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A5 ) ) ) ).

% snd_image_Sigma
thf(fact_932_snd__image__Sigma,axiom,
    ! [A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
      = ( comple6221860311599351331on_val @ ( image_4386470036779017822on_val @ B5 @ A5 ) ) ) ).

% snd_image_Sigma
thf(fact_933_transp__trans,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( transp6142674538225245873at_nat @ top_to4669805908274784177at_nat @ R )
      = ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R ) ) ) ) ).

% transp_trans
thf(fact_934_transp__trans,axiom,
    ! [R: nat > nat > $o] :
      ( ( transp_on_nat @ top_top_set_nat @ R )
      = ( trans_on_nat @ top_top_set_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R ) ) ) ) ).

% transp_trans
thf(fact_935_trans__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R_A )
     => ( ( trans_on_nat @ top_top_set_nat @ R_B )
       => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_936_trans__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R_A )
     => ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( trans_8787906625838855558at_nat @ top_to7745748144976627212at_nat @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_937_trans__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( trans_on_nat @ top_top_set_nat @ R_B )
       => ( trans_690021572951380832at_nat @ top_to2567003362141017830at_nat @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_938_trans__lex__prod,axiom,
    ! [R_A: set_Pr8693737435421807431at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_A )
     => ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R_B )
       => ( trans_613433433080944135at_nat @ top_to8454300755403694231at_nat @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% trans_lex_prod
thf(fact_939_trans__onD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] :
      ( ( trans_3212976107463897212on_val @ A5 @ R )
     => ( ( member2879030624002546611on_val @ X @ A5 )
       => ( ( member2879030624002546611on_val @ Y @ A5 )
         => ( ( member2879030624002546611on_val @ Z2 @ A5 )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ Z2 ) @ R )
               => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_940_trans__onD,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,Z2: produc859450856879609959at_nat] :
      ( ( trans_613433433080944135at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ X @ A5 )
       => ( ( member8206827879206165904at_nat @ Y @ A5 )
         => ( ( member8206827879206165904at_nat @ Z2 @ A5 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ Z2 ) @ R )
               => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_941_trans__onD,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,Z2: produc3843707927480180839at_nat] :
      ( ( trans_4802246114490097159at_nat @ A5 @ R )
     => ( ( member8757157785044589968at_nat @ X @ A5 )
       => ( ( member8757157785044589968at_nat @ Y @ A5 )
         => ( ( member8757157785044589968at_nat @ Z2 @ A5 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ Z2 ) @ R )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_942_trans__onD,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ A5 @ R )
     => ( ( member6273874620674182256at_nat @ X @ A5 )
       => ( ( member6273874620674182256at_nat @ Y @ A5 )
         => ( ( member6273874620674182256at_nat @ Z2 @ A5 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z2 ) @ R )
               => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_943_trans__onD,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ A5 @ R )
     => ( ( member2643936169264416010at_nat @ X @ A5 )
       => ( ( member2643936169264416010at_nat @ Y @ A5 )
         => ( ( member2643936169264416010at_nat @ Z2 @ A5 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ R )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_944_trans__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
      ( ( trans_on_nat @ A5 @ R )
     => ( ( member_nat @ X @ A5 )
       => ( ( member_nat @ Y @ A5 )
         => ( ( member_nat @ Z2 @ A5 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ R )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_945_trans__onD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( ( member8440522571783428010at_nat @ Y @ A5 )
         => ( ( member8440522571783428010at_nat @ Z2 @ A5 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_946_trans__onI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X3 @ A5 )
         => ( ( member2879030624002546611on_val @ Y3 @ A5 )
           => ( ( member2879030624002546611on_val @ Z3 @ A5 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
               => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Z3 ) @ R )
                 => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_3212976107463897212on_val @ A5 @ R ) ) ).

% trans_onI
thf(fact_947_trans__onI,axiom,
    ! [A5: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat,Z3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A5 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A5 )
           => ( ( member8206827879206165904at_nat @ Z3 @ A5 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ R )
               => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ Z3 ) @ R )
                 => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_613433433080944135at_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_948_trans__onI,axiom,
    ! [A5: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A5 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A5 )
           => ( ( member8757157785044589968at_nat @ Z3 @ A5 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R )
               => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ Z3 ) @ R )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_4802246114490097159at_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_949_trans__onI,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A5 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A5 )
           => ( ( member6273874620674182256at_nat @ Z3 @ A5 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R )
               => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z3 ) @ R )
                 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_707035017233543399at_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_950_trans__onI,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A5 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A5 )
           => ( ( member2643936169264416010at_nat @ Z3 @ A5 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z3 ) @ R )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_3732288485672914945at_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_951_trans__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z3: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( ( member_nat @ Y3 @ A5 )
           => ( ( member_nat @ Z3 @ A5 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_on_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_952_trans__onI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A5 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A5 )
           => ( ( member8440522571783428010at_nat @ Z3 @ A5 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z3 ) @ R )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z3 ) @ R ) ) ) ) ) )
     => ( trans_3241703717017304737at_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_953_trans__on__def,axiom,
    ( trans_707035017233543399at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
        ! [X4: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X4 @ A6 )
         => ! [Y7: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y7 @ A6 )
             => ! [Z5: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ Z5 @ A6 )
                 => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R3 )
                   => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ Z5 ) @ R3 )
                     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Z5 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_954_trans__on__def,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ A6 )
         => ! [Y7: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y7 @ A6 )
             => ! [Z5: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Z5 @ A6 )
                 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R3 )
                   => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Z5 ) @ R3 )
                     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Z5 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_955_trans__on__def,axiom,
    ( trans_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A6 )
         => ! [Y7: nat] :
              ( ( member_nat @ Y7 @ A6 )
             => ! [Z5: nat] :
                  ( ( member_nat @ Z5 @ A6 )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R3 )
                   => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Z5 ) @ R3 )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z5 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_956_trans__on__def,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A6 )
         => ! [Y7: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y7 @ A6 )
             => ! [Z5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z5 @ A6 )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R3 )
                   => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Z5 ) @ R3 )
                     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Z5 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_957_UN__Times__distrib,axiom,
    ! [E: nat > set_nat,F3: nat > set_nat,A5: set_nat,B5: set_nat] :
      ( ( comple5685304695842803022at_nat
        @ ( image_5423882813909314213at_nat
          @ ( produc8197505143624133779at_nat
            @ ^ [A4: nat,B4: nat] :
                ( produc457027306803732586at_nat @ ( E @ A4 )
                @ ^ [Uu: nat] : ( F3 @ B4 ) ) )
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : B5 ) ) )
      = ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ E @ A5 ) )
        @ ^ [Uu: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F3 @ B5 ) ) ) ) ).

% UN_Times_distrib
thf(fact_958_Sigma__Union,axiom,
    ! [X5: set_set_nat,B5: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ X5 ) @ B5 )
      = ( comple5685304695842803022at_nat
        @ ( image_6521575952113991828at_nat
          @ ^ [A6: set_nat] : ( produc457027306803732586at_nat @ A6 @ B5 )
          @ X5 ) ) ) ).

% Sigma_Union
thf(fact_959_trans__on__lex__prod,axiom,
    ! [A5: set_nat,R_A: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( trans_on_nat @ A5 @ R_A )
     => ( ( trans_3241703717017304737at_nat @ B5 @ R_B )
       => ( trans_8787906625838855558at_nat
          @ ( produc1809337555817847783at_nat @ A5
            @ ^ [Uu: nat] : B5 )
          @ ( lex_pr8029265285556086080at_nat @ R_A @ R_B ) ) ) ) ).

% trans_on_lex_prod
thf(fact_960_trans__on__lex__prod,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B5: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3241703717017304737at_nat @ A5 @ R_A )
     => ( ( trans_on_nat @ B5 @ R_B )
       => ( trans_690021572951380832at_nat
          @ ( produc7672662199629908489at_nat @ A5
            @ ^ [Uu: product_prod_nat_nat] : B5 )
          @ ( lex_pr4669217892513370978at_nat @ R_A @ R_B ) ) ) ) ).

% trans_on_lex_prod
thf(fact_961_trans__on__lex__prod,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R_A: set_Pr8693737435421807431at_nat,B5: set_Pr1261947904930325089at_nat,R_B: set_Pr8693737435421807431at_nat] :
      ( ( trans_3241703717017304737at_nat @ A5 @ R_A )
     => ( ( trans_3241703717017304737at_nat @ B5 @ R_B )
       => ( trans_613433433080944135at_nat
          @ ( produc2761391749766926216at_nat @ A5
            @ ^ [Uu: product_prod_nat_nat] : B5 )
          @ ( lex_pr8801849515957261039at_nat @ R_A @ R_B ) ) ) ) ).

% trans_on_lex_prod
thf(fact_962_trans__on__lex__prod,axiom,
    ! [A5: set_nat,R_A: set_Pr1261947904930325089at_nat,B5: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ A5 @ R_A )
     => ( ( trans_on_nat @ B5 @ R_B )
       => ( trans_3241703717017304737at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : B5 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% trans_on_lex_prod
thf(fact_963_transD,axiom,
    ! [R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z2 ) @ R )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_964_transD,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_965_transD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_966_transD,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transD
thf(fact_967_transE,axiom,
    ! [R: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z2 ) @ R )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_968_transE,axiom,
    ! [R: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ R )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_969_transE,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_970_transE,axiom,
    ! [R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R ) ) ) ) ).

% transE
thf(fact_971_transI,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z3 ) @ R )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R ) ) ).

% transI
thf(fact_972_transI,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z3 ) @ R )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% transI
thf(fact_973_transI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% transI
thf(fact_974_transI,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z3 ) @ R )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z3 ) @ R ) ) )
     => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% transI
thf(fact_975_transp__on__trans__on__eq,axiom,
    ! [A5: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( transp5165774385813125367at_nat @ A5
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_707035017233543399at_nat @ A5 @ R ) ) ).

% transp_on_trans_on_eq
thf(fact_976_transp__on__trans__on__eq,axiom,
    ! [A5: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( transp4951929671385591825at_nat @ A5
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_3732288485672914945at_nat @ A5 @ R ) ) ).

% transp_on_trans_on_eq
thf(fact_977_transp__on__trans__on__eq,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( transp_on_nat @ A5
        @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_on_nat @ A5 @ R ) ) ).

% transp_on_trans_on_eq
thf(fact_978_transp__on__trans__on__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( transp6142674538225245873at_nat @ A5
        @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_3241703717017304737at_nat @ A5 @ R ) ) ).

% transp_on_trans_on_eq
thf(fact_979_transp__trans__eq,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ( transp5165774385813125367at_nat @ top_to1754375979021635447at_nat
        @ ^ [X4: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R ) ) ).

% transp_trans_eq
thf(fact_980_transp__trans__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( transp4951929671385591825at_nat @ top_to7629004291339433233at_nat
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R ) ) ).

% transp_trans_eq
thf(fact_981_transp__trans__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( transp_on_nat @ top_top_set_nat
        @ ^ [X4: nat,Y7: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% transp_trans_eq
thf(fact_982_transp__trans__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( transp6142674538225245873at_nat @ top_to4669805908274784177at_nat
        @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y7 ) @ R ) )
      = ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R ) ) ).

% transp_trans_eq
thf(fact_983_under__incr,axiom,
    ! [R: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R )
       => ( ord_le7966958417815640519at_nat @ ( order_5971310966627442348at_nat @ R @ A ) @ ( order_5971310966627442348at_nat @ R @ B ) ) ) ) ).

% under_incr
thf(fact_984_under__incr,axiom,
    ! [R: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
       => ( ord_le2077887516847798113at_nat @ ( order_8557040927991152966at_nat @ R @ A ) @ ( order_8557040927991152966at_nat @ R @ B ) ) ) ) ).

% under_incr
thf(fact_985_under__incr,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( ord_less_eq_set_nat @ ( order_under_nat @ R @ A ) @ ( order_under_nat @ R @ B ) ) ) ) ).

% under_incr
thf(fact_986_under__incr,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
       => ( ord_le3146513528884898305at_nat @ ( order_6448319830299018214at_nat @ R @ A ) @ ( order_6448319830299018214at_nat @ R @ B ) ) ) ) ).

% under_incr
thf(fact_987_trans__join,axiom,
    ! [R: set_Pr7459493094073627847at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R )
      = ( ! [X4: produc1319942482725812455at_nat] :
            ( ( member1466754251312161552at_nat @ X4 @ R )
           => ( produc9020218426428501292_nat_o
              @ ^ [Y7: set_Pr4329608150637261639at_nat,Y12: set_Pr4329608150637261639at_nat] :
                ! [Z5: produc1319942482725812455at_nat] :
                  ( ( member1466754251312161552at_nat @ Z5 @ R )
                 => ( produc9020218426428501292_nat_o
                    @ ^ [Y22: set_Pr4329608150637261639at_nat,Aa: set_Pr4329608150637261639at_nat] :
                        ( ( Y12 = Y22 )
                       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ Aa ) @ R ) )
                    @ Z5 ) )
              @ X4 ) ) ) ) ).

% trans_join
thf(fact_988_trans__join,axiom,
    ! [R: set_Pr4329608150637261639at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R )
      = ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ R )
           => ( produc410239310623530412_nat_o
              @ ^ [Y7: set_Pr1261947904930325089at_nat,Y12: set_Pr1261947904930325089at_nat] :
                ! [Z5: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ Z5 @ R )
                 => ( produc410239310623530412_nat_o
                    @ ^ [Y22: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat] :
                        ( ( Y12 = Y22 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Aa ) @ R ) )
                    @ Z5 ) )
              @ X4 ) ) ) ) ).

% trans_join
thf(fact_989_trans__join,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
      = ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ R )
           => ( produc6590410687421337004_nat_o
              @ ^ [Y7: product_prod_nat_nat,Y12: product_prod_nat_nat] :
                ! [Z5: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ Z5 @ R )
                 => ( produc6590410687421337004_nat_o
                    @ ^ [Y22: product_prod_nat_nat,Aa: product_prod_nat_nat] :
                        ( ( Y12 = Y22 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Aa ) @ R ) )
                    @ Z5 ) )
              @ X4 ) ) ) ) ).

% trans_join
thf(fact_990_trans__join,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ R )
           => ( produc6081775807080527818_nat_o
              @ ^ [Y7: nat,Y12: nat] :
                ! [Z5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z5 @ R )
                 => ( produc6081775807080527818_nat_o
                    @ ^ [Y22: nat,Aa: nat] :
                        ( ( Y12 = Y22 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Aa ) @ R ) )
                    @ Z5 ) )
              @ X4 ) ) ) ) ).

% trans_join
thf(fact_991_trans__singleton,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ bot_bo4948859079157340979at_nat ) ) ).

% trans_singleton
thf(fact_992_trans__singleton,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ bot_bo228742789529271731at_nat ) ) ).

% trans_singleton
thf(fact_993_trans__singleton,axiom,
    ! [A: nat] : ( trans_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ bot_bo2099793752762293965at_nat ) ) ).

% trans_singleton
thf(fact_994_trans__singleton,axiom,
    ! [A: product_prod_nat_nat] : ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ bot_bo5327735625951526323at_nat ) ) ).

% trans_singleton
thf(fact_995_insert__subsetI,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,X5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ X @ A5 )
     => ( ( ord_le6900622908513426738on_val @ X5 @ A5 )
       => ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X @ X5 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_996_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( ord_le3146513528884898305at_nat @ X5 @ A5 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_997_insert__subsetI,axiom,
    ! [X: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X @ A5 )
     => ( ( ord_le3000389064537975527at_nat @ X5 @ A5 )
       => ( ord_le3000389064537975527at_nat @ ( insert5050368324300391991at_nat @ X @ X5 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_998_insert__subsetI,axiom,
    ! [X: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A5 )
     => ( ( ord_le1268244103169919719at_nat @ X5 @ A5 )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ X5 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_999_Ball__Collect,axiom,
    ( ball_P2871712132005019634on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,P4: produc9003781588162623004on_val > $o] : ( ord_le6900622908513426738on_val @ A6 @ ( collec7475294502966882545on_val @ P4 ) ) ) ) ).

% Ball_Collect
thf(fact_1000_Ball__Collect,axiom,
    ( ball_P8096063237992195499at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,P4: product_prod_nat_nat > $o] : ( ord_le3146513528884898305at_nat @ A6 @ ( collec3392354462482085612at_nat @ P4 ) ) ) ) ).

% Ball_Collect
thf(fact_1001_Ball__Collect,axiom,
    ( ball_nat
    = ( ^ [A6: set_nat,P4: nat > $o] : ( ord_less_eq_set_nat @ A6 @ ( collect_nat @ P4 ) ) ) ) ).

% Ball_Collect
thf(fact_1002_under__def,axiom,
    ( order_2832490146329081463on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A4: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ A4 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1003_under__def,axiom,
    ( order_6448319830299018214at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A4: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ A4 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1004_under__def,axiom,
    ( order_5971310966627442348at_nat
    = ( ^ [R3: set_Pr7459493094073627847at_nat,A4: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B4 @ A4 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1005_under__def,axiom,
    ( order_8557040927991152966at_nat
    = ( ^ [R3: set_Pr4329608150637261639at_nat,A4: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ A4 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1006_under__def,axiom,
    ( order_under_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A4: nat] :
          ( collect_nat
          @ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A4 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1007_refl__on__singleton,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( refl_o6272981002401706822at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ bot_bo4948859079157340979at_nat ) ) ).

% refl_on_singleton
thf(fact_1008_refl__on__singleton,axiom,
    ! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% refl_on_singleton
thf(fact_1009_refl__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% refl_on_singleton
thf(fact_1010_refl__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( refl_o7122697186039111552at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% refl_on_singleton
thf(fact_1011_refl__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( refl_o7446464703481863270at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% refl_on_singleton
thf(fact_1012_trans__on__join,axiom,
    ( trans_3212976107463897212on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X4: produc1317171644805857433on_val] :
          ( ( member6505152944228695362on_val @ X4 @ R3 )
         => ( produc1845727980445322802_val_o
            @ ^ [Y7: produc9003781588162623004on_val,Y12: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ Y7 @ A6 )
               => ( ( member2879030624002546611on_val @ Y12 @ A6 )
                 => ! [Z5: produc1317171644805857433on_val] :
                      ( ( member6505152944228695362on_val @ Z5 @ R3 )
                     => ( produc1845727980445322802_val_o
                        @ ^ [Y22: produc9003781588162623004on_val,Aa: produc9003781588162623004on_val] :
                            ( ( Y12 = Y22 )
                           => ( ( member2879030624002546611on_val @ Aa @ A6 )
                             => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1013_trans__on__join,axiom,
    ( trans_613433433080944135at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat] :
        ! [X4: produc349518998152878311at_nat] :
          ( ( member8062223511168850704at_nat @ X4 @ R3 )
         => ( produc3603361296207914284_nat_o
            @ ^ [Y7: produc859450856879609959at_nat,Y12: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y7 @ A6 )
               => ( ( member8206827879206165904at_nat @ Y12 @ A6 )
                 => ! [Z5: produc349518998152878311at_nat] :
                      ( ( member8062223511168850704at_nat @ Z5 @ R3 )
                     => ( produc3603361296207914284_nat_o
                        @ ^ [Y22: produc859450856879609959at_nat,Aa: produc859450856879609959at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8206827879206165904at_nat @ Aa @ A6 )
                             => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1014_trans__on__join,axiom,
    ( trans_4802246114490097159at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
        ! [X4: produc1906917888121415911at_nat] :
          ( ( member6009668390874684176at_nat @ X4 @ R3 )
         => ( produc1561160019221259564_nat_o
            @ ^ [Y7: produc3843707927480180839at_nat,Y12: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y7 @ A6 )
               => ( ( member8757157785044589968at_nat @ Y12 @ A6 )
                 => ! [Z5: produc1906917888121415911at_nat] :
                      ( ( member6009668390874684176at_nat @ Z5 @ R3 )
                     => ( produc1561160019221259564_nat_o
                        @ ^ [Y22: produc3843707927480180839at_nat,Aa: produc3843707927480180839at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8757157785044589968at_nat @ Aa @ A6 )
                             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1015_trans__on__join,axiom,
    ( trans_707035017233543399at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
        ! [X4: produc1319942482725812455at_nat] :
          ( ( member1466754251312161552at_nat @ X4 @ R3 )
         => ( produc9020218426428501292_nat_o
            @ ^ [Y7: set_Pr4329608150637261639at_nat,Y12: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Y7 @ A6 )
               => ( ( member6273874620674182256at_nat @ Y12 @ A6 )
                 => ! [Z5: produc1319942482725812455at_nat] :
                      ( ( member1466754251312161552at_nat @ Z5 @ R3 )
                     => ( produc9020218426428501292_nat_o
                        @ ^ [Y22: set_Pr4329608150637261639at_nat,Aa: set_Pr4329608150637261639at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member6273874620674182256at_nat @ Aa @ A6 )
                             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1016_trans__on__join,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X4 @ R3 )
         => ( produc410239310623530412_nat_o
            @ ^ [Y7: set_Pr1261947904930325089at_nat,Y12: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y7 @ A6 )
               => ( ( member2643936169264416010at_nat @ Y12 @ A6 )
                 => ! [Z5: produc3843707927480180839at_nat] :
                      ( ( member8757157785044589968at_nat @ Z5 @ R3 )
                     => ( produc410239310623530412_nat_o
                        @ ^ [Y22: set_Pr1261947904930325089at_nat,Aa: set_Pr1261947904930325089at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member2643936169264416010at_nat @ Aa @ A6 )
                             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1017_trans__on__join,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
        ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ R3 )
         => ( produc6590410687421337004_nat_o
            @ ^ [Y7: product_prod_nat_nat,Y12: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y7 @ A6 )
               => ( ( member8440522571783428010at_nat @ Y12 @ A6 )
                 => ! [Z5: produc859450856879609959at_nat] :
                      ( ( member8206827879206165904at_nat @ Z5 @ R3 )
                     => ( produc6590410687421337004_nat_o
                        @ ^ [Y22: product_prod_nat_nat,Aa: product_prod_nat_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8440522571783428010at_nat @ Aa @ A6 )
                             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1018_trans__on__join,axiom,
    ( trans_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ R3 )
         => ( produc6081775807080527818_nat_o
            @ ^ [Y7: nat,Y12: nat] :
                ( ( member_nat @ Y7 @ A6 )
               => ( ( member_nat @ Y12 @ A6 )
                 => ! [Z5: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ Z5 @ R3 )
                     => ( produc6081775807080527818_nat_o
                        @ ^ [Y22: nat,Aa: nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member_nat @ Aa @ A6 )
                             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Aa ) @ R3 ) ) )
                        @ Z5 ) ) ) )
            @ X4 ) ) ) ) ).

% trans_on_join
thf(fact_1019_refl__on__def_H,axiom,
    ( refl_o6679066329051764061on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
          ( ! [X4: produc1317171644805857433on_val] :
              ( ( member6505152944228695362on_val @ X4 @ R3 )
             => ( produc1845727980445322802_val_o
                @ ^ [Y7: produc9003781588162623004on_val,Z5: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ Y7 @ A6 )
                    & ( member2879030624002546611on_val @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X4 @ A6 )
             => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1020_refl__on__def_H,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
          ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ R3 )
             => ( produc6590410687421337004_nat_o
                @ ^ [Y7: product_prod_nat_nat,Z5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y7 @ A6 )
                    & ( member8440522571783428010at_nat @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1021_refl__on__def_H,axiom,
    ( refl_o6679620997798956134at_nat
    = ( ^ [A6: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat] :
          ( ! [X4: produc349518998152878311at_nat] :
              ( ( member8062223511168850704at_nat @ X4 @ R3 )
             => ( produc3603361296207914284_nat_o
                @ ^ [Y7: produc859450856879609959at_nat,Z5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Y7 @ A6 )
                    & ( member8206827879206165904at_nat @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A6 )
             => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1022_refl__on__def_H,axiom,
    ( refl_o7446464703481863270at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
          ( ! [X4: produc1906917888121415911at_nat] :
              ( ( member6009668390874684176at_nat @ X4 @ R3 )
             => ( produc1561160019221259564_nat_o
                @ ^ [Y7: produc3843707927480180839at_nat,Z5: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Y7 @ A6 )
                    & ( member8757157785044589968at_nat @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ A6 )
             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1023_refl__on__def_H,axiom,
    ( refl_o6272981002401706822at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
          ( ! [X4: produc1319942482725812455at_nat] :
              ( ( member1466754251312161552at_nat @ X4 @ R3 )
             => ( produc9020218426428501292_nat_o
                @ ^ [Y7: set_Pr4329608150637261639at_nat,Z5: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ Y7 @ A6 )
                    & ( member6273874620674182256at_nat @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X4 @ A6 )
             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1024_refl__on__def_H,axiom,
    ( refl_o6955724340487615712at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
          ( ! [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ R3 )
             => ( produc410239310623530412_nat_o
                @ ^ [Y7: set_Pr1261947904930325089at_nat,Z5: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Y7 @ A6 )
                    & ( member2643936169264416010at_nat @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X4 @ A6 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1025_refl__on__def_H,axiom,
    ( refl_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
          ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ R3 )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y7: nat,Z5: nat] :
                    ( ( member_nat @ Y7 @ A6 )
                    & ( member_nat @ Z5 @ A6 ) )
                @ X4 ) )
          & ! [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_1026_Sigma__def,axiom,
    ( produc6258361011560675592at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,B8: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X4: set_Pr4329608150637261639at_nat] :
                ( comple7342073961973170484at_nat
                @ ( image_1124969593608450885at_nat
                  @ ^ [Y7: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y7 ) @ bot_bo4948859079157340979at_nat )
                  @ ( B8 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1027_Sigma__def,axiom,
    ( produc6030673636645808891t_char
    = ( ^ [A6: set_nat,B8: nat > set_list_char] :
          ( comple9178978156834306527t_char
          @ ( image_7918911184113263195t_char
            @ ^ [X4: nat] :
                ( comple9178978156834306527t_char
                @ ( image_366428816823531018t_char
                  @ ^ [Y7: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X4 @ Y7 ) @ bot_bo1414094081888755934t_char )
                  @ ( B8 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1028_Sigma__def,axiom,
    ( produc1366239611153933861on_val
    = ( ^ [A6: set_Pr3433046674199179443r_prat,B8: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
          ( comple5258588689616361253on_val
          @ ( image_3306202642189553376on_val
            @ ^ [X4: produc1457211279475724562t_char > prat] :
                ( comple5258588689616361253on_val
                @ ( image_8136294080383932403on_val
                  @ ^ [Y7: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X4 @ Y7 ) @ bot_bo6722054719108133734on_val )
                  @ ( B8 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1029_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A6: set_nat,B8: nat > set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X4: nat] :
                ( comple5685304695842803022at_nat
                @ ( image_7178329752028323786at_nat
                  @ ^ [Y7: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ Y7 ) @ bot_bo2099793752762293965at_nat )
                  @ ( B8 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1030_Sigma__def,axiom,
    ( produc7726050920695093128at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,B8: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X4: set_Pr1261947904930325089at_nat] :
                ( comple4364388969068068532at_nat
                @ ( image_6898982319499645995at_nat
                  @ ^ [Y7: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y7 ) @ bot_bo228742789529271731at_nat )
                  @ ( B8 @ X4 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_1031_Id__on__def,axiom,
    ( id_on_3275838214495744996at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X4: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X4 @ X4 ) @ bot_bo4948859079157340979at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_1032_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A6: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ bot_bo2099793752762293965at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_1033_Id__on__def,axiom,
    ( id_on_6713504163309144702at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X4: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X4 @ X4 ) @ bot_bo228742789529271731at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_1034_linear__order__on__singleton,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( order_1878278401035043452at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ bot_bo4948859079157340979at_nat ) ) ).

% linear_order_on_singleton
thf(fact_1035_linear__order__on__singleton,axiom,
    ! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% linear_order_on_singleton
thf(fact_1036_linear__order__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% linear_order_on_singleton
thf(fact_1037_linear__order__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( order_7480980593162297270at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% linear_order_on_singleton
thf(fact_1038_linear__order__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( order_2711500226769404828at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% linear_order_on_singleton
thf(fact_1039_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( produc3090783324798197870at_nat @ ( insert8211810215607154385at_nat @ A @ A5 )
        @ ^ [Uu: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
      = ( insert8243672688981495325at_nat @ ( produc8594291983450332477at_nat @ A @ B )
        @ ( sup_su1595509577888697473at_nat
          @ ( produc3090783324798197870at_nat @ A5
            @ ^ [Uu: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
          @ ( produc3090783324798197870at_nat @ ( insert8211810215607154385at_nat @ A @ A5 )
            @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1040_insert__Times__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( produc8743256393616062242at_nat @ ( insert9069300056098147895at_nat @ A @ A5 )
        @ ^ [Uu: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
      = ( insert373714669040926161at_nat @ ( produc5023393015413421041at_nat @ A @ B )
        @ ( sup_su6071048768109290037at_nat
          @ ( produc8743256393616062242at_nat @ A5
            @ ^ [Uu: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
          @ ( produc8743256393616062242at_nat @ ( insert9069300056098147895at_nat @ A @ A5 )
            @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1041_insert__Times__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( produc4441300235185154312at_nat @ ( insert9069300056098147895at_nat @ A @ A5 )
        @ ^ [Uu: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
      = ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B )
        @ ( sup_su6664651903789602459at_nat
          @ ( produc4441300235185154312at_nat @ A5
            @ ^ [Uu: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
          @ ( produc4441300235185154312at_nat @ ( insert9069300056098147895at_nat @ A @ A5 )
            @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1042_insert__Times__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat,B: set_Pr4329608150637261639at_nat,B5: set_se5479201105039950375at_nat] :
      ( ( produc6258361011560675592at_nat @ ( insert2290795007606087959at_nat @ A @ A5 )
        @ ^ [Uu: set_Pr4329608150637261639at_nat] : ( insert2290795007606087959at_nat @ B @ B5 ) )
      = ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B )
        @ ( sup_su5395584912770621083at_nat
          @ ( produc6258361011560675592at_nat @ A5
            @ ^ [Uu: set_Pr4329608150637261639at_nat] : ( insert2290795007606087959at_nat @ B @ B5 ) )
          @ ( produc6258361011560675592at_nat @ ( insert2290795007606087959at_nat @ A @ A5 )
            @ ^ [Uu: set_Pr4329608150637261639at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1043_insert__Times__insert,axiom,
    ! [A: nat,A5: set_nat,B: list_char,B5: set_list_char] :
      ( ( produc6030673636645808891t_char @ ( insert_nat @ A @ A5 )
        @ ^ [Uu: nat] : ( insert_list_char @ B @ B5 ) )
      = ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B )
        @ ( sup_su7899656352754981958t_char
          @ ( produc6030673636645808891t_char @ A5
            @ ^ [Uu: nat] : ( insert_list_char @ B @ B5 ) )
          @ ( produc6030673636645808891t_char @ ( insert_nat @ A @ A5 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1044_insert__Times__insert,axiom,
    ! [A: produc1457211279475724562t_char > prat,A5: set_Pr3433046674199179443r_prat,B: produc1457211279475724562t_char > option_val,B5: set_Pr1004501353197068112on_val] :
      ( ( produc1366239611153933861on_val @ ( insert8113562578550770211r_prat @ A @ A5 )
        @ ^ [Uu: produc1457211279475724562t_char > prat] : ( insert216038912599248202on_val @ B @ B5 ) )
      = ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B )
        @ ( sup_su7506060154676209150on_val
          @ ( produc1366239611153933861on_val @ A5
            @ ^ [Uu: produc1457211279475724562t_char > prat] : ( insert216038912599248202on_val @ B @ B5 ) )
          @ ( produc1366239611153933861on_val @ ( insert8113562578550770211r_prat @ A @ A5 )
            @ ^ [Uu: produc1457211279475724562t_char > prat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1045_insert__Times__insert,axiom,
    ! [A: nat,A5: set_nat,B: nat,B5: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A5 )
        @ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A5 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1046_insert__Times__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( insert9200635055090092081at_nat @ A @ A5 )
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : ( insert9200635055090092081at_nat @ B @ B5 ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B )
        @ ( sup_su5525570899277871387at_nat
          @ ( produc7726050920695093128at_nat @ A5
            @ ^ [Uu: set_Pr1261947904930325089at_nat] : ( insert9200635055090092081at_nat @ B @ B5 ) )
          @ ( produc7726050920695093128at_nat @ ( insert9200635055090092081at_nat @ A @ A5 )
            @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1047_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( produc2761391749766926216at_nat @ ( insert8211810215607154385at_nat @ A @ A5 )
        @ ^ [Uu: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
      = ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B )
        @ ( sup_su718114333110466843at_nat
          @ ( produc2761391749766926216at_nat @ A5
            @ ^ [Uu: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
          @ ( produc2761391749766926216at_nat @ ( insert8211810215607154385at_nat @ A @ A5 )
            @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_1048_Id__onI,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ A5 )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ ( id_on_1505035389332359999on_val @ A5 ) ) ) ).

% Id_onI
thf(fact_1049_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A5 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_1050_Id__onI,axiom,
    ! [A: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A5 )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( id_on_7917584626360399108at_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_1051_Id__onI,axiom,
    ! [A: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A5 )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( id_on_7869290023980668164at_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_1052_Id__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ A @ A5 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( id_on_3275838214495744996at_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_1053_Id__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A5 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( id_on_6713504163309144702at_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_1054_Id__onI,axiom,
    ! [A: nat,A5: set_nat] :
      ( ( member_nat @ A @ A5 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A5 ) ) ) ).

% Id_onI
thf(fact_1055_Times__Un__distrib1,axiom,
    ! [A5: set_nat,B5: set_nat,C3: set_nat] :
      ( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ A5 @ B5 )
        @ ^ [Uu: nat] : C3 )
      = ( sup_su6327502436637775413at_nat
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu: nat] : C3 )
        @ ( produc457027306803732586at_nat @ B5
          @ ^ [Uu: nat] : C3 ) ) ) ).

% Times_Un_distrib1
thf(fact_1056_Times__Un__distrib1,axiom,
    ! [A5: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat,C3: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ A5 @ B5 )
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : C3 )
      = ( sup_su5525570899277871387at_nat
        @ ( produc7726050920695093128at_nat @ A5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : C3 )
        @ ( produc7726050920695093128at_nat @ B5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : C3 ) ) ) ).

% Times_Un_distrib1
thf(fact_1057_Times__Un__distrib1,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( produc2761391749766926216at_nat @ ( sup_su6327502436637775413at_nat @ A5 @ B5 )
        @ ^ [Uu: product_prod_nat_nat] : C3 )
      = ( sup_su718114333110466843at_nat
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu: product_prod_nat_nat] : C3 )
        @ ( produc2761391749766926216at_nat @ B5
          @ ^ [Uu: product_prod_nat_nat] : C3 ) ) ) ).

% Times_Un_distrib1
thf(fact_1058_Sigma__Un__distrib2,axiom,
    ! [I2: set_nat,A5: nat > set_nat,B5: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ I2
        @ ^ [I: nat] : ( sup_sup_set_nat @ ( A5 @ I ) @ ( B5 @ I ) ) )
      = ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I2 @ A5 ) @ ( produc457027306803732586at_nat @ I2 @ B5 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1059_Sigma__Un__distrib2,axiom,
    ! [I2: set_se7855581050983116737at_nat,A5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ I2
        @ ^ [I: set_Pr1261947904930325089at_nat] : ( sup_su3642409539654194069at_nat @ ( A5 @ I ) @ ( B5 @ I ) ) )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I2 @ A5 ) @ ( produc7726050920695093128at_nat @ I2 @ B5 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1060_Sigma__Un__distrib2,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat > set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( produc2761391749766926216at_nat @ I2
        @ ^ [I: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( A5 @ I ) @ ( B5 @ I ) ) )
      = ( sup_su718114333110466843at_nat @ ( produc2761391749766926216at_nat @ I2 @ A5 ) @ ( produc2761391749766926216at_nat @ I2 @ B5 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1061_Sigma__Un__distrib1,axiom,
    ! [I2: set_nat,J2: set_nat,C3: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ I2 @ J2 ) @ C3 )
      = ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I2 @ C3 ) @ ( produc457027306803732586at_nat @ J2 @ C3 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1062_Sigma__Un__distrib1,axiom,
    ! [I2: set_se7855581050983116737at_nat,J2: set_se7855581050983116737at_nat,C3: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ I2 @ J2 ) @ C3 )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I2 @ C3 ) @ ( produc7726050920695093128at_nat @ J2 @ C3 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1063_Sigma__Un__distrib1,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,J2: set_Pr1261947904930325089at_nat,C3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( produc2761391749766926216at_nat @ ( sup_su6327502436637775413at_nat @ I2 @ J2 ) @ C3 )
      = ( sup_su718114333110466843at_nat @ ( produc2761391749766926216at_nat @ I2 @ C3 ) @ ( produc2761391749766926216at_nat @ J2 @ C3 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1064_Id__on__iff,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( id_on_1505035389332359999on_val @ A5 ) )
      = ( ( X = Y )
        & ( member2879030624002546611on_val @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1065_Id__on__iff,axiom,
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ ( id_on_7917584626360399108at_nat @ A5 ) )
      = ( ( X = Y )
        & ( member8206827879206165904at_nat @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1066_Id__on__iff,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ ( id_on_7869290023980668164at_nat @ A5 ) )
      = ( ( X = Y )
        & ( member8757157785044589968at_nat @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1067_Id__on__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( id_on_2554058798563519774at_nat @ A5 ) )
      = ( ( X = Y )
        & ( member8440522571783428010at_nat @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1068_Id__on__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( id_on_3275838214495744996at_nat @ A5 ) )
      = ( ( X = Y )
        & ( member6273874620674182256at_nat @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1069_Id__on__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( id_on_6713504163309144702at_nat @ A5 ) )
      = ( ( X = Y )
        & ( member2643936169264416010at_nat @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1070_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A5 ) )
      = ( ( X = Y )
        & ( member_nat @ X @ A5 ) ) ) ).

% Id_on_iff
thf(fact_1071_Id__on__eqI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( A = B )
     => ( ( member2879030624002546611on_val @ A @ A5 )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( id_on_1505035389332359999on_val @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1072_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( A = B )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1073_Id__on__eqI,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( A = B )
     => ( ( member8206827879206165904at_nat @ A @ A5 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( id_on_7917584626360399108at_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1074_Id__on__eqI,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( A = B )
     => ( ( member8757157785044589968at_nat @ A @ A5 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ ( id_on_7869290023980668164at_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1075_Id__on__eqI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A5: set_se5479201105039950375at_nat] :
      ( ( A = B )
     => ( ( member6273874620674182256at_nat @ A @ A5 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( id_on_3275838214495744996at_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1076_Id__on__eqI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A5: set_se7855581050983116737at_nat] :
      ( ( A = B )
     => ( ( member2643936169264416010at_nat @ A @ A5 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( id_on_6713504163309144702at_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1077_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A5: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A5 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A5 ) ) ) ) ).

% Id_on_eqI
thf(fact_1078_Id__onE,axiom,
    ! [C: produc1317171644805857433on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ C @ ( id_on_1505035389332359999on_val @ A5 ) )
     => ~ ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A5 )
           => ( C
             != ( produc4958730763592889809on_val @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1079_Id__onE,axiom,
    ! [C: produc349518998152878311at_nat,A5: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C @ ( id_on_7917584626360399108at_nat @ A5 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A5 )
           => ( C
             != ( produc4662710985925991255at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1080_Id__onE,axiom,
    ! [C: produc1906917888121415911at_nat,A5: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ C @ ( id_on_7869290023980668164at_nat @ A5 ) )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A5 )
           => ( C
             != ( produc2335810615602371415at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1081_Id__onE,axiom,
    ! [C: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( id_on_2554058798563519774at_nat @ A5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ( C
             != ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1082_Id__onE,axiom,
    ! [C: produc1319942482725812455at_nat,A5: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ C @ ( id_on_3275838214495744996at_nat @ A5 ) )
     => ~ ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A5 )
           => ( C
             != ( produc9060074326276436823at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1083_Id__onE,axiom,
    ! [C: produc3843707927480180839at_nat,A5: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( id_on_6713504163309144702at_nat @ A5 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A5 )
           => ( C
             != ( produc2922128104949294807at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1084_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ( C
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_1085_Id__on__def_H,axiom,
    ! [A5: produc9003781588162623004on_val > $o] :
      ( ( id_on_1505035389332359999on_val @ ( collec7475294502966882545on_val @ A5 ) )
      = ( collec5166231387257825412on_val
        @ ( produc1845727980445322802_val_o
          @ ^ [X4: produc9003781588162623004on_val,Y7: produc9003781588162623004on_val] :
              ( ( X4 = Y7 )
              & ( A5 @ X4 ) ) ) ) ) ).

% Id_on_def'
thf(fact_1086_Id__on__def_H,axiom,
    ! [A5: product_prod_nat_nat > $o] :
      ( ( id_on_2554058798563519774at_nat @ ( collec3392354462482085612at_nat @ A5 ) )
      = ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X4: product_prod_nat_nat,Y7: product_prod_nat_nat] :
              ( ( X4 = Y7 )
              & ( A5 @ X4 ) ) ) ) ) ).

% Id_on_def'
thf(fact_1087_Id__on__def_H,axiom,
    ! [A5: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A5 ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( X4 = Y7 )
              & ( A5 @ X4 ) ) ) ) ) ).

% Id_on_def'
thf(fact_1088_Field__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( field_1428494348145443988at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R ) )
      = ( sup_su474284620922681339at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R ) ) ) ).

% Field_insert
thf(fact_1089_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R ) ) ) ).

% Field_insert
thf(fact_1090_Field__insert,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).

% Field_insert
thf(fact_1091_Field__insert,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( field_3044959742847878580at_nat @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R ) )
      = ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( field_3044959742847878580at_nat @ R ) ) ) ).

% Field_insert
thf(fact_1092_Field__insert,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) )
      = ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R ) ) ) ).

% Field_insert
thf(fact_1093_Field__insert,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,R: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R ) )
      = ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R ) ) ) ).

% Field_insert
thf(fact_1094_prod__set__simps_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( basic_5713354656523905413at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) ).

% prod_set_simps(2)
thf(fact_1095_prod__set__simps_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_484852565060450053at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_1096_prod__set__simps_I2_J,axiom,
    ! [X: nat,Y: list_char] :
      ( ( basic_2224790070303523448t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( insert_list_char @ Y @ bot_bo8503570744482963627t_char ) ) ).

% prod_set_simps(2)
thf(fact_1097_prod__set__simps_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( basic_snds_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( insert_nat @ Y @ bot_bot_set_nat ) ) ).

% prod_set_simps(2)
thf(fact_1098_prod__set__simps_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( basic_8841519627501779048on_val @ ( produc74620467203698198on_val @ X @ Y ) )
      = ( insert216038912599248202on_val @ Y @ bot_bo5671083314777306084on_val ) ) ).

% prod_set_simps(2)
thf(fact_1099_prod__set__simps_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( basic_1274512288150691911at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ).

% prod_set_simps(1)
thf(fact_1100_prod__set__simps_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_3661821009137050567at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(1)
thf(fact_1101_prod__set__simps_I1_J,axiom,
    ! [X: nat,Y: list_char] :
      ( ( basic_4688923273072336698t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% prod_set_simps(1)
thf(fact_1102_prod__set__simps_I1_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( basic_fsts_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% prod_set_simps(1)
thf(fact_1103_prod__set__simps_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: produc1457211279475724562t_char > option_val] :
      ( ( basic_7826031515394172198on_val @ ( produc74620467203698198on_val @ X @ Y ) )
      = ( insert8113562578550770211r_prat @ X @ bot_bo3164680191535668511r_prat ) ) ).

% prod_set_simps(1)
thf(fact_1104_Range__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( range_7218127889571902005at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R ) )
      = ( insert2290795007606087959at_nat @ B @ ( range_7218127889571902005at_nat @ R ) ) ) ).

% Range_insert
thf(fact_1105_Range__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( range_880292732526890933at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_880292732526890933at_nat @ R ) ) ) ).

% Range_insert
thf(fact_1106_Range__insert,axiom,
    ! [A: nat,B: list_char,R: set_Pr2507339680178222706t_char] :
      ( ( range_nat_list_char @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B ) @ R ) )
      = ( insert_list_char @ B @ ( range_nat_list_char @ R ) ) ) ).

% Range_insert
thf(fact_1107_Range__insert,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( insert_nat @ B @ ( range_nat_nat @ R ) ) ) ).

% Range_insert
thf(fact_1108_Range__insert,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( range_8542210083549373368on_val @ ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B ) @ R ) )
      = ( insert216038912599248202on_val @ B @ ( range_8542210083549373368on_val @ R ) ) ) ).

% Range_insert
thf(fact_1109_Domain__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( domain6426273929576919500at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R ) )
      = ( insert2290795007606087959at_nat @ A @ ( domain6426273929576919500at_nat @ R ) ) ) ).

% Domain_insert
thf(fact_1110_Domain__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( domain8802589944165416012at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R ) )
      = ( insert9200635055090092081at_nat @ A @ ( domain8802589944165416012at_nat @ R ) ) ) ).

% Domain_insert
thf(fact_1111_Domain__insert,axiom,
    ! [A: nat,B: list_char,R: set_Pr2507339680178222706t_char] :
      ( ( domain_nat_list_char @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B ) @ R ) )
      = ( insert_nat @ A @ ( domain_nat_list_char @ R ) ) ) ).

% Domain_insert
thf(fact_1112_Domain__insert,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( insert_nat @ A @ ( domain_nat_nat @ R ) ) ) ).

% Domain_insert
thf(fact_1113_Domain__insert,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( domain6100248181835736161on_val @ ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B ) @ R ) )
      = ( insert8113562578550770211r_prat @ A @ ( domain6100248181835736161on_val @ R ) ) ) ).

% Domain_insert
thf(fact_1114_Domain__Collect__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( domain6100248181835736161on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P2 ) ) )
      = ( collec7093475260167521854r_prat
        @ ^ [X4: produc1457211279475724562t_char > prat] :
          ? [X7: produc1457211279475724562t_char > option_val] : ( P2 @ X4 @ X7 ) ) ) ).

% Domain_Collect_case_prod
thf(fact_1115_Domain__Collect__case__prod,axiom,
    ! [P2: nat > nat > $o] :
      ( ( domain_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
      = ( collect_nat
        @ ^ [X4: nat] :
          ? [X7: nat] : ( P2 @ X4 @ X7 ) ) ) ).

% Domain_Collect_case_prod
thf(fact_1116_Range__Collect__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( range_8542210083549373368on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P2 ) ) )
      = ( collec8455114619259388271on_val
        @ ^ [Y7: produc1457211279475724562t_char > option_val] :
          ? [X4: produc1457211279475724562t_char > prat] : ( P2 @ X4 @ Y7 ) ) ) ).

% Range_Collect_case_prod
thf(fact_1117_Range__Collect__case__prod,axiom,
    ! [P2: nat > nat > $o] :
      ( ( range_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
      = ( collect_nat
        @ ^ [Y7: nat] :
          ? [X4: nat] : ( P2 @ X4 @ Y7 ) ) ) ).

% Range_Collect_case_prod
thf(fact_1118_FieldI2,axiom,
    ! [I3: produc9003781588162623004on_val,J3: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I3 @ J3 ) @ R2 )
     => ( member2879030624002546611on_val @ J3 @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI2
thf(fact_1119_FieldI2,axiom,
    ! [I3: produc859450856879609959at_nat,J3: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J3 ) @ R2 )
     => ( member8206827879206165904at_nat @ J3 @ ( field_3044959742847878580at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_1120_FieldI2,axiom,
    ! [I3: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J3 ) @ R2 )
     => ( member8757157785044589968at_nat @ J3 @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_1121_FieldI2,axiom,
    ! [I3: product_prod_nat_nat,J3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J3 ) @ R2 )
     => ( member8440522571783428010at_nat @ J3 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_1122_FieldI2,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J3: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J3 ) @ R2 )
     => ( member6273874620674182256at_nat @ J3 @ ( field_1428494348145443988at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_1123_FieldI2,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J3 ) @ R2 )
     => ( member2643936169264416010at_nat @ J3 @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_1124_FieldI2,axiom,
    ! [I3: nat,J3: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J3 ) @ R2 )
     => ( member_nat @ J3 @ ( field_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_1125_FieldI1,axiom,
    ! [I3: produc9003781588162623004on_val,J3: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I3 @ J3 ) @ R2 )
     => ( member2879030624002546611on_val @ I3 @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI1
thf(fact_1126_FieldI1,axiom,
    ! [I3: produc859450856879609959at_nat,J3: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J3 ) @ R2 )
     => ( member8206827879206165904at_nat @ I3 @ ( field_3044959742847878580at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_1127_FieldI1,axiom,
    ! [I3: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J3 ) @ R2 )
     => ( member8757157785044589968at_nat @ I3 @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_1128_FieldI1,axiom,
    ! [I3: product_prod_nat_nat,J3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J3 ) @ R2 )
     => ( member8440522571783428010at_nat @ I3 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_1129_FieldI1,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J3: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J3 ) @ R2 )
     => ( member6273874620674182256at_nat @ I3 @ ( field_1428494348145443988at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_1130_FieldI1,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J3 ) @ R2 )
     => ( member2643936169264416010at_nat @ I3 @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_1131_FieldI1,axiom,
    ! [I3: nat,J3: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J3 ) @ R2 )
     => ( member_nat @ I3 @ ( field_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_1132_Domain_Ocases,axiom,
    ! [A: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R ) )
     => ~ ! [B3: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_1133_Domain_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R ) )
     => ~ ! [B3: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_1134_Domain_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
     => ~ ! [B3: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_1135_Domain_Ocases,axiom,
    ! [A: nat,R: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R ) )
     => ~ ! [B3: list_char] :
            ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_1136_Domain_Ocases,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R ) )
     => ~ ! [B3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_1137_Domain_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
     => ~ ! [B3: produc1457211279475724562t_char > option_val] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_1138_Domain_Osimps,axiom,
    ! [A: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R ) )
      = ( ? [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
            ( ( A = A4 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_1139_Domain_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R ) )
      = ( ? [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
            ( ( A = A4 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A4 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_1140_Domain_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
      = ( ? [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
            ( ( A = A4 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A4 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_1141_Domain_Osimps,axiom,
    ! [A: nat,R: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R ) )
      = ( ? [A4: nat,B4: list_char] :
            ( ( A = A4 )
            & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A4 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_1142_Domain_Osimps,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R ) )
      = ( ? [A4: nat,B4: nat] :
            ( ( A = A4 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_1143_Domain_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
      = ( ? [A4: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
            ( ( A = A4 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A4 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_1144_Domain_ODomainI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
     => ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_1145_Domain_ODomainI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R )
     => ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_1146_Domain_ODomainI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R )
     => ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_1147_Domain_ODomainI,axiom,
    ! [A: nat,B: list_char,R: set_Pr2507339680178222706t_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R )
     => ( member_nat @ A @ ( domain_nat_list_char @ R ) ) ) ).

% Domain.DomainI
thf(fact_1148_Domain_ODomainI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member_nat @ A @ ( domain_nat_nat @ R ) ) ) ).

% Domain.DomainI
thf(fact_1149_Domain_ODomainI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) ) ) ).

% Domain.DomainI
thf(fact_1150_DomainE,axiom,
    ! [A: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R ) )
     => ~ ! [B3: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_1151_DomainE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R ) )
     => ~ ! [B3: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_1152_DomainE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R ) )
     => ~ ! [B3: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_1153_DomainE,axiom,
    ! [A: nat,R: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R ) )
     => ~ ! [B3: list_char] :
            ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_1154_DomainE,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R ) )
     => ~ ! [B3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_1155_DomainE,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
     => ~ ! [B3: produc1457211279475724562t_char > option_val] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_1156_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P2 @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K ) ) )
       => ( P2 @ M ) ) ) ).

% nat_descend_induct
thf(fact_1157_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y7: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y7 @ N )
                & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_1158_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y7: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y7 @ N )
                & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y7 @ N )
              & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_1159_natLeq__on__well__order__on,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y7 @ N )
              & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_1160_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_1161_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_1162_wf__pred__nat,axiom,
    wf_nat @ pred_nat ).

% wf_pred_nat
thf(fact_1163_trans__less__than,axiom,
    trans_on_nat @ top_top_set_nat @ less_than ).

% trans_less_than
thf(fact_1164_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_1165_wf__less__than,axiom,
    wf_nat @ less_than ).

% wf_less_than
thf(fact_1166_irrefl__less__than,axiom,
    irrefl_on_nat @ top_top_set_nat @ less_than ).

% irrefl_less_than
thf(fact_1167_asym__less__than,axiom,
    asym_on_nat @ top_top_set_nat @ less_than ).

% asym_less_than
thf(fact_1168_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1169_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z2 ) ) ).

% pair_less_iff1
thf(fact_1170_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1171_trans__pair__less,axiom,
    trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ fun_pair_less ).

% trans_pair_less
thf(fact_1172_wf__pair__less,axiom,
    wf_Pro7803398752247294826at_nat @ fun_pair_less ).

% wf_pair_less
thf(fact_1173_pair__less__def,axiom,
    ( fun_pair_less
    = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).

% pair_less_def
thf(fact_1174_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1175_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1176_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1177_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_1178_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_1179_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X5 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_1180_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_1181_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1182_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_1183_min__strict__def,axiom,
    ( fun_min_strict
    = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).

% min_strict_def
thf(fact_1184_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_1185_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_1186_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y8: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y8 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y8 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ Y8 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_1187_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1188_smax__emptyI,axiom,
    ! [Y8: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y8 )
     => ( ( Y8 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y8 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_1189_max__strict__def,axiom,
    ( fun_max_strict
    = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).

% max_strict_def
thf(fact_1190_less__than__def,axiom,
    ( less_than
    = ( transi6264000038957366511cl_nat @ pred_nat ) ) ).

% less_than_def
thf(fact_1191_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_1192_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N ) ) ).

% less_eq
thf(fact_1193_pair__leq__def,axiom,
    ( fun_pair_leq
    = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).

% pair_leq_def
thf(fact_1194_total__pair__less,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A5 @ fun_pair_less ) ).

% total_pair_less
thf(fact_1195_total__on__less__than,axiom,
    ! [A5: set_nat] : ( total_on_nat @ A5 @ less_than ) ).

% total_on_less_than
thf(fact_1196_total__less__than,axiom,
    total_on_nat @ top_top_set_nat @ less_than ).

% total_less_than
thf(fact_1197_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y7 @ N )
              & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_1198_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y7 @ N )
              & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_1199_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) )
          @ ^ [Uu: nat] :
              ( collect_nat
              @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y7: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y7 @ N )
              & ( ord_less_eq_nat @ X4 @ Y7 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_1200_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_1201_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M2: nat,N2: nat] :
            ( N2
            = ( suc @ M2 ) ) ) ) ) ).

% pred_nat_def
thf(fact_1202_less__nat__rel,axiom,
    ( ord_less_nat
    = ( transi2163837189807498211lp_nat
      @ ^ [M2: nat,N2: nat] :
          ( N2
          = ( suc @ M2 ) ) ) ) ).

% less_nat_rel

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( rep_state @ x )
    = ( produc74620467203698198on_val @ ( get_m @ x ) @ ( get_h @ x ) ) ) ).

%------------------------------------------------------------------------------