TPTP Problem File: SLH0367^1.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_00446_016107__7358538_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2035 ( 594 unt; 756 typ;   0 def)
%            Number of atoms       : 3984 (1303 equ;   0 cnn)
%            Maximal formula atoms :   14 (   3 avg)
%            Number of connectives : 14028 ( 478   ~;  46   |; 306   &;11493   @)
%                                         (   0 <=>;1705  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :   71 (  70 usr)
%            Number of type conns  : 4614 (4614   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  689 ( 686 usr;  33 con; 0-5 aty)
%            Number of variables   : 4436 ( 782   ^;3512   !; 142   ?;4436   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:08:40.975
%------------------------------------------------------------------------------
% Could-be-implicit typings (70)
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    produc2489117125269924006t_char: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_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,
    produc2951589610084125259on_val: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_Mt__Nat__Onat_J,type,
    produc8366770746848206027al_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Set__Oset_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_J_J,type,
    set_se6878779149560807558on_val: $tType ).

thf(ty_n_t__Product____Type__Oprod_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,
    produc3766087481356336504r_prat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Nat__Onat_J,type,
    produc4488441066493689938at_nat: $tType ).

thf(ty_n_t__Option__Ooption_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_J,type,
    option5747939881698552298on_val: $tType ).

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

thf(ty_n_t__Set__Oset_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_J,type,
    set_Pr1004501353197068112on_val: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__PartialHeapSA__Ostate_J_Mt__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    produc3142500478612311029_state: $tType ).

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

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

thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Mask__Oval_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    produc5122021904355189735on_val: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_op4508134149509766951at_nat: $tType ).

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

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

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

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

thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    option4927543243414619207at_nat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    set_option_state: $tType ).

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

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Mask__Oval_J_J,type,
    set_option_val: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    set_option_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    option_state: $tType ).

thf(ty_n_t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    set_state: $tType ).

thf(ty_n_t__Option__Ooption_It__Mask__Oval_J,type,
    option_val: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Mask__Oval_J,type,
    set_val: $tType ).

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

thf(ty_n_t__PartialHeapSA__Ostate,type,
    state: $tType ).

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

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

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

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

% Explicit typings (686)
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Nat__Onat,type,
    bNF_Ca5159621140400407487al_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_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_Ca4299268644204761867on_val: set_Pr7649655576376376530on_val > set_Pr4305388979797279737on_val > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Ca968750328013420230at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_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,type,
    bNF_Ca8411853759642212370al_nat: set_Pr4305388979797279737on_val > ( produc9003781588162623004on_val > nat ) > $o ).

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__Nat__Onat,type,
    bNF_Gr_nat_nat: set_nat > ( nat > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Def_OGr_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,type,
    bNF_Gr8368824372578450268on_val: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > set_Pr4305388979797279737on_val ).

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_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__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,
    bNF_Gr193424670081168899at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_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_Gr5132772187013160586on_val: set_Pr4305388979797279737on_val > ( produc1317171644805857433on_val > produc1757238238866670357on_val ) > produc1317171644805857433on_val > produc1757238238866670357on_val > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    bNF_Gr2052668271496221660r_prat: set_Pr4305388979797279737on_val > ( produc1317171644805857433on_val > produc3336119888008106215r_prat ) > produc1317171644805857433on_val > produc3336119888008106215r_prat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_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_Mt__Nat__Onat_J,type,
    bNF_Gr9125604968117925025al_nat: set_Pr1142523028707314192at_nat > ( produc8271722145941708506at_nat > produc8366770746848206027al_nat ) > produc8271722145941708506at_nat > produc8366770746848206027al_nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_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_Mt__Nat__Onat_J,type,
    bNF_Gr5945892418093068284at_nat: set_Pr1142523028707314192at_nat > ( produc8271722145941708506at_nat > produc4488441066493689938at_nat ) > produc8271722145941708506at_nat > produc4488441066493689938at_nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_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_001t__Product____Type__Oprod_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_Gr2855355261350141231on_val: set_Pr6637471833410209410on_val > ( produc5414607237814678604on_val > produc2951589610084125259on_val ) > produc5414607237814678604on_val > produc2951589610084125259on_val > $o ).

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    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 ).

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    bNF_re4208021347924984265on_val: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ).

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    bNF_re3147323125767581259at_nat: ( ( produc1457211279475724562t_char > prat ) > nat > $o ) > ( ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > ( nat > product_prod_nat_nat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > ( nat > nat > product_prod_nat_nat ) > $o ).

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    bNF_re5744771098293236043on_val: ( ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > $o ) > ( ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ) > $o ).

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    bNF_re431696514778167572on_val: ( nat > ( produc1457211279475724562t_char > option_val ) > $o ) > ( product_prod_nat_nat > produc9003781588162623004on_val > $o ) > ( nat > product_prod_nat_nat ) > ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ).

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    bNF_re8571388671295005273on_val: ( nat > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( nat > product_prod_nat_nat ) > ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ) > ( nat > nat > product_prod_nat_nat ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ).

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    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 ).

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    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

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    bNF_re3696488243098009855at_nat: ( nat > nat > $o ) > ( ( nat > product_prod_nat_nat ) > ( nat > product_prod_nat_nat ) > $o ) > ( nat > nat > product_prod_nat_nat ) > ( nat > nat > product_prod_nat_nat ) > $o ).

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    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_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 ).

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    bNF_re901410630616637503at_nat: ( nat > nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > product_prod_nat_nat ) > ( nat > product_prod_nat_nat ) > $o ).

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    bNF_re2010443014418883223on_val: ( nat > produc9003781588162623004on_val > $o ) > ( ( nat > product_prod_nat_nat ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > $o ) > ( nat > nat > product_prod_nat_nat ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ) > $o ).

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    bNF_re1120834113401173856on_val: ( produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o ) > ( produc1317171644805857433on_val > produc9003781588162623004on_val > $o ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ).

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    bNF_re6243659927171069681on_val: ( produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re4791974009433356543at_nat: ( produc9003781588162623004on_val > nat > $o ) > ( ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( nat > product_prod_nat_nat ) > $o ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( nat > nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re583272441465906419at_nat: ( produc9003781588162623004on_val > nat > $o ) > ( produc1317171644805857433on_val > product_prod_nat_nat > $o ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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_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_Mt__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J,type,
    bNF_re1796254245058062999on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > ( ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > $o ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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_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_re1164559784078602061on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > ( ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    bNF_re5965109370979550551r_prat: ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_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_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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,type,
    bNF_re3825428800981762519on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > ( produc1317171644805857433on_val > produc1317171644805857433on_val > $o ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > $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_Ovimage2p_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_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,type,
    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 ).

thf(sy_c_BNF__Def_Ovimage2p_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Nat__Onat_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__Nat__Onat_001_Eo,type,
    bNF_vi2516110938607903234_nat_o: ( ( produc1457211279475724562t_char > prat ) > nat ) > ( ( produc1457211279475724562t_char > option_val ) > nat ) > ( nat > nat > $o ) > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_BNF__Def_Ovimage2p_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Nat__Onat_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,type,
    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 ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_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_Gr6158647150560195831on_val: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > set_Pr7649655576376376530on_val ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_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_Gr3695791879626193176at_nat: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > nat ) > ( produc9003781588162623004on_val > nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_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_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_Gr828556658099662512on_val: set_Pr7649655576376376530on_val > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > set_Pr4305388979797279737on_val ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_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_Gr1122962726781069062on_val: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > produc1457211279475724562t_char > prat ) > ( product_prod_nat_nat > produc1457211279475724562t_char > option_val ) > set_Pr7649655576376376530on_val ).

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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_n2711240574518091068at_nat: ( nat > nat ) > ( product_prod_nat_nat > nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Mask__Oval_J_001t__Mask__Oval_001t__Mask__Oval,type,
    comp_o5604524432708427236al_val: ( option_val > val ) > ( val > option_val ) > val > val ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__PartialHeapSA__Ostate_001t__PartialHeapSA__Ostate,type,
    comp_o4916166309651032803_state: ( option_state > state ) > ( state > option_state ) > state > state ).

thf(sy_c_Fun_Ocomp_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_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__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,
    comp_P3343515995226764714on_val: ( produc1757238238866670357on_val > produc1457211279475724562t_char > option_val ) > ( produc9003781588162623004on_val > produc1757238238866670357on_val ) > produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ).

thf(sy_c_Fun_Ocomp_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__PosRat__Oprat_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__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,
    comp_P1319385742190247241on_val: ( produc7776263825685906734r_prat > produc1457211279475724562t_char > option_val ) > ( produc9003781588162623004on_val > produc7776263825685906734r_prat ) > produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ).

thf(sy_c_Fun_Ocomp_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__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,
    comp_P1591131706424975134on_val: ( produc7776263825685906734r_prat > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc7776263825685906734r_prat ) > produc9003781588162623004on_val > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Ocomp_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__Nat__Onat,type,
    comp_P6187037784939234283al_nat: ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > ( nat > produc9003781588162623004on_val ) > nat > produc1457211279475724562t_char > option_val ).

thf(sy_c_Fun_Ocomp_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__PosRat__Oprat_J_J,type,
    comp_P8287346254047880777r_prat: ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > ( produc7776263825685906734r_prat > produc9003781588162623004on_val ) > produc7776263825685906734r_prat > produc1457211279475724562t_char > option_val ).

thf(sy_c_Fun_Ocomp_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__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,
    comp_P291491979669821239on_val: ( produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > produc9003781588162623004on_val > produc1457211279475724562t_char > option_val ).

thf(sy_c_Fun_Ocomp_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_001t__Nat__Onat,type,
    comp_P4447563945128311908at_nat: ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( nat > produc9003781588162623004on_val ) > nat > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Ocomp_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_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__PosRat__Oprat_J_J,type,
    comp_P4800886777526708674r_prat: ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( produc7776263825685906734r_prat > produc9003781588162623004on_val ) > produc7776263825685906734r_prat > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Ocomp_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_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,
    comp_P6028404540003424944on_val: ( produc9003781588162623004on_val > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val ) > produc9003781588162623004on_val > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Ocomp_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,
    comp_P5489260687716092971on_val: ( produc3336119888008106215r_prat > produc1457211279475724562t_char > prat ) > ( produc9003781588162623004on_val > produc3336119888008106215r_prat ) > produc9003781588162623004on_val > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_P1955610211387027840at_nat: ( product_prod_nat_nat > nat ) > ( nat > product_prod_nat_nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_P1567445206330693457at_nat: ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_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,type,
    comp_P3366439835523679408on_val: ( produc1317171644805857433on_val > produc9003781588162623004on_val ) > ( produc9003781588162623004on_val > produc1317171644805857433on_val ) > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_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_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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,type,
    comp_P7221957804074544389on_val: ( produc1317171644805857433on_val > produc9003781588162623004on_val ) > ( produc1317171644805857433on_val > produc1317171644805857433on_val ) > produc1317171644805857433on_val > produc9003781588162623004on_val ).

thf(sy_c_Fun_Ofun__upd_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Option__Ooption_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_J,type,
    fun_up1601731230164360576on_val: ( ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ) > ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val > ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
    fun_up1493157387958331631on_nat: ( nat > option_nat ) > nat > option_nat > nat > option_nat ).

thf(sy_c_Fun_Ofun__upd_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__Option__Ooption_It__Mask__Oval_J,type,
    fun_up8559965269643824108on_val: ( produc9003781588162623004on_val > option_val ) > produc9003781588162623004on_val > option_val > produc9003781588162623004on_val > option_val ).

thf(sy_c_Fun_Ofun__upd_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__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    fun_up6681822530636708645_state: ( produc9003781588162623004on_val > option_state ) > produc9003781588162623004on_val > option_state > produc9003781588162623004on_val > option_state ).

thf(sy_c_Fun_Ofun__upd_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__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    fun_up273046073882939271on_val: ( produc9003781588162623004on_val > option8887860577033306220on_val ) > produc9003781588162623004on_val > option8887860577033306220on_val > produc9003781588162623004on_val > option8887860577033306220on_val ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    fun_up2801401594293523459on_val: ( product_prod_nat_nat > option_val ) > product_prod_nat_nat > option_val > product_prod_nat_nat > option_val ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    fun_up2914124688375849550_state: ( product_prod_nat_nat > option_state ) > product_prod_nat_nat > option_state > product_prod_nat_nat > option_state ).

thf(sy_c_Fun_Oid_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J,type,
    id_Pro6476562945676181316r_prat: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Oid_001t__Mask__Oval,type,
    id_val: val > val ).

thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

thf(sy_c_Fun_Oid_001t__PartialHeapSA__Ostate,type,
    id_state: state > state ).

thf(sy_c_Fun_Oid_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,
    id_Pro1402184692240358763on_val: produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    id_Pro7559728917991913458at_nat: product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Fun_Oid_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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,type,
    id_Pro2088550933252328842on_val: produc1317171644805857433on_val > produc1317171644805857433on_val ).

thf(sy_c_Fun__Def_Oin__rel_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,
    fun_in3588502033776859947on_val: set_Pr7649655576376376530on_val > ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_Fun__Def_Oin__rel_001t__Nat__Onat_001t__Nat__Onat,type,
    fun_in_rel_nat_nat: set_Pr1261947904930325089at_nat > nat > nat > $o ).

thf(sy_c_Fun__Def_Oin__rel_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,type,
    fun_in8416620500172528508on_val: set_Pr4305388979797279737on_val > produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_Fun__Def_Oreduction__pair_001t__Nat__Onat,type,
    fun_re1951171424197210636ir_nat: produc3843707927480180839at_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    minus_9160195963678339755on_val: set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J,type,
    minus_7211471524202483680on_val: set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Mask__Oval_J,type,
    if_option_val: $o > option_val > option_val > option_val ).

thf(sy_c_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    if_option_state: $o > option_state > option_state > option_state ).

thf(sy_c_If_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    if_set1193336150724294796on_val: $o > set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val ).

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

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J,type,
    inf_in4072335026415195879on_val: set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_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,type,
    sup_su7506060154676209150on_val: set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J,type,
    sup_su3890347010539237325on_val: set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val ).

thf(sy_c_Map_Odom_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,
    dom_Pr4152820065153570843on_val: ( ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ) > set_Pr3433046674199179443r_prat ).

thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Mask__Oval,type,
    dom_nat_val: ( nat > option_val ) > set_nat ).

thf(sy_c_Map_Odom_001t__Nat__Onat_001t__Nat__Onat,type,
    dom_nat_nat: ( nat > option_nat ) > set_nat ).

thf(sy_c_Map_Odom_001t__Nat__Onat_001t__PartialHeapSA__Ostate,type,
    dom_nat_state: ( nat > option_state ) > set_nat ).

thf(sy_c_Map_Odom_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__Mask__Oval,type,
    dom_Pr7381289304162206737al_val: ( produc9003781588162623004on_val > option_val ) > set_Pr7649655576376376530on_val ).

thf(sy_c_Map_Odom_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__PartialHeapSA__Ostate,type,
    dom_Pr4448331044451638570_state: ( produc9003781588162623004on_val > option_state ) > set_Pr7649655576376376530on_val ).

thf(sy_c_Map_Odom_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,type,
    dom_Pr2742230247756046988on_val: ( produc9003781588162623004on_val > option8887860577033306220on_val ) > set_Pr7649655576376376530on_val ).

thf(sy_c_Map_Odom_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Mask__Oval,type,
    dom_Pr1716353499919422226at_val: ( product_prod_nat_nat > option_val ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Map_Odom_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__PartialHeapSA__Ostate,type,
    dom_Pr7687107951487557865_state: ( product_prod_nat_nat > option_state ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Map_Ograph_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,
    graph_1288227317702120175on_val: ( ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ) > set_Pr7649655576376376530on_val ).

thf(sy_c_Map_Ograph_001t__Nat__Onat_001t__Nat__Onat,type,
    graph_nat_nat: ( nat > option_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Map_Ograph_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,type,
    graph_6108330199704490680on_val: ( produc9003781588162623004on_val > option8887860577033306220on_val ) > set_Pr4305388979797279737on_val ).

thf(sy_c_Map_Omap__add_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
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thf(sy_c_Map_Oran_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,
    ran_Pr878516737976613822on_val: ( ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ) > set_Pr1004501353197068112on_val ).

thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Nat__Onat,type,
    ran_nat_nat: ( nat > option_nat ) > set_nat ).

thf(sy_c_Map_Orestrict__map_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,
    restri7594233908965391805on_val: ( ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ) > set_Pr3433046674199179443r_prat > ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val ).

thf(sy_c_Map_Orestrict__map_001t__Nat__Onat_001t__Nat__Onat,type,
    restrict_map_nat_nat: ( nat > option_nat ) > set_nat > nat > option_nat ).

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    restri5251746167140823987al_val: ( produc9003781588162623004on_val > option_val ) > set_Pr7649655576376376530on_val > produc9003781588162623004on_val > option_val ).

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    restri5336766904122854152_state: ( produc9003781588162623004on_val > option_state ) > set_Pr7649655576376376530on_val > produc9003781588162623004on_val > option_state ).

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thf(sy_c_Map_Orestrict__map_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Mask__Oval,type,
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thf(sy_c_Mask_Oadd__masks_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Mask_Oadd__masks__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Mask_Oempty__mask_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(sy_c_Mask_Ofull__mask,type,
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    ord_le3852884270828607079_val_o: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__PartialHeapSA__Ostate_M_062_It__PartialHeapSA__Ostate_M_Eo_J_J,type,
    ord_le4344933087531911346tate_o: ( state > state > $o ) > ( state > state > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__PartialHeapSA__Ostate_M_Eo_J,type,
    ord_less_eq_state_o: ( state > $o ) > ( state > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_M_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_M_Eo_J_J,type,
    ord_le616650860478780142_val_o: ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_M_Eo_J,type,
    ord_le6140313152057869779_val_o: ( produc9003781588162623004on_val > $o ) > ( produc9003781588162623004on_val > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_M_062_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_M_Eo_J_J,type,
    ord_le2319172193063290574_val_o: ( produc1317171644805857433on_val > produc1317171644805857433on_val > $o ) > ( produc1317171644805857433on_val > produc1317171644805857433on_val > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_M_Eo_J,type,
    ord_le8528861841627966596_val_o: ( produc1317171644805857433on_val > $o ) > ( produc1317171644805857433on_val > $o ) > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_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__PosRat__Oprat_J_J_J,type,
    ord_le2122960673345045316r_prat: set_Pr2871993341207995108r_prat > set_Pr2871993341207995108r_prat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    ord_le6900622908513426738on_val: set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__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_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_J,type,
    ord_le7224449410216889241on_val: set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_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_J,type,
    top_to3326603054294128128on_val: set_Pr1004501353197068112on_val ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_J,type,
    top_to6849109582029272579r_prat: set_Pr3433046674199179443r_prat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Mask__Oval_J,type,
    top_top_set_val: set_val ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Mask__Oval_J_J,type,
    top_to853123400021307383on_val: set_option_val ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
    top_to8920198386146353926on_nat: set_option_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__PartialHeapSA__Ostate_J_J,type,
    top_to7666338855062656496_state: set_option_state ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__PartialHeapSA__Ostate_J,type,
    top_top_set_state: set_state ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    top_to1561438921946393474on_val: set_Pr7649655576376376530on_val ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_PartialHeapSA_Oadd__states,type,
    add_states: produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val ).

thf(sy_c_PartialHeapSA_Oadd__states__rel,type,
    add_states_rel: produc1317171644805857433on_val > produc1317171644805857433on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible,type,
    compatible: produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__heaps,type,
    compatible_heaps: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options_001t__Mask__Oval,type,
    compat7209469420100717861ns_val: option_val > option_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options_001t__PartialHeapSA__Ostate,type,
    compat2278460363914054422_state: option_state > option_state > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__Mask__Oval,type,
    compat8294034345482142370el_val: produc5122021904355189735on_val > produc5122021904355189735on_val > $o ).

thf(sy_c_PartialHeapSA_Ocompatible__options__rel_001t__PartialHeapSA__Ostate,type,
    compat1844074344818064729_state: produc3142500478612311029_state > produc3142500478612311029_state > $o ).

thf(sy_c_PartialHeapSA_Ocore,type,
    core: state > state ).

thf(sy_c_PartialHeapSA_Oempty__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
    empty_7512026568161686585ar_val: produc1457211279475724562t_char > option_val ).

thf(sy_c_PartialHeapSA_Oget__h,type,
    get_h: state > produc1457211279475724562t_char > option_val ).

thf(sy_c_PartialHeapSA_Oget__h__rel,type,
    get_h_rel: state > state > $o ).

thf(sy_c_PartialHeapSA_Oget__m,type,
    get_m: state > produc1457211279475724562t_char > prat ).

thf(sy_c_PartialHeapSA_Oget__m__rel,type,
    get_m_rel: state > state > $o ).

thf(sy_c_PartialHeapSA_Olarger__heap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Mask__Oval,type,
    larger1532547405891354081ar_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Oplus,type,
    plus: state > state > option_state ).

thf(sy_c_PartialHeapSA_Ostable,type,
    stable: state > $o ).

thf(sy_c_PartialHeapSA_Ostate_OAbs__state,type,
    abs_state: produc9003781588162623004on_val > state ).

thf(sy_c_PartialHeapSA_Ostate_ORep__state,type,
    rep_state: state > produc9003781588162623004on_val ).

thf(sy_c_PartialHeapSA_Ounit,type,
    unit: state ).

thf(sy_c_PartialHeapSA_Ovalid__heap,type,
    valid_heap: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ).

thf(sy_c_PartialHeapSA_Ovalid__state,type,
    valid_state: produc9003781588162623004on_val > $o ).

thf(sy_c_PartialHeapSA_Ovalid__state__rel,type,
    valid_state_rel: produc9003781588162623004on_val > produc9003781588162623004on_val > $o ).

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

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

thf(sy_c_Product__Type_OPair_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,
    produc8008623296994167373on_val: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > produc1757238238866670357on_val ).

thf(sy_c_Product__Type_OPair_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__PosRat__Oprat_J,type,
    produc1897429902272487136r_prat: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > prat ) > produc7776263825685906734r_prat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    produc74620467203698198on_val: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val ).

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

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

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

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

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_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,
    produc487954823221557125on_val: nat > produc9003781588162623004on_val > produc2138146677468684109on_val ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Mask__Oval_J_001t__Option__Ooption_It__Mask__Oval_J,type,
    produc904852209897547607on_val: option_val > option_val > produc5122021904355189735on_val ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    produc9160152616361873709_state: option_state > option_state > produc3142500478612311029_state ).

thf(sy_c_Product__Type_OPair_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,type,
    produc6190288198252369029al_nat: produc9003781588162623004on_val > nat > produc4583794713730567757al_nat ).

thf(sy_c_Product__Type_OPair_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,type,
    produc4958730763592889809on_val: produc9003781588162623004on_val > produc9003781588162623004on_val > produc1317171644805857433on_val ).

thf(sy_c_Product__Type_OPair_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_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2486738652603726348at_nat: produc9003781588162623004on_val > product_prod_nat_nat > produc8271722145941708506at_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_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,
    produc1110776065552527814on_val: product_prod_nat_nat > produc9003781588162623004on_val > produc5414607237814678604on_val ).

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    lex_pr5802446539711250333on_val: set_Pr1261947904930325089at_nat > set_Pr4305388979797279737on_val > set_Pr2391776111385609287on_val ).

thf(sy_c_Wellfounded_Olex__prod_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,type,
    lex_pr2281407877887286429al_nat: set_Pr4305388979797279737on_val > set_Pr1261947904930325089at_nat > set_Pr6788513813867545671al_nat ).

thf(sy_c_Wellfounded_Olex__prod_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,type,
    lex_pr8992908938815505897on_val: set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val > set_Pr4590028993149660487on_val ).

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

thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_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,
    max_ex7543004642404436129on_val: set_Pr4305388979797279737on_val > set_Pr8539644059184709221on_val ).

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

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

thf(sy_c_Wellfounded_Omax__extp_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,
    max_ex1577223295445257289on_val: ( produc9003781588162623004on_val > produc9003781588162623004on_val > $o ) > set_Pr7649655576376376530on_val > set_Pr7649655576376376530on_val > $o ).

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

thf(sy_c_Wellfounded_Omeasure_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,
    measur6236884747139477332on_val: ( produc9003781588162623004on_val > nat ) > set_Pr4305388979797279737on_val ).

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

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

thf(sy_c_Wellfounded_Omlex__prod_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,
    mlex_p4921486899325709005on_val: ( produc9003781588162623004on_val > nat ) > set_Pr4305388979797279737on_val > set_Pr4305388979797279737on_val ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Wellfounded_Owf_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,
    wf_Pro3300669915054815219on_val: set_Pr4305388979797279737on_val > $o ).

thf(sy_c_Wellfounded_Owf_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_Wellfounded_Owf_001t__Set__Oset_It__Nat__Onat_J,type,
    wf_set_nat: set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_Wfrec_Osame__fst_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    same_f8138205353876899841on_val: ( ( produc1457211279475724562t_char > prat ) > $o ) > ( ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val ) > set_Pr4305388979797279737on_val ).

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

thf(sy_c_Wfrec_Osame__fst_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,type,
    same_f7521981028318775590on_val: ( produc9003781588162623004on_val > $o ) > ( produc9003781588162623004on_val > set_Pr4305388979797279737on_val ) > set_Pr4590028993149660487on_val ).

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    member798016022416105634on_val: ( nat > produc9003781588162623004on_val ) > set_na3731733139948239937on_val > $o ).

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

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J,type,
    member1577473829074448177on_val: ( produc1457211279475724562t_char > option_val ) > set_Pr1004501353197068112on_val > $o ).

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

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

thf(sy_c_member_001t__Mask__Oval,type,
    member_val: val > set_val > $o ).

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

thf(sy_c_member_001t__Option__Ooption_It__Mask__Oval_J,type,
    member_option_val: option_val > set_option_val > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
    member_option_nat: option_nat > set_option_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__PartialHeapSA__Ostate_J,type,
    member_option_state: option_state > set_option_state > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J,type,
    member4602090469187516291on_val: option8887860577033306220on_val > set_op6240181410663953058on_val > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member3954567711264315760at_nat: option4927543243414619207at_nat > set_op4508134149509766951at_nat > $o ).

thf(sy_c_member_001t__PartialHeapSA__Ostate,type,
    member_state: state > set_state > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__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,
    member5502714215428086462on_val: produc1757238238866670357on_val > set_Pr4698884584343881589on_val > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__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__PosRat__Oprat_J_J,type,
    member1651512861525830341r_prat: produc7776263825685906734r_prat > set_Pr2871993341207995108r_prat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J,type,
    member2879030624002546611on_val: produc9003781588162623004on_val > set_Pr7649655576376376530on_val > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__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_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,type,
    member6505152944228695362on_val: produc1317171644805857433on_val > set_Pr4305388979797279737on_val > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member7906376990877558257at_nat: produc8271722145941708506at_nat > set_Pr1142523028707314192at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_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,type,
    member5049262082750528355on_val: produc5414607237814678604on_val > set_Pr6637471833410209410on_val > $o ).

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

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    member6187517401309038224on_val: produc6411138178144226151on_val > set_Pr2391776111385609287on_val > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_Mt__Nat__Onat_J_J,type,
    member7614044688163767952al_nat: produc7837665464998955879al_nat > set_Pr6788513813867545671al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_Mt__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_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_J,type,
    member2023868432420333456on_val: produc3517408837545735783on_val > set_Pr4590028993149660487on_val > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_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_Mt__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_J,type,
    member806545433280400814on_val: produc2447758447588595973on_val > set_Pr8539644059184709221on_val > $o ).

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

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

% Relevant facts (1271)
thf(fact_0_valid__pre__unit,axiom,
    valid_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ).

% valid_pre_unit
thf(fact_1_unit__def,axiom,
    ( unit
    = ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ empty_7512026568161686585ar_val ) ) ) ).

% unit_def
thf(fact_2_core__defined_I1_J,axiom,
    ! [Phi: state] :
      ( ( get_h @ ( core @ Phi ) )
      = ( get_h @ Phi ) ) ).

% core_defined(1)
thf(fact_3_state__ext,axiom,
    ! [A: state,B: state] :
      ( ( ( get_h @ A )
        = ( get_h @ B ) )
     => ( ( ( get_m @ A )
          = ( get_m @ B ) )
       => ( A = B ) ) ) ).

% state_ext
thf(fact_4_empty__heap__def,axiom,
    ( empty_7512026568161686585ar_val
    = ( ^ [Hl: produc1457211279475724562t_char] : none_val ) ) ).

% empty_heap_def
thf(fact_5_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_6_core__def,axiom,
    ( core
    = ( ^ [Phi2: state] : ( abs_state @ ( produc74620467203698198on_val @ empty_3446695950879338768t_char @ ( get_h @ Phi2 ) ) ) ) ) ).

% core_def
thf(fact_7_get__h__m,axiom,
    ( rep_state
    = ( ^ [X2: state] : ( produc74620467203698198on_val @ ( get_m @ X2 ) @ ( get_h @ X2 ) ) ) ) ).

% get_h_m
thf(fact_8_get__pre_I1_J,axiom,
    ( get_h
    = ( ^ [X2: state] : ( produc2319605628138516840on_val @ ( rep_state @ X2 ) ) ) ) ).

% get_pre(1)
thf(fact_9_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_10_get__h_Osimps,axiom,
    ( get_h
    = ( ^ [A2: state] : ( produc2319605628138516840on_val @ ( rep_state @ A2 ) ) ) ) ).

% get_h.simps
thf(fact_11_PartialHeapSA_Ostable__def,axiom,
    ( stable
    = ( ^ [Phi2: state] :
        ! [Hl: produc1457211279475724562t_char] :
          ( ( ppos @ ( get_m @ Phi2 @ Hl ) )
          = ( ( get_h @ Phi2 @ Hl )
           != none_val ) ) ) ) ).

% PartialHeapSA.stable_def
thf(fact_12_core__defined_I2_J,axiom,
    ! [Phi: state] :
      ( ( get_m @ ( core @ Phi ) )
      = empty_3446695950879338768t_char ) ).

% core_defined(2)
thf(fact_13_Rep__state__inject,axiom,
    ! [X: state,Y: state] :
      ( ( ( rep_state @ X )
        = ( rep_state @ Y ) )
      = ( X = Y ) ) ).

% Rep_state_inject
thf(fact_14_Rep__state__inverse,axiom,
    ! [X: state] :
      ( ( abs_state @ ( rep_state @ X ) )
      = X ) ).

% Rep_state_inverse
thf(fact_15_prod_Oinject,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val,Y1: produc1457211279475724562t_char > prat,Y2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ X1 @ X22 )
        = ( produc74620467203698198on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_16_prod_Oinject,axiom,
    ! [X1: produc9003781588162623004on_val,X22: produc9003781588162623004on_val,Y1: produc9003781588162623004on_val,Y2: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ X1 @ X22 )
        = ( produc4958730763592889809on_val @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_17_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A3: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A3: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B )
        = ( produc4958730763592889809on_val @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_sndI,axiom,
    ! [X: produc1317171644805857433on_val,Y: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
      ( ( X
        = ( produc4958730763592889809on_val @ Y @ Z ) )
     => ( ( produc437328469718437375on_val @ X )
        = Z ) ) ).

% sndI
thf(fact_22_sndI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z ) )
     => ( ( product_snd_nat_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_23_sndI,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
      ( ( X
        = ( produc74620467203698198on_val @ Y @ Z ) )
     => ( ( produc2319605628138516840on_val @ X )
        = Z ) ) ).

% sndI
thf(fact_24_snd__eqD,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( ( produc437328469718437375on_val @ ( produc4958730763592889809on_val @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_25_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_26_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_27_snd__conv,axiom,
    ! [X1: produc9003781588162623004on_val,X22: produc9003781588162623004on_val] :
      ( ( produc437328469718437375on_val @ ( produc4958730763592889809on_val @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_28_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_29_snd__conv,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] :
      ( ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_30_eq__snd__iff,axiom,
    ! [B: produc9003781588162623004on_val,P: produc1317171644805857433on_val] :
      ( ( B
        = ( produc437328469718437375on_val @ P ) )
      = ( ? [A2: produc9003781588162623004on_val] :
            ( P
            = ( produc4958730763592889809on_val @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_31_eq__snd__iff,axiom,
    ! [B: nat,P: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P ) )
      = ( ? [A2: nat] :
            ( P
            = ( product_Pair_nat_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_32_eq__snd__iff,axiom,
    ! [B: produc1457211279475724562t_char > option_val,P: produc9003781588162623004on_val] :
      ( ( B
        = ( produc2319605628138516840on_val @ P ) )
      = ( ? [A2: produc1457211279475724562t_char > prat] :
            ( P
            = ( produc74620467203698198on_val @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_33_valid__stateI,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ( valid_mask @ Pi2 )
     => ( ! [Hl2: produc1457211279475724562t_char] :
            ( ( ppos @ ( Pi2 @ Hl2 ) )
           => ( ( H2 @ Hl2 )
             != none_val ) )
       => ( valid_state @ ( produc74620467203698198on_val @ Pi2 @ H2 ) ) ) ) ).

% valid_stateI
thf(fact_34_valid__heapI,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi2 @ Hl2 ) )
         => ( ( H2 @ Hl2 )
           != none_val ) )
     => ( valid_heap @ Pi2 @ H2 ) ) ).

% valid_heapI
thf(fact_35_valid__heap__def,axiom,
    ( valid_heap
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,H3: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char] :
          ( ( ppos @ ( Pi3 @ Hl ) )
         => ( ( H3 @ Hl )
           != none_val ) ) ) ) ).

% valid_heap_def
thf(fact_36_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_37_valid__empty,axiom,
    valid_mask @ empty_3446695950879338768t_char ).

% valid_empty
thf(fact_38_add__states_Ocases,axiom,
    ! [X: produc1317171644805857433on_val] :
      ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val,Pi4: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
          ( X
         != ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi4 @ H4 ) ) ) ).

% add_states.cases
thf(fact_39_valid__state_Osimps,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ Pi2 @ H2 ) )
      = ( ( valid_mask @ Pi2 )
        & ( valid_heap @ Pi2 @ H2 ) ) ) ).

% valid_state.simps
thf(fact_40_valid__state_Oelims_I1_J,axiom,
    ! [X: produc9003781588162623004on_val,Y: $o] :
      ( ( ( valid_state @ X )
        = Y )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ( Y
              = ( ~ ( ( valid_mask @ Pi )
                    & ( valid_heap @ Pi @ H ) ) ) ) ) ) ).

% valid_state.elims(1)
thf(fact_41_valid__state_Oelims_I2_J,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ( valid_state @ X )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ~ ( ( valid_mask @ Pi )
                & ( valid_heap @ Pi @ H ) ) ) ) ).

% valid_state.elims(2)
thf(fact_42_valid__state_Oelims_I3_J,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ( ( valid_mask @ Pi )
              & ( valid_heap @ Pi @ H ) ) ) ) ).

% valid_state.elims(3)
thf(fact_43_prod__induct3,axiom,
    ! [P2: produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > option_val] : ( P2 @ ( produc4958730763592889809on_val @ A4 @ ( produc74620467203698198on_val @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_44_prod__cases3,axiom,
    ! [Y: produc1317171644805857433on_val] :
      ~ ! [A4: produc9003781588162623004on_val,B3: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc4958730763592889809on_val @ A4 @ ( produc74620467203698198on_val @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_45_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_46_mem__Collect__eq,axiom,
    ! [A: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ A @ ( collec7475294502966882545on_val @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( collec7475294502966882545on_val
        @ ^ [X2: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_49_Collect__cong,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P2 )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_50_Collect__cong,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ! [X3: produc9003781588162623004on_val] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec7475294502966882545on_val @ P2 )
        = ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_cong
thf(fact_51_Pair__inject,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A3: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val] :
      ( ( ( produc74620467203698198on_val @ A @ B )
        = ( produc74620467203698198on_val @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_52_Pair__inject,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A3: produc9003781588162623004on_val,B2: produc9003781588162623004on_val] :
      ( ( ( produc4958730763592889809on_val @ A @ B )
        = ( produc4958730763592889809on_val @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_53_Pair__inject,axiom,
    ! [A: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_54_prod__cases,axiom,
    ! [P2: produc9003781588162623004on_val > $o,P: produc9003781588162623004on_val] :
      ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] : ( P2 @ ( produc74620467203698198on_val @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_55_prod__cases,axiom,
    ! [P2: produc1317171644805857433on_val > $o,P: produc1317171644805857433on_val] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] : ( P2 @ ( produc4958730763592889809on_val @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_56_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_57_surj__pair,axiom,
    ! [P: produc9003781588162623004on_val] :
    ? [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
      ( P
      = ( produc74620467203698198on_val @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_58_surj__pair,axiom,
    ! [P: produc1317171644805857433on_val] :
    ? [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
      ( P
      = ( produc4958730763592889809on_val @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_59_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_60_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ~ ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( Y
         != ( produc74620467203698198on_val @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_61_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1317171644805857433on_val] :
      ~ ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
          ( Y
         != ( produc4958730763592889809on_val @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_62_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A4: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_63_valid__mask__full__mask,axiom,
    valid_mask @ full_mask ).

% valid_mask_full_mask
thf(fact_64_pre__get__h_Osimps,axiom,
    pre_ge1486375753531218896on_val = produc2319605628138516840on_val ).

% pre_get_h.simps
thf(fact_65_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_66_accp__induct__rule,axiom,
    ! [R: state > state > $o,A: state,P2: state > $o] :
      ( ( accp_state @ R @ A )
     => ( ! [X3: state] :
            ( ( accp_state @ R @ X3 )
           => ( ! [Y4: state] :
                  ( ( R @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_67_accp__induct__rule,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( accp_P4405556307444930259on_val @ R @ A )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( accp_P4405556307444930259on_val @ R @ X3 )
           => ( ! [Y4: produc9003781588162623004on_val] :
                  ( ( R @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_68_accp__induct__rule,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val,P2: produc1317171644805857433on_val > $o] :
      ( ( accp_P6797916352740768290on_val @ R @ A )
     => ( ! [X3: produc1317171644805857433on_val] :
            ( ( accp_P6797916352740768290on_val @ R @ X3 )
           => ( ! [Y4: produc1317171644805857433on_val] :
                  ( ( R @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_69_not__accp__down,axiom,
    ! [R2: state > state > $o,X: state] :
      ( ~ ( accp_state @ R2 @ X )
     => ~ ! [Z2: state] :
            ( ( R2 @ Z2 @ X )
           => ( accp_state @ R2 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_70_not__accp__down,axiom,
    ! [R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val] :
      ( ~ ( accp_P4405556307444930259on_val @ R2 @ X )
     => ~ ! [Z2: produc9003781588162623004on_val] :
            ( ( R2 @ Z2 @ X )
           => ( accp_P4405556307444930259on_val @ R2 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_71_not__accp__down,axiom,
    ! [R2: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val] :
      ( ~ ( accp_P6797916352740768290on_val @ R2 @ X )
     => ~ ! [Z2: produc1317171644805857433on_val] :
            ( ( R2 @ Z2 @ X )
           => ( accp_P6797916352740768290on_val @ R2 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_72_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_73_accp__downward,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,B: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( accp_P4405556307444930259on_val @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_P4405556307444930259on_val @ R @ A ) ) ) ).

% accp_downward
thf(fact_74_accp__downward,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,B: produc1317171644805857433on_val,A: produc1317171644805857433on_val] :
      ( ( accp_P6797916352740768290on_val @ R @ B )
     => ( ( R @ A @ B )
       => ( accp_P6797916352740768290on_val @ R @ A ) ) ) ).

% accp_downward
thf(fact_75_accp__induct,axiom,
    ! [R: state > state > $o,A: state,P2: state > $o] :
      ( ( accp_state @ R @ A )
     => ( ! [X3: state] :
            ( ( accp_state @ R @ X3 )
           => ( ! [Y4: state] :
                  ( ( R @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_76_accp__induct,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( accp_P4405556307444930259on_val @ R @ A )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( accp_P4405556307444930259on_val @ R @ X3 )
           => ( ! [Y4: produc9003781588162623004on_val] :
                  ( ( R @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_77_accp__induct,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val,P2: produc1317171644805857433on_val > $o] :
      ( ( accp_P6797916352740768290on_val @ R @ A )
     => ( ! [X3: produc1317171644805857433on_val] :
            ( ( accp_P6797916352740768290on_val @ R @ X3 )
           => ( ! [Y4: produc1317171644805857433on_val] :
                  ( ( R @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_78_accpI,axiom,
    ! [R: state > state > $o,X: state] :
      ( ! [Y3: state] :
          ( ( R @ Y3 @ X )
         => ( accp_state @ R @ Y3 ) )
     => ( accp_state @ R @ X ) ) ).

% accpI
thf(fact_79_accpI,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val] :
      ( ! [Y3: produc9003781588162623004on_val] :
          ( ( R @ Y3 @ X )
         => ( accp_P4405556307444930259on_val @ R @ Y3 ) )
     => ( accp_P4405556307444930259on_val @ R @ X ) ) ).

% accpI
thf(fact_80_accpI,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val] :
      ( ! [Y3: produc1317171644805857433on_val] :
          ( ( R @ Y3 @ X )
         => ( accp_P6797916352740768290on_val @ R @ Y3 ) )
     => ( accp_P6797916352740768290on_val @ R @ X ) ) ).

% accpI
thf(fact_81_accp_Ocases,axiom,
    ! [R: state > state > $o,A: state] :
      ( ( accp_state @ R @ A )
     => ! [Y4: state] :
          ( ( R @ Y4 @ A )
         => ( accp_state @ R @ Y4 ) ) ) ).

% accp.cases
thf(fact_82_accp_Ocases,axiom,
    ! [R: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( accp_P4405556307444930259on_val @ R @ A )
     => ! [Y4: produc9003781588162623004on_val] :
          ( ( R @ Y4 @ A )
         => ( accp_P4405556307444930259on_val @ R @ Y4 ) ) ) ).

% accp.cases
thf(fact_83_accp_Ocases,axiom,
    ! [R: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A: produc1317171644805857433on_val] :
      ( ( accp_P6797916352740768290on_val @ R @ A )
     => ! [Y4: produc1317171644805857433on_val] :
          ( ( R @ Y4 @ A )
         => ( accp_P6797916352740768290on_val @ R @ Y4 ) ) ) ).

% accp.cases
thf(fact_84_accp_Osimps,axiom,
    ( accp_state
    = ( ^ [R3: state > state > $o,A2: state] :
        ? [X2: state] :
          ( ( A2 = X2 )
          & ! [Y5: state] :
              ( ( R3 @ Y5 @ X2 )
             => ( accp_state @ R3 @ Y5 ) ) ) ) ) ).

% accp.simps
thf(fact_85_accp_Osimps,axiom,
    ( accp_P4405556307444930259on_val
    = ( ^ [R3: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A2: produc9003781588162623004on_val] :
        ? [X2: produc9003781588162623004on_val] :
          ( ( A2 = X2 )
          & ! [Y5: produc9003781588162623004on_val] :
              ( ( R3 @ Y5 @ X2 )
             => ( accp_P4405556307444930259on_val @ R3 @ Y5 ) ) ) ) ) ).

% accp.simps
thf(fact_86_accp_Osimps,axiom,
    ( accp_P6797916352740768290on_val
    = ( ^ [R3: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,A2: produc1317171644805857433on_val] :
        ? [X2: produc1317171644805857433on_val] :
          ( ( A2 = X2 )
          & ! [Y5: produc1317171644805857433on_val] :
              ( ( R3 @ Y5 @ X2 )
             => ( accp_P6797916352740768290on_val @ R3 @ Y5 ) ) ) ) ) ).

% accp.simps
thf(fact_87_pre__get__h_Opelims,axiom,
    ! [X: produc1317171644805857433on_val,Y: produc9003781588162623004on_val] :
      ( ( ( pre_ge5139353195879713175on_val @ X )
        = Y )
     => ( ( accp_P6797916352740768290on_val @ pre_ge7436079647097740950on_val @ X )
       => ~ ( ( Y
              = ( produc437328469718437375on_val @ X ) )
           => ~ ( accp_P6797916352740768290on_val @ pre_ge7436079647097740950on_val @ X ) ) ) ) ).

% pre_get_h.pelims
thf(fact_88_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_89_valid__state_Opelims_I3_J,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ~ ( valid_state @ X )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H ) )
               => ( ( valid_mask @ Pi )
                  & ( valid_heap @ Pi @ H ) ) ) ) ) ) ).

% valid_state.pelims(3)
thf(fact_90_valid__state_Opelims_I2_J,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ( valid_state @ X )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H ) )
               => ~ ( ( valid_mask @ Pi )
                    & ( valid_heap @ Pi @ H ) ) ) ) ) ) ).

% valid_state.pelims(2)
thf(fact_91_valid__state_Opelims_I1_J,axiom,
    ! [X: produc9003781588162623004on_val,Y: $o] :
      ( ( ( valid_state @ X )
        = Y )
     => ( ( accp_P4405556307444930259on_val @ valid_state_rel @ X )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ( ( Y
                  = ( ( valid_mask @ Pi )
                    & ( valid_heap @ Pi @ H ) ) )
               => ~ ( accp_P4405556307444930259on_val @ valid_state_rel @ ( produc74620467203698198on_val @ Pi @ H ) ) ) ) ) ) ).

% valid_state.pelims(1)
thf(fact_92_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: nat,A3: produc9003781588162623004on_val,B2: nat,R: set_Pr4305388979797279737on_val,S: set_Pr1261947904930325089at_nat] :
      ( ( member7614044688163767952al_nat @ ( produc2229095497775398359al_nat @ ( produc6190288198252369029al_nat @ A @ B ) @ ( produc6190288198252369029al_nat @ A3 @ B2 ) ) @ ( lex_pr2281407877887286429al_nat @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A3 ) @ R )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_93_in__lex__prod,axiom,
    ! [A: nat,B: produc9003781588162623004on_val,A3: nat,B2: produc9003781588162623004on_val,R: set_Pr1261947904930325089at_nat,S: set_Pr4305388979797279737on_val] :
      ( ( member6187517401309038224on_val @ ( produc2605982121015836119on_val @ ( produc487954823221557125on_val @ A @ B ) @ ( produc487954823221557125on_val @ A3 @ B2 ) ) @ ( lex_pr5802446539711250333on_val @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A3 ) @ R )
        | ( ( A = A3 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_94_in__lex__prod,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A3: produc9003781588162623004on_val,B2: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4958730763592889809on_val @ A3 @ B2 ) ) @ ( lex_pr8992908938815505897on_val @ R @ S ) )
      = ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A3 ) @ R )
        | ( ( A = A3 )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_95_in__lex__prod,axiom,
    ! [A: nat,B: nat,A3: 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 @ A3 @ B2 ) ) @ ( lex_prod_nat_nat @ R @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A3 ) @ R )
        | ( ( A = A3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_96_in__lex__prod,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A3: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr4698884584343881589on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( produc74620467203698198on_val @ A3 @ B2 ) ) @ ( lex_pr8780192377291449342on_val @ R @ S ) )
      = ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ A3 ) @ R )
        | ( ( A = A3 )
          & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B @ B2 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_97_same__fstI,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,Y6: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val,R2: ( produc1457211279475724562t_char > prat ) > set_Pr4698884584343881589on_val] :
      ( ( P2 @ X )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y6 @ Y ) @ ( R2 @ X ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ X @ Y6 ) @ ( produc74620467203698198on_val @ X @ Y ) ) @ ( same_f8138205353876899841on_val @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_98_same__fstI,axiom,
    ! [P2: produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val,Y6: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R2: produc9003781588162623004on_val > set_Pr4305388979797279737on_val] :
      ( ( P2 @ X )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y6 @ Y ) @ ( R2 @ X ) )
       => ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ X @ Y6 ) @ ( produc4958730763592889809on_val @ X @ Y ) ) @ ( same_f7521981028318775590on_val @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_99_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y6: nat,Y: nat,R2: nat > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R2 @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P2 @ R2 ) ) ) ) ).

% same_fstI
thf(fact_100_valid__larger__mask,axiom,
    ( valid_mask
    = ( greate7680731506090375811t_char @ full_mask ) ) ).

% valid_larger_mask
thf(fact_101_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_102_ssubst__Pair__rhs,axiom,
    ! [R: produc9003781588162623004on_val,S: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,S2: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_103_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_104_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_105_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_106_curryI,axiom,
    ! [F: produc1317171644805857433on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( F @ ( produc4958730763592889809on_val @ A @ B ) )
     => ( produc4800927984219439315_val_o @ F @ A @ B ) ) ).

% curryI
thf(fact_107_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_108_prod_Ocollapse,axiom,
    ! [Prod: produc1317171644805857433on_val] :
      ( ( produc4958730763592889809on_val @ ( produc3129717363260770749on_val @ Prod ) @ ( produc437328469718437375on_val @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_109_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_110_prod_Ocollapse,axiom,
    ! [Prod: produc9003781588162623004on_val] :
      ( ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_111_greater__mask__properties_I3_J,axiom,
    ! [Pi5: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ Pi5 @ Pi2 )
        & ( greate7680731506090375811t_char @ Pi2 @ Pi5 ) )
     => ( Pi2 = Pi5 ) ) ).

% greater_mask_properties(3)
thf(fact_112_greater__mask__properties_I2_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ A @ B )
        & ( greate7680731506090375811t_char @ B @ C2 ) )
     => ( greate7680731506090375811t_char @ A @ C2 ) ) ).

% greater_mask_properties(2)
thf(fact_113_greater__mask__properties_I1_J,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi2 @ Pi2 ) ).

% greater_mask_properties(1)
thf(fact_114_eq__fst__iff,axiom,
    ! [A: produc9003781588162623004on_val,P: produc1317171644805857433on_val] :
      ( ( A
        = ( produc3129717363260770749on_val @ P ) )
      = ( ? [B4: produc9003781588162623004on_val] :
            ( P
            = ( produc4958730763592889809on_val @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_115_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_116_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_117_fst__conv,axiom,
    ! [X1: produc9003781588162623004on_val,X22: produc9003781588162623004on_val] :
      ( ( produc3129717363260770749on_val @ ( produc4958730763592889809on_val @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_118_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_119_fst__conv,axiom,
    ! [X1: produc1457211279475724562t_char > prat,X22: produc1457211279475724562t_char > option_val] :
      ( ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_120_fst__eqD,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,A: produc9003781588162623004on_val] :
      ( ( ( produc3129717363260770749on_val @ ( produc4958730763592889809on_val @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_121_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_122_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_123_fstI,axiom,
    ! [X: produc1317171644805857433on_val,Y: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
      ( ( X
        = ( produc4958730763592889809on_val @ Y @ Z ) )
     => ( ( produc3129717363260770749on_val @ X )
        = Y ) ) ).

% fstI
thf(fact_124_fstI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z ) )
     => ( ( product_fst_nat_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_125_fstI,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > option_val] :
      ( ( X
        = ( produc74620467203698198on_val @ Y @ Z ) )
     => ( ( produc5239619271231976746on_val @ X )
        = Y ) ) ).

% fstI
thf(fact_126_prod__eq__iff,axiom,
    ( ( ^ [Y7: produc9003781588162623004on_val,Z3: produc9003781588162623004on_val] : ( Y7 = Z3 ) )
    = ( ^ [S3: produc9003781588162623004on_val,T: produc9003781588162623004on_val] :
          ( ( ( produc5239619271231976746on_val @ S3 )
            = ( produc5239619271231976746on_val @ T ) )
          & ( ( produc2319605628138516840on_val @ S3 )
            = ( produc2319605628138516840on_val @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_127_prod__eqI,axiom,
    ! [P: produc9003781588162623004on_val,Q2: produc9003781588162623004on_val] :
      ( ( ( produc5239619271231976746on_val @ P )
        = ( produc5239619271231976746on_val @ Q2 ) )
     => ( ( ( produc2319605628138516840on_val @ P )
          = ( produc2319605628138516840on_val @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_128_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_129_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_130_curryE,axiom,
    ! [F: produc1317171644805857433on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc4800927984219439315_val_o @ F @ A @ B )
     => ( F @ ( produc4958730763592889809on_val @ A @ B ) ) ) ).

% curryE
thf(fact_131_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_132_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_133_curryD,axiom,
    ! [F: produc1317171644805857433on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc4800927984219439315_val_o @ F @ A @ B )
     => ( F @ ( produc4958730763592889809on_val @ A @ B ) ) ) ).

% curryD
thf(fact_134_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_135_surjective__pairing,axiom,
    ! [T2: produc1317171644805857433on_val] :
      ( T2
      = ( produc4958730763592889809on_val @ ( produc3129717363260770749on_val @ T2 ) @ ( produc437328469718437375on_val @ T2 ) ) ) ).

% surjective_pairing
thf(fact_136_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_137_surjective__pairing,axiom,
    ! [T2: produc9003781588162623004on_val] :
      ( T2
      = ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ T2 ) @ ( produc2319605628138516840on_val @ T2 ) ) ) ).

% surjective_pairing
thf(fact_138_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1317171644805857433on_val] :
      ( Prod
      = ( produc4958730763592889809on_val @ ( produc3129717363260770749on_val @ Prod ) @ ( produc437328469718437375on_val @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_139_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_140_prod_Oexhaust__sel,axiom,
    ! [Prod: produc9003781588162623004on_val] :
      ( Prod
      = ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ Prod ) @ ( produc2319605628138516840on_val @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_141_get__m_Osimps,axiom,
    ( get_m
    = ( ^ [A2: state] : ( produc5239619271231976746on_val @ ( rep_state @ A2 ) ) ) ) ).

% get_m.simps
thf(fact_142_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_143_get__pre_I2_J,axiom,
    ( get_m
    = ( ^ [X2: state] : ( produc5239619271231976746on_val @ ( rep_state @ X2 ) ) ) ) ).

% get_pre(2)
thf(fact_144_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,A: produc1317171644805857433on_val] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc4958730763592889809on_val @ X @ Y ) )
       => ( P2 @ ( produc3129717363260770749on_val @ A ) @ ( produc437328469718437375on_val @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_145_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_146_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_147_conjI__realizer,axiom,
    ! [P2: produc9003781588162623004on_val > $o,P: produc9003781588162623004on_val,Q: produc9003781588162623004on_val > $o,Q2: produc9003781588162623004on_val] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc3129717363260770749on_val @ ( produc4958730763592889809on_val @ P @ Q2 ) ) )
          & ( Q @ ( produc437328469718437375on_val @ ( produc4958730763592889809on_val @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_148_conjI__realizer,axiom,
    ! [P2: nat > $o,P: nat,Q: nat > $o,Q2: nat] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_149_conjI__realizer,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,P: produc1457211279475724562t_char > prat,Q: ( produc1457211279475724562t_char > option_val ) > $o,Q2: produc1457211279475724562t_char > option_val] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc5239619271231976746on_val @ ( produc74620467203698198on_val @ P @ Q2 ) ) )
          & ( Q @ ( produc2319605628138516840on_val @ ( produc74620467203698198on_val @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_150_exI__realizer,axiom,
    ! [P2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,Y: produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc437328469718437375on_val @ ( produc4958730763592889809on_val @ X @ Y ) ) @ ( produc3129717363260770749on_val @ ( produc4958730763592889809on_val @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_151_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_152_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_153_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_154_prod_Oswap__def,axiom,
    ( produc5819413633627552064r_prat
    = ( ^ [P3: produc7776263825685906734r_prat] : ( produc74620467203698198on_val @ ( produc4142415063207305778r_prat @ P3 ) @ ( produc7062428706300765684r_prat @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_155_prod_Oswap__def,axiom,
    ( produc8960132906472719217on_val
    = ( ^ [P3: produc1317171644805857433on_val] : ( produc4958730763592889809on_val @ ( produc437328469718437375on_val @ P3 ) @ ( produc3129717363260770749on_val @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_156_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_157_prod_Oswap__def,axiom,
    ( produc3996604198558763126on_val
    = ( ^ [P3: produc9003781588162623004on_val] : ( produc1897429902272487136r_prat @ ( produc2319605628138516840on_val @ P3 ) @ ( produc5239619271231976746on_val @ P3 ) ) ) ) ).

% prod.swap_def
thf(fact_158_fst__swap,axiom,
    ! [X: produc7776263825685906734r_prat] :
      ( ( produc5239619271231976746on_val @ ( produc5819413633627552064r_prat @ X ) )
      = ( produc4142415063207305778r_prat @ X ) ) ).

% fst_swap
thf(fact_159_fst__swap,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ( produc7062428706300765684r_prat @ ( produc3996604198558763126on_val @ X ) )
      = ( produc2319605628138516840on_val @ X ) ) ).

% fst_swap
thf(fact_160_snd__swap,axiom,
    ! [X: produc9003781588162623004on_val] :
      ( ( produc4142415063207305778r_prat @ ( produc3996604198558763126on_val @ X ) )
      = ( produc5239619271231976746on_val @ X ) ) ).

% snd_swap
thf(fact_161_snd__swap,axiom,
    ! [X: produc7776263825685906734r_prat] :
      ( ( produc2319605628138516840on_val @ ( produc5819413633627552064r_prat @ X ) )
      = ( produc7062428706300765684r_prat @ X ) ) ).

% snd_swap
thf(fact_162_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_163_pre__get__m_Osimps,axiom,
    pre_ge8481574585653477131on_val = produc5239619271231976746on_val ).

% pre_get_m.simps
thf(fact_164_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_165_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_166_swap__simp,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( produc8960132906472719217on_val @ ( produc4958730763592889809on_val @ X @ Y ) )
      = ( produc4958730763592889809on_val @ Y @ X ) ) ).

% swap_simp
thf(fact_167_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_168_pre__get__m_Opelims,axiom,
    ! [X: produc1317171644805857433on_val,Y: produc9003781588162623004on_val] :
      ( ( ( pre_ge6899797224320663452on_val @ X )
        = Y )
     => ( ( accp_P6797916352740768290on_val @ pre_ge1694281482089598673on_val @ X )
       => ~ ( ( Y
              = ( produc3129717363260770749on_val @ X ) )
           => ~ ( accp_P6797916352740768290on_val @ pre_ge1694281482089598673on_val @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_169_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_170_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_171_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_172_image2__eqI,axiom,
    ! [B: produc1457211279475724562t_char > prat,F: product_prod_nat_nat > produc1457211279475724562t_char > prat,X: product_prod_nat_nat,C2: produc1457211279475724562t_char > option_val,G: product_prod_nat_nat > produc1457211279475724562t_char > option_val,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ C2 ) @ ( bNF_Gr1122962726781069062on_val @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_173_image2__eqI,axiom,
    ! [B: produc1457211279475724562t_char > prat,F: produc9003781588162623004on_val > produc1457211279475724562t_char > prat,X: produc9003781588162623004on_val,C2: produc1457211279475724562t_char > option_val,G: produc9003781588162623004on_val > produc1457211279475724562t_char > option_val,A5: set_Pr7649655576376376530on_val] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member2879030624002546611on_val @ X @ A5 )
         => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ C2 ) @ ( bNF_Gr6158647150560195831on_val @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_174_image2__eqI,axiom,
    ! [B: produc9003781588162623004on_val,F: product_prod_nat_nat > produc9003781588162623004on_val,X: product_prod_nat_nat,C2: produc9003781588162623004on_val,G: product_prod_nat_nat > produc9003781588162623004on_val,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ C2 ) @ ( bNF_Gr2163572297590636769on_val @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_175_image2__eqI,axiom,
    ! [B: produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,X: produc9003781588162623004on_val,C2: produc9003781588162623004on_val,G: produc9003781588162623004on_val > produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member2879030624002546611on_val @ X @ A5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ C2 ) @ ( bNF_Gr828556658099662512on_val @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_176_image2__eqI,axiom,
    ! [B: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,C2: nat,G: product_prod_nat_nat > nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr8954129895687480649at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_177_image2__eqI,axiom,
    ! [B: nat,F: produc9003781588162623004on_val > nat,X: produc9003781588162623004on_val,C2: nat,G: produc9003781588162623004on_val > nat,A5: set_Pr7649655576376376530on_val] :
      ( ( B
        = ( F @ X ) )
     => ( ( C2
          = ( G @ X ) )
       => ( ( member2879030624002546611on_val @ X @ A5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ ( bNF_Gr3695791879626193176at_nat @ A5 @ F @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_178_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_179_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_180_mult__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,P: prat] :
      ( ( greate7680731506090375811t_char @ A @ B )
     => ( greate7680731506090375811t_char @ ( multiply_mask @ P @ A ) @ ( multiply_mask @ P @ B ) ) ) ).

% mult_greater
thf(fact_181_update__trans,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat,Pi5: produc1457211279475724562t_char > prat,Pi6: produc1457211279475724562t_char > prat,Dm: produc1457211279475724562t_char > $o] :
      ( ( greate7680731506090375811t_char @ Pi2 @ Pi5 )
     => ( ( greate7680731506090375811t_char @ Pi5 @ Pi6 )
       => ( ( update5099441844846592839t_char @ ( update5099441844846592839t_char @ Dm @ Pi2 @ Pi5 ) @ Pi5 @ Pi6 )
          = ( update5099441844846592839t_char @ Dm @ Pi2 @ Pi6 ) ) ) ) ).

% update_trans
thf(fact_182_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_183_curry__case__prod,axiom,
    ! [F: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_184_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat,P: produc9003781588162623004on_val] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc3420870819319363678at_nat @ C2 @ P ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P
              = ( produc74620467203698198on_val @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_185_mem__case__prodE,axiom,
    ! [Z: produc9003781588162623004on_val,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr7649655576376376530on_val,P: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Z @ ( produc4471415964250050581on_val @ C2 @ P ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P
              = ( produc74620467203698198on_val @ X3 @ Y3 ) )
           => ~ ( member2879030624002546611on_val @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_186_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr1261947904930325089at_nat,P: produc1317171644805857433on_val] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc5971559172007353083at_nat @ C2 @ P ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( P
              = ( produc4958730763592889809on_val @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_187_mem__case__prodE,axiom,
    ! [Z: produc9003781588162623004on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr7649655576376376530on_val,P: produc1317171644805857433on_val] :
      ( ( member2879030624002546611on_val @ Z @ ( produc643424051632668344on_val @ C2 @ P ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( P
              = ( produc4958730763592889809on_val @ X3 @ Y3 ) )
           => ~ ( member2879030624002546611on_val @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_188_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat,P: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C2 @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_189_mem__case__prodE,axiom,
    ! [Z: produc9003781588162623004on_val,C2: nat > nat > set_Pr7649655576376376530on_val,P: product_prod_nat_nat] :
      ( ( member2879030624002546611on_val @ Z @ ( produc9171785124108960544on_val @ C2 @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member2879030624002546611on_val @ Z @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_190_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_191_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_192_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_193_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_194_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_195_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_196_split__cong,axiom,
    ! [Q2: 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 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F @ P )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_197_csquare__fstOp__sndOp,axiom,
    ! [F: ( ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o ) > produc9003781588162623004on_val > $o,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_cs2794013118909459695on_val @ ( collec7475294502966882545on_val @ ( F @ ( relcom8684624306960484107on_val @ P2 @ Q ) ) ) @ produc5358294431189718917r_prat @ produc5239619271231976746on_val @ ( bNF_fs7282908041383899532on_val @ P2 @ Q ) @ ( bNF_sn6607638108379713230on_val @ P2 @ Q ) ) ).

% csquare_fstOp_sndOp
thf(fact_198_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,Q: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_cs5223968648578849398on_val @ ( collec7475294502966882545on_val @ ( F @ ( relcom7279225663578213656on_val @ P2 @ Q ) ) ) @ produc2319605628138516840on_val @ produc6116476718443979321on_val @ ( bNF_fs7489134050367444823on_val @ P2 @ Q ) @ ( bNF_sn3802693149471413141on_val @ P2 @ Q ) ) ).

% csquare_fstOp_sndOp
thf(fact_199_valid__state__decompose,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val] :
      ( ( valid_state @ ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ A @ B ) @ H2 ) )
     => ( valid_state @ ( produc74620467203698198on_val @ A @ H2 ) ) ) ).

% valid_state_decompose
thf(fact_200_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_201_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_202_pair__in__swap__image,axiom,
    ! [Y: produc9003781588162623004on_val,X: produc9003781588162623004on_val,A5: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ ( image_7325970737918433093on_val @ produc8960132906472719217on_val @ A5 ) )
      = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ A5 ) ) ).

% pair_in_swap_image
thf(fact_203_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_204_snd__sndOp,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( produc2319605628138516840on_val
      = ( comp_P291491979669821239on_val @ produc2319605628138516840on_val @ ( bNF_sn6607638108379713230on_val @ P2 @ Q ) ) ) ).

% snd_sndOp
thf(fact_205_fst__fstOp,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( produc5239619271231976746on_val
      = ( comp_P6028404540003424944on_val @ produc5239619271231976746on_val @ ( bNF_fs7489134050367444823on_val @ P2 @ Q ) ) ) ).

% fst_fstOp
thf(fact_206_add__masks__cancellative,axiom,
    ! [B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B @ C2 )
        = ( add_ma6586698345353345811t_char @ B @ D ) )
     => ( C2 = D ) ) ).

% add_masks_cancellative
thf(fact_207_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B4 @ A2 ) ) ) ).

% add_masks_comm
thf(fact_208_add__masks__asso,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B ) @ C2 )
      = ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C2 ) ) ) ).

% add_masks_asso
thf(fact_209_upper__valid,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C2 ) )
       => ( ( valid_mask @ B )
          & ( valid_mask @ C2 ) ) ) ) ).

% upper_valid
thf(fact_210_upper__valid__aux,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C2 ) )
       => ( valid_mask @ B ) ) ) ).

% upper_valid_aux
thf(fact_211_sndOp__in,axiom,
    ! [Ac2: produc9003781588162623004on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member2879030624002546611on_val @ Ac2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( relcom8684624306960484107on_val @ P2 @ Q ) ) ) )
     => ( member2879030624002546611on_val @ ( bNF_sn6607638108379713230on_val @ P2 @ Q @ Ac2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ Q ) ) ) ) ).

% sndOp_in
thf(fact_212_sndOp__in,axiom,
    ! [Ac2: product_prod_nat_nat,P2: nat > nat > $o,Q: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_sn8916957246138178918at_nat @ P2 @ Q @ Ac2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q ) ) ) ) ).

% sndOp_in
thf(fact_213_fstOp__in,axiom,
    ! [Ac2: produc9003781588162623004on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member2879030624002546611on_val @ Ac2 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( relcom7279225663578213656on_val @ P2 @ Q ) ) ) )
     => ( member2879030624002546611on_val @ ( bNF_fs7489134050367444823on_val @ P2 @ Q @ Ac2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_214_fstOp__in,axiom,
    ! [Ac2: product_prod_nat_nat,P2: nat > nat > $o,Q: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ Ac2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( relcompp_nat_nat_nat @ P2 @ Q ) ) ) )
     => ( member8440522571783428010at_nat @ ( bNF_fs6799373675401686564at_nat @ P2 @ Q @ Ac2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) ) ) ).

% fstOp_in
thf(fact_215_mult__distr__masks,axiom,
    ! [A: prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B @ C2 ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B ) @ ( multiply_mask @ A @ C2 ) ) ) ).

% mult_distr_masks
thf(fact_216_const__sum__mask__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ A @ B )
        = ( add_ma6586698345353345811t_char @ C2 @ D ) )
     => ( ( greate7680731506090375811t_char @ A @ C2 )
       => ( greate7680731506090375811t_char @ D @ B ) ) ) ).

% const_sum_mask_greater
thf(fact_217_greater__mask__decomp,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > prat] :
      ( ( greate7680731506090375811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C2 ) )
     => ? [A1: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char > prat] :
          ( ( A
            = ( add_ma6586698345353345811t_char @ A1 @ A22 ) )
          & ( greate7680731506090375811t_char @ A1 @ B )
          & ( greate7680731506090375811t_char @ A22 @ C2 ) ) ) ).

% greater_mask_decomp
thf(fact_218_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi7: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat] :
        ? [R3: produc1457211279475724562t_char > prat] :
          ( Pi7
          = ( add_ma6586698345353345811t_char @ Pi3 @ R3 ) ) ) ) ).

% greater_mask_def
thf(fact_219_minus__empty,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] :
      ( Pi2
      = ( add_ma6586698345353345811t_char @ Pi2 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_220_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_221_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_222_compatible__def,axiom,
    ( compatible
    = ( ^ [Phi2: produc9003781588162623004on_val,Phi3: produc9003781588162623004on_val] :
          ( ( compatible_heaps @ ( produc2319605628138516840on_val @ Phi2 ) @ ( produc2319605628138516840on_val @ Phi3 ) )
          & ( valid_mask @ ( add_ma6586698345353345811t_char @ ( produc5239619271231976746on_val @ Phi2 ) @ ( produc5239619271231976746on_val @ Phi3 ) ) ) ) ) ) ).

% compatible_def
thf(fact_223_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ Z @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_Gr4383009045007556383on_val @ A5 @ F ) ) ) )
     => ( ( comp_P123150374798284520on_val @ F @ produc5239619271231976746on_val @ Z )
        = ( produc2319605628138516840on_val @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_224_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: product_prod_nat_nat,A5: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A5 @ F ) ) ) )
     => ( ( comp_n2711240574518091068at_nat @ F @ product_fst_nat_nat @ Z )
        = ( product_snd_nat_nat @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_225_Collect__split__mono__strong,axiom,
    ! [X4: set_Pr3433046674199179443r_prat,A5: set_Pr7649655576376376530on_val,Y8: set_Pr1004501353197068112on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( X4
        = ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ A5 ) )
     => ( ( Y8
          = ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ A5 ) )
       => ( ! [X3: produc1457211279475724562t_char > prat] :
              ( ( member2336342582916103932r_prat @ X3 @ X4 )
             => ! [Xa: produc1457211279475724562t_char > option_val] :
                  ( ( member1577473829074448177on_val @ Xa @ Y8 )
                 => ( ( P2 @ X3 @ Xa )
                   => ( Q @ X3 @ Xa ) ) ) )
         => ( ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ P2 ) ) )
           => ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_226_Collect__split__mono__strong,axiom,
    ! [X4: set_nat,A5: set_Pr1261947904930325089at_nat,Y8: set_nat,P2: nat > nat > $o,Q: nat > nat > $o] :
      ( ( X4
        = ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A5 ) )
     => ( ( Y8
          = ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A5 ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ X4 )
             => ! [Xa: nat] :
                  ( ( member_nat @ Xa @ Y8 )
                 => ( ( P2 @ X3 @ Xa )
                   => ( Q @ X3 @ Xa ) ) ) )
         => ( ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ P2 ) ) )
           => ( ord_le3146513528884898305at_nat @ A5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_227_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_228_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_229_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_230_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_231_map__prod__simp,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc8229809728907007980on_val @ F @ G @ ( produc74620467203698198on_val @ A @ B ) )
      = ( produc74620467203698198on_val @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_232_map__prod__simp,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val,G: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( produc4187076267714688945on_val @ F @ G @ ( produc74620467203698198on_val @ A @ B ) )
      = ( produc4958730763592889809on_val @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_233_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_234_map__prod__simp,axiom,
    ! [F: produc9003781588162623004on_val > produc1457211279475724562t_char > prat,G: produc9003781588162623004on_val > produc1457211279475724562t_char > option_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc3887529299458208887on_val @ F @ G @ ( produc4958730763592889809on_val @ A @ B ) )
      = ( produc74620467203698198on_val @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_235_map__prod__simp,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,G: produc9003781588162623004on_val > produc9003781588162623004on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc5488679577617650794on_val @ F @ G @ ( produc4958730763592889809on_val @ A @ B ) )
      = ( produc4958730763592889809on_val @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_236_map__prod__simp,axiom,
    ! [F: produc9003781588162623004on_val > nat,G: produc9003781588162623004on_val > nat,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc1289258586576348626al_nat @ F @ G @ ( produc4958730763592889809on_val @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_237_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_238_map__prod__simp,axiom,
    ! [F: nat > produc9003781588162623004on_val,G: nat > produc9003781588162623004on_val,A: nat,B: nat] :
      ( ( produc1409138104002980818on_val @ F @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc4958730763592889809on_val @ ( F @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_239_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_240_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_241_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_242_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_243_apfst__conv,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( produc3054129509749704210on_val @ F @ ( produc4958730763592889809on_val @ X @ Y ) )
      = ( produc4958730763592889809on_val @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_244_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_245_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_246_apsnd__conv,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( produc4354335345776199764on_val @ F @ ( produc4958730763592889809on_val @ X @ Y ) )
      = ( produc4958730763592889809on_val @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_247_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_248_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_249_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_250_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_251_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_252_map__prod__imageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R2: set_Pr7649655576376376530on_val,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R2 )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ ( F @ A ) @ ( G @ B ) ) @ ( image_4170889944715819327on_val @ ( produc8229809728907007980on_val @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_253_map__prod__imageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R2: set_Pr7649655576376376530on_val,F: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val,G: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R2 )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ A ) @ ( G @ B ) ) @ ( image_4249794487667743926on_val @ ( produc4187076267714688945on_val @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_254_map__prod__imageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R2: set_Pr7649655576376376530on_val,F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > nat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_3154293121060339486at_nat @ ( produc43609526583453465al_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_255_map__prod__imageI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc1457211279475724562t_char > prat,G: produc9003781588162623004on_val > produc1457211279475724562t_char > option_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R2 )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ ( F @ A ) @ ( G @ B ) ) @ ( image_149554504156965488on_val @ ( produc3887529299458208887on_val @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_256_map__prod__imageI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,G: produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R2 )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ A ) @ ( G @ B ) ) @ ( image_7325970737918433093on_val @ ( produc5488679577617650794on_val @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_257_map__prod__imageI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > nat,G: produc9003781588162623004on_val > nat] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A ) @ ( G @ B ) ) @ ( image_2009239599916048301at_nat @ ( produc1289258586576348626al_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_258_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > produc1457211279475724562t_char > prat,G: nat > produc1457211279475724562t_char > option_val] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ ( F @ A ) @ ( G @ B ) ) @ ( image_1778330534009140952on_val @ ( produc7620138996006934647on_val @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_259_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,F: nat > produc9003781588162623004on_val,G: nat > produc9003781588162623004on_val] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ A ) @ ( G @ B ) ) @ ( image_6967685473649684701on_val @ ( produc1409138104002980818on_val @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_260_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_261_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_262_apsnd__apfst,axiom,
    ! [F: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val,G: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
      ( ( produc4375914351996332626on_val @ F @ ( produc1063798742662147279on_val @ G @ X ) )
      = ( produc4958730763592889809on_val @ ( G @ ( produc5239619271231976746on_val @ X ) ) @ ( F @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_263_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_264_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_265_apfst__apsnd,axiom,
    ! [F: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val,G: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
      ( ( produc6059101059804679249on_val @ F @ ( produc7115516623398600349r_prat @ G @ X ) )
      = ( produc4958730763592889809on_val @ ( F @ ( produc5239619271231976746on_val @ X ) ) @ ( G @ ( produc2319605628138516840on_val @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_266_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_267_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_268_ssubsetI,axiom,
    ! [A5: set_Pr4305388979797279737on_val,B5: set_Pr4305388979797279737on_val] :
      ( ! [Pi: produc9003781588162623004on_val,H: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi @ H ) @ A5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Pi @ H ) @ B5 ) )
     => ( ord_le7224449410216889241on_val @ A5 @ B5 ) ) ).

% ssubsetI
thf(fact_269_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_270_prod__fun__imageE,axiom,
    ! [C2: produc9003781588162623004on_val,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat,G: ( produc1457211279475724562t_char > option_val ) > produc1457211279475724562t_char > option_val,R2: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ C2 @ ( image_4170889944715819327on_val @ ( produc8229809728907007980on_val @ F @ G ) @ R2 ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( C2
              = ( produc74620467203698198on_val @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_271_prod__fun__imageE,axiom,
    ! [C2: produc9003781588162623004on_val,F: produc9003781588162623004on_val > produc1457211279475724562t_char > prat,G: produc9003781588162623004on_val > produc1457211279475724562t_char > option_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ C2 @ ( image_149554504156965488on_val @ ( produc3887529299458208887on_val @ F @ G ) @ R2 ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( C2
              = ( produc74620467203698198on_val @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_272_prod__fun__imageE,axiom,
    ! [C2: produc9003781588162623004on_val,F: nat > produc1457211279475724562t_char > prat,G: nat > produc1457211279475724562t_char > option_val,R2: set_Pr1261947904930325089at_nat] :
      ( ( member2879030624002546611on_val @ C2 @ ( image_1778330534009140952on_val @ ( produc7620138996006934647on_val @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( produc74620467203698198on_val @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_273_prod__fun__imageE,axiom,
    ! [C2: produc1317171644805857433on_val,F: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val,G: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val,R2: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ C2 @ ( image_4249794487667743926on_val @ ( produc4187076267714688945on_val @ F @ G ) @ R2 ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( C2
              = ( produc4958730763592889809on_val @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_274_prod__fun__imageE,axiom,
    ! [C2: produc1317171644805857433on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val,G: produc9003781588162623004on_val > produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ C2 @ ( image_7325970737918433093on_val @ ( produc5488679577617650794on_val @ F @ G ) @ R2 ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( C2
              = ( produc4958730763592889809on_val @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_275_prod__fun__imageE,axiom,
    ! [C2: produc1317171644805857433on_val,F: nat > produc9003781588162623004on_val,G: nat > produc9003781588162623004on_val,R2: set_Pr1261947904930325089at_nat] :
      ( ( member6505152944228695362on_val @ C2 @ ( image_6967685473649684701on_val @ ( produc1409138104002980818on_val @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( produc4958730763592889809on_val @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_276_prod__fun__imageE,axiom,
    ! [C2: product_prod_nat_nat,F: ( produc1457211279475724562t_char > prat ) > nat,G: ( produc1457211279475724562t_char > option_val ) > nat,R2: set_Pr7649655576376376530on_val] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_3154293121060339486at_nat @ ( produc43609526583453465al_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_277_prod__fun__imageE,axiom,
    ! [C2: product_prod_nat_nat,F: produc9003781588162623004on_val > nat,G: produc9003781588162623004on_val > nat,R2: set_Pr4305388979797279737on_val] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_2009239599916048301at_nat @ ( produc1289258586576348626al_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_278_prod__fun__imageE,axiom,
    ! [C2: product_prod_nat_nat,F: nat > nat,G: nat > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ G ) @ R2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C2
              = ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_279_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ Z @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( bNF_Gr4383009045007556383on_val @ A5 @ F ) ) ) )
     => ( member2336342582916103932r_prat @ ( produc5239619271231976746on_val @ Z ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_280_Collect__case__prod__Grp__in,axiom,
    ! [Z: product_prod_nat_nat,A5: set_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A5 @ F ) ) ) )
     => ( member_nat @ ( product_fst_nat_nat @ Z ) @ A5 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_281_plus__ab__defined,axiom,
    ! [Phi: state,Phi4: state] :
      ( ( ( plus @ Phi @ Phi4 )
       != none_state )
      = ( ( compatible_heaps @ ( get_h @ Phi ) @ ( get_h @ Phi4 ) )
        & ( valid_mask @ ( add_ma6586698345353345811t_char @ ( get_m @ Phi ) @ ( get_m @ Phi4 ) ) ) ) ) ).

% plus_ab_defined
thf(fact_282_compatibleI__old,axiom,
    ! [Phi: produc9003781588162623004on_val,Phi4: produc9003781588162623004on_val] :
      ( ! [Hl2: produc1457211279475724562t_char,X3: val,Y3: val] :
          ( ( ( ( produc2319605628138516840on_val @ Phi @ Hl2 )
              = ( some_val @ X3 ) )
            & ( ( produc2319605628138516840on_val @ Phi4 @ Hl2 )
              = ( some_val @ Y3 ) ) )
         => ( X3 = Y3 ) )
     => ( ( valid_mask @ ( add_ma6586698345353345811t_char @ ( produc5239619271231976746on_val @ Phi ) @ ( produc5239619271231976746on_val @ Phi4 ) ) )
       => ( compatible @ Phi @ Phi4 ) ) ) ).

% compatibleI_old
thf(fact_283_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_284_subrelI,axiom,
    ! [R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ S ) )
     => ( ord_le7224449410216889241on_val @ R @ S ) ) ).

% subrelI
thf(fact_285_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_286_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_287_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_288_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_289_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_290_relChain__def,axiom,
    ( bNF_Ca8411853759642212370al_nat
    = ( ^ [R3: set_Pr4305388979797279737on_val,As: produc9003781588162623004on_val > nat] :
        ! [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I @ J ) @ R3 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_291_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_292_Collect__case__prod__in__rel__leI,axiom,
    ! [X4: set_Pr7649655576376376530on_val,Y8: set_Pr7649655576376376530on_val] :
      ( ( ord_le6900622908513426738on_val @ X4 @ Y8 )
     => ( ord_le6900622908513426738on_val @ X4 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( fun_in3588502033776859947on_val @ Y8 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_293_Collect__case__prod__in__rel__leI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X4 @ Y8 )
     => ( ord_le3146513528884898305at_nat @ X4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y8 ) ) ) ) ) ).

% Collect_case_prod_in_rel_leI
thf(fact_294_option_Oinject,axiom,
    ! [X22: val,Y2: val] :
      ( ( ( some_val @ X22 )
        = ( some_val @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_295_option_Oinject,axiom,
    ! [X22: state,Y2: state] :
      ( ( ( some_state @ X22 )
        = ( some_state @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_296_not__None__eq,axiom,
    ! [X: option_val] :
      ( ( X != none_val )
      = ( ? [Y5: val] :
            ( X
            = ( some_val @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_297_not__None__eq,axiom,
    ! [X: option_state] :
      ( ( X != none_state )
      = ( ? [Y5: state] :
            ( X
            = ( some_state @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_298_not__Some__eq,axiom,
    ! [X: option_val] :
      ( ( ! [Y5: val] :
            ( X
           != ( some_val @ Y5 ) ) )
      = ( X = none_val ) ) ).

% not_Some_eq
thf(fact_299_not__Some__eq,axiom,
    ! [X: option_state] :
      ( ( ! [Y5: state] :
            ( X
           != ( some_state @ Y5 ) ) )
      = ( X = none_state ) ) ).

% not_Some_eq
thf(fact_300_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_301_accp__subset,axiom,
    ! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( ord_le616650860478780142_val_o @ R1 @ R22 )
     => ( ord_le6140313152057869779_val_o @ ( accp_P4405556307444930259on_val @ R22 ) @ ( accp_P4405556307444930259on_val @ R1 ) ) ) ).

% accp_subset
thf(fact_302_accp__subset,axiom,
    ! [R1: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,R22: produc1317171644805857433on_val > produc1317171644805857433on_val > $o] :
      ( ( ord_le2319172193063290574_val_o @ R1 @ R22 )
     => ( ord_le8528861841627966596_val_o @ ( accp_P6797916352740768290on_val @ R22 ) @ ( accp_P6797916352740768290on_val @ R1 ) ) ) ).

% accp_subset
thf(fact_303_commutative,axiom,
    ( plus
    = ( ^ [A2: state,B4: state] : ( plus @ B4 @ A2 ) ) ) ).

% commutative
thf(fact_304_option_Odistinct_I1_J,axiom,
    ! [X22: val] :
      ( none_val
     != ( some_val @ X22 ) ) ).

% option.distinct(1)
thf(fact_305_option_Odistinct_I1_J,axiom,
    ! [X22: state] :
      ( none_state
     != ( some_state @ X22 ) ) ).

% option.distinct(1)
thf(fact_306_option_OdiscI,axiom,
    ! [Option: option_val,X22: val] :
      ( ( Option
        = ( some_val @ X22 ) )
     => ( Option != none_val ) ) ).

% option.discI
thf(fact_307_option_OdiscI,axiom,
    ! [Option: option_state,X22: state] :
      ( ( Option
        = ( some_state @ X22 ) )
     => ( Option != none_state ) ) ).

% option.discI
thf(fact_308_option_Oexhaust,axiom,
    ! [Y: option_val] :
      ( ( Y != none_val )
     => ~ ! [X23: val] :
            ( Y
           != ( some_val @ X23 ) ) ) ).

% option.exhaust
thf(fact_309_option_Oexhaust,axiom,
    ! [Y: option_state] :
      ( ( Y != none_state )
     => ~ ! [X23: state] :
            ( Y
           != ( some_state @ X23 ) ) ) ).

% option.exhaust
thf(fact_310_split__option__ex,axiom,
    ( ( ^ [P5: option_val > $o] :
        ? [X5: option_val] : ( P5 @ X5 ) )
    = ( ^ [P4: option_val > $o] :
          ( ( P4 @ none_val )
          | ? [X2: val] : ( P4 @ ( some_val @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_311_split__option__ex,axiom,
    ( ( ^ [P5: option_state > $o] :
        ? [X5: option_state] : ( P5 @ X5 ) )
    = ( ^ [P4: option_state > $o] :
          ( ( P4 @ none_state )
          | ? [X2: state] : ( P4 @ ( some_state @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_312_split__option__all,axiom,
    ( ( ^ [P5: option_val > $o] :
        ! [X5: option_val] : ( P5 @ X5 ) )
    = ( ^ [P4: option_val > $o] :
          ( ( P4 @ none_val )
          & ! [X2: val] : ( P4 @ ( some_val @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_313_split__option__all,axiom,
    ( ( ^ [P5: option_state > $o] :
        ! [X5: option_state] : ( P5 @ X5 ) )
    = ( ^ [P4: option_state > $o] :
          ( ( P4 @ none_state )
          & ! [X2: state] : ( P4 @ ( some_state @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_314_combine__options__cases,axiom,
    ! [X: option_val,P2: option_val > option_val > $o,Y: option_val] :
      ( ( ( X = none_val )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_val )
         => ( P2 @ X @ Y ) )
       => ( ! [A4: val,B3: val] :
              ( ( X
                = ( some_val @ A4 ) )
             => ( ( Y
                  = ( some_val @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_315_combine__options__cases,axiom,
    ! [X: option_val,P2: option_val > option_state > $o,Y: option_state] :
      ( ( ( X = none_val )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_state )
         => ( P2 @ X @ Y ) )
       => ( ! [A4: val,B3: state] :
              ( ( X
                = ( some_val @ A4 ) )
             => ( ( Y
                  = ( some_state @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_316_combine__options__cases,axiom,
    ! [X: option_state,P2: option_state > option_val > $o,Y: option_val] :
      ( ( ( X = none_state )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_val )
         => ( P2 @ X @ Y ) )
       => ( ! [A4: state,B3: val] :
              ( ( X
                = ( some_state @ A4 ) )
             => ( ( Y
                  = ( some_val @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_317_combine__options__cases,axiom,
    ! [X: option_state,P2: option_state > option_state > $o,Y: option_state] :
      ( ( ( X = none_state )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_state )
         => ( P2 @ X @ Y ) )
       => ( ! [A4: state,B3: state] :
              ( ( X
                = ( some_state @ A4 ) )
             => ( ( Y
                  = ( some_state @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_318_compatible__options_Ocases,axiom,
    ! [X: produc5122021904355189735on_val] :
      ( ! [A4: val,B3: val] :
          ( X
         != ( produc904852209897547607on_val @ ( some_val @ A4 ) @ ( some_val @ B3 ) ) )
     => ( ! [Uv: option_val] :
            ( X
           != ( produc904852209897547607on_val @ none_val @ Uv ) )
       => ~ ! [Uu: option_val] :
              ( X
             != ( produc904852209897547607on_val @ Uu @ none_val ) ) ) ) ).

% compatible_options.cases
thf(fact_319_compatible__options_Ocases,axiom,
    ! [X: produc3142500478612311029_state] :
      ( ! [A4: state,B3: state] :
          ( X
         != ( produc9160152616361873709_state @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) )
     => ( ! [Uv: option_state] :
            ( X
           != ( produc9160152616361873709_state @ none_state @ Uv ) )
       => ~ ! [Uu: option_state] :
              ( X
             != ( produc9160152616361873709_state @ Uu @ none_state ) ) ) ) ).

% compatible_options.cases
thf(fact_320_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,Z2: state] :
            ( ( D2 @ X3 )
           => ( ( R2 @ Z2 @ X3 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X3: state] :
                ( ( D2 @ X3 )
               => ( ! [Z4: state] :
                      ( ( R2 @ Z4 @ X3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_321_accp__subset__induct,axiom,
    ! [D2: produc9003781588162623004on_val > $o,R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,X: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( ord_le6140313152057869779_val_o @ D2 @ ( accp_P4405556307444930259on_val @ R2 ) )
     => ( ! [X3: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] :
            ( ( D2 @ X3 )
           => ( ( R2 @ Z2 @ X3 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X3: produc9003781588162623004on_val] :
                ( ( D2 @ X3 )
               => ( ! [Z4: produc9003781588162623004on_val] :
                      ( ( R2 @ Z4 @ X3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_322_accp__subset__induct,axiom,
    ! [D2: produc1317171644805857433on_val > $o,R2: produc1317171644805857433on_val > produc1317171644805857433on_val > $o,X: produc1317171644805857433on_val,P2: produc1317171644805857433on_val > $o] :
      ( ( ord_le8528861841627966596_val_o @ D2 @ ( accp_P6797916352740768290on_val @ R2 ) )
     => ( ! [X3: produc1317171644805857433on_val,Z2: produc1317171644805857433on_val] :
            ( ( D2 @ X3 )
           => ( ( R2 @ Z2 @ X3 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X3: produc1317171644805857433on_val] :
                ( ( D2 @ X3 )
               => ( ! [Z4: produc1317171644805857433on_val] :
                      ( ( R2 @ Z4 @ X3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_323_compatible__heapsI,axiom,
    ! [H2: produc1457211279475724562t_char > option_val,H5: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char,A4: val,B3: val] :
          ( ( ( H2 @ Hl2 )
            = ( some_val @ A4 ) )
         => ( ( ( H5 @ Hl2 )
              = ( some_val @ B3 ) )
           => ( A4 = B3 ) ) )
     => ( compatible_heaps @ H2 @ H5 ) ) ).

% compatible_heapsI
thf(fact_324_in__rel__def,axiom,
    ( fun_in3588502033776859947on_val
    = ( ^ [R4: set_Pr7649655576376376530on_val,X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_325_in__rel__def,axiom,
    ( fun_in8416620500172528508on_val
    = ( ^ [R4: set_Pr4305388979797279737on_val,X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_326_in__rel__def,axiom,
    ( fun_in_rel_nat_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R4 ) ) ) ).

% in_rel_def
thf(fact_327_in__rel__Collect__case__prod__eq,axiom,
    ! [X4: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( fun_in3588502033776859947on_val @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ X4 ) ) )
      = X4 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_328_in__rel__Collect__case__prod__eq,axiom,
    ! [X4: nat > nat > $o] :
      ( ( fun_in_rel_nat_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ X4 ) ) )
      = X4 ) ).

% in_rel_Collect_case_prod_eq
thf(fact_329_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_330_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_331_Collect__case__prod__in__rel__leE,axiom,
    ! [X4: set_Pr7649655576376376530on_val,Y8: set_Pr7649655576376376530on_val] :
      ( ( ord_le6900622908513426738on_val @ X4 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ ( fun_in3588502033776859947on_val @ Y8 ) ) ) )
     => ( ord_le6900622908513426738on_val @ X4 @ Y8 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_332_Collect__case__prod__in__rel__leE,axiom,
    ! [X4: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X4 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( fun_in_rel_nat_nat @ Y8 ) ) ) )
     => ( ord_le3146513528884898305at_nat @ X4 @ Y8 ) ) ).

% Collect_case_prod_in_rel_leE
thf(fact_333_plus__def,axiom,
    ( plus
    = ( ^ [A2: state,B4: state] : ( if_option_state @ ( compatible @ ( rep_state @ A2 ) @ ( rep_state @ B4 ) ) @ ( some_state @ ( abs_state @ ( add_states @ ( rep_state @ A2 ) @ ( rep_state @ B4 ) ) ) ) @ none_state ) ) ) ).

% plus_def
thf(fact_334_compatible__options_Oelims_I2_J,axiom,
    ! [X: option_state,Xa2: option_state] :
      ( ( compat2278460363914054422_state @ X @ Xa2 )
     => ( ! [A4: state] :
            ( ( X
              = ( some_state @ A4 ) )
           => ! [B3: state] :
                ( ( Xa2
                  = ( some_state @ B3 ) )
               => ( A4 != B3 ) ) )
       => ( ( X != none_state )
         => ( Xa2 = none_state ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_335_compatible__options_Oelims_I2_J,axiom,
    ! [X: option_val,Xa2: option_val] :
      ( ( compat7209469420100717861ns_val @ X @ Xa2 )
     => ( ! [A4: val] :
            ( ( X
              = ( some_val @ A4 ) )
           => ! [B3: val] :
                ( ( Xa2
                  = ( some_val @ B3 ) )
               => ( A4 != B3 ) ) )
       => ( ( X != none_val )
         => ( Xa2 = none_val ) ) ) ) ).

% compatible_options.elims(2)
thf(fact_336_compatible__options_Oelims_I1_J,axiom,
    ! [X: option_state,Xa2: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X @ Xa2 )
        = Y )
     => ( ! [A4: state] :
            ( ( X
              = ( some_state @ A4 ) )
           => ! [B3: state] :
                ( ( Xa2
                  = ( some_state @ B3 ) )
               => ( Y
                  = ( A4 != B3 ) ) ) )
       => ( ( ( X = none_state )
           => ~ Y )
         => ~ ( ( Xa2 = none_state )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_337_compatible__options_Oelims_I1_J,axiom,
    ! [X: option_val,Xa2: option_val,Y: $o] :
      ( ( ( compat7209469420100717861ns_val @ X @ Xa2 )
        = Y )
     => ( ! [A4: val] :
            ( ( X
              = ( some_val @ A4 ) )
           => ! [B3: val] :
                ( ( Xa2
                  = ( some_val @ B3 ) )
               => ( Y
                  = ( A4 != B3 ) ) ) )
       => ( ( ( X = none_val )
           => ~ Y )
         => ~ ( ( Xa2 = none_val )
             => ~ Y ) ) ) ) ).

% compatible_options.elims(1)
thf(fact_338_positivity,axiom,
    ! [A: state,B: state,C2: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ C2 ) )
     => ( ( ( some_state @ C2 )
          = ( plus @ C2 @ C2 ) )
       => ( ( some_state @ A )
          = ( plus @ A @ A ) ) ) ) ).

% positivity
thf(fact_339_asso1,axiom,
    ! [A: state,B: state,Ab: state,C2: state,Bc: state] :
      ( ( ( ( plus @ A @ B )
          = ( some_state @ Ab ) )
        & ( ( plus @ B @ C2 )
          = ( some_state @ Bc ) ) )
     => ( ( plus @ Ab @ C2 )
        = ( plus @ A @ Bc ) ) ) ).

% asso1
thf(fact_340_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_341_GrD2,axiom,
    ! [X: produc9003781588162623004on_val,Fx: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Fx ) @ ( bNF_Gr8368824372578450268on_val @ A5 @ F ) )
     => ( ( F @ X )
        = Fx ) ) ).

% GrD2
thf(fact_342_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_343_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_344_GrD1,axiom,
    ! [X: produc9003781588162623004on_val,Fx: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Fx ) @ ( bNF_Gr8368824372578450268on_val @ A5 @ F ) )
     => ( member2879030624002546611on_val @ X @ A5 ) ) ).

% GrD1
thf(fact_345_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_346_compatible__options_Osimps_I1_J,axiom,
    ! [A: state,B: state] :
      ( ( compat2278460363914054422_state @ ( some_state @ A ) @ ( some_state @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_347_compatible__options_Osimps_I1_J,axiom,
    ! [A: val,B: val] :
      ( ( compat7209469420100717861ns_val @ ( some_val @ A ) @ ( some_val @ B ) )
      = ( A = B ) ) ).

% compatible_options.simps(1)
thf(fact_348_compatible__options_Oelims_I3_J,axiom,
    ! [X: option_state,Xa2: option_state] :
      ( ~ ( compat2278460363914054422_state @ X @ Xa2 )
     => ~ ! [A4: state] :
            ( ( X
              = ( some_state @ A4 ) )
           => ! [B3: state] :
                ( ( Xa2
                  = ( some_state @ B3 ) )
               => ( A4 = B3 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_349_compatible__options_Oelims_I3_J,axiom,
    ! [X: option_val,Xa2: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X @ Xa2 )
     => ~ ! [A4: val] :
            ( ( X
              = ( some_val @ A4 ) )
           => ! [B3: val] :
                ( ( Xa2
                  = ( some_val @ B3 ) )
               => ( A4 = B3 ) ) ) ) ).

% compatible_options.elims(3)
thf(fact_350_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_state] : ( compat2278460363914054422_state @ none_state @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_351_compatible__options_Osimps_I2_J,axiom,
    ! [Uv2: option_val] : ( compat7209469420100717861ns_val @ none_val @ Uv2 ) ).

% compatible_options.simps(2)
thf(fact_352_compatible__options_Osimps_I3_J,axiom,
    ! [Uu2: option_state] : ( compat2278460363914054422_state @ Uu2 @ none_state ) ).

% compatible_options.simps(3)
thf(fact_353_compatible__options_Osimps_I3_J,axiom,
    ! [Uu2: option_val] : ( compat7209469420100717861ns_val @ Uu2 @ none_val ) ).

% compatible_options.simps(3)
thf(fact_354_core__is__smaller,axiom,
    ( some_state
    = ( ^ [X2: state] : ( plus @ X2 @ ( core @ X2 ) ) ) ) ).

% core_is_smaller
thf(fact_355_core__is__pure,axiom,
    ! [X: state] :
      ( ( some_state @ ( core @ X ) )
      = ( plus @ ( core @ X ) @ ( core @ X ) ) ) ).

% core_is_pure
thf(fact_356_cancellative,axiom,
    ! [A: state,B: state,X: state,Y: state] :
      ( ( ( some_state @ A )
        = ( plus @ B @ X ) )
     => ( ( ( some_state @ A )
          = ( plus @ B @ Y ) )
       => ( ( ( core @ X )
            = ( core @ Y ) )
         => ( X = Y ) ) ) ) ).

% cancellative
thf(fact_357_core__sum,axiom,
    ! [C2: state,A: state,B: state] :
      ( ( ( some_state @ C2 )
        = ( plus @ A @ B ) )
     => ( ( some_state @ ( core @ C2 ) )
        = ( plus @ ( core @ A ) @ ( core @ B ) ) ) ) ).

% core_sum
thf(fact_358_core__max,axiom,
    ! [X: state,C2: state] :
      ( ( ( some_state @ X )
        = ( plus @ X @ C2 ) )
     => ? [R5: state] :
          ( ( some_state @ ( core @ X ) )
          = ( plus @ C2 @ R5 ) ) ) ).

% core_max
thf(fact_359_compatible__heaps__def,axiom,
    ( compatible_heaps
    = ( ^ [H3: produc1457211279475724562t_char > option_val,H6: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char] : ( compat7209469420100717861ns_val @ ( H3 @ Hl ) @ ( H6 @ Hl ) ) ) ) ).

% compatible_heaps_def
thf(fact_360_asso2,axiom,
    ! [A: state,B: state,Ab: state,C2: state] :
      ( ( ( ( plus @ A @ B )
          = ( some_state @ Ab ) )
        & ( ( plus @ B @ C2 )
          = none_state ) )
     => ( ( plus @ Ab @ C2 )
        = none_state ) ) ).

% asso2
thf(fact_361_plus__charact_I1_J,axiom,
    ! [A: state,B: state,X: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X ) )
     => ( ( get_m @ X )
        = ( add_ma6586698345353345811t_char @ ( get_m @ A ) @ ( get_m @ B ) ) ) ) ).

% plus_charact(1)
thf(fact_362_compatible__options_Opelims_I1_J,axiom,
    ! [X: option_state,Xa2: option_state,Y: $o] :
      ( ( ( compat2278460363914054422_state @ X @ Xa2 )
        = Y )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ Xa2 ) )
       => ( ! [A4: state] :
              ( ( X
                = ( some_state @ A4 ) )
             => ! [B3: state] :
                  ( ( Xa2
                    = ( some_state @ B3 ) )
                 => ( ( Y
                      = ( A4 = B3 ) )
                   => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) ) ) ) )
         => ( ( ( X = none_state )
             => ( Y
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) ) )
           => ~ ( ( Xa2 = none_state )
               => ( Y
                 => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ none_state ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_363_compatible__options_Opelims_I1_J,axiom,
    ! [X: option_val,Xa2: option_val,Y: $o] :
      ( ( ( compat7209469420100717861ns_val @ X @ Xa2 )
        = Y )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ Xa2 ) )
       => ( ! [A4: val] :
              ( ( X
                = ( some_val @ A4 ) )
             => ! [B3: val] :
                  ( ( Xa2
                    = ( some_val @ B3 ) )
                 => ( ( Y
                      = ( A4 = B3 ) )
                   => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A4 ) @ ( some_val @ B3 ) ) ) ) ) )
         => ( ( ( X = none_val )
             => ( Y
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa2 ) ) ) )
           => ~ ( ( Xa2 = none_val )
               => ( Y
                 => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ none_val ) ) ) ) ) ) ) ) ).

% compatible_options.pelims(1)
thf(fact_364_compatible__options_Opelims_I2_J,axiom,
    ! [X: option_state,Xa2: option_state] :
      ( ( compat2278460363914054422_state @ X @ Xa2 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ Xa2 ) )
       => ( ! [A4: state] :
              ( ( X
                = ( some_state @ A4 ) )
             => ! [B3: state] :
                  ( ( Xa2
                    = ( some_state @ B3 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) )
                   => ( A4 != B3 ) ) ) )
         => ( ( ( X = none_state )
             => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ none_state @ Xa2 ) ) )
           => ~ ( ( Xa2 = none_state )
               => ~ ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ none_state ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_365_compatible__options_Opelims_I2_J,axiom,
    ! [X: option_val,Xa2: option_val] :
      ( ( compat7209469420100717861ns_val @ X @ Xa2 )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ Xa2 ) )
       => ( ! [A4: val] :
              ( ( X
                = ( some_val @ A4 ) )
             => ! [B3: val] :
                  ( ( Xa2
                    = ( some_val @ B3 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A4 ) @ ( some_val @ B3 ) ) )
                   => ( A4 != B3 ) ) ) )
         => ( ( ( X = none_val )
             => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ none_val @ Xa2 ) ) )
           => ~ ( ( Xa2 = none_val )
               => ~ ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ none_val ) ) ) ) ) ) ) ).

% compatible_options.pelims(2)
thf(fact_366_compatible__options_Opelims_I3_J,axiom,
    ! [X: option_state,Xa2: option_state] :
      ( ~ ( compat2278460363914054422_state @ X @ Xa2 )
     => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ X @ Xa2 ) )
       => ~ ! [A4: state] :
              ( ( X
                = ( some_state @ A4 ) )
             => ! [B3: state] :
                  ( ( Xa2
                    = ( some_state @ B3 ) )
                 => ( ( accp_P7183347322223220350_state @ compat1844074344818064729_state @ ( produc9160152616361873709_state @ ( some_state @ A4 ) @ ( some_state @ B3 ) ) )
                   => ( A4 = B3 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_367_compatible__options_Opelims_I3_J,axiom,
    ! [X: option_val,Xa2: option_val] :
      ( ~ ( compat7209469420100717861ns_val @ X @ Xa2 )
     => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ X @ Xa2 ) )
       => ~ ! [A4: val] :
              ( ( X
                = ( some_val @ A4 ) )
             => ! [B3: val] :
                  ( ( Xa2
                    = ( some_val @ B3 ) )
                 => ( ( accp_P4269981079083053040on_val @ compat8294034345482142370el_val @ ( produc904852209897547607on_val @ ( some_val @ A4 ) @ ( some_val @ B3 ) ) )
                   => ( A4 = B3 ) ) ) ) ) ) ).

% compatible_options.pelims(3)
thf(fact_368_add__states_Opelims,axiom,
    ! [X: produc9003781588162623004on_val,Xa2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X @ Xa2 )
        = Y )
     => ( ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ X @ Xa2 ) )
       => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
              ( ( X
                = ( produc74620467203698198on_val @ Pi @ H ) )
             => ! [Pi4: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
                  ( ( Xa2
                    = ( produc74620467203698198on_val @ Pi4 @ H4 ) )
                 => ( ( Y
                      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi4 ) @ ( map_ad2908576761495157028ar_val @ H @ H4 ) ) )
                   => ~ ( accp_P6797916352740768290on_val @ add_states_rel @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Pi @ H ) @ ( produc74620467203698198on_val @ Pi4 @ H4 ) ) ) ) ) ) ) ) ).

% add_states.pelims
thf(fact_369_add__states_Oelims,axiom,
    ! [X: produc9003781588162623004on_val,Xa2: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( ( add_states @ X @ Xa2 )
        = Y )
     => ~ ! [Pi: produc1457211279475724562t_char > prat,H: produc1457211279475724562t_char > option_val] :
            ( ( X
              = ( produc74620467203698198on_val @ Pi @ H ) )
           => ! [Pi4: produc1457211279475724562t_char > prat,H4: produc1457211279475724562t_char > option_val] :
                ( ( Xa2
                  = ( produc74620467203698198on_val @ Pi4 @ H4 ) )
               => ( Y
                 != ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi @ Pi4 ) @ ( map_ad2908576761495157028ar_val @ H @ H4 ) ) ) ) ) ) ).

% add_states.elims
thf(fact_370_add__states_Osimps,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat,H2: produc1457211279475724562t_char > option_val,Pi5: produc1457211279475724562t_char > prat,H5: produc1457211279475724562t_char > option_val] :
      ( ( add_states @ ( produc74620467203698198on_val @ Pi2 @ H2 ) @ ( produc74620467203698198on_val @ Pi5 @ H5 ) )
      = ( produc74620467203698198on_val @ ( add_ma6586698345353345811t_char @ Pi2 @ Pi5 ) @ ( map_ad2908576761495157028ar_val @ H2 @ H5 ) ) ) ).

% add_states.simps
thf(fact_371_compatible__heaps__sum,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C2: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( compatible_heaps @ A @ C2 )
       => ( compatible_heaps @ A @ ( map_ad2908576761495157028ar_val @ B @ C2 ) ) ) ) ).

% compatible_heaps_sum
thf(fact_372_compatible__heaps__comm,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( compatible_heaps @ A @ B )
     => ( ( map_ad2908576761495157028ar_val @ A @ B )
        = ( map_ad2908576761495157028ar_val @ B @ A ) ) ) ).

% compatible_heaps_comm
thf(fact_373_plus__charact_I2_J,axiom,
    ! [A: state,B: state,X: state] :
      ( ( ( plus @ A @ B )
        = ( some_state @ X ) )
     => ( ( get_h @ X )
        = ( map_ad2908576761495157028ar_val @ ( get_h @ A ) @ ( get_h @ B ) ) ) ) ).

% plus_charact(2)
thf(fact_374_map__add__eq__empty__iff,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( map_ad2908576761495157028ar_val @ F @ G )
        = ( ^ [X2: produc1457211279475724562t_char] : none_val ) )
      = ( ( F
          = ( ^ [X2: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X2: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% map_add_eq_empty_iff
thf(fact_375_map__add__None,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = none_val )
      = ( ( ( N @ K )
          = none_val )
        & ( ( M @ K )
          = none_val ) ) ) ).

% map_add_None
thf(fact_376_map__add__Some__iff,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X ) )
      = ( ( ( N @ K )
          = ( some_val @ X ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X ) ) ) ) ) ).

% map_add_Some_iff
thf(fact_377_map__add__SomeD,axiom,
    ! [M: produc1457211279475724562t_char > option_val,N: produc1457211279475724562t_char > option_val,K: produc1457211279475724562t_char,X: val] :
      ( ( ( map_ad2908576761495157028ar_val @ M @ N @ K )
        = ( some_val @ X ) )
     => ( ( ( N @ K )
          = ( some_val @ X ) )
        | ( ( ( N @ K )
            = none_val )
          & ( ( M @ K )
            = ( some_val @ X ) ) ) ) ) ).

% map_add_SomeD
thf(fact_378_larger__heaps__sum__ineq,axiom,
    ! [A3: produc1457211279475724562t_char > option_val,A: produc1457211279475724562t_char > option_val,B2: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A3 @ A )
     => ( ( larger1532547405891354081ar_val @ B2 @ B )
       => ( ( compatible_heaps @ A3 @ B2 )
         => ( larger1532547405891354081ar_val @ ( map_ad2908576761495157028ar_val @ A3 @ B2 ) @ ( map_ad2908576761495157028ar_val @ A @ B ) ) ) ) ) ).

% larger_heaps_sum_ineq
thf(fact_379_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_380_case__prod__Pair,axiom,
    ( ( produc4192255817509754207on_val @ produc74620467203698198on_val )
    = id_Pro1402184692240358763on_val ) ).

% case_prod_Pair
thf(fact_381_case__prod__Pair,axiom,
    ( ( produc7846207541802040627on_val @ produc4958730763592889809on_val )
    = id_Pro2088550933252328842on_val ) ).

% case_prod_Pair
thf(fact_382_case__prod__Pair,axiom,
    ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat )
    = id_Pro7559728917991913458at_nat ) ).

% case_prod_Pair
thf(fact_383_larger__heap__trans,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C2: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ B @ C2 )
       => ( larger1532547405891354081ar_val @ A @ C2 ) ) ) ).

% larger_heap_trans
thf(fact_384_larger__heap__refl,axiom,
    ! [H2: produc1457211279475724562t_char > option_val] : ( larger1532547405891354081ar_val @ H2 @ H2 ) ).

% larger_heap_refl
thf(fact_385_larger__heap__anti,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ B @ A )
       => ( A = B ) ) ) ).

% larger_heap_anti
thf(fact_386_larger__heap__def,axiom,
    ( larger1532547405891354081ar_val
    = ( ^ [H6: produc1457211279475724562t_char > option_val,H3: produc1457211279475724562t_char > option_val] :
        ! [Hl: produc1457211279475724562t_char,X2: val] :
          ( ( ( H3 @ Hl )
            = ( some_val @ X2 ) )
         => ( ( H6 @ Hl )
            = ( some_val @ X2 ) ) ) ) ) ).

% larger_heap_def
thf(fact_387_larger__heapI,axiom,
    ! [H2: produc1457211279475724562t_char > option_val,H5: produc1457211279475724562t_char > option_val] :
      ( ! [Hl2: produc1457211279475724562t_char,X3: val] :
          ( ( ( H2 @ Hl2 )
            = ( some_val @ X3 ) )
         => ( ( H5 @ Hl2 )
            = ( some_val @ X3 ) ) )
     => ( larger1532547405891354081ar_val @ H5 @ H2 ) ) ).

% larger_heapI
thf(fact_388_larger__compatible__sum__heaps,axiom,
    ! [A: produc1457211279475724562t_char > option_val,X: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,Y: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ X )
     => ( ( larger1532547405891354081ar_val @ B @ Y )
       => ( ( compatible_heaps @ A @ B )
         => ( compatible_heaps @ X @ Y ) ) ) ) ).

% larger_compatible_sum_heaps
thf(fact_389_larger__heap__comp,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C2: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( compatible_heaps @ A @ C2 )
       => ( compatible_heaps @ B @ C2 ) ) ) ).

% larger_heap_comp
thf(fact_390_larger__heap__plus,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > option_val,C2: produc1457211279475724562t_char > option_val] :
      ( ( larger1532547405891354081ar_val @ A @ B )
     => ( ( larger1532547405891354081ar_val @ A @ C2 )
       => ( larger1532547405891354081ar_val @ A @ ( map_ad2908576761495157028ar_val @ B @ C2 ) ) ) ) ).

% larger_heap_plus
thf(fact_391_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_392_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_393_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_394_mult__add__states,axiom,
    ! [A: prat,B: prat,Pi2: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ ( padd @ A @ B ) @ Pi2 )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ Pi2 ) @ ( multiply_mask @ B @ Pi2 ) ) ) ).

% mult_add_states
thf(fact_395_comp__the__Some,axiom,
    ( ( comp_o5604524432708427236al_val @ the_val @ some_val )
    = id_val ) ).

% comp_the_Some
thf(fact_396_comp__the__Some,axiom,
    ( ( comp_o4916166309651032803_state @ the_state @ some_state )
    = id_state ) ).

% comp_the_Some
thf(fact_397_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_398_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_399_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_400_option_Ocollapse,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( ( some_val @ ( the_val @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_401_option_Ocollapse,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( ( some_state @ ( the_state @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_402_range__fst,axiom,
    ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ top_to1561438921946393474on_val )
    = top_to6849109582029272579r_prat ) ).

% range_fst
thf(fact_403_range__snd,axiom,
    ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ top_to1561438921946393474on_val )
    = top_to3326603054294128128on_val ) ).

% range_snd
thf(fact_404_option_Osel,axiom,
    ! [X22: val] :
      ( ( the_val @ ( some_val @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_405_option_Osel,axiom,
    ! [X22: state] :
      ( ( the_state @ ( some_state @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_406_option_Oexpand,axiom,
    ! [Option: option_val,Option2: option_val] :
      ( ( ( Option = none_val )
        = ( Option2 = none_val ) )
     => ( ( ( Option != none_val )
         => ( ( Option2 != none_val )
           => ( ( the_val @ Option )
              = ( the_val @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_407_option_Oexpand,axiom,
    ! [Option: option_state,Option2: option_state] :
      ( ( ( Option = none_state )
        = ( Option2 = none_state ) )
     => ( ( ( Option != none_state )
         => ( ( Option2 != none_state )
           => ( ( the_state @ Option )
              = ( the_state @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_408_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi7: produc1457211279475724562t_char > prat,Pi3: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( padd @ ( Pi7 @ Hl ) @ ( Pi3 @ Hl ) ) ) ) ).

% add_masks.simps
thf(fact_409_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_410_Id__onE,axiom,
    ! [C2: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( id_on_2554058798563519774at_nat @ A5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ( C2
             != ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_411_Id__onE,axiom,
    ! [C2: produc1317171644805857433on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ C2 @ ( id_on_1505035389332359999on_val @ A5 ) )
     => ~ ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A5 )
           => ( C2
             != ( produc4958730763592889809on_val @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_412_Id__onE,axiom,
    ! [C2: product_prod_nat_nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( id_on_nat @ A5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ( C2
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_413_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_414_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_415_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_416_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_417_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_418_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_419_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_420_option_Oexhaust__sel,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
     => ( Option
        = ( some_val @ ( the_val @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_421_option_Oexhaust__sel,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
     => ( Option
        = ( some_state @ ( the_state @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_422_eq__alt,axiom,
    ( ( ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 ) )
    = ( bNF_Grp_nat_nat @ top_top_set_nat @ id_nat ) ) ).

% eq_alt
thf(fact_423_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_424_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_425_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_426_asymI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
         => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R ) )
     => ( asym_o6325306393319210090on_val @ top_to1561438921946393474on_val @ R ) ) ).

% asymI
thf(fact_427_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_428_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_429_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_430_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_431_asym__on__def,axiom,
    ( asym_o6325306393319210090on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ A6 )
         => ! [Y5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y5 @ A6 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 )
               => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_432_asym__on__def,axiom,
    ( asym_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R3 ) ) ) ) ) ) ).

% asym_on_def
thf(fact_433_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_434_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_435_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_436_None__notin__image__Some,axiom,
    ! [A5: set_val] :
      ~ ( member_option_val @ none_val @ ( image_val_option_val @ some_val @ A5 ) ) ).

% None_notin_image_Some
thf(fact_437_None__notin__image__Some,axiom,
    ! [A5: set_state] :
      ~ ( member_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ A5 ) ) ).

% None_notin_image_Some
thf(fact_438_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_439_asym__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( asym_o6325306393319210090on_val @ top_to1561438921946393474on_val @ R )
      = ( ! [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_440_asym__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R )
      = ( ! [X2: nat,Y5: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R ) ) ) ) ).

% asym_iff
thf(fact_441_asymD,axiom,
    ! [R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( asym_o6325306393319210090on_val @ top_to1561438921946393474on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
       => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R ) ) ) ).

% asymD
thf(fact_442_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_443_notin__range__Some,axiom,
    ! [X: option_val] :
      ( ( ~ ( member_option_val @ X @ ( image_val_option_val @ some_val @ top_top_set_val ) ) )
      = ( X = none_val ) ) ).

% notin_range_Some
thf(fact_444_notin__range__Some,axiom,
    ! [X: option_state] :
      ( ( ~ ( member_option_state @ X @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) )
      = ( X = none_state ) ) ).

% notin_range_Some
thf(fact_445_notin__range__Some,axiom,
    ! [X: option_nat] :
      ( ( ~ ( member_option_nat @ X @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
      = ( X = none_nat ) ) ).

% notin_range_Some
thf(fact_446_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_447_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_448_symp__conversep,axiom,
    ! [R2: nat > nat > $o] :
      ( ( symp_on_nat @ top_top_set_nat @ R2 )
      = ( ord_le2646555220125990790_nat_o @ ( conversep_nat_nat @ R2 ) @ R2 ) ) ).

% symp_conversep
thf(fact_449_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_450_fun_Orel__compp__Grp,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( bNF_re2896540935372247726on_val
        @ ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 )
        @ R2 )
      = ( relcom7349770684181336927on_val
        @ ( conver8203838583239980984r_prat
          @ ( bNF_Gr5444735849185432863r_prat
            @ ( collec2477365322772234464on_val
              @ ^ [X2: nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ X2 @ top_top_set_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) )
            @ ( comp_P4447563945128311908at_nat @ produc5239619271231976746on_val ) ) )
        @ ( bNF_Gr2620532623635037156on_val
          @ ( collec2477365322772234464on_val
            @ ^ [X2: nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ X2 @ top_top_set_nat ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) )
          @ ( comp_P6187037784939234283al_nat @ produc2319605628138516840on_val ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_451_fun_Orel__compp__Grp,axiom,
    ! [R2: nat > nat > $o] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 )
        @ R2 )
      = ( relcom8186637903969514807at_nat
        @ ( conver3498139987538047730at_nat
          @ ( bNF_Gr3388551082464497497at_nat
            @ ( collec8718012497556192979at_nat
              @ ^ [X2: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X2 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) )
            @ ( comp_P1955610211387027840at_nat @ product_fst_nat_nat ) ) )
        @ ( bNF_Gr3388551082464497497at_nat
          @ ( collec8718012497556192979at_nat
            @ ^ [X2: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X2 @ top_top_set_nat ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) )
          @ ( comp_P1955610211387027840at_nat @ product_snd_nat_nat ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_452_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
        @ ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 )
        @ R2
        @ A
        @ B )
      = ( ? [Z5: nat > produc9003781588162623004on_val] :
            ( ( member798016022416105634on_val @ Z5
              @ ( collec2477365322772234464on_val
                @ ^ [X2: nat > produc9003781588162623004on_val] : ( ord_le6900622908513426738on_val @ ( image_5220263881187111155on_val @ X2 @ 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_453_fun_Oin__rel,axiom,
    ! [R2: nat > nat > $o,A: nat > nat,B: nat > nat] :
      ( ( bNF_re5653821019739307937at_nat
        @ ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 )
        @ R2
        @ A
        @ B )
      = ( ? [Z5: nat > product_prod_nat_nat] :
            ( ( member6743353555990375057at_nat @ Z5
              @ ( collec8718012497556192979at_nat
                @ ^ [X2: nat > product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ X2 @ 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_454_UNIV__option__conv,axiom,
    ( top_to853123400021307383on_val
    = ( insert_option_val @ none_val @ ( image_val_option_val @ some_val @ top_top_set_val ) ) ) ).

% UNIV_option_conv
thf(fact_455_UNIV__option__conv,axiom,
    ( top_to7666338855062656496_state
    = ( insert_option_state @ none_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) ) ) ).

% UNIV_option_conv
thf(fact_456_UNIV__option__conv,axiom,
    ( top_to8920198386146353926on_nat
    = ( insert_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) ) ).

% UNIV_option_conv
thf(fact_457_split__part,axiom,
    ! [P2: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A2: nat,B4: nat] :
            ( P2
            & ( Q @ A2 @ B4 ) ) )
      = ( ^ [Ab2: product_prod_nat_nat] :
            ( P2
            & ( produc6081775807080527818_nat_o @ Q @ Ab2 ) ) ) ) ).

% split_part
thf(fact_458_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_459_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_460_case__prodI,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( F @ A @ B )
     => ( produc1845727980445322802_val_o @ F @ ( produc4958730763592889809on_val @ A @ B ) ) ) ).

% case_prodI
thf(fact_461_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_462_case__prodI2,axiom,
    ! [P: produc9003781588162623004on_val,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( ( P
            = ( produc74620467203698198on_val @ A4 @ B3 ) )
         => ( C2 @ A4 @ B3 ) )
     => ( produc5248837549992124053_val_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_463_case__prodI2,axiom,
    ! [P: produc1317171644805857433on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
          ( ( P
            = ( produc4958730763592889809on_val @ A4 @ B3 ) )
         => ( C2 @ A4 @ B3 ) )
     => ( produc1845727980445322802_val_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_464_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C2: nat > nat > $o] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( C2 @ A4 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_465_empty__eq__map__add__iff,axiom,
    ! [F: produc1457211279475724562t_char > option_val,G: produc1457211279475724562t_char > option_val] :
      ( ( ( ^ [X2: produc1457211279475724562t_char] : none_val )
        = ( map_ad2908576761495157028ar_val @ F @ G ) )
      = ( ( F
          = ( ^ [X2: produc1457211279475724562t_char] : none_val ) )
        & ( G
          = ( ^ [X2: produc1457211279475724562t_char] : none_val ) ) ) ) ).

% empty_eq_map_add_iff
thf(fact_466_map__add__empty,axiom,
    ! [M: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val @ M
        @ ^ [X2: produc1457211279475724562t_char] : none_val )
      = M ) ).

% map_add_empty
thf(fact_467_empty__map__add,axiom,
    ! [M: produc1457211279475724562t_char > option_val] :
      ( ( map_ad2908576761495157028ar_val
        @ ^ [X2: produc1457211279475724562t_char] : none_val
        @ M )
      = M ) ).

% empty_map_add
thf(fact_468_case__swap,axiom,
    ! [F: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y5: nat,X2: nat] : ( F @ X2 @ Y5 )
        @ ( product_swap_nat_nat @ P ) )
      = ( produc6081775807080527818_nat_o @ F @ P ) ) ).

% case_swap
thf(fact_469_mem__case__prodI2,axiom,
    ! [P: produc9003781588162623004on_val,Z: product_prod_nat_nat,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat] :
      ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( ( P
            = ( produc74620467203698198on_val @ A4 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc3420870819319363678at_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_470_mem__case__prodI2,axiom,
    ! [P: produc9003781588162623004on_val,Z: produc9003781588162623004on_val,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr7649655576376376530on_val] :
      ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
          ( ( P
            = ( produc74620467203698198on_val @ A4 @ B3 ) )
         => ( member2879030624002546611on_val @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member2879030624002546611on_val @ Z @ ( produc4471415964250050581on_val @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_471_mem__case__prodI2,axiom,
    ! [P: produc1317171644805857433on_val,Z: product_prod_nat_nat,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
          ( ( P
            = ( produc4958730763592889809on_val @ A4 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5971559172007353083at_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_472_mem__case__prodI2,axiom,
    ! [P: produc1317171644805857433on_val,Z: produc9003781588162623004on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
          ( ( P
            = ( produc4958730763592889809on_val @ A4 @ B3 ) )
         => ( member2879030624002546611on_val @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member2879030624002546611on_val @ Z @ ( produc643424051632668344on_val @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_473_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_474_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: produc9003781588162623004on_val,C2: nat > nat > set_Pr7649655576376376530on_val] :
      ( ! [A4: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A4 @ B3 ) )
         => ( member2879030624002546611on_val @ Z @ ( C2 @ A4 @ B3 ) ) )
     => ( member2879030624002546611on_val @ Z @ ( produc9171785124108960544on_val @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_475_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr1261947904930325089at_nat,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc3420870819319363678at_nat @ C2 @ ( produc74620467203698198on_val @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_476_mem__case__prodI,axiom,
    ! [Z: produc9003781588162623004on_val,C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > set_Pr7649655576376376530on_val,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val] :
      ( ( member2879030624002546611on_val @ Z @ ( C2 @ A @ B ) )
     => ( member2879030624002546611on_val @ Z @ ( produc4471415964250050581on_val @ C2 @ ( produc74620467203698198on_val @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_477_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr1261947904930325089at_nat,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5971559172007353083at_nat @ C2 @ ( produc4958730763592889809on_val @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_478_mem__case__prodI,axiom,
    ! [Z: produc9003781588162623004on_val,C2: produc9003781588162623004on_val > produc9003781588162623004on_val > set_Pr7649655576376376530on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Z @ ( C2 @ A @ B ) )
     => ( member2879030624002546611on_val @ Z @ ( produc643424051632668344on_val @ C2 @ ( produc4958730763592889809on_val @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_479_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C2: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C2 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_480_mem__case__prodI,axiom,
    ! [Z: produc9003781588162623004on_val,C2: nat > nat > set_Pr7649655576376376530on_val,A: nat,B: nat] :
      ( ( member2879030624002546611on_val @ Z @ ( C2 @ A @ B ) )
     => ( member2879030624002546611on_val @ Z @ ( produc9171785124108960544on_val @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_481_pair__imageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A5: set_Pr7649655576376376530on_val,F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > product_prod_nat_nat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_3154293121060339486at_nat @ ( produc7975119609531489022at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_482_pair__imageI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,A5: set_Pr7649655576376376530on_val,F: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ A5 )
     => ( member2879030624002546611on_val @ ( F @ A @ B ) @ ( image_4170889944715819327on_val @ ( produc4192255817509754207on_val @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_483_pair__imageI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val > product_prod_nat_nat] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_2009239599916048301at_nat @ ( produc4859565875879145883at_nat @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_484_pair__imageI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ A5 )
     => ( member2879030624002546611on_val @ ( F @ A @ B ) @ ( image_149554504156965488on_val @ ( produc2874708884313821186on_val @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_485_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_486_pair__imageI,axiom,
    ! [A: nat,B: nat,A5: set_Pr1261947904930325089at_nat,F: nat > nat > produc9003781588162623004on_val] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A5 )
     => ( member2879030624002546611on_val @ ( F @ A @ B ) @ ( image_1778330534009140952on_val @ ( produc5050726480593351786on_val @ F ) @ A5 ) ) ) ).

% pair_imageI
thf(fact_487_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_488_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_val] :
      ( ( Option != none_val )
      = ( case_option_o_val @ $false
        @ ^ [Uu3: val] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_489_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_state] :
      ( ( Option != none_state )
      = ( case_option_o_state @ $false
        @ ^ [Uu3: state] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_490_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_val] :
      ( ( Option = none_val )
      = ( case_option_o_val @ $true
        @ ^ [Uu3: val] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_491_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_state] :
      ( ( Option = none_state )
      = ( case_option_o_state @ $true
        @ ^ [Uu3: state] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_492_case__optionE,axiom,
    ! [P2: $o,Q: val > $o,X: option_val] :
      ( ( case_option_o_val @ P2 @ Q @ X )
     => ( ( ( X = none_val )
         => ~ P2 )
       => ~ ! [Y3: val] :
              ( ( X
                = ( some_val @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_493_case__optionE,axiom,
    ! [P2: $o,Q: state > $o,X: option_state] :
      ( ( case_option_o_state @ P2 @ Q @ X )
     => ( ( ( X = none_state )
         => ~ P2 )
       => ~ ! [Y3: state] :
              ( ( X
                = ( some_state @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_494_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_495_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_496_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_497_image__Collect__subsetI,axiom,
    ! [P2: produc9003781588162623004on_val > $o,F: produc9003781588162623004on_val > produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ! [X3: produc9003781588162623004on_val] :
          ( ( P2 @ X3 )
         => ( member2879030624002546611on_val @ ( F @ X3 ) @ B5 ) )
     => ( ord_le6900622908513426738on_val @ ( image_4170889944715819327on_val @ F @ ( collec7475294502966882545on_val @ P2 ) ) @ B5 ) ) ).

% image_Collect_subsetI
thf(fact_498_subset__CollectI,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,Q: 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 )
           => ( ( Q @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_499_subset__CollectI,axiom,
    ! [B5: set_Pr7649655576376376530on_val,A5: set_Pr7649655576376376530on_val,Q: produc9003781588162623004on_val > $o,P2: produc9003781588162623004on_val > $o] :
      ( ( ord_le6900622908513426738on_val @ B5 @ A5 )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ B5 )
           => ( ( Q @ X3 )
             => ( P2 @ X3 ) ) )
       => ( ord_le6900622908513426738on_val
          @ ( collec7475294502966882545on_val
            @ ^ [X2: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec7475294502966882545on_val
            @ ^ [X2: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_500_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
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_501_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
            @ ^ [X2: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_502_prop__restrict,axiom,
    ! [X: product_prod_nat_nat,Z6: set_Pr1261947904930325089at_nat,X4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ Z6 )
     => ( ( ord_le3146513528884898305at_nat @ Z6
          @ ( collec3392354462482085612at_nat
            @ ^ [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_503_prop__restrict,axiom,
    ! [X: produc9003781588162623004on_val,Z6: set_Pr7649655576376376530on_val,X4: set_Pr7649655576376376530on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ X @ Z6 )
     => ( ( ord_le6900622908513426738on_val @ Z6
          @ ( collec7475294502966882545on_val
            @ ^ [X2: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X ) ) ) ).

% prop_restrict
thf(fact_504_Collect__restrict,axiom,
    ! [X4: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_505_Collect__restrict,axiom,
    ! [X4: set_Pr7649655576376376530on_val,P2: produc9003781588162623004on_val > $o] :
      ( ord_le6900622908513426738on_val
      @ ( collec7475294502966882545on_val
        @ ^ [X2: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_506_pred__equals__eq2,axiom,
    ! [R2: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
      ( ( ( ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R2 ) )
        = ( ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_507_pred__equals__eq2,axiom,
    ! [R2: set_Pr4305388979797279737on_val,S4: set_Pr4305388979797279737on_val] :
      ( ( ( ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R2 ) )
        = ( ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_508_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) )
        = ( ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_509_snd__def,axiom,
    ( produc2319605628138516840on_val
    = ( produc985966260317727453on_val
      @ ^ [X12: produc1457211279475724562t_char > prat,X24: produc1457211279475724562t_char > option_val] : X24 ) ) ).

% snd_def
thf(fact_510_fst__def,axiom,
    ( produc5239619271231976746on_val
    = ( produc5787256515117077072r_prat
      @ ^ [X12: produc1457211279475724562t_char > prat,X24: produc1457211279475724562t_char > option_val] : X12 ) ) ).

% fst_def
thf(fact_511_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_512_case__prodD,axiom,
    ! [F: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( produc1845727980445322802_val_o @ F @ ( produc4958730763592889809on_val @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_513_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_514_case__prodE,axiom,
    ! [C2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P: produc9003781588162623004on_val] :
      ( ( produc5248837549992124053_val_o @ C2 @ P )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( P
              = ( produc74620467203698198on_val @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_515_case__prodE,axiom,
    ! [C2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,P: produc1317171644805857433on_val] :
      ( ( produc1845727980445322802_val_o @ C2 @ P )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( P
              = ( produc4958730763592889809on_val @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_516_case__prodE,axiom,
    ! [C2: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C2 @ P )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_517_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X12: nat,X24: nat] : ( H2 @ ( F @ X12 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_518_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu3: nat,Uv3: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_519_case__prod__Pair__iden,axiom,
    ! [P: produc9003781588162623004on_val] :
      ( ( produc4192255817509754207on_val @ produc74620467203698198on_val @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_520_case__prod__Pair__iden,axiom,
    ! [P: produc1317171644805857433on_val] :
      ( ( produc7846207541802040627on_val @ produc4958730763592889809on_val @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_521_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_522_case__prodE2,axiom,
    ! [Q: $o > $o,P2: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P2 @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_523_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X2: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_524_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_525_insert__subsetI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,X4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( ord_le3146513528884898305at_nat @ X4 @ A5 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ X4 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_526_insert__subsetI,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,X4: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ X @ A5 )
     => ( ( ord_le6900622908513426738on_val @ X4 @ A5 )
       => ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ X @ X4 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_527_exE__realizer,axiom,
    ! [P2: nat > nat > $o,P: product_prod_nat_nat,Q: $o > $o,F: nat > nat > $o] :
      ( ( P2 @ ( product_snd_nat_nat @ P ) @ ( product_fst_nat_nat @ P ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P2 @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc6081775807080527818_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_528_case__prod__beta_H,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F2: nat > nat > $o,X2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X2 ) @ ( product_snd_nat_nat @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_529_case__prod__unfold,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [C3: nat > nat > $o,P3: product_prod_nat_nat] : ( C3 @ ( product_fst_nat_nat @ P3 ) @ ( product_snd_nat_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_530_pred__subset__eq2,axiom,
    ! [R2: set_Pr7649655576376376530on_val,S4: set_Pr7649655576376376530on_val] :
      ( ( ord_le3852884270828607079_val_o
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R2 )
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le6900622908513426738on_val @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_531_pred__subset__eq2,axiom,
    ! [R2: set_Pr4305388979797279737on_val,S4: set_Pr4305388979797279737on_val] :
      ( ( ord_le616650860478780142_val_o
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R2 )
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le7224449410216889241on_val @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_532_pred__subset__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 )
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_533_lex__prod__def,axiom,
    ( lex_pr8992908938815505897on_val
    = ( ^ [Ra: set_Pr4305388979797279737on_val,Rb: set_Pr4305388979797279737on_val] :
          ( collec4536212101028033490on_val
          @ ( produc3489267648137303980_val_o
            @ ( produc7816547785737266186_val_o
              @ ^ [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
                  ( produc1845727980445322802_val_o
                  @ ^ [A7: produc9003781588162623004on_val,B6: produc9003781588162623004on_val] :
                      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_534_lex__prod__def,axiom,
    ( lex_pr2281407877887286429al_nat
    = ( ^ [Ra: set_Pr4305388979797279737on_val,Rb: set_Pr1261947904930325089at_nat] :
          ( collec1578930132947420882al_nat
          @ ( produc3370014461711855276_nat_o
            @ ( produc5559079767204974858_nat_o
              @ ^ [A2: produc9003781588162623004on_val,B4: nat] :
                  ( produc5057885034201482878_nat_o
                  @ ^ [A7: produc9003781588162623004on_val,B6: nat] :
                      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_535_lex__prod__def,axiom,
    ( lex_pr5802446539711250333on_val
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4305388979797279737on_val] :
          ( collec152402846092691154on_val
          @ ( produc6953174998001118892_val_o
            @ ( produc6040135174426305290_val_o
              @ ^ [A2: nat,B4: produc9003781588162623004on_val] :
                  ( produc1906844705236591486_val_o
                  @ ^ [A7: nat,B6: produc9003781588162623004on_val] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_536_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A2: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A7 ) @ Ra )
                      | ( ( A2 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_537_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
                  @ ^ [X2: nat,Y5: nat] :
                      ( ( X6 = X2 )
                      & ( P4 @ X2 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y9 @ Y5 ) @ ( R4 @ X2 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_538_predicate2__transferD,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( bNF_re2295284346247405049_val_o @ R1
        @ ( bNF_re6067813816088466256al_o_o @ R22
          @ ^ [Y7: $o,Z3: $o] : ( Y7 = Z3 ) )
        @ P2
        @ Q )
     => ( ( member2879030624002546611on_val @ A @ A5 )
       => ( ( member2879030624002546611on_val @ B @ B5 )
         => ( ( ord_le6900622908513426738on_val @ A5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R1 ) ) )
           => ( ( ord_le6900622908513426738on_val @ B5 @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R22 ) ) )
             => ( ( P2 @ ( produc5239619271231976746on_val @ A ) @ ( produc5239619271231976746on_val @ B ) )
                = ( Q @ ( produc2319605628138516840on_val @ A ) @ ( produc2319605628138516840on_val @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_539_predicate2__transferD,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R22: nat > nat > $o,P2: ( produc1457211279475724562t_char > prat ) > nat > $o,Q: ( 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
          @ ^ [Y7: $o,Z3: $o] : ( Y7 = Z3 ) )
        @ P2
        @ Q )
     => ( ( 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 ) )
                = ( Q @ ( produc2319605628138516840on_val @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_540_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,P2: nat > ( produc1457211279475724562t_char > prat ) > $o,Q: 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
          @ ^ [Y7: $o,Z3: $o] : ( Y7 = Z3 ) )
        @ P2
        @ Q )
     => ( ( 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 ) )
                = ( Q @ ( product_snd_nat_nat @ A ) @ ( produc2319605628138516840on_val @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_541_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,P2: nat > nat > $o,Q: 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
          @ ^ [Y7: $o,Z3: $o] : ( Y7 = Z3 ) )
        @ P2
        @ Q )
     => ( ( 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 ) )
                = ( Q @ ( product_snd_nat_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_542_fst__diag__id,axiom,
    ! [Z: produc9003781588162623004on_val] :
      ( ( comp_P3366439835523679408on_val @ produc3129717363260770749on_val
        @ ^ [X2: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ X2 @ X2 )
        @ Z )
      = ( id_Pro1402184692240358763on_val @ Z ) ) ).

% fst_diag_id
thf(fact_543_fst__diag__id,axiom,
    ! [Z: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat
        @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ X2 )
        @ Z )
      = ( id_nat @ Z ) ) ).

% fst_diag_id
thf(fact_544_snd__diag__id,axiom,
    ! [Z: produc9003781588162623004on_val] :
      ( ( comp_P3366439835523679408on_val @ produc437328469718437375on_val
        @ ^ [X2: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ X2 @ X2 )
        @ Z )
      = ( id_Pro1402184692240358763on_val @ Z ) ) ).

% snd_diag_id
thf(fact_545_snd__diag__id,axiom,
    ! [Z: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat
        @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ X2 )
        @ Z )
      = ( id_nat @ Z ) ) ).

% snd_diag_id
thf(fact_546_fst__diag__fst,axiom,
    ( ( comp_P5489260687716092971on_val @ produc2193917115357690435r_prat
      @ ( comp_P4016055770202289235on_val
        @ ^ [X2: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ X2 @ X2 )
        @ produc5239619271231976746on_val ) )
    = produc5239619271231976746on_val ) ).

% fst_diag_fst
thf(fact_547_snd__diag__snd,axiom,
    ( ( comp_P3343515995226764714on_val @ produc1305954679496617083on_val
      @ ( comp_P3324790395486970996on_val
        @ ^ [X2: produc1457211279475724562t_char > option_val] : ( produc8008623296994167373on_val @ X2 @ X2 )
        @ produc2319605628138516840on_val ) )
    = produc2319605628138516840on_val ) ).

% snd_diag_snd
thf(fact_548_snd__fst__flip,axiom,
    ( produc437328469718437375on_val
    = ( comp_P7221957804074544389on_val @ produc3129717363260770749on_val
      @ ( produc7846207541802040627on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_549_snd__fst__flip,axiom,
    ( product_snd_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_fst_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_550_snd__fst__flip,axiom,
    ( produc4142415063207305778r_prat
    = ( comp_P4800886777526708674r_prat @ produc5239619271231976746on_val
      @ ( produc4110721796449090601on_val
        @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_551_snd__fst__flip,axiom,
    ( produc2319605628138516840on_val
    = ( comp_P1319385742190247241on_val @ produc7062428706300765684r_prat
      @ ( produc2964738055033037937r_prat
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_552_fst__snd__flip,axiom,
    ( produc3129717363260770749on_val
    = ( comp_P7221957804074544389on_val @ produc437328469718437375on_val
      @ ( produc7846207541802040627on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_553_fst__snd__flip,axiom,
    ( product_fst_nat_nat
    = ( comp_P1567445206330693457at_nat @ product_snd_nat_nat
      @ ( produc2626176000494625587at_nat
        @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_554_fst__snd__flip,axiom,
    ( produc5239619271231976746on_val
    = ( comp_P1591131706424975134on_val @ produc4142415063207305778r_prat
      @ ( produc2964738055033037937r_prat
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_555_fst__snd__flip,axiom,
    ( produc7062428706300765684r_prat
    = ( comp_P8287346254047880777r_prat @ produc2319605628138516840on_val
      @ ( produc4110721796449090601on_val
        @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_556_fst__diag__snd,axiom,
    ( ( comp_P3343515995226764714on_val @ produc6116476718443979321on_val
      @ ( comp_P3324790395486970996on_val
        @ ^ [X2: produc1457211279475724562t_char > option_val] : ( produc8008623296994167373on_val @ X2 @ X2 )
        @ produc2319605628138516840on_val ) )
    = produc2319605628138516840on_val ) ).

% fst_diag_snd
thf(fact_557_snd__diag__fst,axiom,
    ( ( comp_P5489260687716092971on_val @ produc5358294431189718917r_prat
      @ ( comp_P4016055770202289235on_val
        @ ^ [X2: produc1457211279475724562t_char > prat] : ( produc1164665353665699927r_prat @ X2 @ X2 )
        @ produc5239619271231976746on_val ) )
    = produc5239619271231976746on_val ) ).

% snd_diag_fst
thf(fact_558_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
            @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( produc1897429902272487136r_prat @ Y5 @ X2 ) )
          @ A5 )
        @ ( collec6247776740490166275r_prat @ ( produc5226666187743293259prat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_559_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
            @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ Y5 @ X2 ) )
          @ A5 )
        @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_560_flip__pred,axiom,
    ! [A5: set_Pr4305388979797279737on_val,R2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( ord_le7224449410216889241on_val @ A5 @ ( collec5166231387257825412on_val @ ( produc1845727980445322802_val_o @ ( conver1343519957138882863on_val @ R2 ) ) ) )
     => ( ord_le7224449410216889241on_val
        @ ( image_7325970737918433093on_val
          @ ( produc7846207541802040627on_val
            @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ Y5 @ X2 ) )
          @ A5 )
        @ ( collec5166231387257825412on_val @ ( produc1845727980445322802_val_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_561_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
            @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) )
          @ A5 )
        @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R2 ) ) ) ) ).

% flip_pred
thf(fact_562_these__insert__Some,axiom,
    ! [X: val,A5: set_option_val] :
      ( ( these_val @ ( insert_option_val @ ( some_val @ X ) @ A5 ) )
      = ( insert_val @ X @ ( these_val @ A5 ) ) ) ).

% these_insert_Some
thf(fact_563_these__insert__Some,axiom,
    ! [X: state,A5: set_option_state] :
      ( ( these_state @ ( insert_option_state @ ( some_state @ X ) @ A5 ) )
      = ( insert_state @ X @ ( these_state @ A5 ) ) ) ).

% these_insert_Some
thf(fact_564_these__insert__None,axiom,
    ! [A5: set_option_val] :
      ( ( these_val @ ( insert_option_val @ none_val @ A5 ) )
      = ( these_val @ A5 ) ) ).

% these_insert_None
thf(fact_565_these__insert__None,axiom,
    ! [A5: set_option_state] :
      ( ( these_state @ ( insert_option_state @ none_state @ A5 ) )
      = ( these_state @ A5 ) ) ).

% these_insert_None
thf(fact_566_Some__image__these__eq,axiom,
    ! [A5: set_option_val] :
      ( ( image_val_option_val @ some_val @ ( these_val @ A5 ) )
      = ( collect_option_val
        @ ^ [X2: option_val] :
            ( ( member_option_val @ X2 @ A5 )
            & ( X2 != none_val ) ) ) ) ).

% Some_image_these_eq
thf(fact_567_Some__image__these__eq,axiom,
    ! [A5: set_option_state] :
      ( ( image_6076465424260689483_state @ some_state @ ( these_state @ A5 ) )
      = ( collect_option_state
        @ ^ [X2: option_state] :
            ( ( member_option_state @ X2 @ A5 )
            & ( X2 != none_state ) ) ) ) ).

% Some_image_these_eq
thf(fact_568_these__image__Some__eq,axiom,
    ! [A5: set_val] :
      ( ( these_val @ ( image_val_option_val @ some_val @ A5 ) )
      = A5 ) ).

% these_image_Some_eq
thf(fact_569_these__image__Some__eq,axiom,
    ! [A5: set_state] :
      ( ( these_state @ ( image_6076465424260689483_state @ some_state @ A5 ) )
      = A5 ) ).

% these_image_Some_eq
thf(fact_570_asymp__asym__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( asymp_3982633545749505022on_val @ top_to1561438921946393474on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( asym_o6325306393319210090on_val @ top_to1561438921946393474on_val @ R ) ) ).

% asymp_asym_eq
thf(fact_571_asymp__asym__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ top_top_set_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( asym_on_nat @ top_top_set_nat @ R ) ) ).

% asymp_asym_eq
thf(fact_572_Option_Othese__def,axiom,
    ( these_val
    = ( ^ [A6: set_option_val] :
          ( image_option_val_val @ the_val
          @ ( collect_option_val
            @ ^ [X2: option_val] :
                ( ( member_option_val @ X2 @ A6 )
                & ( X2 != none_val ) ) ) ) ) ) ).

% Option.these_def
thf(fact_573_Option_Othese__def,axiom,
    ( these_state
    = ( ^ [A6: set_option_state] :
          ( image_3532137647693456075_state @ the_state
          @ ( collect_option_state
            @ ^ [X2: option_state] :
                ( ( member_option_state @ X2 @ A6 )
                & ( X2 != none_state ) ) ) ) ) ) ).

% Option.these_def
thf(fact_574_in__these__eq,axiom,
    ! [X: product_prod_nat_nat,A5: set_op4508134149509766951at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( these_8061402112538453438at_nat @ A5 ) )
      = ( member3954567711264315760at_nat @ ( some_P7363390416028606310at_nat @ X ) @ A5 ) ) ).

% in_these_eq
thf(fact_575_in__these__eq,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_op6240181410663953058on_val] :
      ( ( member2879030624002546611on_val @ X @ ( these_3919539662624406175on_val @ A5 ) )
      = ( member4602090469187516291on_val @ ( some_P2581650809551911671on_val @ X ) @ A5 ) ) ).

% in_these_eq
thf(fact_576_in__these__eq,axiom,
    ! [X: val,A5: set_option_val] :
      ( ( member_val @ X @ ( these_val @ A5 ) )
      = ( member_option_val @ ( some_val @ X ) @ A5 ) ) ).

% in_these_eq
thf(fact_577_in__these__eq,axiom,
    ! [X: state,A5: set_option_state] :
      ( ( member_state @ X @ ( these_state @ A5 ) )
      = ( member_option_state @ ( some_state @ X ) @ A5 ) ) ).

% in_these_eq
thf(fact_578_asymp__on__asym__on__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ( asymp_3982633545749505022on_val @ A5
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( asym_o6325306393319210090on_val @ A5 @ R ) ) ).

% asymp_on_asym_on_eq
thf(fact_579_asymp__on__asym__on__eq,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ A5
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( asym_on_nat @ A5 @ R ) ) ).

% asymp_on_asym_on_eq
thf(fact_580_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
            @ ^ [Uu3: nat] : B5 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% asym_on_lex_prod
thf(fact_581_subset__fst__snd,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ord_le6900622908513426738on_val @ A5
      @ ( produc1366239611153933861on_val @ ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ A5 )
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ A5 ) ) ) ).

% subset_fst_snd
thf(fact_582_combine__options__def,axiom,
    ( combine_options_val
    = ( ^ [F2: val > val > val,X2: option_val,Y5: option_val] :
          ( case_o7794706883426356101al_val @ Y5
          @ ^ [Z5: val] :
              ( case_o7794706883426356101al_val @ ( some_val @ Z5 )
              @ ^ [Aa: val] : ( some_val @ ( F2 @ Z5 @ Aa ) )
              @ Y5 )
          @ X2 ) ) ) ).

% combine_options_def
thf(fact_583_combine__options__def,axiom,
    ( combin3633514924185701883_state
    = ( ^ [F2: state > state > state,X2: option_state,Y5: option_state] :
          ( case_o8916064052540797265_state @ Y5
          @ ^ [Z5: state] :
              ( case_o8916064052540797265_state @ ( some_state @ Z5 )
              @ ^ [Aa: state] : ( some_state @ ( F2 @ Z5 @ Aa ) )
              @ Y5 )
          @ X2 ) ) ) ).

% combine_options_def
thf(fact_584_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_585_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B: produc9003781588162623004on_val,B5: product_prod_nat_nat > set_Pr7649655576376376530on_val] :
      ( ( member8440522571783428010at_nat @ A @ A5 )
     => ( ( member2879030624002546611on_val @ B @ ( B5 @ A ) )
       => ( member5049262082750528355on_val @ ( produc1110776065552527814on_val @ A @ B ) @ ( produc7035663349334832981on_val @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_586_SigmaI,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B: product_prod_nat_nat,B5: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ( member2879030624002546611on_val @ A @ A5 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member7906376990877558257at_nat @ ( produc2486738652603726348at_nat @ A @ B ) @ ( produc8411625936386031515at_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_587_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_588_SigmaI,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val,B5: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ A @ A5 )
     => ( ( member2879030624002546611on_val @ B @ ( B5 @ A ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4067394217145614850on_val @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_589_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_590_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_591_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: produc9003781588162623004on_val,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr7649655576376376530on_val] :
      ( ( member5049262082750528355on_val @ ( produc1110776065552527814on_val @ A @ B ) @ ( produc7035663349334832981on_val @ A5 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A5 )
        & ( member2879030624002546611on_val @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_592_mem__Sigma__iff,axiom,
    ! [A: produc9003781588162623004on_val,B: product_prod_nat_nat,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ( member7906376990877558257at_nat @ ( produc2486738652603726348at_nat @ A @ B ) @ ( produc8411625936386031515at_nat @ A5 @ B5 ) )
      = ( ( member2879030624002546611on_val @ A @ A5 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_593_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_594_mem__Sigma__iff,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4067394217145614850on_val @ A5 @ B5 ) )
      = ( ( member2879030624002546611on_val @ A @ A5 )
        & ( member2879030624002546611on_val @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_595_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_596_combine__options__simps_I3_J,axiom,
    ! [F: val > val > val,A: val,B: val] :
      ( ( combine_options_val @ F @ ( some_val @ A ) @ ( some_val @ B ) )
      = ( some_val @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_597_combine__options__simps_I3_J,axiom,
    ! [F: state > state > state,A: state,B: state] :
      ( ( combin3633514924185701883_state @ F @ ( some_state @ A ) @ ( some_state @ B ) )
      = ( some_state @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_598_combine__options__simps_I2_J,axiom,
    ! [F: val > val > val,X: option_val] :
      ( ( combine_options_val @ F @ X @ none_val )
      = X ) ).

% combine_options_simps(2)
thf(fact_599_combine__options__simps_I2_J,axiom,
    ! [F: state > state > state,X: option_state] :
      ( ( combin3633514924185701883_state @ F @ X @ none_state )
      = X ) ).

% combine_options_simps(2)
thf(fact_600_combine__options__simps_I1_J,axiom,
    ! [F: val > val > val,Y: option_val] :
      ( ( combine_options_val @ F @ none_val @ Y )
      = Y ) ).

% combine_options_simps(1)
thf(fact_601_combine__options__simps_I1_J,axiom,
    ! [F: state > state > state,Y: option_state] :
      ( ( combin3633514924185701883_state @ F @ none_state @ Y )
      = Y ) ).

% combine_options_simps(1)
thf(fact_602_Collect__case__prod,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [A2: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( P2 @ A2 )
              & ( Q @ B4 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu3: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_603_Collect__case__prod,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( collec7293267056801676193on_val
        @ ( produc3458391012123660517_val_o
          @ ^ [A2: product_prod_nat_nat,B4: produc9003781588162623004on_val] :
              ( ( P2 @ A2 )
              & ( Q @ B4 ) ) ) )
      = ( produc7035663349334832981on_val @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [Uu3: product_prod_nat_nat] : ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_case_prod
thf(fact_604_Collect__case__prod,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec927009928073930287at_nat
        @ ( produc8420946574280188703_nat_o
          @ ^ [A2: produc9003781588162623004on_val,B4: product_prod_nat_nat] :
              ( ( P2 @ A2 )
              & ( Q @ B4 ) ) ) )
      = ( produc8411625936386031515at_nat @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [Uu3: produc9003781588162623004on_val] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_605_Collect__case__prod,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > $o] :
      ( ( collec5166231387257825412on_val
        @ ( produc1845727980445322802_val_o
          @ ^ [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
              ( ( P2 @ A2 )
              & ( Q @ B4 ) ) ) )
      = ( produc4067394217145614850on_val @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [Uu3: produc9003781588162623004on_val] : ( collec7475294502966882545on_val @ Q ) ) ) ).

% Collect_case_prod
thf(fact_606_Collect__case__prod,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( collec7475294502966882545on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
              ( ( P2 @ A2 )
              & ( Q @ B4 ) ) ) )
      = ( produc1366239611153933861on_val @ ( collec7093475260167521854r_prat @ P2 )
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( collec8455114619259388271on_val @ Q ) ) ) ).

% Collect_case_prod
thf(fact_607_Collect__case__prod,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A2: nat,B4: nat] :
              ( ( P2 @ A2 )
              & ( Q @ B4 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
        @ ^ [Uu3: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_608_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu3: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_609_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_610_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: produc9003781588162623004on_val,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr7649655576376376530on_val] :
      ( ( member5049262082750528355on_val @ ( produc1110776065552527814on_val @ A @ B ) @ ( produc7035663349334832981on_val @ A5 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A5 )
         => ~ ( member2879030624002546611on_val @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_611_SigmaE2,axiom,
    ! [A: produc9003781588162623004on_val,B: product_prod_nat_nat,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ( member7906376990877558257at_nat @ ( produc2486738652603726348at_nat @ A @ B ) @ ( produc8411625936386031515at_nat @ A5 @ B5 ) )
     => ~ ( ( member2879030624002546611on_val @ A @ A5 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_612_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_613_SigmaE2,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4067394217145614850on_val @ A5 @ B5 ) )
     => ~ ( ( member2879030624002546611on_val @ A @ A5 )
         => ~ ( member2879030624002546611on_val @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_614_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_615_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_616_SigmaD2,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4067394217145614850on_val @ A5 @ B5 ) )
     => ( member2879030624002546611on_val @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_617_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_618_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_619_SigmaD1,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4067394217145614850on_val @ A5 @ B5 ) )
     => ( member2879030624002546611on_val @ A @ A5 ) ) ).

% SigmaD1
thf(fact_620_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_621_SigmaE,axiom,
    ! [C2: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( produc2761391749766926216at_nat @ A5 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_622_SigmaE,axiom,
    ! [C2: produc5414607237814678604on_val,A5: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr7649655576376376530on_val] :
      ( ( member5049262082750528355on_val @ C2 @ ( produc7035663349334832981on_val @ A5 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A5 )
           => ! [Y3: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc1110776065552527814on_val @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_623_SigmaE,axiom,
    ! [C2: produc8271722145941708506at_nat,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ( member7906376990877558257at_nat @ C2 @ ( produc8411625936386031515at_nat @ A5 @ B5 ) )
     => ~ ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A5 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc2486738652603726348at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_624_SigmaE,axiom,
    ! [C2: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,B5: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ C2 @ ( produc1366239611153933861on_val @ A5 @ B5 ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X3 @ A5 )
           => ! [Y3: produc1457211279475724562t_char > option_val] :
                ( ( member1577473829074448177on_val @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc74620467203698198on_val @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_625_SigmaE,axiom,
    ! [C2: produc1317171644805857433on_val,A5: set_Pr7649655576376376530on_val,B5: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ C2 @ ( produc4067394217145614850on_val @ A5 @ B5 ) )
     => ~ ! [X3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X3 @ A5 )
           => ! [Y3: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( produc4958730763592889809on_val @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_626_SigmaE,axiom,
    ! [C2: product_prod_nat_nat,A5: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( produc457027306803732586at_nat @ A5 @ B5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A5 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C2
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_627_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] :
              ( ( P2 @ X2 )
              & ( Q @ X2 @ Y5 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X2: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_628_Collect__case__prod__Sigma,axiom,
    ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > produc9003781588162623004on_val > $o] :
      ( ( collec7293267056801676193on_val
        @ ( produc3458391012123660517_val_o
          @ ^ [X2: product_prod_nat_nat,Y5: produc9003781588162623004on_val] :
              ( ( P2 @ X2 )
              & ( Q @ X2 @ Y5 ) ) ) )
      = ( produc7035663349334832981on_val @ ( collec3392354462482085612at_nat @ P2 )
        @ ^ [X2: product_prod_nat_nat] : ( collec7475294502966882545on_val @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_629_Collect__case__prod__Sigma,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > product_prod_nat_nat > $o] :
      ( ( collec927009928073930287at_nat
        @ ( produc8420946574280188703_nat_o
          @ ^ [X2: produc9003781588162623004on_val,Y5: product_prod_nat_nat] :
              ( ( P2 @ X2 )
              & ( Q @ X2 @ Y5 ) ) ) )
      = ( produc8411625936386031515at_nat @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [X2: produc9003781588162623004on_val] : ( collec3392354462482085612at_nat @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_630_Collect__case__prod__Sigma,axiom,
    ! [P2: produc9003781588162623004on_val > $o,Q: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( collec5166231387257825412on_val
        @ ( produc1845727980445322802_val_o
          @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] :
              ( ( P2 @ X2 )
              & ( Q @ X2 @ Y5 ) ) ) )
      = ( produc4067394217145614850on_val @ ( collec7475294502966882545on_val @ P2 )
        @ ^ [X2: produc9003781588162623004on_val] : ( collec7475294502966882545on_val @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_631_Collect__case__prod__Sigma,axiom,
    ! [P2: ( produc1457211279475724562t_char > prat ) > $o,Q: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( collec7475294502966882545on_val
        @ ( produc5248837549992124053_val_o
          @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] :
              ( ( P2 @ X2 )
              & ( Q @ X2 @ Y5 ) ) ) )
      = ( produc1366239611153933861on_val @ ( collec7093475260167521854r_prat @ P2 )
        @ ^ [X2: produc1457211279475724562t_char > prat] : ( collec8455114619259388271on_val @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_632_Collect__case__prod__Sigma,axiom,
    ! [P2: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y5: nat] :
              ( ( P2 @ X2 )
              & ( Q @ X2 @ Y5 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P2 )
        @ ^ [X2: nat] : ( collect_nat @ ( Q @ X2 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_633_mem__Times__iff,axiom,
    ! [X: produc859450856879609959at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu3: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ X ) @ A5 )
        & ( member8440522571783428010at_nat @ ( produc6408287024330202629at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_634_mem__Times__iff,axiom,
    ! [X: produc5414607237814678604on_val,A5: set_Pr1261947904930325089at_nat,B5: set_Pr7649655576376376530on_val] :
      ( ( member5049262082750528355on_val @ X
        @ ( produc7035663349334832981on_val @ A5
          @ ^ [Uu3: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc947599988113312474on_val @ X ) @ A5 )
        & ( member2879030624002546611on_val @ ( produc1696119027951677976on_val @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_635_mem__Times__iff,axiom,
    ! [X: produc8271722145941708506at_nat,A5: set_Pr7649655576376376530on_val,B5: set_Pr1261947904930325089at_nat] :
      ( ( member7906376990877558257at_nat @ X
        @ ( produc8411625936386031515at_nat @ A5
          @ ^ [Uu3: produc9003781588162623004on_val] : B5 ) )
      = ( ( member2879030624002546611on_val @ ( produc2323562575164511008at_nat @ X ) @ A5 )
        & ( member8440522571783428010at_nat @ ( produc3072081615002876510at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_636_mem__Times__iff,axiom,
    ! [X: produc1317171644805857433on_val,A5: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ X
        @ ( produc4067394217145614850on_val @ A5
          @ ^ [Uu3: produc9003781588162623004on_val] : B5 ) )
      = ( ( member2879030624002546611on_val @ ( produc3129717363260770749on_val @ X ) @ A5 )
        & ( member2879030624002546611on_val @ ( produc437328469718437375on_val @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_637_mem__Times__iff,axiom,
    ! [X: product_prod_nat_nat,A5: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu3: nat] : B5 ) )
      = ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A5 )
        & ( member_nat @ ( product_snd_nat_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_638_mem__Times__iff,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr3433046674199179443r_prat,B5: set_Pr1004501353197068112on_val] :
      ( ( member2879030624002546611on_val @ X
        @ ( produc1366239611153933861on_val @ A5
          @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) )
      = ( ( member2336342582916103932r_prat @ ( produc5239619271231976746on_val @ X ) @ A5 )
        & ( member1577473829074448177on_val @ ( produc2319605628138516840on_val @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_639_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
          @ ^ [Uu3: produc1457211279475724562t_char > option_val] : B5 ) )
      = ( produc1366239611153933861on_val @ B5
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : A5 ) ) ).

% swap_product
thf(fact_640_swap__product,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( image_7325970737918433093on_val
        @ ( produc7846207541802040627on_val
          @ ^ [I: produc9003781588162623004on_val,J: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ J @ I ) )
        @ ( produc4067394217145614850on_val @ A5
          @ ^ [Uu3: produc9003781588162623004on_val] : B5 ) )
      = ( produc4067394217145614850on_val @ B5
        @ ^ [Uu3: produc9003781588162623004on_val] : A5 ) ) ).

% swap_product
thf(fact_641_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
          @ ^ [Uu3: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ B5
        @ ^ [Uu3: nat] : A5 ) ) ).

% swap_product
thf(fact_642_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
          @ ^ [Uu3: nat] : B5 ) ) ) ).

% member_product
thf(fact_643_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
          @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) ) ) ).

% member_product
thf(fact_644_snd__image__times,axiom,
    ! [A5: set_Pr3433046674199179443r_prat,B5: set_Pr1004501353197068112on_val] :
      ( ( ( A5 = bot_bo3164680191535668511r_prat )
       => ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) )
          = bot_bo5671083314777306084on_val ) )
      & ( ( A5 != bot_bo3164680191535668511r_prat )
       => ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_645_fst__image__times,axiom,
    ! [B5: set_Pr1004501353197068112on_val,A5: set_Pr3433046674199179443r_prat] :
      ( ( ( B5 = bot_bo5671083314777306084on_val )
       => ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) )
          = bot_bo3164680191535668511r_prat ) )
      & ( ( B5 != bot_bo5671083314777306084on_val )
       => ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val
            @ ( produc1366239611153933861on_val @ A5
              @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) )
          = A5 ) ) ) ).

% fst_image_times
thf(fact_646_inv__image__def,axiom,
    ( inv_im7322137579843302848al_nat
    = ( ^ [R3: set_Pr4305388979797279737on_val,F2: nat > produc9003781588162623004on_val] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F2 @ X2 ) @ ( F2 @ Y5 ) ) @ R3 ) ) ) ) ) ).

% inv_image_def
thf(fact_647_inv__image__def,axiom,
    ( inv_image_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,F2: nat > nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X2 ) @ ( F2 @ Y5 ) ) @ R3 ) ) ) ) ) ).

% inv_image_def
thf(fact_648_in__inv__image,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( inv_im8279329903829788684on_val @ R @ F ) )
      = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_649_in__inv__image,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R: set_Pr1261947904930325089at_nat,F: produc9003781588162623004on_val > nat] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( inv_im1619804204812490944on_val @ R @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_650_in__inv__image,axiom,
    ! [X: nat,Y: nat,R: set_Pr4305388979797279737on_val,F: nat > produc9003781588162623004on_val] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im7322137579843302848al_nat @ R @ F ) )
      = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).

% in_inv_image
thf(fact_651_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_652_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec7475294502966882545on_val
            @ ( produc5248837549992124053_val_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] : P2 ) )
          = top_to1561438921946393474on_val ) )
      & ( ~ P2
       => ( ( collec7475294502966882545on_val
            @ ( produc5248837549992124053_val_o
              @ ^ [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] : P2 ) )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Collect_const_case_prod
thf(fact_653_Collect__const__case__prod,axiom,
    ! [P2: $o] :
      ( ( P2
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B4: nat] : P2 ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P2
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B4: nat] : P2 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_654_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_655_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_656_these__empty__eq,axiom,
    ! [B5: set_option_val] :
      ( ( ( these_val @ B5 )
        = bot_bot_set_val )
      = ( ( B5 = bot_bo5906790151302476563on_val )
        | ( B5
          = ( insert_option_val @ none_val @ bot_bo5906790151302476563on_val ) ) ) ) ).

% these_empty_eq
thf(fact_657_these__empty__eq,axiom,
    ! [B5: set_option_state] :
      ( ( ( these_state @ B5 )
        = bot_bot_set_state )
      = ( ( B5 = bot_bo710180891245420500_state )
        | ( B5
          = ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_empty_eq
thf(fact_658_these__not__empty__eq,axiom,
    ! [B5: set_option_val] :
      ( ( ( these_val @ B5 )
       != bot_bot_set_val )
      = ( ( B5 != bot_bo5906790151302476563on_val )
        & ( B5
         != ( insert_option_val @ none_val @ bot_bo5906790151302476563on_val ) ) ) ) ).

% these_not_empty_eq
thf(fact_659_these__not__empty__eq,axiom,
    ! [B5: set_option_state] :
      ( ( ( these_state @ B5 )
       != bot_bot_set_state )
      = ( ( B5 != bot_bo710180891245420500_state )
        & ( B5
         != ( insert_option_state @ none_state @ bot_bo710180891245420500_state ) ) ) ) ).

% these_not_empty_eq
thf(fact_660_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
        @ ^ [X2: produc1457211279475724562t_char > prat] :
            ( ( member2336342582916103932r_prat @ X2 @ A5 )
            & ( ( B5 @ X2 )
             != bot_bo5671083314777306084on_val ) ) ) ) ).

% fst_image_Sigma
thf(fact_661_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_662_prod__set__simps_I2_J,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( basic_2713455076908730111on_val @ ( produc4958730763592889809on_val @ X @ Y ) )
      = ( insert3030518898131756108on_val @ Y @ bot_bo6722054719108133734on_val ) ) ).

% prod_set_simps(2)
thf(fact_663_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_664_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_665_prod__set__simps_I1_J,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( basic_2381348828333783745on_val @ ( produc4958730763592889809on_val @ X @ Y ) )
      = ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) ) ).

% prod_set_simps(1)
thf(fact_666_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_667_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 )
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( insert216038912599248202on_val @ B @ B5 ) )
      = ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ A @ B )
        @ ( sup_su7506060154676209150on_val
          @ ( produc1366239611153933861on_val @ A5
            @ ^ [Uu3: produc1457211279475724562t_char > prat] : ( insert216038912599248202on_val @ B @ B5 ) )
          @ ( produc1366239611153933861on_val @ ( insert8113562578550770211r_prat @ A @ A5 )
            @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_668_insert__Times__insert,axiom,
    ! [A: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( produc4067394217145614850on_val @ ( insert3030518898131756108on_val @ A @ A5 )
        @ ^ [Uu3: produc9003781588162623004on_val] : ( insert3030518898131756108on_val @ B @ B5 ) )
      = ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B )
        @ ( sup_su3890347010539237325on_val
          @ ( produc4067394217145614850on_val @ A5
            @ ^ [Uu3: produc9003781588162623004on_val] : ( insert3030518898131756108on_val @ B @ B5 ) )
          @ ( produc4067394217145614850on_val @ ( insert3030518898131756108on_val @ A @ A5 )
            @ ^ [Uu3: produc9003781588162623004on_val] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_669_insert__Times__insert,axiom,
    ! [A: nat,A5: set_nat,B: nat,B5: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A5 )
        @ ^ [Uu3: nat] : ( insert_nat @ B @ B5 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu3: nat] : ( insert_nat @ B @ B5 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A5 )
            @ ^ [Uu3: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_670_prod__set__defs_I2_J,axiom,
    ( basic_8841519627501779048on_val
    = ( ^ [P3: produc9003781588162623004on_val] : ( insert216038912599248202on_val @ ( produc2319605628138516840on_val @ P3 ) @ bot_bo5671083314777306084on_val ) ) ) ).

% prod_set_defs(2)
thf(fact_671_fsts_Ointros,axiom,
    ! [P: produc9003781588162623004on_val] : ( member2336342582916103932r_prat @ ( produc5239619271231976746on_val @ P ) @ ( basic_7826031515394172198on_val @ P ) ) ).

% fsts.intros
thf(fact_672_fsts_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > prat,P: produc9003781588162623004on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( basic_7826031515394172198on_val @ P ) )
      = ( A
        = ( produc5239619271231976746on_val @ P ) ) ) ).

% fsts.simps
thf(fact_673_fsts_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > prat,P: produc9003781588162623004on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( basic_7826031515394172198on_val @ P ) )
     => ( A
        = ( produc5239619271231976746on_val @ P ) ) ) ).

% fsts.cases
thf(fact_674_snds_Ointros,axiom,
    ! [P: produc9003781588162623004on_val] : ( member1577473829074448177on_val @ ( produc2319605628138516840on_val @ P ) @ ( basic_8841519627501779048on_val @ P ) ) ).

% snds.intros
thf(fact_675_snds_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > option_val,P: produc9003781588162623004on_val] :
      ( ( member1577473829074448177on_val @ A @ ( basic_8841519627501779048on_val @ P ) )
      = ( A
        = ( produc2319605628138516840on_val @ P ) ) ) ).

% snds.simps
thf(fact_676_snds_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > option_val,P: produc9003781588162623004on_val] :
      ( ( member1577473829074448177on_val @ A @ ( basic_8841519627501779048on_val @ P ) )
     => ( A
        = ( produc2319605628138516840on_val @ P ) ) ) ).

% snds.cases
thf(fact_677_prod__set__defs_I1_J,axiom,
    ( basic_7826031515394172198on_val
    = ( ^ [P3: produc9003781588162623004on_val] : ( insert8113562578550770211r_prat @ ( produc5239619271231976746on_val @ P3 ) @ bot_bo3164680191535668511r_prat ) ) ) ).

% prod_set_defs(1)
thf(fact_678_Field__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( field_9010718613293825679on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( sup_su7506060154676209150on_val @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( field_9010718613293825679on_val @ R ) ) ) ).

% Field_insert
thf(fact_679_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_680_refl__on__singleton,axiom,
    ! [X: produc9003781588162623004on_val] : ( refl_o6679066329051764061on_val @ ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X @ X ) @ bot_bo402267660617762917on_val ) ) ).

% refl_on_singleton
thf(fact_681_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_682_Id__on__def,axiom,
    ( id_on_1505035389332359999on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val] :
          ( comple3779562415432662246on_val
          @ ( image_3026073218995088406on_val
            @ ^ [X2: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X2 @ X2 ) @ bot_bo402267660617762917on_val )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_683_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A6: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X2: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ bot_bo2099793752762293965at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_684_linear__order__on__singleton,axiom,
    ! [X: produc9003781588162623004on_val] : ( order_6061556925426544295on_val @ ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X @ X ) @ bot_bo402267660617762917on_val ) ) ).

% linear_order_on_singleton
thf(fact_685_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_686_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_687_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_688_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_689_FieldI2,axiom,
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 )
     => ( member8440522571783428010at_nat @ J2 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_690_FieldI2,axiom,
    ! [I2: produc9003781588162623004on_val,J2: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I2 @ J2 ) @ R2 )
     => ( member2879030624002546611on_val @ J2 @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI2
thf(fact_691_FieldI2,axiom,
    ! [I2: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 )
     => ( member_nat @ J2 @ ( field_nat @ R2 ) ) ) ).

% FieldI2
thf(fact_692_FieldI1,axiom,
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 )
     => ( member8440522571783428010at_nat @ I2 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_693_FieldI1,axiom,
    ! [I2: produc9003781588162623004on_val,J2: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I2 @ J2 ) @ R2 )
     => ( member2879030624002546611on_val @ I2 @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% FieldI1
thf(fact_694_FieldI1,axiom,
    ! [I2: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 )
     => ( member_nat @ I2 @ ( field_nat @ R2 ) ) ) ).

% FieldI1
thf(fact_695_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_696_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_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_reflI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ X3 ) @ R )
     => ( refl_o6679066329051764061on_val @ top_to1561438921946393474on_val @ R ) ) ).

% reflI
thf(fact_705_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_706_reflD,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( refl_o6679066329051764061on_val @ top_to1561438921946393474on_val @ R )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ).

% reflD
thf(fact_707_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_708_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_709_refl__on__def,axiom,
    ( refl_o6679066329051764061on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
          ( ( ord_le7224449410216889241on_val @ R3
            @ ( produc4067394217145614850on_val @ A6
              @ ^ [Uu3: produc9003781588162623004on_val] : A6 ) )
          & ! [X2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X2 @ A6 )
             => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ X2 ) @ R3 ) ) ) ) ) ).

% refl_on_def
thf(fact_710_refl__on__def,axiom,
    ( refl_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ R3
            @ ( produc457027306803732586at_nat @ A6
              @ ^ [Uu3: nat] : A6 ) )
          & ! [X2: nat] :
              ( ( member_nat @ X2 @ A6 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R3 ) ) ) ) ) ).

% refl_on_def
thf(fact_711_refl__onI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu3: 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_712_refl__onI,axiom,
    ! [R: set_Pr4305388979797279737on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( ord_le7224449410216889241on_val @ R
        @ ( produc4067394217145614850on_val @ A5
          @ ^ [Uu3: 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_713_refl__onI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A5: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu3: 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_714_Sigma__def,axiom,
    ( produc1366239611153933861on_val
    = ( ^ [A6: set_Pr3433046674199179443r_prat,B7: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
          ( comple5258588689616361253on_val
          @ ( image_3306202642189553376on_val
            @ ^ [X2: produc1457211279475724562t_char > prat] :
                ( comple5258588689616361253on_val
                @ ( image_8136294080383932403on_val
                  @ ^ [Y5: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ bot_bo6722054719108133734on_val )
                  @ ( B7 @ X2 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_715_Sigma__def,axiom,
    ( produc4067394217145614850on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,B7: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
          ( comple3779562415432662246on_val
          @ ( image_3026073218995088406on_val
            @ ^ [X2: produc9003781588162623004on_val] :
                ( comple3779562415432662246on_val
                @ ( image_3026073218995088406on_val
                  @ ^ [Y5: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ bot_bo402267660617762917on_val )
                  @ ( B7 @ X2 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_716_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A6: set_nat,B7: nat > set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X2: nat] :
                ( comple5685304695842803022at_nat
                @ ( image_7178329752028323786at_nat
                  @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ bot_bo2099793752762293965at_nat )
                  @ ( B7 @ X2 ) ) )
            @ A6 ) ) ) ) ).

% Sigma_def
thf(fact_717_cofinal__def,axiom,
    ( bNF_Ca4299268644204761867on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ ( field_9010718613293825679on_val @ R3 ) )
         => ? [Y5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y5 @ A6 )
              & ( X2 != Y5 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_718_cofinal__def,axiom,
    ( bNF_Ca5159621140400407487al_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ ( field_nat @ R3 ) )
         => ? [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
              & ( X2 != Y5 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% cofinal_def
thf(fact_719_relation__of__def,axiom,
    ( order_8899622344125824590on_val
    = ( ^ [P4: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A6: set_Pr7649655576376376530on_val] :
          ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ^ [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
                ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 )
                  @ ( produc4067394217145614850on_val @ A6
                    @ ^ [Uu3: produc9003781588162623004on_val] : A6 ) )
                & ( P4 @ A2 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_720_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P4: nat > nat > $o,A6: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B4: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 )
                  @ ( produc457027306803732586at_nat @ A6
                    @ ^ [Uu3: nat] : A6 ) )
                & ( P4 @ A2 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_721_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] :
              ( ( B4 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_722_aboveS__def,axiom,
    ( order_86566288019506545on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A2: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] :
              ( ( B4 != A2 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_723_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B4: nat] :
              ( ( B4 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_724_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( order_2579646547231075838on_val @ ( field_9010718613293825679on_val @ R ) @ R )
        = ( ! [A6: set_Pr7649655576376376530on_val] :
              ( ( ord_le6900622908513426738on_val @ A6 @ ( field_9010718613293825679on_val @ R ) )
             => ( ( A6 != bot_bo6722054719108133734on_val )
               => ? [X2: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ X2 @ A6 )
                    & ! [Y5: produc9003781588162623004on_val] :
                        ( ( member2879030624002546611on_val @ Y5 @ A6 )
                       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_725_Linear__order__Well__order__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
        = ( ! [A6: set_nat] :
              ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R ) )
             => ( ( A6 != bot_bot_set_nat )
               => ? [X2: nat] :
                    ( ( member_nat @ X2 @ A6 )
                    & ! [Y5: nat] :
                        ( ( member_nat @ Y5 @ A6 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_726_well__order__on__domain,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( order_8331247224601271455at_nat @ A5 @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
       => ( ( member8440522571783428010at_nat @ A @ A5 )
          & ( member8440522571783428010at_nat @ B @ A5 ) ) ) ) ).

% well_order_on_domain
thf(fact_727_well__order__on__domain,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_2579646547231075838on_val @ A5 @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ( member2879030624002546611on_val @ A @ A5 )
          & ( member2879030624002546611on_val @ B @ A5 ) ) ) ) ).

% well_order_on_domain
thf(fact_728_well__order__on__domain,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_2888998067076097458on_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
       => ( ( member_nat @ A @ A5 )
          & ( member_nat @ B @ A5 ) ) ) ) ).

% well_order_on_domain
thf(fact_729_underS__incl__iff,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( ord_le3146513528884898305at_nat @ ( order_4046151456452760519at_nat @ R @ A ) @ ( order_4046151456452760519at_nat @ R @ B ) )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_730_underS__incl__iff,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ord_le6900622908513426738on_val @ ( order_8621009067429368022on_val @ R @ A ) @ ( order_8621009067429368022on_val @ R @ B ) )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_731_underS__incl__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).

% underS_incl_iff
thf(fact_732_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_733_Range__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( range_6350497433844811695on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( insert3030518898131756108on_val @ B @ ( range_6350497433844811695on_val @ R ) ) ) ).

% Range_insert
thf(fact_734_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_735_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_736_Domain__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( domain6215983837468562630on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( insert3030518898131756108on_val @ A @ ( domain6215983837468562630on_val @ R ) ) ) ).

% Domain_insert
thf(fact_737_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_738_Range__iff,axiom,
    ! [A: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ A @ ( range_8542210083549373368on_val @ R ) )
      = ( ? [Y5: produc1457211279475724562t_char > prat] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Y5 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_739_Range__iff,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( range_6350497433844811695on_val @ R ) )
      = ( ? [Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_740_Range__iff,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R ) )
      = ( ? [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ A ) @ R ) ) ) ).

% Range_iff
thf(fact_741_RangeE,axiom,
    ! [B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ B @ ( range_8542210083549373368on_val @ R ) )
     => ~ ! [A4: produc1457211279475724562t_char > prat] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A4 @ B ) @ R ) ) ).

% RangeE
thf(fact_742_RangeE,axiom,
    ! [B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ B @ ( range_6350497433844811695on_val @ R ) )
     => ~ ! [A4: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ B ) @ R ) ) ).

% RangeE
thf(fact_743_RangeE,axiom,
    ! [B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ B @ ( range_nat_nat @ R ) )
     => ~ ! [A4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B ) @ R ) ) ).

% RangeE
thf(fact_744_Range_Ointros,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( member1577473829074448177on_val @ B @ ( range_8542210083549373368on_val @ R ) ) ) ).

% Range.intros
thf(fact_745_Range_Ointros,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( member2879030624002546611on_val @ B @ ( range_6350497433844811695on_val @ R ) ) ) ).

% Range.intros
thf(fact_746_Range_Ointros,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member_nat @ B @ ( range_nat_nat @ R ) ) ) ).

% Range.intros
thf(fact_747_Range_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ A @ ( range_8542210083549373368on_val @ R ) )
      = ( ? [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
            ( ( A = B4 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A2 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_748_Range_Osimps,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( range_6350497433844811695on_val @ R ) )
      = ( ? [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
            ( ( A = B4 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_749_Range_Osimps,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R ) )
      = ( ? [A2: nat,B4: nat] :
            ( ( A = B4 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R ) ) ) ) ).

% Range.simps
thf(fact_750_Range_Ocases,axiom,
    ! [A: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member1577473829074448177on_val @ A @ ( range_8542210083549373368on_val @ R ) )
     => ~ ! [A4: produc1457211279475724562t_char > prat] :
            ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A4 @ A ) @ R ) ) ).

% Range.cases
thf(fact_751_Range_Ocases,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( range_6350497433844811695on_val @ R ) )
     => ~ ! [A4: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ A ) @ R ) ) ).

% Range.cases
thf(fact_752_Range_Ocases,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R ) )
     => ~ ! [A4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A ) @ R ) ) ).

% Range.cases
thf(fact_753_Domain__iff,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
      = ( ? [Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ Y5 ) @ R ) ) ) ).

% Domain_iff
thf(fact_754_Domain__iff,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
      = ( ? [Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ Y5 ) @ R ) ) ) ).

% Domain_iff
thf(fact_755_Domain__iff,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R ) )
      = ( ? [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y5 ) @ R ) ) ) ).

% Domain_iff
thf(fact_756_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_757_DomainE,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
     => ~ ! [B3: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B3 ) @ R ) ) ).

% DomainE
thf(fact_758_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_759_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_760_Domain_ODomainI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) ) ) ).

% Domain.DomainI
thf(fact_761_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_762_Domain_Osimps,axiom,
    ! [A: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member2336342582916103932r_prat @ A @ ( domain6100248181835736161on_val @ R ) )
      = ( ? [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
            ( ( A = A2 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A2 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_763_Domain_Osimps,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
      = ( ? [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
            ( ( A = A2 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_764_Domain_Osimps,axiom,
    ! [A: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R ) )
      = ( ? [A2: nat,B4: nat] :
            ( ( A = A2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R ) ) ) ) ).

% Domain.simps
thf(fact_765_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_766_Domain_Ocases,axiom,
    ! [A: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member2879030624002546611on_val @ A @ ( domain6215983837468562630on_val @ R ) )
     => ~ ! [B3: produc9003781588162623004on_val] :
            ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B3 ) @ R ) ) ).

% Domain.cases
thf(fact_767_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_768_underS__E,axiom,
    ! [I2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,J2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ I2 @ ( order_4046151456452760519at_nat @ R2 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 ) ) ) ).

% underS_E
thf(fact_769_underS__E,axiom,
    ! [I2: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,J2: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ I2 @ ( order_8621009067429368022on_val @ R2 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I2 @ J2 ) @ R2 ) ) ) ).

% underS_E
thf(fact_770_underS__E,axiom,
    ! [I2: nat,R2: set_Pr1261947904930325089at_nat,J2: nat] :
      ( ( member_nat @ I2 @ ( order_underS_nat @ R2 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 ) ) ) ).

% underS_E
thf(fact_771_underS__I,axiom,
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( I2 != J2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 )
       => ( member8440522571783428010at_nat @ I2 @ ( order_4046151456452760519at_nat @ R2 @ J2 ) ) ) ) ).

% underS_I
thf(fact_772_underS__I,axiom,
    ! [I2: produc9003781588162623004on_val,J2: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( I2 != J2 )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I2 @ J2 ) @ R2 )
       => ( member2879030624002546611on_val @ I2 @ ( order_8621009067429368022on_val @ R2 @ J2 ) ) ) ) ).

% underS_I
thf(fact_773_underS__I,axiom,
    ! [I2: nat,J2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( I2 != J2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 )
       => ( member_nat @ I2 @ ( order_underS_nat @ R2 @ J2 ) ) ) ) ).

% underS_I
thf(fact_774_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,J2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ I2 @ ( order_4046151456452760519at_nat @ R2 @ J2 ) )
     => ( member8440522571783428010at_nat @ I2 @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_775_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [I2: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,J2: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ I2 @ ( order_8621009067429368022on_val @ R2 @ J2 ) )
     => ( member2879030624002546611on_val @ I2 @ ( field_9010718613293825679on_val @ R2 ) ) ) ).

% BNF_Least_Fixpoint.underS_Field
thf(fact_776_underS__def,axiom,
    ( order_4046151456452760519at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] :
              ( ( B4 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ A2 ) @ R3 ) ) ) ) ) ).

% underS_def
thf(fact_777_underS__def,axiom,
    ( order_8621009067429368022on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A2: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] :
              ( ( B4 != A2 )
              & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ A2 ) @ R3 ) ) ) ) ) ).

% underS_def
thf(fact_778_underS__def,axiom,
    ( order_underS_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B4: nat] :
              ( ( B4 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A2 ) @ R3 ) ) ) ) ) ).

% underS_def
thf(fact_779_fst__eq__Domain,axiom,
    ! [R2: set_Pr7649655576376376530on_val] :
      ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ R2 )
      = ( domain6100248181835736161on_val @ R2 ) ) ).

% fst_eq_Domain
thf(fact_780_Domain__fst,axiom,
    ( domain6100248181835736161on_val
    = ( image_3067683418878703216r_prat @ produc5239619271231976746on_val ) ) ).

% Domain_fst
thf(fact_781_snd__eq__Range,axiom,
    ! [R2: set_Pr7649655576376376530on_val] :
      ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ R2 )
      = ( range_8542210083549373368on_val @ R2 ) ) ).

% snd_eq_Range
thf(fact_782_Range__snd,axiom,
    ( range_8542210083549373368on_val
    = ( image_2330660378567450301on_val @ produc2319605628138516840on_val ) ) ).

% Range_snd
thf(fact_783_graph__map__upd,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val] :
      ( ( graph_1288227317702120175on_val @ ( fun_up1601731230164360576on_val @ M @ K @ ( some_P7669353234731512949on_val @ V ) ) )
      = ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ ( fun_up1601731230164360576on_val @ M @ K @ none_P1334921381875262201on_val ) ) ) ) ).

% graph_map_upd
thf(fact_784_graph__map__upd,axiom,
    ! [M: produc9003781588162623004on_val > option8887860577033306220on_val,K: produc9003781588162623004on_val,V: produc9003781588162623004on_val] :
      ( ( graph_6108330199704490680on_val @ ( fun_up273046073882939271on_val @ M @ K @ ( some_P2581650809551911671on_val @ V ) ) )
      = ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ ( fun_up273046073882939271on_val @ M @ K @ none_P8728355543607141243on_val ) ) ) ) ).

% graph_map_upd
thf(fact_785_graph__map__upd,axiom,
    ! [M: nat > option_nat,K: nat,V: nat] :
      ( ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ ( some_nat @ V ) ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ none_nat ) ) ) ) ).

% graph_map_upd
thf(fact_786_Rangep__Range__eq,axiom,
    ! [R: set_Pr7649655576376376530on_val] :
      ( ( rangep8247911282204559858on_val
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: produc1457211279475724562t_char > option_val] : ( member1577473829074448177on_val @ X2 @ ( range_8542210083549373368on_val @ R ) ) ) ) ).

% Rangep_Range_eq
thf(fact_787_Rangep__Range__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( rangep745334838155002613on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X2 @ ( range_6350497433844811695on_val @ R ) ) ) ) ).

% Rangep_Range_eq
thf(fact_788_Rangep__Range__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( rangep_nat_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: nat] : ( member_nat @ X2 @ ( range_nat_nat @ R ) ) ) ) ).

% Rangep_Range_eq
thf(fact_789_Range__def,axiom,
    ( range_8542210083549373368on_val
    = ( ^ [R3: set_Pr7649655576376376530on_val] :
          ( collec8455114619259388271on_val
          @ ( rangep8247911282204559858on_val
            @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% Range_def
thf(fact_790_Range__def,axiom,
    ( range_6350497433844811695on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
          ( collec7475294502966882545on_val
          @ ( rangep745334838155002613on_val
            @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% Range_def
thf(fact_791_Range__def,axiom,
    ( range_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ( rangep_nat_nat
            @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% Range_def
thf(fact_792_fun__upd__None__if__notin__dom,axiom,
    ! [K: product_prod_nat_nat,M: product_prod_nat_nat > option_val] :
      ( ~ ( member8440522571783428010at_nat @ K @ ( dom_Pr1716353499919422226at_val @ M ) )
     => ( ( fun_up2801401594293523459on_val @ M @ K @ none_val )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_793_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option_val] :
      ( ~ ( member2879030624002546611on_val @ K @ ( dom_Pr7381289304162206737al_val @ M ) )
     => ( ( fun_up8559965269643824108on_val @ M @ K @ none_val )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_794_fun__upd__None__if__notin__dom,axiom,
    ! [K: product_prod_nat_nat,M: product_prod_nat_nat > option_state] :
      ( ~ ( member8440522571783428010at_nat @ K @ ( dom_Pr7687107951487557865_state @ M ) )
     => ( ( fun_up2914124688375849550_state @ M @ K @ none_state )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_795_fun__upd__None__if__notin__dom,axiom,
    ! [K: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option_state] :
      ( ~ ( member2879030624002546611on_val @ K @ ( dom_Pr4448331044451638570_state @ M ) )
     => ( ( fun_up6681822530636708645_state @ M @ K @ none_state )
        = M ) ) ).

% fun_upd_None_if_notin_dom
thf(fact_796_domIff,axiom,
    ! [A: product_prod_nat_nat,M: product_prod_nat_nat > option_val] :
      ( ( member8440522571783428010at_nat @ A @ ( dom_Pr1716353499919422226at_val @ M ) )
      = ( ( M @ A )
       != none_val ) ) ).

% domIff
thf(fact_797_domIff,axiom,
    ! [A: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option_val] :
      ( ( member2879030624002546611on_val @ A @ ( dom_Pr7381289304162206737al_val @ M ) )
      = ( ( M @ A )
       != none_val ) ) ).

% domIff
thf(fact_798_domIff,axiom,
    ! [A: product_prod_nat_nat,M: product_prod_nat_nat > option_state] :
      ( ( member8440522571783428010at_nat @ A @ ( dom_Pr7687107951487557865_state @ M ) )
      = ( ( M @ A )
       != none_state ) ) ).

% domIff
thf(fact_799_domIff,axiom,
    ! [A: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option_state] :
      ( ( member2879030624002546611on_val @ A @ ( dom_Pr4448331044451638570_state @ M ) )
      = ( ( M @ A )
       != none_state ) ) ).

% domIff
thf(fact_800_graph__domD,axiom,
    ! [X: product_prod_nat_nat,M: nat > option_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( graph_nat_nat @ M ) )
     => ( member_nat @ ( product_fst_nat_nat @ X ) @ ( dom_nat_nat @ M ) ) ) ).

% graph_domD
thf(fact_801_graph__domD,axiom,
    ! [X: produc9003781588162623004on_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
      ( ( member2879030624002546611on_val @ X @ ( graph_1288227317702120175on_val @ M ) )
     => ( member2336342582916103932r_prat @ ( produc5239619271231976746on_val @ X ) @ ( dom_Pr4152820065153570843on_val @ M ) ) ) ).

% graph_domD
thf(fact_802_fst__graph__eq__dom,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
      ( ( image_3067683418878703216r_prat @ produc5239619271231976746on_val @ ( graph_1288227317702120175on_val @ M ) )
      = ( dom_Pr4152820065153570843on_val @ M ) ) ).

% fst_graph_eq_dom
thf(fact_803_dom__def,axiom,
    ( dom_Pr1716353499919422226at_val
    = ( ^ [M2: product_prod_nat_nat > option_val] :
          ( collec3392354462482085612at_nat
          @ ^ [A2: product_prod_nat_nat] :
              ( ( M2 @ A2 )
             != none_val ) ) ) ) ).

% dom_def
thf(fact_804_dom__def,axiom,
    ( dom_Pr7381289304162206737al_val
    = ( ^ [M2: produc9003781588162623004on_val > option_val] :
          ( collec7475294502966882545on_val
          @ ^ [A2: produc9003781588162623004on_val] :
              ( ( M2 @ A2 )
             != none_val ) ) ) ) ).

% dom_def
thf(fact_805_dom__def,axiom,
    ( dom_Pr7687107951487557865_state
    = ( ^ [M2: product_prod_nat_nat > option_state] :
          ( collec3392354462482085612at_nat
          @ ^ [A2: product_prod_nat_nat] :
              ( ( M2 @ A2 )
             != none_state ) ) ) ) ).

% dom_def
thf(fact_806_dom__def,axiom,
    ( dom_Pr4448331044451638570_state
    = ( ^ [M2: produc9003781588162623004on_val > option_state] :
          ( collec7475294502966882545on_val
          @ ^ [A2: produc9003781588162623004on_val] :
              ( ( M2 @ A2 )
             != none_state ) ) ) ) ).

% dom_def
thf(fact_807_graph__eq__to__snd__dom,axiom,
    ( graph_1288227317702120175on_val
    = ( ^ [M2: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
          ( image_1459058096221225002on_val
          @ ^ [X2: produc1457211279475724562t_char > prat] : ( produc74620467203698198on_val @ X2 @ ( the_Pr646328391962797958on_val @ ( M2 @ X2 ) ) )
          @ ( dom_Pr4152820065153570843on_val @ M2 ) ) ) ) ).

% graph_eq_to_snd_dom
thf(fact_808_graph__eq__to__snd__dom,axiom,
    ( graph_6108330199704490680on_val
    = ( ^ [M2: produc9003781588162623004on_val > option8887860577033306220on_val] :
          ( image_4249794487667743926on_val
          @ ^ [X2: produc9003781588162623004on_val] : ( produc4958730763592889809on_val @ X2 @ ( the_Pr3639693038709580936on_val @ ( M2 @ X2 ) ) )
          @ ( dom_Pr2742230247756046988on_val @ M2 ) ) ) ) ).

% graph_eq_to_snd_dom
thf(fact_809_graph__eq__to__snd__dom,axiom,
    ( graph_nat_nat
    = ( ^ [M2: nat > option_nat] :
          ( image_5846123807819985514at_nat
          @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ ( the_nat @ ( M2 @ X2 ) ) )
          @ ( dom_nat_nat @ M2 ) ) ) ) ).

% graph_eq_to_snd_dom
thf(fact_810_in__graphD,axiom,
    ! [K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ M ) )
     => ( ( M @ K )
        = ( some_P7669353234731512949on_val @ V ) ) ) ).

% in_graphD
thf(fact_811_in__graphD,axiom,
    ! [K: produc9003781588162623004on_val,V: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option8887860577033306220on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ M ) )
     => ( ( M @ K )
        = ( some_P2581650809551911671on_val @ V ) ) ) ).

% in_graphD
thf(fact_812_in__graphD,axiom,
    ! [K: nat,V: nat,M: nat > option_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ M ) )
     => ( ( M @ K )
        = ( some_nat @ V ) ) ) ).

% in_graphD
thf(fact_813_in__graphI,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val] :
      ( ( ( M @ K )
        = ( some_P7669353234731512949on_val @ V ) )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ M ) ) ) ).

% in_graphI
thf(fact_814_in__graphI,axiom,
    ! [M: produc9003781588162623004on_val > option8887860577033306220on_val,K: produc9003781588162623004on_val,V: produc9003781588162623004on_val] :
      ( ( ( M @ K )
        = ( some_P2581650809551911671on_val @ V ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ M ) ) ) ).

% in_graphI
thf(fact_815_in__graphI,axiom,
    ! [M: nat > option_nat,K: nat,V: nat] :
      ( ( ( M @ K )
        = ( some_nat @ V ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ M ) ) ) ).

% in_graphI
thf(fact_816_graph__ranD,axiom,
    ! [X: product_prod_nat_nat,M: nat > option_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( graph_nat_nat @ M ) )
     => ( member_nat @ ( product_snd_nat_nat @ X ) @ ( ran_nat_nat @ M ) ) ) ).

% graph_ranD
thf(fact_817_graph__ranD,axiom,
    ! [X: produc9003781588162623004on_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
      ( ( member2879030624002546611on_val @ X @ ( graph_1288227317702120175on_val @ M ) )
     => ( member1577473829074448177on_val @ ( produc2319605628138516840on_val @ X ) @ ( ran_Pr878516737976613822on_val @ M ) ) ) ).

% graph_ranD
thf(fact_818_snd__graph__ran,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val] :
      ( ( image_2330660378567450301on_val @ produc2319605628138516840on_val @ ( graph_1288227317702120175on_val @ M ) )
      = ( ran_Pr878516737976613822on_val @ M ) ) ).

% snd_graph_ran
thf(fact_819_graph__fun__upd__None,axiom,
    ! [M: nat > option_nat,K: nat] :
      ( ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ none_nat ) )
      = ( collec3392354462482085612at_nat
        @ ^ [E: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ E @ ( graph_nat_nat @ M ) )
            & ( ( product_fst_nat_nat @ E )
             != K ) ) ) ) ).

% graph_fun_upd_None
thf(fact_820_graph__fun__upd__None,axiom,
    ! [M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,K: produc1457211279475724562t_char > prat] :
      ( ( graph_1288227317702120175on_val @ ( fun_up1601731230164360576on_val @ M @ K @ none_P1334921381875262201on_val ) )
      = ( collec7475294502966882545on_val
        @ ^ [E: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ E @ ( graph_1288227317702120175on_val @ M ) )
            & ( ( produc5239619271231976746on_val @ E )
             != K ) ) ) ) ).

% graph_fun_upd_None
thf(fact_821_finite__Map__induct,axiom,
    ! [M: product_prod_nat_nat > option_val,P2: ( product_prod_nat_nat > option_val ) > $o] :
      ( ( finite6177210948735845034at_nat @ ( dom_Pr1716353499919422226at_val @ M ) )
     => ( ( P2
          @ ^ [X2: product_prod_nat_nat] : none_val )
       => ( ! [K2: product_prod_nat_nat,V2: val,M3: product_prod_nat_nat > option_val] :
              ( ( finite6177210948735845034at_nat @ ( dom_Pr1716353499919422226at_val @ M3 ) )
             => ( ~ ( member8440522571783428010at_nat @ K2 @ ( dom_Pr1716353499919422226at_val @ M3 ) )
               => ( ( P2 @ M3 )
                 => ( P2 @ ( fun_up2801401594293523459on_val @ M3 @ K2 @ ( some_val @ V2 ) ) ) ) ) )
         => ( P2 @ M ) ) ) ) ).

% finite_Map_induct
thf(fact_822_finite__Map__induct,axiom,
    ! [M: produc9003781588162623004on_val > option_val,P2: ( produc9003781588162623004on_val > option_val ) > $o] :
      ( ( finite8324237431073730739on_val @ ( dom_Pr7381289304162206737al_val @ M ) )
     => ( ( P2
          @ ^ [X2: produc9003781588162623004on_val] : none_val )
       => ( ! [K2: produc9003781588162623004on_val,V2: val,M3: produc9003781588162623004on_val > option_val] :
              ( ( finite8324237431073730739on_val @ ( dom_Pr7381289304162206737al_val @ M3 ) )
             => ( ~ ( member2879030624002546611on_val @ K2 @ ( dom_Pr7381289304162206737al_val @ M3 ) )
               => ( ( P2 @ M3 )
                 => ( P2 @ ( fun_up8559965269643824108on_val @ M3 @ K2 @ ( some_val @ V2 ) ) ) ) ) )
         => ( P2 @ M ) ) ) ) ).

% finite_Map_induct
thf(fact_823_finite__Map__induct,axiom,
    ! [M: product_prod_nat_nat > option_state,P2: ( product_prod_nat_nat > option_state ) > $o] :
      ( ( finite6177210948735845034at_nat @ ( dom_Pr7687107951487557865_state @ M ) )
     => ( ( P2
          @ ^ [X2: product_prod_nat_nat] : none_state )
       => ( ! [K2: product_prod_nat_nat,V2: state,M3: product_prod_nat_nat > option_state] :
              ( ( finite6177210948735845034at_nat @ ( dom_Pr7687107951487557865_state @ M3 ) )
             => ( ~ ( member8440522571783428010at_nat @ K2 @ ( dom_Pr7687107951487557865_state @ M3 ) )
               => ( ( P2 @ M3 )
                 => ( P2 @ ( fun_up2914124688375849550_state @ M3 @ K2 @ ( some_state @ V2 ) ) ) ) ) )
         => ( P2 @ M ) ) ) ) ).

% finite_Map_induct
thf(fact_824_finite__Map__induct,axiom,
    ! [M: produc9003781588162623004on_val > option_state,P2: ( produc9003781588162623004on_val > option_state ) > $o] :
      ( ( finite8324237431073730739on_val @ ( dom_Pr4448331044451638570_state @ M ) )
     => ( ( P2
          @ ^ [X2: produc9003781588162623004on_val] : none_state )
       => ( ! [K2: produc9003781588162623004on_val,V2: state,M3: produc9003781588162623004on_val > option_state] :
              ( ( finite8324237431073730739on_val @ ( dom_Pr4448331044451638570_state @ M3 ) )
             => ( ~ ( member2879030624002546611on_val @ K2 @ ( dom_Pr4448331044451638570_state @ M3 ) )
               => ( ( P2 @ M3 )
                 => ( P2 @ ( fun_up6681822530636708645_state @ M3 @ K2 @ ( some_state @ V2 ) ) ) ) ) )
         => ( P2 @ M ) ) ) ) ).

% finite_Map_induct
thf(fact_825_wf__map__prod__image__Dom__Ran,axiom,
    ! [R: set_Pr8693737435421807431at_nat,F: product_prod_nat_nat > nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R )
     => ( ! [A4: product_prod_nat_nat,A8: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A4 @ ( domain2386162569883866188at_nat @ R ) )
           => ( ( member8440522571783428010at_nat @ A8 @ ( range_4455291928337903541at_nat @ R ) )
             => ( ( ( F @ A4 )
                  = ( F @ A8 ) )
               => ( A4 = A8 ) ) ) )
       => ( wf_nat @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F @ F ) @ R ) ) ) ) ).

% wf_map_prod_image_Dom_Ran
thf(fact_826_wf__map__prod__image__Dom__Ran,axiom,
    ! [R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > nat] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ! [A4: produc9003781588162623004on_val,A8: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ A4 @ ( domain6215983837468562630on_val @ R ) )
           => ( ( member2879030624002546611on_val @ A8 @ ( range_6350497433844811695on_val @ R ) )
             => ( ( ( F @ A4 )
                  = ( F @ A8 ) )
               => ( A4 = A8 ) ) ) )
       => ( wf_nat @ ( image_2009239599916048301at_nat @ ( produc1289258586576348626al_nat @ F @ F ) @ R ) ) ) ) ).

% wf_map_prod_image_Dom_Ran
thf(fact_827_wf__map__prod__image__Dom__Ran,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( wf_nat @ R )
     => ( ! [A4: nat,A8: nat] :
            ( ( member_nat @ A4 @ ( domain_nat_nat @ R ) )
           => ( ( member_nat @ A8 @ ( range_nat_nat @ R ) )
             => ( ( ( F @ A4 )
                  = ( F @ A8 ) )
               => ( A4 = A8 ) ) ) )
       => ( wf_nat @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F @ F ) @ R ) ) ) ) ).

% wf_map_prod_image_Dom_Ran
thf(fact_828_prod_Orel__compp__Grp,axiom,
    ( basic_7417786039168870389on_val
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R23: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
          ( relcom3521699623761503666on_val
          @ ( conver75858737322263939r_prat
            @ ( bNF_Gr2052668271496221660r_prat
              @ ( collec5166231387257825412on_val
                @ ^ [X2: produc1317171644805857433on_val] :
                    ( ( ord_le6900622908513426738on_val @ ( basic_2381348828333783745on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R12 ) ) )
                    & ( ord_le6900622908513426738on_val @ ( basic_2713455076908730111on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R23 ) ) ) ) )
              @ ( produc1002149148328127798r_prat @ produc5239619271231976746on_val @ produc5239619271231976746on_val ) ) )
          @ ( bNF_Gr5132772187013160586on_val
            @ ( collec5166231387257825412on_val
              @ ^ [X2: produc1317171644805857433on_val] :
                  ( ( ord_le6900622908513426738on_val @ ( basic_2381348828333783745on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R12 ) ) )
                  & ( ord_le6900622908513426738on_val @ ( basic_2713455076908730111on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R23 ) ) ) ) )
            @ ( produc1134234647556835942on_val @ produc2319605628138516840on_val @ produc2319605628138516840on_val ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_829_prod_Orel__compp__Grp,axiom,
    ( basic_400063943313148570at_nat
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R23: nat > nat > $o] :
          ( relcom625270308737498028al_nat
          @ ( conver4492206232696316067at_nat
            @ ( bNF_Gr5945892418093068284at_nat
              @ ( collec927009928073930287at_nat
                @ ^ [X2: produc8271722145941708506at_nat] :
                    ( ( ord_le6900622908513426738on_val @ ( basic_2415657071401982236at_nat @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R12 ) ) )
                    & ( ord_le3146513528884898305at_nat @ ( basic_8506799250368921950at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) ) ) )
              @ ( produc8335123919915634002at_nat @ produc5239619271231976746on_val @ product_fst_nat_nat ) ) )
          @ ( bNF_Gr9125604968117925025al_nat
            @ ( collec927009928073930287at_nat
              @ ^ [X2: produc8271722145941708506at_nat] :
                  ( ( ord_le6900622908513426738on_val @ ( basic_2415657071401982236at_nat @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R12 ) ) )
                  & ( ord_le3146513528884898305at_nat @ ( basic_8506799250368921950at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) ) ) )
            @ ( produc5420676907050541347at_nat @ produc2319605628138516840on_val @ product_snd_nat_nat ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_830_prod_Orel__compp__Grp,axiom,
    ( basic_9214219793883178326on_val
    = ( ^ [R12: nat > nat > $o,R23: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
          ( relcom650432926881473428on_val
          @ ( conver2313424737844633659r_prat
            @ ( bNF_Gr3767110923241385876r_prat
              @ ( collec7293267056801676193on_val
                @ ^ [X2: produc5414607237814678604on_val] :
                    ( ( ord_le3146513528884898305at_nat @ ( basic_1039694484350783702on_val @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
                    & ( ord_le6900622908513426738on_val @ ( basic_7130836663317723416on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R23 ) ) ) ) )
              @ ( produc4911518146750293274r_prat @ product_fst_nat_nat @ produc5239619271231976746on_val ) ) )
          @ ( bNF_Gr2855355261350141231on_val
            @ ( collec7293267056801676193on_val
              @ ^ [X2: produc5414607237814678604on_val] :
                  ( ( ord_le3146513528884898305at_nat @ ( basic_1039694484350783702on_val @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
                  & ( ord_le6900622908513426738on_val @ ( basic_7130836663317723416on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R23 ) ) ) ) )
            @ ( produc3522203662121252755on_val @ product_snd_nat_nat @ produc2319605628138516840on_val ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_831_prod_Orel__compp__Grp,axiom,
    ( basic_5328504652464829177at_nat
    = ( ^ [R12: nat > nat > $o,R23: nat > nat > $o] :
          ( relcom6553484135723333994at_nat
          @ ( conver9077181057519864143at_nat
            @ ( bNF_Gr8820210792972196264at_nat
              @ ( collec7088162979684241874at_nat
                @ ^ [X2: produc859450856879609959at_nat] :
                    ( ( ord_le3146513528884898305at_nat @ ( basic_1775256158546629575at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
                    & ( ord_le3146513528884898305at_nat @ ( basic_2061583120998029061at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) ) ) )
              @ ( produc5156754000680454454at_nat @ product_fst_nat_nat @ product_fst_nat_nat ) ) )
          @ ( bNF_Gr8820210792972196264at_nat
            @ ( collec7088162979684241874at_nat
              @ ^ [X2: produc859450856879609959at_nat] :
                  ( ( ord_le3146513528884898305at_nat @ ( basic_1775256158546629575at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
                  & ( ord_le3146513528884898305at_nat @ ( basic_2061583120998029061at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) ) ) )
            @ ( produc5156754000680454454at_nat @ product_snd_nat_nat @ product_snd_nat_nat ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_832_rel__prod__inject,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R22: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,C2: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > option_val] :
      ( ( basic_1679859423659471403on_val @ R1 @ R22 @ ( produc74620467203698198on_val @ A @ B ) @ ( produc74620467203698198on_val @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_833_rel__prod__inject,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > $o,R22: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,C2: produc9003781588162623004on_val,D: produc9003781588162623004on_val] :
      ( ( basic_1859977137585984946on_val @ R1 @ R22 @ ( produc74620467203698198on_val @ A @ B ) @ ( produc4958730763592889809on_val @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_834_rel__prod__inject,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > nat > $o,R22: ( produc1457211279475724562t_char > option_val ) > nat > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,C2: nat,D: nat] :
      ( ( basic_7417112278364389146al_nat @ R1 @ R22 @ ( produc74620467203698198on_val @ A @ B ) @ ( product_Pair_nat_nat @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_835_rel__prod__inject,axiom,
    ! [R1: produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o,R22: produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,C2: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > option_val] :
      ( ( basic_1560430169329504888on_val @ R1 @ R22 @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc74620467203698198on_val @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_836_rel__prod__inject,axiom,
    ! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,C2: produc9003781588162623004on_val,D: produc9003781588162623004on_val] :
      ( ( basic_4773232560198594089on_val @ R1 @ R22 @ ( produc4958730763592889809on_val @ A @ B ) @ ( produc4958730763592889809on_val @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_837_rel__prod__inject,axiom,
    ! [R1: produc9003781588162623004on_val > nat > $o,R22: produc9003781588162623004on_val > nat > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,C2: nat,D: nat] :
      ( ( basic_2945586035346441617al_nat @ R1 @ R22 @ ( produc4958730763592889809on_val @ A @ B ) @ ( product_Pair_nat_nat @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_838_rel__prod__inject,axiom,
    ! [R1: nat > ( produc1457211279475724562t_char > prat ) > $o,R22: nat > ( produc1457211279475724562t_char > option_val ) > $o,A: nat,B: nat,C2: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > option_val] :
      ( ( basic_5770269710933094520on_val @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( produc74620467203698198on_val @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_839_rel__prod__inject,axiom,
    ! [R1: nat > produc9003781588162623004on_val > $o,R22: nat > produc9003781588162623004on_val > $o,A: nat,B: nat,C2: produc9003781588162623004on_val,D: produc9003781588162623004on_val] :
      ( ( basic_3065465552773073809on_val @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( produc4958730763592889809on_val @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_840_rel__prod__inject,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,A: nat,B: nat,C2: nat,D: nat] :
      ( ( basic_5328504652464829177at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C2 @ D ) )
      = ( ( R1 @ A @ C2 )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_841_wf__empty,axiom,
    wf_nat @ bot_bo2099793752762293965at_nat ).

% wf_empty
thf(fact_842_wf__lex__prod,axiom,
    ! [Ra2: set_Pr1261947904930325089at_nat,Rb2: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ Ra2 )
     => ( ( wf_nat @ Rb2 )
       => ( wf_Pro7803398752247294826at_nat @ ( lex_prod_nat_nat @ Ra2 @ Rb2 ) ) ) ) ).

% wf_lex_prod
thf(fact_843_wf__inv__image,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( wf_nat @ R )
     => ( wf_nat @ ( inv_image_nat_nat @ R @ F ) ) ) ).

% wf_inv_image
thf(fact_844_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_845_snd__transfer,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B5: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_re1164559784078602061on_val @ ( basic_1679859423659471403on_val @ A5 @ B5 ) @ B5 @ produc2319605628138516840on_val @ produc2319605628138516840on_val ) ).

% snd_transfer
thf(fact_846_wf__subset,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( ( ord_le3146513528884898305at_nat @ P @ R )
       => ( wf_nat @ P ) ) ) ).

% wf_subset
thf(fact_847_fst__transfer,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B5: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_re5965109370979550551r_prat @ ( basic_1679859423659471403on_val @ A5 @ B5 ) @ A5 @ produc5239619271231976746on_val @ produc5239619271231976746on_val ) ).

% fst_transfer
thf(fact_848_rel__prod_Ocases,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R22: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val] :
      ( ( basic_1679859423659471403on_val @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > option_val] :
            ( ( A12
              = ( produc74620467203698198on_val @ A4 @ C ) )
           => ! [D3: produc1457211279475724562t_char > option_val] :
                ( ( A23
                  = ( produc74620467203698198on_val @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_849_rel__prod_Ocases,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > $o,R22: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > $o,A12: produc9003781588162623004on_val,A23: produc1317171644805857433on_val] :
      ( ( basic_1859977137585984946on_val @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: produc1457211279475724562t_char > prat,B3: produc9003781588162623004on_val,C: produc1457211279475724562t_char > option_val] :
            ( ( A12
              = ( produc74620467203698198on_val @ A4 @ C ) )
           => ! [D3: produc9003781588162623004on_val] :
                ( ( A23
                  = ( produc4958730763592889809on_val @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_850_rel__prod_Ocases,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > nat > $o,R22: ( produc1457211279475724562t_char > option_val ) > nat > $o,A12: produc9003781588162623004on_val,A23: product_prod_nat_nat] :
      ( ( basic_7417112278364389146al_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: produc1457211279475724562t_char > prat,B3: nat,C: produc1457211279475724562t_char > option_val] :
            ( ( A12
              = ( produc74620467203698198on_val @ A4 @ C ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_851_rel__prod_Ocases,axiom,
    ! [R1: produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o,R22: produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o,A12: produc1317171644805857433on_val,A23: produc9003781588162623004on_val] :
      ( ( basic_1560430169329504888on_val @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: produc9003781588162623004on_val,B3: produc1457211279475724562t_char > prat,C: produc9003781588162623004on_val] :
            ( ( A12
              = ( produc4958730763592889809on_val @ A4 @ C ) )
           => ! [D3: produc1457211279475724562t_char > option_val] :
                ( ( A23
                  = ( produc74620467203698198on_val @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_852_rel__prod_Ocases,axiom,
    ! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A12: produc1317171644805857433on_val,A23: produc1317171644805857433on_val] :
      ( ( basic_4773232560198594089on_val @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,C: produc9003781588162623004on_val] :
            ( ( A12
              = ( produc4958730763592889809on_val @ A4 @ C ) )
           => ! [D3: produc9003781588162623004on_val] :
                ( ( A23
                  = ( produc4958730763592889809on_val @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_853_rel__prod_Ocases,axiom,
    ! [R1: produc9003781588162623004on_val > nat > $o,R22: produc9003781588162623004on_val > nat > $o,A12: produc1317171644805857433on_val,A23: product_prod_nat_nat] :
      ( ( basic_2945586035346441617al_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: produc9003781588162623004on_val,B3: nat,C: produc9003781588162623004on_val] :
            ( ( A12
              = ( produc4958730763592889809on_val @ A4 @ C ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_854_rel__prod_Ocases,axiom,
    ! [R1: nat > ( produc1457211279475724562t_char > prat ) > $o,R22: nat > ( produc1457211279475724562t_char > option_val ) > $o,A12: product_prod_nat_nat,A23: produc9003781588162623004on_val] :
      ( ( basic_5770269710933094520on_val @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: nat,B3: produc1457211279475724562t_char > prat,C: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A4 @ C ) )
           => ! [D3: produc1457211279475724562t_char > option_val] :
                ( ( A23
                  = ( produc74620467203698198on_val @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_855_rel__prod_Ocases,axiom,
    ! [R1: nat > produc9003781588162623004on_val > $o,R22: nat > produc9003781588162623004on_val > $o,A12: product_prod_nat_nat,A23: produc1317171644805857433on_val] :
      ( ( basic_3065465552773073809on_val @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: nat,B3: produc9003781588162623004on_val,C: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A4 @ C ) )
           => ! [D3: produc9003781588162623004on_val] :
                ( ( A23
                  = ( produc4958730763592889809on_val @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_856_rel__prod_Ocases,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,A12: product_prod_nat_nat,A23: product_prod_nat_nat] :
      ( ( basic_5328504652464829177at_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A4: nat,B3: nat,C: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A4 @ C ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B3 @ D3 ) )
               => ( ( R1 @ A4 @ B3 )
                 => ~ ( R22 @ C @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_857_rel__prod_Osimps,axiom,
    ( basic_1679859423659471403on_val
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R23: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,A13: produc9003781588162623004on_val,A24: produc9003781588162623004on_val] :
        ? [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat,C3: produc1457211279475724562t_char > option_val,D4: produc1457211279475724562t_char > option_val] :
          ( ( A13
            = ( produc74620467203698198on_val @ A2 @ C3 ) )
          & ( A24
            = ( produc74620467203698198on_val @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_858_rel__prod_Osimps,axiom,
    ( basic_1859977137585984946on_val
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > $o,R23: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > $o,A13: produc9003781588162623004on_val,A24: produc1317171644805857433on_val] :
        ? [A2: produc1457211279475724562t_char > prat,B4: produc9003781588162623004on_val,C3: produc1457211279475724562t_char > option_val,D4: produc9003781588162623004on_val] :
          ( ( A13
            = ( produc74620467203698198on_val @ A2 @ C3 ) )
          & ( A24
            = ( produc4958730763592889809on_val @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_859_rel__prod_Osimps,axiom,
    ( basic_7417112278364389146al_nat
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > nat > $o,R23: ( produc1457211279475724562t_char > option_val ) > nat > $o,A13: produc9003781588162623004on_val,A24: product_prod_nat_nat] :
        ? [A2: produc1457211279475724562t_char > prat,B4: nat,C3: produc1457211279475724562t_char > option_val,D4: nat] :
          ( ( A13
            = ( produc74620467203698198on_val @ A2 @ C3 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_860_rel__prod_Osimps,axiom,
    ( basic_1560430169329504888on_val
    = ( ^ [R12: produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o,R23: produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o,A13: produc1317171644805857433on_val,A24: produc9003781588162623004on_val] :
        ? [A2: produc9003781588162623004on_val,B4: produc1457211279475724562t_char > prat,C3: produc9003781588162623004on_val,D4: produc1457211279475724562t_char > option_val] :
          ( ( A13
            = ( produc4958730763592889809on_val @ A2 @ C3 ) )
          & ( A24
            = ( produc74620467203698198on_val @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_861_rel__prod_Osimps,axiom,
    ( basic_4773232560198594089on_val
    = ( ^ [R12: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,R23: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A13: produc1317171644805857433on_val,A24: produc1317171644805857433on_val] :
        ? [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val,C3: produc9003781588162623004on_val,D4: produc9003781588162623004on_val] :
          ( ( A13
            = ( produc4958730763592889809on_val @ A2 @ C3 ) )
          & ( A24
            = ( produc4958730763592889809on_val @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_862_rel__prod_Osimps,axiom,
    ( basic_2945586035346441617al_nat
    = ( ^ [R12: produc9003781588162623004on_val > nat > $o,R23: produc9003781588162623004on_val > nat > $o,A13: produc1317171644805857433on_val,A24: product_prod_nat_nat] :
        ? [A2: produc9003781588162623004on_val,B4: nat,C3: produc9003781588162623004on_val,D4: nat] :
          ( ( A13
            = ( produc4958730763592889809on_val @ A2 @ C3 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_863_rel__prod_Osimps,axiom,
    ( basic_5770269710933094520on_val
    = ( ^ [R12: nat > ( produc1457211279475724562t_char > prat ) > $o,R23: nat > ( produc1457211279475724562t_char > option_val ) > $o,A13: product_prod_nat_nat,A24: produc9003781588162623004on_val] :
        ? [A2: nat,B4: produc1457211279475724562t_char > prat,C3: nat,D4: produc1457211279475724562t_char > option_val] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C3 ) )
          & ( A24
            = ( produc74620467203698198on_val @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_864_rel__prod_Osimps,axiom,
    ( basic_3065465552773073809on_val
    = ( ^ [R12: nat > produc9003781588162623004on_val > $o,R23: nat > produc9003781588162623004on_val > $o,A13: product_prod_nat_nat,A24: produc1317171644805857433on_val] :
        ? [A2: nat,B4: produc9003781588162623004on_val,C3: nat,D4: produc9003781588162623004on_val] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C3 ) )
          & ( A24
            = ( produc4958730763592889809on_val @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_865_rel__prod_Osimps,axiom,
    ( basic_5328504652464829177at_nat
    = ( ^ [R12: nat > nat > $o,R23: nat > nat > $o,A13: product_prod_nat_nat,A24: product_prod_nat_nat] :
        ? [A2: nat,B4: nat,C3: nat,D4: nat] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C3 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B4 @ D4 ) )
          & ( R12 @ A2 @ B4 )
          & ( R23 @ C3 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_866_rel__prod_Ointros,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,R22: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,C2: produc1457211279475724562t_char > option_val,D: produc1457211279475724562t_char > option_val] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_1679859423659471403on_val @ R1 @ R22 @ ( produc74620467203698198on_val @ A @ C2 ) @ ( produc74620467203698198on_val @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_867_rel__prod_Ointros,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > $o,A: produc1457211279475724562t_char > prat,B: produc9003781588162623004on_val,R22: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > $o,C2: produc1457211279475724562t_char > option_val,D: produc9003781588162623004on_val] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_1859977137585984946on_val @ R1 @ R22 @ ( produc74620467203698198on_val @ A @ C2 ) @ ( produc4958730763592889809on_val @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_868_rel__prod_Ointros,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > nat > $o,A: produc1457211279475724562t_char > prat,B: nat,R22: ( produc1457211279475724562t_char > option_val ) > nat > $o,C2: produc1457211279475724562t_char > option_val,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_7417112278364389146al_nat @ R1 @ R22 @ ( produc74620467203698198on_val @ A @ C2 ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_869_rel__prod_Ointros,axiom,
    ! [R1: produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o,A: produc9003781588162623004on_val,B: produc1457211279475724562t_char > prat,R22: produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o,C2: produc9003781588162623004on_val,D: produc1457211279475724562t_char > option_val] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_1560430169329504888on_val @ R1 @ R22 @ ( produc4958730763592889809on_val @ A @ C2 ) @ ( produc74620467203698198on_val @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_870_rel__prod_Ointros,axiom,
    ! [R1: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R22: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,C2: produc9003781588162623004on_val,D: produc9003781588162623004on_val] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_4773232560198594089on_val @ R1 @ R22 @ ( produc4958730763592889809on_val @ A @ C2 ) @ ( produc4958730763592889809on_val @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_871_rel__prod_Ointros,axiom,
    ! [R1: produc9003781588162623004on_val > nat > $o,A: produc9003781588162623004on_val,B: nat,R22: produc9003781588162623004on_val > nat > $o,C2: produc9003781588162623004on_val,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_2945586035346441617al_nat @ R1 @ R22 @ ( produc4958730763592889809on_val @ A @ C2 ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_872_rel__prod_Ointros,axiom,
    ! [R1: nat > ( produc1457211279475724562t_char > prat ) > $o,A: nat,B: produc1457211279475724562t_char > prat,R22: nat > ( produc1457211279475724562t_char > option_val ) > $o,C2: nat,D: produc1457211279475724562t_char > option_val] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_5770269710933094520on_val @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C2 ) @ ( produc74620467203698198on_val @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_873_rel__prod_Ointros,axiom,
    ! [R1: nat > produc9003781588162623004on_val > $o,A: nat,B: produc9003781588162623004on_val,R22: nat > produc9003781588162623004on_val > $o,C2: nat,D: produc9003781588162623004on_val] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_3065465552773073809on_val @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C2 ) @ ( produc4958730763592889809on_val @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_874_rel__prod_Ointros,axiom,
    ! [R1: nat > nat > $o,A: nat,B: nat,R22: nat > nat > $o,C2: nat,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C2 @ D )
       => ( basic_5328504652464829177at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C2 ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_875_Pair__transfer,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B5: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_re4208021347924984265on_val @ A5 @ ( bNF_re4251900314791790925on_val @ B5 @ ( basic_1679859423659471403on_val @ A5 @ B5 ) ) @ produc74620467203698198on_val @ produc74620467203698198on_val ) ).

% Pair_transfer
thf(fact_876_Pair__transfer,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > produc9003781588162623004on_val > $o,B5: ( produc1457211279475724562t_char > option_val ) > produc9003781588162623004on_val > $o] : ( bNF_re5744771098293236043on_val @ A5 @ ( bNF_re4595520508589708070on_val @ B5 @ ( basic_1859977137585984946on_val @ A5 @ B5 ) ) @ produc74620467203698198on_val @ produc4958730763592889809on_val ) ).

% Pair_transfer
thf(fact_877_Pair__transfer,axiom,
    ! [A5: ( produc1457211279475724562t_char > prat ) > nat > $o,B5: ( produc1457211279475724562t_char > option_val ) > nat > $o] : ( bNF_re3147323125767581259at_nat @ A5 @ ( bNF_re3875731290923947482at_nat @ B5 @ ( basic_7417112278364389146al_nat @ A5 @ B5 ) ) @ produc74620467203698198on_val @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_878_Pair__transfer,axiom,
    ! [A5: produc9003781588162623004on_val > ( produc1457211279475724562t_char > prat ) > $o,B5: produc9003781588162623004on_val > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_re6243659927171069681on_val @ A5 @ ( bNF_re1120834113401173856on_val @ B5 @ ( basic_1560430169329504888on_val @ A5 @ B5 ) ) @ produc4958730763592889809on_val @ produc74620467203698198on_val ) ).

% Pair_transfer
thf(fact_879_Pair__transfer,axiom,
    ! [A5: produc9003781588162623004on_val > produc9003781588162623004on_val > $o,B5: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] : ( bNF_re1796254245058062999on_val @ A5 @ ( bNF_re3825428800981762519on_val @ B5 @ ( basic_4773232560198594089on_val @ A5 @ B5 ) ) @ produc4958730763592889809on_val @ produc4958730763592889809on_val ) ).

% Pair_transfer
thf(fact_880_Pair__transfer,axiom,
    ! [A5: produc9003781588162623004on_val > nat > $o,B5: produc9003781588162623004on_val > nat > $o] : ( bNF_re4791974009433356543at_nat @ A5 @ ( bNF_re583272441465906419at_nat @ B5 @ ( basic_2945586035346441617al_nat @ A5 @ B5 ) ) @ produc4958730763592889809on_val @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_881_Pair__transfer,axiom,
    ! [A5: nat > ( produc1457211279475724562t_char > prat ) > $o,B5: nat > ( produc1457211279475724562t_char > option_val ) > $o] : ( bNF_re8571388671295005273on_val @ A5 @ ( bNF_re431696514778167572on_val @ B5 @ ( basic_5770269710933094520on_val @ A5 @ B5 ) ) @ product_Pair_nat_nat @ produc74620467203698198on_val ) ).

% Pair_transfer
thf(fact_882_Pair__transfer,axiom,
    ! [A5: nat > produc9003781588162623004on_val > $o,B5: nat > produc9003781588162623004on_val > $o] : ( bNF_re2010443014418883223on_val @ A5 @ ( bNF_re3336399577128851747on_val @ B5 @ ( basic_3065465552773073809on_val @ A5 @ B5 ) ) @ product_Pair_nat_nat @ produc4958730763592889809on_val ) ).

% Pair_transfer
thf(fact_883_Pair__transfer,axiom,
    ! [A5: nat > nat > $o,B5: nat > nat > $o] : ( bNF_re3696488243098009855at_nat @ A5 @ ( bNF_re901410630616637503at_nat @ B5 @ ( basic_5328504652464829177at_nat @ A5 @ B5 ) ) @ product_Pair_nat_nat @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_884_wf__def,axiom,
    ( wf_Pro3300669915054815219on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
        ! [P4: produc9003781588162623004on_val > $o] :
          ( ! [X2: produc9003781588162623004on_val] :
              ( ! [Y5: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R3 )
                 => ( P4 @ Y5 ) )
             => ( P4 @ X2 ) )
         => ! [X7: produc9003781588162623004on_val] : ( P4 @ X7 ) ) ) ) ).

% wf_def
thf(fact_885_wf__def,axiom,
    ( wf_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
        ! [P4: nat > $o] :
          ( ! [X2: nat] :
              ( ! [Y5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R3 )
                 => ( P4 @ Y5 ) )
             => ( P4 @ X2 ) )
         => ! [X7: nat] : ( P4 @ X7 ) ) ) ) ).

% wf_def
thf(fact_886_wfE__min,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ Q )
       => ~ ! [Z2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Z2 @ Q )
             => ~ ! [Y4: product_prod_nat_nat] :
                    ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z2 ) @ R2 )
                   => ~ ( member8440522571783428010at_nat @ Y4 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_887_wfE__min,axiom,
    ! [R2: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Q: set_Pr7649655576376376530on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R2 )
     => ( ( member2879030624002546611on_val @ X @ Q )
       => ~ ! [Z2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Z2 @ Q )
             => ~ ! [Y4: produc9003781588162623004on_val] :
                    ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Z2 ) @ R2 )
                   => ~ ( member2879030624002546611on_val @ Y4 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_888_wfE__min,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Q: set_nat] :
      ( ( wf_nat @ R2 )
     => ( ( member_nat @ X @ Q )
       => ~ ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Q )
             => ~ ! [Y4: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z2 ) @ R2 )
                   => ~ ( member_nat @ Y4 @ Q ) ) ) ) ) ).

% wfE_min
thf(fact_889_wfI__min,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Q4: set_Pr1261947904930325089at_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ Q4 )
         => ? [Xa3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa3 @ Q4 )
              & ! [Y3: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Xa3 ) @ R2 )
                 => ~ ( member8440522571783428010at_nat @ Y3 @ Q4 ) ) ) )
     => ( wf_Pro7803398752247294826at_nat @ R2 ) ) ).

% wfI_min
thf(fact_890_wfI__min,axiom,
    ! [R2: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Q4: set_Pr7649655576376376530on_val] :
          ( ( member2879030624002546611on_val @ X3 @ Q4 )
         => ? [Xa3: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Xa3 @ Q4 )
              & ! [Y3: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Xa3 ) @ R2 )
                 => ~ ( member2879030624002546611on_val @ Y3 @ Q4 ) ) ) )
     => ( wf_Pro3300669915054815219on_val @ R2 ) ) ).

% wfI_min
thf(fact_891_wfI__min,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Q4: set_nat] :
          ( ( member_nat @ X3 @ Q4 )
         => ? [Xa3: nat] :
              ( ( member_nat @ Xa3 @ Q4 )
              & ! [Y3: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa3 ) @ R2 )
                 => ~ ( member_nat @ Y3 @ Q4 ) ) ) )
     => ( wf_nat @ R2 ) ) ).

% wfI_min
thf(fact_892_wfUNIVI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [P6: produc9003781588162623004on_val > $o,X3: produc9003781588162623004on_val] :
          ( ! [Xa3: produc9003781588162623004on_val] :
              ( ! [Y3: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Xa3 ) @ R )
                 => ( P6 @ Y3 ) )
             => ( P6 @ Xa3 ) )
         => ( P6 @ X3 ) )
     => ( wf_Pro3300669915054815219on_val @ R ) ) ).

% wfUNIVI
thf(fact_893_wfUNIVI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [P6: nat > $o,X3: nat] :
          ( ! [Xa3: nat] :
              ( ! [Y3: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa3 ) @ R )
                 => ( P6 @ Y3 ) )
             => ( P6 @ Xa3 ) )
         => ( P6 @ X3 ) )
     => ( wf_nat @ R ) ) ).

% wfUNIVI
thf(fact_894_wf__asym,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ X ) @ R )
       => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_895_wf__asym,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,X: nat] :
      ( ( wf_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ R ) ) ) ).

% wf_asym
thf(fact_896_wf__induct,axiom,
    ! [R: set_Pr4305388979797279737on_val,P2: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ! [Y4: produc9003781588162623004on_val] :
                ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ X3 ) @ R )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_897_wf__induct,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( wf_nat @ R )
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct
thf(fact_898_wf__irrefl,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_899_wf__irrefl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( wf_nat @ R )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).

% wf_irrefl
thf(fact_900_wf__not__sym,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,X: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ X ) @ R )
       => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_901_wf__not__sym,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,X: nat] :
      ( ( wf_nat @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ R )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ A ) @ R ) ) ) ).

% wf_not_sym
thf(fact_902_wf__not__refl,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_903_wf__not__refl,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( wf_nat @ R )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).

% wf_not_refl
thf(fact_904_wf__eq__minimal,axiom,
    ( wf_Pro7803398752247294826at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat] :
        ! [Q3: set_Pr1261947904930325089at_nat] :
          ( ? [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ Q3 )
         => ? [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ Q3 )
              & ! [Y5: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R3 )
                 => ~ ( member8440522571783428010at_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_905_wf__eq__minimal,axiom,
    ( wf_Pro3300669915054815219on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
        ! [Q3: set_Pr7649655576376376530on_val] :
          ( ? [X2: produc9003781588162623004on_val] : ( member2879030624002546611on_val @ X2 @ Q3 )
         => ? [X2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X2 @ Q3 )
              & ! [Y5: produc9003781588162623004on_val] :
                  ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R3 )
                 => ~ ( member2879030624002546611on_val @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_906_wf__eq__minimal,axiom,
    ( wf_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
        ! [Q3: set_nat] :
          ( ? [X2: nat] : ( member_nat @ X2 @ Q3 )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ Q3 )
              & ! [Y5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R3 )
                 => ~ ( member_nat @ Y5 @ Q3 ) ) ) ) ) ) ).

% wf_eq_minimal
thf(fact_907_wf__induct__rule,axiom,
    ! [R: set_Pr4305388979797279737on_val,P2: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ! [Y4: produc9003781588162623004on_val] :
                ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ X3 ) @ R )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_908_wf__induct__rule,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( wf_nat @ R )
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wf_induct_rule
thf(fact_909_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4305388979797279737on_val,R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ S )
     => ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X3 ) @ ( F @ Y3 ) ) @ S ) )
       => ( wf_Pro3300669915054815219on_val @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_910_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1261947904930325089at_nat,R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > nat] :
      ( ( wf_nat @ S )
     => ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) @ S ) )
       => ( wf_Pro3300669915054815219on_val @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_911_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr4305388979797279737on_val,R: set_Pr1261947904930325089at_nat,F: nat > produc9003781588162623004on_val] :
      ( ( wf_Pro3300669915054815219on_val @ S )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( F @ X3 ) @ ( F @ Y3 ) ) @ S ) )
       => ( wf_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_912_wf__if__convertible__to__wf,axiom,
    ! [S: set_Pr1261947904930325089at_nat,R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( wf_nat @ S )
     => ( ! [X3: nat,Y3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) @ S ) )
       => ( wf_nat @ R ) ) ) ).

% wf_if_convertible_to_wf
thf(fact_913_finite__range__Some,axiom,
    ( ( finite4212117559080419824on_val @ ( image_val_option_val @ some_val @ top_top_set_val ) )
    = ( finite_finite_val @ top_top_set_val ) ) ).

% finite_range_Some
thf(fact_914_finite__range__Some,axiom,
    ( ( finite3180955649987104801_state @ ( image_6076465424260689483_state @ some_state @ top_top_set_state ) )
    = ( finite_finite_state @ top_top_set_state ) ) ).

% finite_range_Some
thf(fact_915_finite__range__Some,axiom,
    ( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_range_Some
thf(fact_916_wfE__min_H,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,Q: set_Pr1261947904930325089at_nat] :
      ( ( wf_Pro7803398752247294826at_nat @ R2 )
     => ( ( Q != bot_bo2099793752762293965at_nat )
       => ~ ! [Z2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Z2 @ Q )
             => ~ ! [Y4: product_prod_nat_nat] :
                    ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z2 ) @ R2 )
                   => ~ ( member8440522571783428010at_nat @ Y4 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_917_wfE__min_H,axiom,
    ! [R2: set_Pr4305388979797279737on_val,Q: set_Pr7649655576376376530on_val] :
      ( ( wf_Pro3300669915054815219on_val @ R2 )
     => ( ( Q != bot_bo6722054719108133734on_val )
       => ~ ! [Z2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Z2 @ Q )
             => ~ ! [Y4: produc9003781588162623004on_val] :
                    ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ Z2 ) @ R2 )
                   => ~ ( member2879030624002546611on_val @ Y4 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_918_wfE__min_H,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,Q: set_nat] :
      ( ( wf_nat @ R2 )
     => ( ( Q != bot_bot_set_nat )
       => ~ ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Q )
             => ~ ! [Y4: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z2 ) @ R2 )
                   => ~ ( member_nat @ Y4 @ Q ) ) ) ) ) ).

% wfE_min'
thf(fact_919_rel__prod__sel,axiom,
    ( basic_1679859423659471403on_val
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R23: ( produc1457211279475724562t_char > option_val ) > ( produc1457211279475724562t_char > option_val ) > $o,P3: produc9003781588162623004on_val,Q5: produc9003781588162623004on_val] :
          ( ( R12 @ ( produc5239619271231976746on_val @ P3 ) @ ( produc5239619271231976746on_val @ Q5 ) )
          & ( R23 @ ( produc2319605628138516840on_val @ P3 ) @ ( produc2319605628138516840on_val @ Q5 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_920_wf__imp__asym,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( asym_on_nat @ top_top_set_nat @ R ) ) ).

% wf_imp_asym
thf(fact_921_wfI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R
        @ ( produc2761391749766926216at_nat @ A5
          @ ^ [Uu3: product_prod_nat_nat] : B5 ) )
     => ( ! [X3: product_prod_nat_nat,P6: product_prod_nat_nat > $o] :
            ( ! [Xa3: product_prod_nat_nat] :
                ( ! [Y3: product_prod_nat_nat] :
                    ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Xa3 ) @ R )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa3 ) )
           => ( ( member8440522571783428010at_nat @ X3 @ A5 )
             => ( ( member8440522571783428010at_nat @ X3 @ B5 )
               => ( P6 @ X3 ) ) ) )
       => ( wf_Pro7803398752247294826at_nat @ R ) ) ) ).

% wfI
thf(fact_922_wfI,axiom,
    ! [R: set_Pr4305388979797279737on_val,A5: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( ord_le7224449410216889241on_val @ R
        @ ( produc4067394217145614850on_val @ A5
          @ ^ [Uu3: produc9003781588162623004on_val] : B5 ) )
     => ( ! [X3: produc9003781588162623004on_val,P6: produc9003781588162623004on_val > $o] :
            ( ! [Xa3: produc9003781588162623004on_val] :
                ( ! [Y3: produc9003781588162623004on_val] :
                    ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Xa3 ) @ R )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa3 ) )
           => ( ( member2879030624002546611on_val @ X3 @ A5 )
             => ( ( member2879030624002546611on_val @ X3 @ B5 )
               => ( P6 @ X3 ) ) ) )
       => ( wf_Pro3300669915054815219on_val @ R ) ) ) ).

% wfI
thf(fact_923_wfI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A5: set_nat,B5: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R
        @ ( produc457027306803732586at_nat @ A5
          @ ^ [Uu3: nat] : B5 ) )
     => ( ! [X3: nat,P6: nat > $o] :
            ( ! [Xa3: nat] :
                ( ! [Y3: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Xa3 ) @ R )
                   => ( P6 @ Y3 ) )
               => ( P6 @ Xa3 ) )
           => ( ( member_nat @ X3 @ A5 )
             => ( ( member_nat @ X3 @ B5 )
               => ( P6 @ X3 ) ) ) )
       => ( wf_nat @ R ) ) ) ).

% wfI
thf(fact_924_finite__map__freshness,axiom,
    ! [F: nat > option_val] :
      ( ( finite_finite_nat @ ( dom_nat_val @ F ) )
     => ( ~ ( finite_finite_nat @ top_top_set_nat )
       => ? [X3: nat] :
            ( ( F @ X3 )
            = none_val ) ) ) ).

% finite_map_freshness
thf(fact_925_finite__map__freshness,axiom,
    ! [F: nat > option_state] :
      ( ( finite_finite_nat @ ( dom_nat_state @ F ) )
     => ( ~ ( finite_finite_nat @ top_top_set_nat )
       => ? [X3: nat] :
            ( ( F @ X3 )
            = none_state ) ) ) ).

% finite_map_freshness
thf(fact_926_wf__eq__minimal2,axiom,
    ( wf_Pro3300669915054815219on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
        ! [A6: set_Pr7649655576376376530on_val] :
          ( ( ( ord_le6900622908513426738on_val @ A6 @ ( field_9010718613293825679on_val @ R3 ) )
            & ( A6 != bot_bo6722054719108133734on_val ) )
         => ? [X2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X2 @ A6 )
              & ! [Y5: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ Y5 @ A6 )
                 => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_927_wf__eq__minimal2,axiom,
    ( wf_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
        ! [A6: set_nat] :
          ( ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R3 ) )
            & ( A6 != bot_bot_set_nat ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ A6 )
              & ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ A6 )
                 => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R3 ) ) ) ) ) ) ).

% wf_eq_minimal2
thf(fact_928_prod_Oin__rel,axiom,
    ( basic_7417786039168870389on_val
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R23: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A2: produc3336119888008106215r_prat,B4: produc1757238238866670357on_val] :
        ? [Z5: produc1317171644805857433on_val] :
          ( ( member6505152944228695362on_val @ Z5
            @ ( collec5166231387257825412on_val
              @ ^ [X2: produc1317171644805857433on_val] :
                  ( ( ord_le6900622908513426738on_val @ ( basic_2381348828333783745on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R12 ) ) )
                  & ( ord_le6900622908513426738on_val @ ( basic_2713455076908730111on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R23 ) ) ) ) ) )
          & ( ( produc1002149148328127798r_prat @ produc5239619271231976746on_val @ produc5239619271231976746on_val @ Z5 )
            = A2 )
          & ( ( produc1134234647556835942on_val @ produc2319605628138516840on_val @ produc2319605628138516840on_val @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_929_prod_Oin__rel,axiom,
    ( basic_400063943313148570at_nat
    = ( ^ [R12: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,R23: nat > nat > $o,A2: produc4488441066493689938at_nat,B4: produc8366770746848206027al_nat] :
        ? [Z5: produc8271722145941708506at_nat] :
          ( ( member7906376990877558257at_nat @ Z5
            @ ( collec927009928073930287at_nat
              @ ^ [X2: produc8271722145941708506at_nat] :
                  ( ( ord_le6900622908513426738on_val @ ( basic_2415657071401982236at_nat @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R12 ) ) )
                  & ( ord_le3146513528884898305at_nat @ ( basic_8506799250368921950at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) ) ) ) )
          & ( ( produc8335123919915634002at_nat @ produc5239619271231976746on_val @ product_fst_nat_nat @ Z5 )
            = A2 )
          & ( ( produc5420676907050541347at_nat @ produc2319605628138516840on_val @ product_snd_nat_nat @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_930_prod_Oin__rel,axiom,
    ( basic_9214219793883178326on_val
    = ( ^ [R12: nat > nat > $o,R23: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o,A2: produc3766087481356336504r_prat,B4: produc2951589610084125259on_val] :
        ? [Z5: produc5414607237814678604on_val] :
          ( ( member5049262082750528355on_val @ Z5
            @ ( collec7293267056801676193on_val
              @ ^ [X2: produc5414607237814678604on_val] :
                  ( ( ord_le3146513528884898305at_nat @ ( basic_1039694484350783702on_val @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
                  & ( ord_le6900622908513426738on_val @ ( basic_7130836663317723416on_val @ X2 ) @ ( collec7475294502966882545on_val @ ( produc5248837549992124053_val_o @ R23 ) ) ) ) ) )
          & ( ( produc4911518146750293274r_prat @ product_fst_nat_nat @ produc5239619271231976746on_val @ Z5 )
            = A2 )
          & ( ( produc3522203662121252755on_val @ product_snd_nat_nat @ produc2319605628138516840on_val @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_931_prod_Oin__rel,axiom,
    ( basic_5328504652464829177at_nat
    = ( ^ [R12: nat > nat > $o,R23: nat > nat > $o,A2: product_prod_nat_nat,B4: product_prod_nat_nat] :
        ? [Z5: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ Z5
            @ ( collec7088162979684241874at_nat
              @ ^ [X2: produc859450856879609959at_nat] :
                  ( ( ord_le3146513528884898305at_nat @ ( basic_1775256158546629575at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R12 ) ) )
                  & ( ord_le3146513528884898305at_nat @ ( basic_2061583120998029061at_nat @ X2 ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R23 ) ) ) ) ) )
          & ( ( produc5156754000680454454at_nat @ product_fst_nat_nat @ product_fst_nat_nat @ Z5 )
            = A2 )
          & ( ( produc5156754000680454454at_nat @ product_snd_nat_nat @ product_snd_nat_nat @ Z5 )
            = B4 ) ) ) ) ).

% prod.in_rel
thf(fact_932_reduction__pair__lemma,axiom,
    ! [P2: produc3843707927480180839at_nat,R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( fun_re1951171424197210636ir_nat @ P2 )
     => ( ( ord_le3146513528884898305at_nat @ R2 @ ( produc6011863565407846595at_nat @ P2 ) )
       => ( ( ord_le3146513528884898305at_nat @ S4 @ ( produc2937545259423399429at_nat @ P2 ) )
         => ( ( wf_nat @ S4 )
           => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S4 ) ) ) ) ) ) ).

% reduction_pair_lemma
thf(fact_933_fun__upd__None__restrict,axiom,
    ! [X: product_prod_nat_nat,D2: set_Pr1261947904930325089at_nat,M: product_prod_nat_nat > option_val] :
      ( ( ( member8440522571783428010at_nat @ X @ D2 )
       => ( ( fun_up2801401594293523459on_val @ ( restri1810717250645833456at_val @ M @ D2 ) @ X @ none_val )
          = ( restri1810717250645833456at_val @ M @ ( minus_1356011639430497352at_nat @ D2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ D2 )
       => ( ( fun_up2801401594293523459on_val @ ( restri1810717250645833456at_val @ M @ D2 ) @ X @ none_val )
          = ( restri1810717250645833456at_val @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_934_fun__upd__None__restrict,axiom,
    ! [X: produc9003781588162623004on_val,D2: set_Pr7649655576376376530on_val,M: produc9003781588162623004on_val > option_val] :
      ( ( ( member2879030624002546611on_val @ X @ D2 )
       => ( ( fun_up8559965269643824108on_val @ ( restri5251746167140823987al_val @ M @ D2 ) @ X @ none_val )
          = ( restri5251746167140823987al_val @ M @ ( minus_9160195963678339755on_val @ D2 @ ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) ) ) ) )
      & ( ~ ( member2879030624002546611on_val @ X @ D2 )
       => ( ( fun_up8559965269643824108on_val @ ( restri5251746167140823987al_val @ M @ D2 ) @ X @ none_val )
          = ( restri5251746167140823987al_val @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_935_fun__upd__None__restrict,axiom,
    ! [X: product_prod_nat_nat,D2: set_Pr1261947904930325089at_nat,M: product_prod_nat_nat > option_state] :
      ( ( ( member8440522571783428010at_nat @ X @ D2 )
       => ( ( fun_up2914124688375849550_state @ ( restri8243093853081436043_state @ M @ D2 ) @ X @ none_state )
          = ( restri8243093853081436043_state @ M @ ( minus_1356011639430497352at_nat @ D2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ D2 )
       => ( ( fun_up2914124688375849550_state @ ( restri8243093853081436043_state @ M @ D2 ) @ X @ none_state )
          = ( restri8243093853081436043_state @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_936_fun__upd__None__restrict,axiom,
    ! [X: produc9003781588162623004on_val,D2: set_Pr7649655576376376530on_val,M: produc9003781588162623004on_val > option_state] :
      ( ( ( member2879030624002546611on_val @ X @ D2 )
       => ( ( fun_up6681822530636708645_state @ ( restri5336766904122854152_state @ M @ D2 ) @ X @ none_state )
          = ( restri5336766904122854152_state @ M @ ( minus_9160195963678339755on_val @ D2 @ ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) ) ) ) )
      & ( ~ ( member2879030624002546611on_val @ X @ D2 )
       => ( ( fun_up6681822530636708645_state @ ( restri5336766904122854152_state @ M @ D2 ) @ X @ none_state )
          = ( restri5336766904122854152_state @ M @ D2 ) ) ) ) ).

% fun_upd_None_restrict
thf(fact_937_wf__measure,axiom,
    ! [F: nat > nat] : ( wf_nat @ ( measure_nat @ F ) ) ).

% wf_measure
thf(fact_938_restrict__out,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,M: product_prod_nat_nat > option_val] :
      ( ~ ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( restri1810717250645833456at_val @ M @ A5 @ X )
        = none_val ) ) ).

% restrict_out
thf(fact_939_restrict__out,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,M: produc9003781588162623004on_val > option_val] :
      ( ~ ( member2879030624002546611on_val @ X @ A5 )
     => ( ( restri5251746167140823987al_val @ M @ A5 @ X )
        = none_val ) ) ).

% restrict_out
thf(fact_940_restrict__out,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,M: product_prod_nat_nat > option_state] :
      ( ~ ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( restri8243093853081436043_state @ M @ A5 @ X )
        = none_state ) ) ).

% restrict_out
thf(fact_941_restrict__out,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,M: produc9003781588162623004on_val > option_state] :
      ( ~ ( member2879030624002546611on_val @ X @ A5 )
     => ( ( restri5336766904122854152_state @ M @ A5 @ X )
        = none_state ) ) ).

% restrict_out
thf(fact_942_restrict__map__def,axiom,
    ( restri1810717250645833456at_val
    = ( ^ [M2: product_prod_nat_nat > option_val,A6: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( if_option_val @ ( member8440522571783428010at_nat @ X2 @ A6 ) @ ( M2 @ X2 ) @ none_val ) ) ) ).

% restrict_map_def
thf(fact_943_restrict__map__def,axiom,
    ( restri5251746167140823987al_val
    = ( ^ [M2: produc9003781588162623004on_val > option_val,A6: set_Pr7649655576376376530on_val,X2: produc9003781588162623004on_val] : ( if_option_val @ ( member2879030624002546611on_val @ X2 @ A6 ) @ ( M2 @ X2 ) @ none_val ) ) ) ).

% restrict_map_def
thf(fact_944_restrict__map__def,axiom,
    ( restri8243093853081436043_state
    = ( ^ [M2: product_prod_nat_nat > option_state,A6: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( if_option_state @ ( member8440522571783428010at_nat @ X2 @ A6 ) @ ( M2 @ X2 ) @ none_state ) ) ) ).

% restrict_map_def
thf(fact_945_restrict__map__def,axiom,
    ( restri5336766904122854152_state
    = ( ^ [M2: produc9003781588162623004on_val > option_state,A6: set_Pr7649655576376376530on_val,X2: produc9003781588162623004on_val] : ( if_option_state @ ( member2879030624002546611on_val @ X2 @ A6 ) @ ( M2 @ X2 ) @ none_state ) ) ) ).

% restrict_map_def
thf(fact_946_graph__restrictD_I1_J,axiom,
    ! [K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,A5: set_Pr3433046674199179443r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ ( restri7594233908965391805on_val @ M @ A5 ) ) )
     => ( member2336342582916103932r_prat @ K @ A5 ) ) ).

% graph_restrictD(1)
thf(fact_947_graph__restrictD_I1_J,axiom,
    ! [K: produc9003781588162623004on_val,V: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option8887860577033306220on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ ( restri2354629191430751338on_val @ M @ A5 ) ) )
     => ( member2879030624002546611on_val @ K @ A5 ) ) ).

% graph_restrictD(1)
thf(fact_948_graph__restrictD_I1_J,axiom,
    ! [K: nat,V: nat,M: nat > option_nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( restrict_map_nat_nat @ M @ A5 ) ) )
     => ( member_nat @ K @ A5 ) ) ).

% graph_restrictD(1)
thf(fact_949_graph__restrictD_I2_J,axiom,
    ! [K: produc1457211279475724562t_char > prat,V: produc1457211279475724562t_char > option_val,M: ( produc1457211279475724562t_char > prat ) > option5747939881698552298on_val,A5: set_Pr3433046674199179443r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ K @ V ) @ ( graph_1288227317702120175on_val @ ( restri7594233908965391805on_val @ M @ A5 ) ) )
     => ( ( M @ K )
        = ( some_P7669353234731512949on_val @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_950_graph__restrictD_I2_J,axiom,
    ! [K: produc9003781588162623004on_val,V: produc9003781588162623004on_val,M: produc9003781588162623004on_val > option8887860577033306220on_val,A5: set_Pr7649655576376376530on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ K @ V ) @ ( graph_6108330199704490680on_val @ ( restri2354629191430751338on_val @ M @ A5 ) ) )
     => ( ( M @ K )
        = ( some_P2581650809551911671on_val @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_951_graph__restrictD_I2_J,axiom,
    ! [K: nat,V: nat,M: nat > option_nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( restrict_map_nat_nat @ M @ A5 ) ) )
     => ( ( M @ K )
        = ( some_nat @ V ) ) ) ).

% graph_restrictD(2)
thf(fact_952_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( wf_Pro3300669915054815219on_val @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) )
        = ( ! [A6: set_Pr7649655576376376530on_val] :
              ( ( ord_le6900622908513426738on_val @ A6 @ ( field_9010718613293825679on_val @ R ) )
             => ( ( A6 != bot_bo6722054719108133734on_val )
               => ? [X2: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ X2 @ A6 )
                    & ! [Y5: produc9003781588162623004on_val] :
                        ( ( member2879030624002546611on_val @ Y5 @ A6 )
                       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_953_Linear__order__wf__diff__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( wf_nat @ ( minus_1356011639430497352at_nat @ R @ id_nat2 ) )
        = ( ! [A6: set_nat] :
              ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R ) )
             => ( ( A6 != bot_bot_set_nat )
               => ? [X2: nat] :
                    ( ( member_nat @ X2 @ A6 )
                    & ! [Y5: nat] :
                        ( ( member_nat @ Y5 @ A6 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
thf(fact_954_wf__UN,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [I4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I4 @ I3 )
         => ( wf_nat @ ( R @ I4 ) ) )
     => ( ! [I4: product_prod_nat_nat,J3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I4 @ I3 )
           => ( ( member8440522571783428010at_nat @ J3 @ I3 )
             => ( ( ( R @ I4 )
                 != ( R @ J3 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R @ I4 ) ) @ ( range_nat_nat @ ( R @ J3 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ R @ I3 ) ) ) ) ) ).

% wf_UN
thf(fact_955_wf__UN,axiom,
    ! [I3: set_Pr7649655576376376530on_val,R: produc9003781588162623004on_val > set_Pr1261947904930325089at_nat] :
      ( ! [I4: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ I4 @ I3 )
         => ( wf_nat @ ( R @ I4 ) ) )
     => ( ! [I4: produc9003781588162623004on_val,J3: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ I4 @ I3 )
           => ( ( member2879030624002546611on_val @ J3 @ I3 )
             => ( ( ( R @ I4 )
                 != ( R @ J3 ) )
               => ( ( inf_inf_set_nat @ ( domain_nat_nat @ ( R @ I4 ) ) @ ( range_nat_nat @ ( R @ J3 ) ) )
                  = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ ( image_4270514353411284094at_nat @ R @ I3 ) ) ) ) ) ).

% wf_UN
thf(fact_956_IdI,axiom,
    ! [A: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ id_Pro1455714331835775424on_val ) ).

% IdI
thf(fact_957_IdI,axiom,
    ! [A: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ id_nat2 ) ).

% IdI
thf(fact_958_pair__in__Id__conv,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ id_Pro1455714331835775424on_val )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_959_pair__in__Id__conv,axiom,
    ! [A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_960_wf__Int1,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( wf_nat @ ( inf_in2572325071724192079at_nat @ R @ R6 ) ) ) ).

% wf_Int1
thf(fact_961_wf__Int2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,R6: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( wf_nat @ ( inf_in2572325071724192079at_nat @ R6 @ R ) ) ) ).

% wf_Int2
thf(fact_962_IdE,axiom,
    ! [P: produc1317171644805857433on_val] :
      ( ( member6505152944228695362on_val @ P @ id_Pro1455714331835775424on_val )
     => ~ ! [X3: produc9003781588162623004on_val] :
            ( P
           != ( produc4958730763592889809on_val @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_963_IdE,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ P @ id_nat2 )
     => ~ ! [X3: nat] :
            ( P
           != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_964_IdD,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ id_Pro1455714331835775424on_val )
     => ( A = B ) ) ).

% IdD
thf(fact_965_IdD,axiom,
    ! [A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ id_nat2 )
     => ( A = B ) ) ).

% IdD
thf(fact_966_Id__fstsnd__eq,axiom,
    ( id_nat2
    = ( collec3392354462482085612at_nat
      @ ^ [X2: product_prod_nat_nat] :
          ( ( product_fst_nat_nat @ X2 )
          = ( product_snd_nat_nat @ X2 ) ) ) ) ).

% Id_fstsnd_eq
thf(fact_967_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R ) @ R )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R )
            = ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ ( minus_8321449233255521966at_nat @ R @ id_Pro2258643101195443293at_nat ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_968_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( order_6061556925426544295on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
            = ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_969_Linear__order__in__diff__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
            = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat2 ) ) ) ) ) ) ) ).

% Linear_order_in_diff_Id
thf(fact_970_wf__Un,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( ( wf_nat @ S )
       => ( ( ( inf_inf_set_nat @ ( domain_nat_nat @ R ) @ ( range_nat_nat @ S ) )
            = bot_bot_set_nat )
         => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R @ S ) ) ) ) ) ).

% wf_Un
thf(fact_971_wf__Union,axiom,
    ! [R2: set_se7855581050983116737at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ R2 )
         => ( wf_nat @ X3 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ R2 )
           => ! [Xa: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa @ R2 )
               => ( ( X3 != Xa )
                 => ( ( inf_inf_set_nat @ ( domain_nat_nat @ X3 ) @ ( range_nat_nat @ Xa ) )
                    = bot_bot_set_nat ) ) ) )
       => ( wf_nat @ ( comple5685304695842803022at_nat @ R2 ) ) ) ) ).

% wf_Union
thf(fact_972_Total__subset__Id,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( total_4784279244336621976on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( ord_le7224449410216889241on_val @ R @ id_Pro1455714331835775424on_val )
       => ( ( R = bot_bo402267660617762917on_val )
          | ? [A4: produc9003781588162623004on_val] :
              ( R
              = ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A4 @ A4 ) @ bot_bo402267660617762917on_val ) ) ) ) ) ).

% Total_subset_Id
thf(fact_973_Total__subset__Id,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ ( field_nat @ R ) @ R )
     => ( ( ord_le3146513528884898305at_nat @ R @ id_nat2 )
       => ( ( R = bot_bo2099793752762293965at_nat )
          | ? [A4: nat] :
              ( R
              = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Total_subset_Id
thf(fact_974_reduction__pair__def,axiom,
    ( fun_re1951171424197210636ir_nat
    = ( ^ [P4: produc3843707927480180839at_nat] :
          ( ( wf_nat @ ( produc6011863565407846595at_nat @ P4 ) )
          & ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( produc6011863565407846595at_nat @ P4 ) @ ( produc2937545259423399429at_nat @ P4 ) ) @ ( produc6011863565407846595at_nat @ P4 ) ) ) ) ) ).

% reduction_pair_def
thf(fact_975_total__on__lex__prod,axiom,
    ! [A5: set_nat,R_A: set_Pr1261947904930325089at_nat,B5: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ A5 @ R_A )
     => ( ( total_on_nat @ B5 @ R_B )
       => ( total_3592101749530773125at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu3: nat] : B5 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% total_on_lex_prod
thf(fact_976_relcomp_Ocases,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom2642467956565963995on_val @ R @ S ) )
     => ~ ! [B3: produc1457211279475724562t_char > prat] :
            ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A12 @ B3 ) @ R )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B3 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_977_relcomp_Ocases,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom3547033359321004872on_val @ R @ S ) )
     => ~ ! [B3: produc1457211279475724562t_char > option_val] :
            ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ B3 ) @ R )
           => ~ ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B3 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_978_relcomp_Ocases,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( relcom3664992426205683031on_val @ R @ S ) )
     => ~ ! [B3: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ B3 ) @ R )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_979_relcomp_Ocases,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ B3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A23 ) @ S ) ) ) ).

% relcomp.cases
thf(fact_980_relcomp_Osimps,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom2642467956565963995on_val @ R @ S ) )
      = ( ? [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > prat,C3: produc1457211279475724562t_char > option_val] :
            ( ( A12 = A2 )
            & ( A23 = C3 )
            & ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A2 @ B4 ) @ R )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_981_relcomp_Osimps,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( relcom3547033359321004872on_val @ R @ S ) )
      = ( ? [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val,C3: produc1457211279475724562t_char > option_val] :
            ( ( A12 = A2 )
            & ( A23 = C3 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A2 @ B4 ) @ R )
            & ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_982_relcomp_Osimps,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( relcom3664992426205683031on_val @ R @ S ) )
      = ( ? [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val,C3: produc9003781588162623004on_val] :
            ( ( A12 = A2 )
            & ( A23 = C3 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_983_relcomp_Osimps,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
      = ( ? [A2: nat,B4: nat,C3: nat] :
            ( ( A12 = A2 )
            & ( A23 = C3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ C3 ) @ S ) ) ) ) ).

% relcomp.simps
thf(fact_984_relcomp_OrelcompI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,R: set_Pr1685704261205283783r_prat,C2: produc1457211279475724562t_char > option_val,S: set_Pr7649655576376376530on_val] :
      ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ B ) @ R )
     => ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ C2 ) @ S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C2 ) @ ( relcom2642467956565963995on_val @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_985_relcomp_OrelcompI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,C2: produc1457211279475724562t_char > option_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B @ C2 ) @ S )
       => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C2 ) @ ( relcom3547033359321004872on_val @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_986_relcomp_OrelcompI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,C2: produc9003781588162623004on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ C2 ) @ S )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ C2 ) @ ( relcom3664992426205683031on_val @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_987_relcomp_OrelcompI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,C2: nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C2 ) @ S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) ).

% relcomp.relcompI
thf(fact_988_relcompE,axiom,
    ! [Xz: produc9003781588162623004on_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ Xz @ ( relcom2642467956565963995on_val @ R @ S ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > option_val] :
            ( ( Xz
              = ( produc74620467203698198on_val @ X3 @ Z2 ) )
           => ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X3 @ Y3 ) @ R )
             => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_989_relcompE,axiom,
    ! [Xz: produc9003781588162623004on_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ Xz @ ( relcom3547033359321004872on_val @ R @ S ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val,Z2: produc1457211279475724562t_char > option_val] :
            ( ( Xz
              = ( produc74620467203698198on_val @ X3 @ Z2 ) )
           => ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ R )
             => ~ ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_990_relcompE,axiom,
    ! [Xz: produc1317171644805857433on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ Xz @ ( relcom3664992426205683031on_val @ R @ S ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] :
            ( ( Xz
              = ( produc4958730763592889809on_val @ X3 @ Z2 ) )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_991_relcompE,axiom,
    ! [Xz: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [X3: nat,Y3: nat,Z2: nat] :
            ( ( Xz
              = ( product_Pair_nat_nat @ X3 @ Z2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).

% relcompE
thf(fact_992_relcompEpair,axiom,
    ! [A: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > option_val,R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C2 ) @ ( relcom2642467956565963995on_val @ R @ S ) )
     => ~ ! [B3: produc1457211279475724562t_char > prat] :
            ( ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ A @ B3 ) @ R )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B3 @ C2 ) @ S ) ) ) ).

% relcompEpair
thf(fact_993_relcompEpair,axiom,
    ! [A: produc1457211279475724562t_char > prat,C2: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ C2 ) @ ( relcom3547033359321004872on_val @ R @ S ) )
     => ~ ! [B3: produc1457211279475724562t_char > option_val] :
            ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B3 ) @ R )
           => ~ ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ B3 @ C2 ) @ S ) ) ) ).

% relcompEpair
thf(fact_994_relcompEpair,axiom,
    ! [A: produc9003781588162623004on_val,C2: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ C2 ) @ ( relcom3664992426205683031on_val @ R @ S ) )
     => ~ ! [B3: produc9003781588162623004on_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B3 ) @ R )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ C2 ) @ S ) ) ) ).

% relcompEpair
thf(fact_995_relcompEpair,axiom,
    ! [A: nat,C2: nat,R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C2 ) @ ( relcomp_nat_nat_nat @ R @ S ) )
     => ~ ! [B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C2 ) @ S ) ) ) ).

% relcompEpair
thf(fact_996_total__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 )
           => ( ( X3 != Y3 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_3592101749530773125at_nat @ A5 @ R ) ) ).

% total_onI
thf(fact_997_total__onI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X3 @ A5 )
         => ( ( member2879030624002546611on_val @ Y3 @ A5 )
           => ( ( X3 != Y3 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
                | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_4784279244336621976on_val @ A5 @ R ) ) ).

% total_onI
thf(fact_998_total__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( ( member_nat @ Y3 @ A5 )
           => ( ( X3 != Y3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) ) ) )
     => ( total_on_nat @ A5 @ R ) ) ).

% total_onI
thf(fact_999_total__on__def,axiom,
    ( total_4784279244336621976on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ A6 )
         => ! [Y5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y5 @ A6 )
             => ( ( X2 != Y5 )
               => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 )
                  | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_1000_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ( ( X2 != Y5 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_1001_wf__comp__self,axiom,
    ( wf_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] : ( wf_nat @ ( relcomp_nat_nat_nat @ R4 @ R4 ) ) ) ) ).

% wf_comp_self
thf(fact_1002_total__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( total_on_nat @ top_top_set_nat @ R_A )
     => ( ( total_on_nat @ top_top_set_nat @ R_B )
       => ( total_3592101749530773125at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% total_lex_prod
thf(fact_1003_wf__relcomp__compatible,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S4 ) @ ( relcomp_nat_nat_nat @ S4 @ R2 ) )
       => ( wf_nat @ ( relcomp_nat_nat_nat @ S4 @ R2 ) ) ) ) ).

% wf_relcomp_compatible
thf(fact_1004_wf__union__merge,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S4 ) )
      = ( wf_nat @ ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( relcomp_nat_nat_nat @ R2 @ R2 ) @ ( relcomp_nat_nat_nat @ S4 @ R2 ) ) @ S4 ) ) ) ).

% wf_union_merge
thf(fact_1005_relcompp__relcomp__eq,axiom,
    ! [R: set_Pr1685704261205283783r_prat,S: set_Pr7649655576376376530on_val] :
      ( ( relcom8684624306960484107on_val
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X2 @ Y5 ) @ R )
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ S ) )
      = ( ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ ( relcom2642467956565963995on_val @ R @ S ) ) ) ) ).

% relcompp_relcomp_eq
thf(fact_1006_relcompp__relcomp__eq,axiom,
    ! [R: set_Pr7649655576376376530on_val,S: set_Pr4698884584343881589on_val] :
      ( ( relcom7279225663578213656on_val
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R )
        @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > option_val] : ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ X2 @ Y5 ) @ S ) )
      = ( ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ ( relcom3547033359321004872on_val @ R @ S ) ) ) ) ).

% relcompp_relcomp_eq
thf(fact_1007_relcompp__relcomp__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val,S: set_Pr4305388979797279737on_val] :
      ( ( relcom8888620135848005511on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R )
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ S ) )
      = ( ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ ( relcom3664992426205683031on_val @ R @ S ) ) ) ) ).

% relcompp_relcomp_eq
thf(fact_1008_relcompp__relcomp__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( relcompp_nat_nat_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ S ) )
      = ( ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ ( relcomp_nat_nat_nat @ R @ S ) ) ) ) ).

% relcompp_relcomp_eq
thf(fact_1009_totalI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( X3 != Y3 )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
            | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R ) ) )
     => ( total_4784279244336621976on_val @ top_to1561438921946393474on_val @ R ) ) ).

% totalI
thf(fact_1010_totalI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( X3 != Y3 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
            | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R ) ) )
     => ( total_on_nat @ top_top_set_nat @ R ) ) ).

% totalI
thf(fact_1011_wf__union__compatible,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R2 )
     => ( ( wf_nat @ S4 )
       => ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R2 @ S4 ) @ R2 )
         => ( wf_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S4 ) ) ) ) ) ).

% wf_union_compatible
thf(fact_1012_relcomp__def,axiom,
    ( relcom2642467956565963995on_val
    = ( ^ [R3: set_Pr1685704261205283783r_prat,S3: set_Pr7649655576376376530on_val] :
          ( collec7475294502966882545on_val
          @ ( produc5248837549992124053_val_o
            @ ( relcom8684624306960484107on_val
              @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > prat] : ( member1381939458143960592r_prat @ ( produc1164665353665699927r_prat @ X2 @ Y5 ) @ R3 )
              @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_1013_relcomp__def,axiom,
    ( relcom3547033359321004872on_val
    = ( ^ [R3: set_Pr7649655576376376530on_val,S3: set_Pr4698884584343881589on_val] :
          ( collec7475294502966882545on_val
          @ ( produc5248837549992124053_val_o
            @ ( relcom7279225663578213656on_val
              @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R3 )
              @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > option_val] : ( member5502714215428086462on_val @ ( produc8008623296994167373on_val @ X2 @ Y5 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_1014_relcomp__def,axiom,
    ( relcom3664992426205683031on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,S3: set_Pr4305388979797279737on_val] :
          ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ( relcom8888620135848005511on_val
              @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 )
              @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_1015_relcomp__def,axiom,
    ( relcomp_nat_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( relcompp_nat_nat_nat
              @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 )
              @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ S3 ) ) ) ) ) ) ).

% relcomp_def
thf(fact_1016_insert__relcomp__union__fold,axiom,
    ! [S4: set_Pr4698884584343881589on_val,X: produc9003781588162623004on_val,X4: set_Pr7649655576376376530on_val] :
      ( ( finite9151518043362608574on_val @ S4 )
     => ( ( sup_su7506060154676209150on_val @ ( relcom3547033359321004872on_val @ ( insert3030518898131756108on_val @ X @ bot_bo6722054719108133734on_val ) @ S4 ) @ X4 )
        = ( finite7387624949280933791on_val
          @ ( produc6383109249687246501on_val
            @ ^ [W: produc1457211279475724562t_char > option_val,Z5: produc1457211279475724562t_char > option_val,A9: set_Pr7649655576376376530on_val] :
                ( if_set1193336150724294796on_val
                @ ( ( produc2319605628138516840on_val @ X )
                  = W )
                @ ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ X ) @ Z5 ) @ A9 )
                @ A9 ) )
          @ X4
          @ S4 ) ) ) ).

% insert_relcomp_union_fold
thf(fact_1017_insert__relcomp__fold,axiom,
    ! [S4: set_Pr4698884584343881589on_val,X: produc9003781588162623004on_val,R2: set_Pr7649655576376376530on_val] :
      ( ( finite9151518043362608574on_val @ S4 )
     => ( ( relcom3547033359321004872on_val @ ( insert3030518898131756108on_val @ X @ R2 ) @ S4 )
        = ( finite7387624949280933791on_val
          @ ( produc6383109249687246501on_val
            @ ^ [W: produc1457211279475724562t_char > option_val,Z5: produc1457211279475724562t_char > option_val,A9: set_Pr7649655576376376530on_val] :
                ( if_set1193336150724294796on_val
                @ ( ( produc2319605628138516840on_val @ X )
                  = W )
                @ ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ ( produc5239619271231976746on_val @ X ) @ Z5 ) @ A9 )
                @ A9 ) )
          @ ( relcom3547033359321004872on_val @ R2 @ S4 )
          @ S4 ) ) ) ).

% insert_relcomp_fold
thf(fact_1018_Id__on__fold,axiom,
    ! [A5: set_Pr7649655576376376530on_val] :
      ( ( finite8324237431073730739on_val @ A5 )
     => ( ( id_on_1505035389332359999on_val @ A5 )
        = ( finite5491383895224522515on_val
          @ ^ [X2: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X2 @ X2 ) )
          @ bot_bo402267660617762917on_val
          @ A5 ) ) ) ).

% Id_on_fold
thf(fact_1019_Id__on__fold,axiom,
    ! [A5: set_nat] :
      ( ( finite_finite_nat @ A5 )
     => ( ( id_on_nat @ A5 )
        = ( finite3745491028973389255at_nat
          @ ^ [X2: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) )
          @ bot_bo2099793752762293965at_nat
          @ A5 ) ) ) ).

% Id_on_fold
thf(fact_1020_fold__union__pair,axiom,
    ! [B5: set_Pr1004501353197068112on_val,X: produc1457211279475724562t_char > prat,A5: set_Pr7649655576376376530on_val] :
      ( ( finite2752370459137344561on_val @ B5 )
     => ( ( sup_su7506060154676209150on_val
          @ ( comple5258588689616361253on_val
            @ ( image_8136294080383932403on_val
              @ ^ [Y5: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X @ Y5 ) @ bot_bo6722054719108133734on_val )
              @ B5 ) )
          @ A5 )
        = ( finite4470675969487272118on_val
          @ ^ [Y5: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X @ Y5 ) )
          @ A5
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_1021_fold__union__pair,axiom,
    ! [B5: set_Pr7649655576376376530on_val,X: produc9003781588162623004on_val,A5: set_Pr4305388979797279737on_val] :
      ( ( finite8324237431073730739on_val @ B5 )
     => ( ( sup_su3890347010539237325on_val
          @ ( comple3779562415432662246on_val
            @ ( image_3026073218995088406on_val
              @ ^ [Y5: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X @ Y5 ) @ bot_bo402267660617762917on_val )
              @ B5 ) )
          @ A5 )
        = ( finite5491383895224522515on_val
          @ ^ [Y5: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X @ Y5 ) )
          @ A5
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_1022_fold__union__pair,axiom,
    ! [B5: set_nat,X: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( sup_su6327502436637775413at_nat
          @ ( comple5685304695842803022at_nat
            @ ( image_7178329752028323786at_nat
              @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ bot_bo2099793752762293965at_nat )
              @ B5 ) )
          @ A5 )
        = ( finite3745491028973389255at_nat
          @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) )
          @ A5
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_1023_product__fold,axiom,
    ! [A5: set_Pr3433046674199179443r_prat,B5: set_Pr1004501353197068112on_val] :
      ( ( finite9036063106636032508r_prat @ A5 )
     => ( ( finite2752370459137344561on_val @ B5 )
       => ( ( produc1366239611153933861on_val @ A5
            @ ^ [Uu3: produc1457211279475724562t_char > prat] : B5 )
          = ( finite3809360149423730653on_val
            @ ^ [X2: produc1457211279475724562t_char > prat,Z5: set_Pr7649655576376376530on_val] :
                ( finite4470675969487272118on_val
                @ ^ [Y5: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) )
                @ Z5
                @ B5 )
            @ bot_bo6722054719108133734on_val
            @ A5 ) ) ) ) ).

% product_fold
thf(fact_1024_product__fold,axiom,
    ! [A5: set_Pr7649655576376376530on_val,B5: set_Pr7649655576376376530on_val] :
      ( ( finite8324237431073730739on_val @ A5 )
     => ( ( finite8324237431073730739on_val @ B5 )
       => ( ( produc4067394217145614850on_val @ A5
            @ ^ [Uu3: produc9003781588162623004on_val] : B5 )
          = ( finite5491383895224522515on_val
            @ ^ [X2: produc9003781588162623004on_val,Z5: set_Pr4305388979797279737on_val] :
                ( finite5491383895224522515on_val
                @ ^ [Y5: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) )
                @ Z5
                @ B5 )
            @ bot_bo402267660617762917on_val
            @ A5 ) ) ) ) ).

% product_fold
thf(fact_1025_product__fold,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A5 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( produc457027306803732586at_nat @ A5
            @ ^ [Uu3: nat] : B5 )
          = ( finite3745491028973389255at_nat
            @ ^ [X2: nat,Z5: set_Pr1261947904930325089at_nat] :
                ( finite3745491028973389255at_nat
                @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) )
                @ Z5
                @ B5 )
            @ bot_bo2099793752762293965at_nat
            @ A5 ) ) ) ) ).

% product_fold
thf(fact_1026_comp__fun__commute__product__fold,axiom,
    ! [B5: set_Pr1004501353197068112on_val] :
      ( ( finite2752370459137344561on_val @ B5 )
     => ( finite6863617556316062612on_val
        @ ^ [X2: produc1457211279475724562t_char > prat,Z5: set_Pr7649655576376376530on_val] :
            ( finite4470675969487272118on_val
            @ ^ [Y5: produc1457211279475724562t_char > option_val] : ( insert3030518898131756108on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) )
            @ Z5
            @ B5 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1027_comp__fun__commute__product__fold,axiom,
    ! [B5: set_Pr7649655576376376530on_val] :
      ( ( finite8324237431073730739on_val @ B5 )
     => ( finite3060867834099665098on_val
        @ ^ [X2: produc9003781588162623004on_val,Z5: set_Pr4305388979797279737on_val] :
            ( finite5491383895224522515on_val
            @ ^ [Y5: produc9003781588162623004on_val] : ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) )
            @ Z5
            @ B5 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1028_comp__fun__commute__product__fold,axiom,
    ! [B5: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( finite1174777150170982526at_nat
        @ ^ [X2: nat,Z5: set_Pr1261947904930325089at_nat] :
            ( finite3745491028973389255at_nat
            @ ^ [Y5: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) )
            @ Z5
            @ B5 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_1029_bsqr__def,axiom,
    ( bNF_We3507095839090954431on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
          ( collec4536212101028033490on_val
          @ ( produc3489267648137303980_val_o
            @ ( produc7816547785737266186_val_o
              @ ^ [A13: produc9003781588162623004on_val,A24: produc9003781588162623004on_val] :
                  ( produc1845727980445322802_val_o
                  @ ^ [B1: produc9003781588162623004on_val,B22: produc9003781588162623004on_val] :
                      ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ A13 @ ( insert3030518898131756108on_val @ A24 @ ( insert3030518898131756108on_val @ B1 @ ( insert3030518898131756108on_val @ B22 @ bot_bo6722054719108133734on_val ) ) ) ) @ ( field_9010718613293825679on_val @ R3 ) )
                      & ( ( ( A13 = B1 )
                          & ( A24 = B22 ) )
                        | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( bNF_We3765001366693588814on_val @ R3 @ A13 @ A24 ) @ ( bNF_We3765001366693588814on_val @ R3 @ B1 @ B22 ) ) @ ( minus_7211471524202483680on_val @ R3 @ id_Pro1455714331835775424on_val ) )
                        | ( ( ( bNF_We3765001366693588814on_val @ R3 @ A13 @ A24 )
                            = ( bNF_We3765001366693588814on_val @ R3 @ B1 @ B22 ) )
                          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A13 @ B1 ) @ ( minus_7211471524202483680on_val @ R3 @ id_Pro1455714331835775424on_val ) ) )
                        | ( ( ( bNF_We3765001366693588814on_val @ R3 @ A13 @ A24 )
                            = ( bNF_We3765001366693588814on_val @ R3 @ B1 @ B22 ) )
                          & ( A13 = B1 )
                          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A24 @ B22 ) @ ( minus_7211471524202483680on_val @ R3 @ id_Pro1455714331835775424on_val ) ) ) ) ) ) ) ) ) ) ) ).

% bsqr_def
thf(fact_1030_bsqr__def,axiom,
    ( bNF_We1732890022510290291qr_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A13: nat,A24: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [B1: nat,B22: nat] :
                      ( ( ord_less_eq_set_nat @ ( insert_nat @ A13 @ ( insert_nat @ A24 @ ( insert_nat @ B1 @ ( insert_nat @ B22 @ bot_bot_set_nat ) ) ) ) @ ( field_nat @ R3 ) )
                      & ( ( ( A13 = B1 )
                          & ( A24 = B22 ) )
                        | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R3 @ A13 @ A24 ) @ ( bNF_We2380939065524664066x2_nat @ R3 @ B1 @ B22 ) ) @ ( minus_1356011639430497352at_nat @ R3 @ id_nat2 ) )
                        | ( ( ( bNF_We2380939065524664066x2_nat @ R3 @ A13 @ A24 )
                            = ( bNF_We2380939065524664066x2_nat @ R3 @ B1 @ B22 ) )
                          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A13 @ B1 ) @ ( minus_1356011639430497352at_nat @ R3 @ id_nat2 ) ) )
                        | ( ( ( bNF_We2380939065524664066x2_nat @ R3 @ A13 @ A24 )
                            = ( bNF_We2380939065524664066x2_nat @ R3 @ B1 @ B22 ) )
                          & ( A13 = B1 )
                          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A24 @ B22 ) @ ( minus_1356011639430497352at_nat @ R3 @ id_nat2 ) ) ) ) ) ) ) ) ) ) ) ).

% bsqr_def
thf(fact_1031_converse__iff,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A @ B ) @ ( conver385509077141697010on_val @ R ) )
      = ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ A ) @ R ) ) ).

% converse_iff
thf(fact_1032_converse__iff,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr2871993341207995108r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( conver2208318512210485948r_prat @ R ) )
      = ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ B @ A ) @ R ) ) ).

% converse_iff
thf(fact_1033_converse__iff,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( conver3382110228903287541on_val @ R ) )
      = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R ) ) ).

% converse_iff
thf(fact_1034_converse__iff,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( converse_nat_nat @ R ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ).

% converse_iff
thf(fact_1035_converseI,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > prat,R: set_Pr2871993341207995108r_prat] :
      ( ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A @ B ) @ R )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ A ) @ ( conver2208318512210485948r_prat @ R ) ) ) ).

% converseI
thf(fact_1036_converseI,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr7649655576376376530on_val] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ R )
     => ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ B @ A ) @ ( conver385509077141697010on_val @ R ) ) ) ).

% converseI
thf(fact_1037_converseI,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ ( conver3382110228903287541on_val @ R ) ) ) ).

% converseI
thf(fact_1038_converseI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( converse_nat_nat @ R ) ) ) ).

% converseI
thf(fact_1039_converseE,axiom,
    ! [Yx: produc7776263825685906734r_prat,R: set_Pr7649655576376376530on_val] :
      ( ( member1651512861525830341r_prat @ Yx @ ( conver385509077141697010on_val @ R ) )
     => ~ ! [X3: produc1457211279475724562t_char > prat,Y3: produc1457211279475724562t_char > option_val] :
            ( ( Yx
              = ( produc1897429902272487136r_prat @ Y3 @ X3 ) )
           => ~ ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X3 @ Y3 ) @ R ) ) ) ).

% converseE
thf(fact_1040_converseE,axiom,
    ! [Yx: produc9003781588162623004on_val,R: set_Pr2871993341207995108r_prat] :
      ( ( member2879030624002546611on_val @ Yx @ ( conver2208318512210485948r_prat @ R ) )
     => ~ ! [X3: produc1457211279475724562t_char > option_val,Y3: produc1457211279475724562t_char > prat] :
            ( ( Yx
              = ( produc74620467203698198on_val @ Y3 @ X3 ) )
           => ~ ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ X3 @ Y3 ) @ R ) ) ) ).

% converseE
thf(fact_1041_converseE,axiom,
    ! [Yx: produc1317171644805857433on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ Yx @ ( conver3382110228903287541on_val @ R ) )
     => ~ ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
            ( ( Yx
              = ( produc4958730763592889809on_val @ Y3 @ X3 ) )
           => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R ) ) ) ).

% converseE
thf(fact_1042_converseE,axiom,
    ! [Yx: product_prod_nat_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Yx @ ( converse_nat_nat @ R ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Yx
              = ( product_Pair_nat_nat @ Y3 @ X3 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ).

% converseE
thf(fact_1043_converseD,axiom,
    ! [A: produc1457211279475724562t_char > option_val,B: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A @ B ) @ ( conver385509077141697010on_val @ R ) )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ B @ A ) @ R ) ) ).

% converseD
thf(fact_1044_converseD,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > option_val,R: set_Pr2871993341207995108r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A @ B ) @ ( conver2208318512210485948r_prat @ R ) )
     => ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ B @ A ) @ R ) ) ).

% converseD
thf(fact_1045_converseD,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( conver3382110228903287541on_val @ R ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R ) ) ).

% converseD
thf(fact_1046_converseD,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( converse_nat_nat @ R ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ).

% converseD
thf(fact_1047_converse_Osimps,axiom,
    ! [A12: produc1457211279475724562t_char > option_val,A23: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A12 @ A23 ) @ ( conver385509077141697010on_val @ R ) )
      = ( ? [A2: produc1457211279475724562t_char > prat,B4: produc1457211279475724562t_char > option_val] :
            ( ( A12 = B4 )
            & ( A23 = A2 )
            & ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A2 @ B4 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_1048_converse_Osimps,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr2871993341207995108r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( conver2208318512210485948r_prat @ R ) )
      = ( ? [A2: produc1457211279475724562t_char > option_val,B4: produc1457211279475724562t_char > prat] :
            ( ( A12 = B4 )
            & ( A23 = A2 )
            & ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A2 @ B4 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_1049_converse_Osimps,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( conver3382110228903287541on_val @ R ) )
      = ( ? [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] :
            ( ( A12 = B4 )
            & ( A23 = A2 )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_1050_converse_Osimps,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( converse_nat_nat @ R ) )
      = ( ? [A2: nat,B4: nat] :
            ( ( A12 = B4 )
            & ( A23 = A2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R ) ) ) ) ).

% converse.simps
thf(fact_1051_converse_Ocases,axiom,
    ! [A12: produc1457211279475724562t_char > option_val,A23: produc1457211279475724562t_char > prat,R: set_Pr7649655576376376530on_val] :
      ( ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A12 @ A23 ) @ ( conver385509077141697010on_val @ R ) )
     => ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A23 @ A12 ) @ R ) ) ).

% converse.cases
thf(fact_1052_converse_Ocases,axiom,
    ! [A12: produc1457211279475724562t_char > prat,A23: produc1457211279475724562t_char > option_val,R: set_Pr2871993341207995108r_prat] :
      ( ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ A12 @ A23 ) @ ( conver2208318512210485948r_prat @ R ) )
     => ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ A23 @ A12 ) @ R ) ) ).

% converse.cases
thf(fact_1053_converse_Ocases,axiom,
    ! [A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( conver3382110228903287541on_val @ R ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A23 @ A12 ) @ R ) ) ).

% converse.cases
thf(fact_1054_converse_Ocases,axiom,
    ! [A12: nat,A23: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( converse_nat_nat @ R ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A23 @ A12 ) @ R ) ) ).

% converse.cases
thf(fact_1055_bsqr__max2,axiom,
    ! [R: set_Pr4305388979797279737on_val,A12: produc9003781588162623004on_val,A23: produc9003781588162623004on_val,B12: produc9003781588162623004on_val,B23: produc9003781588162623004on_val] :
      ( ( order_2579646547231075838on_val @ ( field_9010718613293825679on_val @ R ) @ R )
     => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A12 @ A23 ) @ ( produc4958730763592889809on_val @ B12 @ B23 ) ) @ ( bNF_We3507095839090954431on_val @ R ) )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( bNF_We3765001366693588814on_val @ R @ A12 @ A23 ) @ ( bNF_We3765001366693588814on_val @ R @ B12 @ B23 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_1056_bsqr__max2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A12: nat,A23: nat,B12: nat,B23: nat] :
      ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A12 @ A23 ) @ ( product_Pair_nat_nat @ B12 @ B23 ) ) @ ( bNF_We1732890022510290291qr_nat @ R ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A12 @ A23 ) @ ( bNF_We2380939065524664066x2_nat @ R @ B12 @ B23 ) ) @ R ) ) ) ).

% bsqr_max2
thf(fact_1057_converse__unfold,axiom,
    ( conver2208318512210485948r_prat
    = ( ^ [R3: set_Pr2871993341207995108r_prat] :
          ( collec7475294502966882545on_val
          @ ( produc5248837549992124053_val_o
            @ ^ [Y5: produc1457211279475724562t_char > prat,X2: produc1457211279475724562t_char > option_val] : ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% converse_unfold
thf(fact_1058_converse__unfold,axiom,
    ( conver385509077141697010on_val
    = ( ^ [R3: set_Pr7649655576376376530on_val] :
          ( collec6247776740490166275r_prat
          @ ( produc5226666187743293259prat_o
            @ ^ [Y5: produc1457211279475724562t_char > option_val,X2: produc1457211279475724562t_char > prat] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% converse_unfold
thf(fact_1059_converse__unfold,axiom,
    ( conver3382110228903287541on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
          ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ^ [Y5: produc9003781588162623004on_val,X2: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% converse_unfold
thf(fact_1060_converse__unfold,axiom,
    ( converse_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [Y5: nat,X2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 ) ) ) ) ) ).

% converse_unfold
thf(fact_1061_conversep__converse__eq,axiom,
    ! [R: set_Pr2871993341207995108r_prat] :
      ( ( conver4900003399366114050r_prat
        @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > prat] : ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ ( conver2208318512210485948r_prat @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_1062_conversep__converse__eq,axiom,
    ! [R: set_Pr7649655576376376530on_val] :
      ( ( conver3077193964297325112on_val
        @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > prat] : ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ X2 @ Y5 ) @ ( conver385509077141697010on_val @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_1063_conversep__converse__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( conver1343519957138882863on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ ( conver3382110228903287541on_val @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_1064_conversep__converse__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( conversep_nat_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ ( converse_nat_nat @ R ) ) ) ) ).

% conversep_converse_eq
thf(fact_1065_Pair__vimage__Sigma,axiom,
    ! [X: produc1457211279475724562t_char > prat,A5: set_Pr3433046674199179443r_prat,F: ( produc1457211279475724562t_char > prat ) > set_Pr1004501353197068112on_val] :
      ( ( ( member2336342582916103932r_prat @ X @ A5 )
       => ( ( vimage2891818772453376755on_val @ ( produc74620467203698198on_val @ X ) @ ( produc1366239611153933861on_val @ A5 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member2336342582916103932r_prat @ X @ A5 )
       => ( ( vimage2891818772453376755on_val @ ( produc74620467203698198on_val @ X ) @ ( produc1366239611153933861on_val @ A5 @ F ) )
          = bot_bo5671083314777306084on_val ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1066_Pair__vimage__Sigma,axiom,
    ! [X: produc9003781588162623004on_val,A5: set_Pr7649655576376376530on_val,F: produc9003781588162623004on_val > set_Pr7649655576376376530on_val] :
      ( ( ( member2879030624002546611on_val @ X @ A5 )
       => ( ( vimage6220219822850736768on_val @ ( produc4958730763592889809on_val @ X ) @ ( produc4067394217145614850on_val @ A5 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member2879030624002546611on_val @ X @ A5 )
       => ( ( vimage6220219822850736768on_val @ ( produc4958730763592889809on_val @ X ) @ ( produc4067394217145614850on_val @ A5 @ F ) )
          = bot_bo6722054719108133734on_val ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1067_Pair__vimage__Sigma,axiom,
    ! [X: nat,A5: set_nat,F: nat > set_nat] :
      ( ( ( member_nat @ X @ A5 )
       => ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X ) @ ( produc457027306803732586at_nat @ A5 @ F ) )
          = ( F @ X ) ) )
      & ( ~ ( member_nat @ X @ A5 )
       => ( ( vimage8013328719654469172at_nat @ ( product_Pair_nat_nat @ X ) @ ( produc457027306803732586at_nat @ A5 @ F ) )
          = bot_bot_set_nat ) ) ) ).

% Pair_vimage_Sigma
thf(fact_1068_vimage__fst,axiom,
    ! [A5: set_Pr3433046674199179443r_prat] :
      ( ( vimage4321719143775365178r_prat @ produc5239619271231976746on_val @ A5 )
      = ( produc1366239611153933861on_val @ A5
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : top_to3326603054294128128on_val ) ) ).

% vimage_fst
thf(fact_1069_vimage__snd,axiom,
    ! [A5: set_Pr1004501353197068112on_val] :
      ( ( vimage1979914069722614899on_val @ produc2319605628138516840on_val @ A5 )
      = ( produc1366239611153933861on_val @ top_to6849109582029272579r_prat
        @ ^ [Uu3: produc1457211279475724562t_char > prat] : A5 ) ) ).

% vimage_snd
thf(fact_1070_converse__def,axiom,
    ( conver2208318512210485948r_prat
    = ( ^ [R3: set_Pr2871993341207995108r_prat] :
          ( collec7475294502966882545on_val
          @ ( produc5248837549992124053_val_o
            @ ( conver4900003399366114050r_prat
              @ ^ [X2: produc1457211279475724562t_char > option_val,Y5: produc1457211279475724562t_char > prat] : ( member1651512861525830341r_prat @ ( produc1897429902272487136r_prat @ X2 @ Y5 ) @ R3 ) ) ) ) ) ) ).

% converse_def
thf(fact_1071_converse__def,axiom,
    ( conver385509077141697010on_val
    = ( ^ [R3: set_Pr7649655576376376530on_val] :
          ( collec6247776740490166275r_prat
          @ ( produc5226666187743293259prat_o
            @ ( conver3077193964297325112on_val
              @ ^ [X2: produc1457211279475724562t_char > prat,Y5: produc1457211279475724562t_char > option_val] : ( member2879030624002546611on_val @ ( produc74620467203698198on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ) ).

% converse_def
thf(fact_1072_converse__def,axiom,
    ( conver3382110228903287541on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val] :
          ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ( conver1343519957138882863on_val
              @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 ) ) ) ) ) ) ).

% converse_def
thf(fact_1073_converse__def,axiom,
    ( converse_nat_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ( conversep_nat_nat
              @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 ) ) ) ) ) ) ).

% converse_def
thf(fact_1074_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,Phi5: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( field_9010718613293825679on_val @ R ) )
       => ( ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) )
              | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ ( minus_7211471524202483680on_val @ R @ id_Pro1455714331835775424on_val ) ) )
           => ( Phi5 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi5 @ A @ B ) )
           => ( Phi5 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_1075_wo__rel_Ocases__Total3,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi5: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
       => ( ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( minus_1356011639430497352at_nat @ R @ id_nat2 ) )
              | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( minus_1356011639430497352at_nat @ R @ id_nat2 ) ) )
           => ( Phi5 @ A @ B ) )
         => ( ( ( A = B )
             => ( Phi5 @ A @ B ) )
           => ( Phi5 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total3
thf(fact_1076_finite__acyclic__wf__converse,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R )
     => ( ( transi3139058337892597825ic_nat @ R )
       => ( wf_nat @ ( converse_nat_nat @ R ) ) ) ) ).

% finite_acyclic_wf_converse
thf(fact_1077_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
         => ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
            = B ) )
        & ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
         => ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_1078_wo__rel_Omax2__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
         => ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
            = B ) )
        & ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
         => ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
            = A ) ) ) ) ).

% wo_rel.max2_def
thf(fact_1079_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr4305388979797279737on_val,P2: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ! [Y4: produc9003781588162623004on_val] :
                ( ( ( Y4 != X3 )
                  & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ X3 ) @ R ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1080_wo__rel_Owell__order__induct,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( ( Y4 != X3 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1081_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ! [X8: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X8 @ ( field_9010718613293825679on_val @ R ) )
         => ! [Xa3: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Xa3 @ ( field_9010718613293825679on_val @ R ) )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X8 @ Xa3 ) @ R )
                | ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Xa3 @ X8 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1082_wo__rel_OTOTALS,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ! [X8: nat] :
          ( ( member_nat @ X8 @ ( field_nat @ R ) )
         => ! [Xa3: nat] :
              ( ( member_nat @ Xa3 @ ( field_nat @ R ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X8 @ Xa3 ) @ R )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X8 ) @ R ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1083_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr8693737435421807431at_nat,J2: product_prod_nat_nat,I2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ J2 @ I2 ) @ R )
          | ( J2 = I2 ) )
       => ( ( member8440522571783428010at_nat @ I2 @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ J2 @ ( field_4706345496283612622at_nat @ R ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1084_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr4305388979797279737on_val,J2: produc9003781588162623004on_val,I2: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ J2 @ I2 ) @ R )
          | ( J2 = I2 ) )
       => ( ( member2879030624002546611on_val @ I2 @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member2879030624002546611on_val @ J2 @ ( field_9010718613293825679on_val @ R ) )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ I2 @ J2 ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1085_wo__rel_Oin__notinI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,J2: nat,I2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J2 @ I2 ) @ R )
          | ( J2 = I2 ) )
       => ( ( member_nat @ I2 @ ( field_nat @ R ) )
         => ( ( member_nat @ J2 @ ( field_nat @ R ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1086_wf__acyclic,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( transi3139058337892597825ic_nat @ R ) ) ).

% wf_acyclic
thf(fact_1087_well__order__induct__imp,axiom,
    ! [R: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ! [X3: product_prod_nat_nat] :
            ( ! [Y4: product_prod_nat_nat] :
                ( ( ( Y4 != X3 )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X3 ) @ R ) )
               => ( ( member8440522571783428010at_nat @ Y4 @ ( field_4706345496283612622at_nat @ R ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member8440522571783428010at_nat @ X3 @ ( field_4706345496283612622at_nat @ R ) )
             => ( P2 @ X3 ) ) )
       => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1088_well__order__induct__imp,axiom,
    ! [R: set_Pr4305388979797279737on_val,P2: produc9003781588162623004on_val > $o,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ! [X3: produc9003781588162623004on_val] :
            ( ! [Y4: produc9003781588162623004on_val] :
                ( ( ( Y4 != X3 )
                  & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y4 @ X3 ) @ R ) )
               => ( ( member2879030624002546611on_val @ Y4 @ ( field_9010718613293825679on_val @ R ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member2879030624002546611on_val @ X3 @ ( field_9010718613293825679on_val @ R ) )
             => ( P2 @ X3 ) ) )
       => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1089_well__order__induct__imp,axiom,
    ! [R: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( ( Y4 != X3 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R ) )
               => ( ( member_nat @ Y4 @ ( field_nat @ R ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member_nat @ X3 @ ( field_nat @ R ) )
             => ( P2 @ X3 ) ) )
       => ( ( member_nat @ A @ ( field_nat @ R ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1090_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( ( bNF_We6793920340776068943at_nat @ R @ A @ B )
              = A )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1091_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
              = A )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1092_wo__rel_Omax2__equals1,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
              = A )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals1
thf(fact_1093_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( ( bNF_We6793920340776068943at_nat @ R @ A @ B )
              = B )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1094_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( ( bNF_We3765001366693588814on_val @ R @ A @ B )
              = B )
            = ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1095_wo__rel_Omax2__equals2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( ( bNF_We2380939065524664066x2_nat @ R @ A @ B )
              = B )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) ) ) ) ) ).

% wo_rel.max2_equals2
thf(fact_1096_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ ( bNF_We6793920340776068943at_nat @ R @ A @ B ) ) @ R )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ ( bNF_We6793920340776068943at_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1097_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1098_wo__rel_Omax2__greater,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R ) ) ) ) ) ).

% wo_rel.max2_greater
thf(fact_1099_finite__acyclic__wf,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R )
     => ( ( transi3139058337892597825ic_nat @ R )
       => ( wf_nat @ R ) ) ) ).

% finite_acyclic_wf
thf(fact_1100_wf__iff__acyclic__if__finite,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R )
     => ( ( wf_nat @ R )
        = ( transi3139058337892597825ic_nat @ R ) ) ) ).

% wf_iff_acyclic_if_finite
thf(fact_1101_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,Phi5: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) @ ( field_9010718613293825679on_val @ R ) )
       => ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
           => ( Phi5 @ A @ B ) )
         => ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ A ) @ R )
             => ( Phi5 @ A @ B ) )
           => ( Phi5 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1102_wo__rel_Ocases__Total,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi5: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R ) )
       => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
           => ( Phi5 @ A @ B ) )
         => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R )
             => ( Phi5 @ A @ B ) )
           => ( Phi5 @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1103_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ ( bNF_We6793920340776068943at_nat @ R @ A @ B ) ) @ R )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ ( bNF_We6793920340776068943at_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( bNF_We6793920340776068943at_nat @ R @ A @ B ) @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1104_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( member2879030624002546611on_val @ A @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ ( field_9010718613293825679on_val @ R ) )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R )
            & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) ) @ R )
            & ( member2879030624002546611on_val @ ( bNF_We3765001366693588814on_val @ R @ A @ B ) @ ( insert3030518898131756108on_val @ A @ ( insert3030518898131756108on_val @ B @ bot_bo6722054719108133734on_val ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1105_wo__rel_Omax2__greater__among,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( member_nat @ A @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ ( field_nat @ R ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) ) @ R )
            & ( member_nat @ ( bNF_We2380939065524664066x2_nat @ R @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% wo_rel.max2_greater_among
thf(fact_1106_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr8693737435421807431at_nat,B5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( bNF_We4841632414944885965at_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1107_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr4305388979797279737on_val,B5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ B @ B5 )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( bNF_We316632424347587920on_val @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1108_wo__rel_Ominim__least,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B5: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
       => ( ( member_nat @ B @ B5 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( bNF_We3275124272252294916im_nat @ R @ B5 ) @ B ) @ R ) ) ) ) ).

% wo_rel.minim_least
thf(fact_1109_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr8693737435421807431at_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( ord_le3146513528884898305at_nat @ B5 @ ( field_4706345496283612622at_nat @ R ) )
       => ( ( member8440522571783428010at_nat @ A @ B5 )
         => ( ! [B3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B3 @ B5 )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We4841632414944885965at_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1110_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr4305388979797279737on_val,B5: set_Pr7649655576376376530on_val,A: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( ord_le6900622908513426738on_val @ B5 @ ( field_9010718613293825679on_val @ R ) )
       => ( ( member2879030624002546611on_val @ A @ B5 )
         => ( ! [B3: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ B3 @ B5 )
               => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We316632424347587920on_val @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1111_wo__rel_Oequals__minim,axiom,
    ! [R: set_Pr1261947904930325089at_nat,B5: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R ) )
       => ( ( member_nat @ A @ B5 )
         => ( ! [B3: nat] :
                ( ( member_nat @ B3 @ B5 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R ) )
           => ( A
              = ( bNF_We3275124272252294916im_nat @ R @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1112_max__ext_Omax__extI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X4 )
     => ( ( finite6177210948735845034at_nat @ Y8 )
       => ( ( Y8 != bot_bo2099793752762293965at_nat )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ X4 )
               => ? [Xa3: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa3 @ Y8 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Xa3 ) @ R2 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y8 ) @ ( max_ex8135407076693332796at_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1113_max__ext_Omax__extI,axiom,
    ! [X4: set_Pr7649655576376376530on_val,Y8: set_Pr7649655576376376530on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( finite8324237431073730739on_val @ X4 )
     => ( ( finite8324237431073730739on_val @ Y8 )
       => ( ( Y8 != bot_bo6722054719108133734on_val )
         => ( ! [X3: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ X3 @ X4 )
               => ? [Xa3: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ Xa3 @ Y8 )
                    & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Xa3 ) @ R2 ) ) )
           => ( member806545433280400814on_val @ ( produc113594951777976637on_val @ X4 @ Y8 ) @ ( max_ex7543004642404436129on_val @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1114_max__ext_Omax__extI,axiom,
    ! [X4: set_nat,Y8: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X4 )
     => ( ( finite_finite_nat @ Y8 )
       => ( ( Y8 != bot_bot_set_nat )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ X4 )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ Y8 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa3 ) @ R2 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X4 @ Y8 ) @ ( max_ext_nat @ R2 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1115_max__ext__wf,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( wf_set_nat @ ( max_ext_nat @ R ) ) ) ).

% max_ext_wf
thf(fact_1116_max__ext_Ocases,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A12 )
         => ( ( finite6177210948735845034at_nat @ A23 )
           => ( ( A23 != bot_bo2099793752762293965at_nat )
             => ~ ! [X8: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X8 @ A12 )
                   => ? [Xa: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa @ A23 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X8 @ Xa ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1117_max__ext_Ocases,axiom,
    ! [A12: set_Pr7649655576376376530on_val,A23: set_Pr7649655576376376530on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member806545433280400814on_val @ ( produc113594951777976637on_val @ A12 @ A23 ) @ ( max_ex7543004642404436129on_val @ R2 ) )
     => ~ ( ( finite8324237431073730739on_val @ A12 )
         => ( ( finite8324237431073730739on_val @ A23 )
           => ( ( A23 != bot_bo6722054719108133734on_val )
             => ~ ! [X8: produc9003781588162623004on_val] :
                    ( ( member2879030624002546611on_val @ X8 @ A12 )
                   => ? [Xa: produc9003781588162623004on_val] :
                        ( ( member2879030624002546611on_val @ Xa @ A23 )
                        & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X8 @ Xa ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1118_max__ext_Ocases,axiom,
    ! [A12: set_nat,A23: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A12 @ A23 ) @ ( max_ext_nat @ R2 ) )
     => ~ ( ( finite_finite_nat @ A12 )
         => ( ( finite_finite_nat @ A23 )
           => ( ( A23 != bot_bot_set_nat )
             => ~ ! [X8: nat] :
                    ( ( member_nat @ X8 @ A12 )
                   => ? [Xa: nat] :
                        ( ( member_nat @ Xa @ A23 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X8 @ Xa ) @ R2 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1119_max__ext_Osimps,axiom,
    ! [A12: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A12 @ A23 ) @ ( max_ex8135407076693332796at_nat @ R2 ) )
      = ( ( finite6177210948735845034at_nat @ A12 )
        & ( finite6177210948735845034at_nat @ A23 )
        & ( A23 != bot_bo2099793752762293965at_nat )
        & ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A12 )
           => ? [Y5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y5 @ A23 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1120_max__ext_Osimps,axiom,
    ! [A12: set_Pr7649655576376376530on_val,A23: set_Pr7649655576376376530on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( member806545433280400814on_val @ ( produc113594951777976637on_val @ A12 @ A23 ) @ ( max_ex7543004642404436129on_val @ R2 ) )
      = ( ( finite8324237431073730739on_val @ A12 )
        & ( finite8324237431073730739on_val @ A23 )
        & ( A23 != bot_bo6722054719108133734on_val )
        & ! [X2: produc9003781588162623004on_val] :
            ( ( member2879030624002546611on_val @ X2 @ A12 )
           => ? [Y5: produc9003781588162623004on_val] :
                ( ( member2879030624002546611on_val @ Y5 @ A23 )
                & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1121_max__ext_Osimps,axiom,
    ! [A12: set_nat,A23: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A12 @ A23 ) @ ( max_ext_nat @ R2 ) )
      = ( ( finite_finite_nat @ A12 )
        & ( finite_finite_nat @ A23 )
        & ( A23 != bot_bot_set_nat )
        & ! [X2: nat] :
            ( ( member_nat @ X2 @ A12 )
           => ? [Y5: nat] :
                ( ( member_nat @ Y5 @ A23 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1122_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr4305388979797279737on_val] :
      ( ( max_ex1577223295445257289on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: set_Pr7649655576376376530on_val,Y5: set_Pr7649655576376376530on_val] : ( member806545433280400814on_val @ ( produc113594951777976637on_val @ X2 @ Y5 ) @ ( max_ex7543004642404436129on_val @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1123_max__extp__max__ext__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: set_nat,Y5: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X2 @ Y5 ) @ ( max_ext_nat @ R2 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1124_max__ext__def,axiom,
    ( max_ex7543004642404436129on_val
    = ( ^ [R4: set_Pr4305388979797279737on_val] :
          ( collec4160121533503351024on_val
          @ ( produc2326311653359198150_val_o
            @ ( max_ex1577223295445257289on_val
              @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R4 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_1125_max__ext__def,axiom,
    ( max_ext_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ( max_extp_nat
              @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R4 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_1126_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr8693737435421807431at_nat,A5: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R )
     => ( ( bNF_We1232559802637082039at_nat @ R @ A5 @ B )
        = ( ( member8440522571783428010at_nat @ B @ A5 )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A5 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ X2 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1127_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr4305388979797279737on_val,A5: set_Pr7649655576376376530on_val,B: produc9003781588162623004on_val] :
      ( ( bNF_We4395177084845204272on_val @ R )
     => ( ( bNF_We5046398979041125094on_val @ R @ A5 @ B )
        = ( ( member2879030624002546611on_val @ B @ A5 )
          & ! [X2: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ X2 @ A5 )
             => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ X2 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1128_wo__rel_OisMinim__def,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A5: set_nat,B: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R )
     => ( ( bNF_We6513959723706032794im_nat @ R @ A5 @ B )
        = ( ( member_nat @ B @ A5 )
          & ! [X2: nat] :
              ( ( member_nat @ X2 @ A5 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ X2 ) @ R ) ) ) ) ) ).

% wo_rel.isMinim_def
thf(fact_1129_min__ext__wf,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( wf_set_nat @ ( min_ext_nat @ R ) ) ) ).

% min_ext_wf
thf(fact_1130_wf,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf
thf(fact_1131_acyclicI__order,axiom,
    ! [R: set_Pr4305388979797279737on_val,F: produc9003781588162623004on_val > nat] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ B3 ) @ R )
         => ( ord_less_nat @ ( F @ B3 ) @ ( F @ A4 ) ) )
     => ( transi5235682513840182669on_val @ R ) ) ).

% acyclicI_order
thf(fact_1132_acyclicI__order,axiom,
    ! [R: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ R )
         => ( ord_less_nat @ ( F @ B3 ) @ ( F @ A4 ) ) )
     => ( transi3139058337892597825ic_nat @ R ) ) ).

% acyclicI_order
thf(fact_1133_trans__singleton,axiom,
    ! [A: produc9003781588162623004on_val] : ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ A ) @ bot_bo402267660617762917on_val ) ) ).

% trans_singleton
thf(fact_1134_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_1135_above__def,axiom,
    ( order_793026671032835073at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_1136_above__def,axiom,
    ( order_646696875765705500on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A2: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_1137_above__def,axiom,
    ( order_above_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R3 ) ) ) ) ).

% above_def
thf(fact_1138_in__measure,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,F: produc9003781588162623004on_val > nat] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( measur6236884747139477332on_val @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_1139_in__measure,axiom,
    ! [X: nat,Y: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_1140_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_1141_transI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Z2 ) @ R )
           => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Z2 ) @ R ) ) )
     => ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R ) ) ).

% transI
thf(fact_1142_transI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% transI
thf(fact_1143_transE,axiom,
    ! [R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
      ( ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
       => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ Z ) @ R )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_1144_transE,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: 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 @ Z ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R ) ) ) ) ).

% transE
thf(fact_1145_transD,axiom,
    ! [R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
      ( ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
       => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ Z ) @ R )
         => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_1146_transD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: 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 @ Z ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R ) ) ) ) ).

% transD
thf(fact_1147_trans__on__def,axiom,
    ( trans_3212976107463897212on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ A6 )
         => ! [Y5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y5 @ A6 )
             => ! [Z5: produc9003781588162623004on_val] :
                  ( ( member2879030624002546611on_val @ Z5 @ A6 )
                 => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 )
                   => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ Z5 ) @ R3 )
                     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Z5 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_1148_trans__on__def,axiom,
    ( trans_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ! [Z5: nat] :
                  ( ( member_nat @ Z5 @ A6 )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 )
                   => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z5 ) @ R3 )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z5 ) @ R3 ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_1149_trans__onI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A5 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A5 )
           => ( ( member8440522571783428010at_nat @ Z2 @ A5 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z2 ) @ R )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z2 ) @ R ) ) ) ) ) )
     => ( trans_3241703717017304737at_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_1150_trans__onI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val,Z2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X3 @ A5 )
         => ( ( member2879030624002546611on_val @ Y3 @ A5 )
           => ( ( member2879030624002546611on_val @ Z2 @ A5 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
               => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ Z2 ) @ R )
                 => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Z2 ) @ R ) ) ) ) ) )
     => ( trans_3212976107463897212on_val @ A5 @ R ) ) ).

% trans_onI
thf(fact_1151_trans__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( member_nat @ X3 @ A5 )
         => ( ( member_nat @ Y3 @ A5 )
           => ( ( member_nat @ Z2 @ A5 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R ) ) ) ) ) )
     => ( trans_on_nat @ A5 @ R ) ) ).

% trans_onI
thf(fact_1152_trans__onD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( ( member8440522571783428010at_nat @ Y @ A5 )
         => ( ( member8440522571783428010at_nat @ Z @ A5 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z ) @ R )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1153_trans__onD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,Z: produc9003781588162623004on_val] :
      ( ( trans_3212976107463897212on_val @ A5 @ R )
     => ( ( member2879030624002546611on_val @ X @ A5 )
       => ( ( member2879030624002546611on_val @ Y @ A5 )
         => ( ( member2879030624002546611on_val @ Z @ A5 )
           => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ Z ) @ R )
               => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1154_trans__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( ( trans_on_nat @ A5 @ R )
     => ( ( member_nat @ X @ A5 )
       => ( ( member_nat @ Y @ A5 )
         => ( ( member_nat @ Z @ A5 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1155_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
            @ ^ [Uu3: nat] : B5 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% trans_on_lex_prod
thf(fact_1156_wf__bounded__measure,axiom,
    ! [R: set_Pr4305388979797279737on_val,Ub: produc9003781588162623004on_val > nat,F: produc9003781588162623004on_val > nat] :
      ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B3 @ A4 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_eq_nat @ ( F @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B3 ) ) ) )
     => ( wf_Pro3300669915054815219on_val @ R ) ) ).

% wf_bounded_measure
thf(fact_1157_wf__bounded__measure,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Ub: nat > nat,F: nat > nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A4 ) @ R )
         => ( ( ord_less_eq_nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_eq_nat @ ( F @ B3 ) @ ( Ub @ A4 ) )
            & ( ord_less_nat @ ( F @ A4 ) @ ( F @ B3 ) ) ) )
     => ( wf_nat @ R ) ) ).

% wf_bounded_measure
thf(fact_1158_transp__on__trans__on__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ( transp6791619540397306988on_val @ A5
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( trans_3212976107463897212on_val @ A5 @ R ) ) ).

% transp_on_trans_on_eq
thf(fact_1159_transp__on__trans__on__eq,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( transp_on_nat @ A5
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( trans_on_nat @ A5 @ R ) ) ).

% transp_on_trans_on_eq
thf(fact_1160_wf__if__measure,axiom,
    ! [P2: nat > $o,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
         => ( ord_less_nat @ ( F @ ( G @ X3 ) ) @ ( F @ X3 ) ) )
     => ( wf_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [Y5: nat,X2: nat] :
                ( ( P2 @ X2 )
                & ( Y5
                  = ( G @ X2 ) ) ) ) ) ) ) ).

% wf_if_measure
thf(fact_1161_transp__trans__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( transp6791619540397306988on_val @ top_to1561438921946393474on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R ) ) ).

% transp_trans_eq
thf(fact_1162_transp__trans__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( transp_on_nat @ top_top_set_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( trans_on_nat @ top_top_set_nat @ R ) ) ).

% transp_trans_eq
thf(fact_1163_wf__finite__segments,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ R )
     => ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R )
       => ( ! [X3: product_prod_nat_nat] :
              ( finite6177210948735845034at_nat
              @ ( collec3392354462482085612at_nat
                @ ^ [Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X3 ) @ R ) ) )
         => ( wf_Pro7803398752247294826at_nat @ R ) ) ) ) ).

% wf_finite_segments
thf(fact_1164_wf__finite__segments,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( irrefl4606354804476895398on_val @ top_to1561438921946393474on_val @ R )
     => ( ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R )
       => ( ! [X3: produc9003781588162623004on_val] :
              ( finite8324237431073730739on_val
              @ ( collec7475294502966882545on_val
                @ ^ [Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X3 ) @ R ) ) )
         => ( wf_Pro3300669915054815219on_val @ R ) ) ) ) ).

% wf_finite_segments
thf(fact_1165_wf__finite__segments,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ( ( trans_on_nat @ top_top_set_nat @ R )
       => ( ! [X3: nat] :
              ( finite_finite_nat
              @ ( collect_nat
                @ ^ [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X3 ) @ R ) ) )
         => ( wf_nat @ R ) ) ) ) ).

% wf_finite_segments
thf(fact_1166_wf__converse,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ( ( trans_on_nat @ top_top_set_nat @ R )
       => ( ( finite6177210948735845034at_nat @ R )
         => ( wf_nat @ ( converse_nat_nat @ R ) ) ) ) ) ).

% wf_converse
thf(fact_1167_underS__incr,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R )
     => ( ( antisy5014944754269107577on_val @ top_to1561438921946393474on_val @ R )
       => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
         => ( ord_le6900622908513426738on_val @ ( order_8621009067429368022on_val @ R @ A ) @ ( order_8621009067429368022on_val @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1168_underS__incr,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R )
     => ( ( antisym_on_nat @ top_top_set_nat @ R )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
         => ( ord_less_eq_set_nat @ ( order_underS_nat @ R @ A ) @ ( order_underS_nat @ R @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1169_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_1170_irrefl__on__lex__prod,axiom,
    ! [A5: set_nat,R_A: set_Pr1261947904930325089at_nat,B5: set_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ A5 @ R_A )
     => ( ( irrefl_on_nat @ B5 @ R_B )
       => ( irrefl7408258699751485431at_nat
          @ ( produc457027306803732586at_nat @ A5
            @ ^ [Uu3: nat] : B5 )
          @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_on_lex_prod
thf(fact_1171_irrefl__on__def,axiom,
    ( irrefl4606354804476895398on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ A6 )
         => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ X2 ) @ R3 ) ) ) ) ).

% irrefl_on_def
thf(fact_1172_irrefl__on__def,axiom,
    ( irrefl_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R3 ) ) ) ) ).

% irrefl_on_def
thf(fact_1173_irrefl__onI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [A4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ A4 @ A5 )
         => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A4 ) @ R ) )
     => ( irrefl7408258699751485431at_nat @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_1174_irrefl__onI,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ! [A4: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ A4 @ A5 )
         => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ A4 ) @ R ) )
     => ( irrefl4606354804476895398on_val @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_1175_irrefl__onI,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A4: nat] :
          ( ( member_nat @ A4 @ A5 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R ) )
     => ( irrefl_on_nat @ A5 @ R ) ) ).

% irrefl_onI
thf(fact_1176_irrefl__onD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( irrefl7408258699751485431at_nat @ A5 @ R )
     => ( ( member8440522571783428010at_nat @ A @ A5 )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_1177_irrefl__onD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val] :
      ( ( irrefl4606354804476895398on_val @ A5 @ R )
     => ( ( member2879030624002546611on_val @ A @ A5 )
       => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_1178_irrefl__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( irrefl_on_nat @ A5 @ R )
     => ( ( member_nat @ A @ A5 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% irrefl_onD
thf(fact_1179_antisym__on__def,axiom,
    ( antisy5014944754269107577on_val
    = ( ^ [A6: set_Pr7649655576376376530on_val,R3: set_Pr4305388979797279737on_val] :
        ! [X2: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ X2 @ A6 )
         => ! [Y5: produc9003781588162623004on_val] :
              ( ( member2879030624002546611on_val @ Y5 @ A6 )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R3 )
               => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y5 @ X2 ) @ R3 )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1180_antisym__on__def,axiom,
    ( antisym_on_nat
    = ( ^ [A6: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R3 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R3 )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1181_antisym__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 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy1245043278253178980at_nat @ A5 @ R ) ) ).

% antisym_onI
thf(fact_1182_antisym__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 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy5014944754269107577on_val @ A5 @ R ) ) ).

% antisym_onI
thf(fact_1183_antisym__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 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisym_on_nat @ A5 @ R ) ) ).

% antisym_onI
thf(fact_1184_antisym__onD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( antisy1245043278253178980at_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 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1185_antisym__onD,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( antisy5014944754269107577on_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 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1186_antisym__onD,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( antisym_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 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1187_irrefl__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R_A )
     => ( ( irrefl_on_nat @ top_top_set_nat @ R_B )
       => ( irrefl7408258699751485431at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% irrefl_lex_prod
thf(fact_1188_antisymI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [X3: produc9003781588162623004on_val,Y3: produc9003781588162623004on_val] :
          ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X3 @ Y3 ) @ R )
         => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y3 @ X3 ) @ R )
           => ( X3 = Y3 ) ) )
     => ( antisy5014944754269107577on_val @ top_to1561438921946393474on_val @ R ) ) ).

% antisymI
thf(fact_1189_antisymI,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 )
           => ( X3 = Y3 ) ) )
     => ( antisym_on_nat @ top_top_set_nat @ R ) ) ).

% antisymI
thf(fact_1190_antisymD,axiom,
    ! [R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( antisy5014944754269107577on_val @ top_to1561438921946393474on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R )
       => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1191_antisymD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( antisym_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 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1192_irreflI,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ! [A4: produc9003781588162623004on_val] :
          ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A4 @ A4 ) @ R )
     => ( irrefl4606354804476895398on_val @ top_to1561438921946393474on_val @ R ) ) ).

% irreflI
thf(fact_1193_irreflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [A4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
     => ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% irreflI
thf(fact_1194_irreflD,axiom,
    ! [R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val] :
      ( ( irrefl4606354804476895398on_val @ top_to1561438921946393474on_val @ R )
     => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_1195_irreflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( irrefl_on_nat @ top_top_set_nat @ R )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R ) ) ).

% irreflD
thf(fact_1196_wf__imp__irrefl,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ R )
     => ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% wf_imp_irrefl
thf(fact_1197_mlex__eq,axiom,
    ( mlex_p4921486899325709005on_val
    = ( ^ [F2: produc9003781588162623004on_val > nat,R4: set_Pr4305388979797279737on_val] :
          ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] :
                ( ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y5 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y5 ) )
                  & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1198_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F2: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] :
                ( ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y5 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y5 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R4 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_1199_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( irrefl7423879665237053122on_val @ top_to1561438921946393474on_val
        @ ^ [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R ) )
      = ( irrefl4606354804476895398on_val @ top_to1561438921946393474on_val @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_1200_irreflp__irrefl__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( irreflp_on_nat @ top_top_set_nat
        @ ^ [A2: nat,B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R ) )
      = ( irrefl_on_nat @ top_top_set_nat @ R ) ) ).

% irreflp_irrefl_eq
thf(fact_1201_under__incr,axiom,
    ! [R: set_Pr4305388979797279737on_val,A: produc9003781588162623004on_val,B: produc9003781588162623004on_val] :
      ( ( trans_3212976107463897212on_val @ top_to1561438921946393474on_val @ R )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R )
       => ( ord_le6900622908513426738on_val @ ( order_2832490146329081463on_val @ R @ A ) @ ( order_2832490146329081463on_val @ R @ B ) ) ) ) ).

% under_incr
thf(fact_1202_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_1203_under__def,axiom,
    ( order_6448319830299018214at_nat
    = ( ^ [R3: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ A2 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1204_under__def,axiom,
    ( order_2832490146329081463on_val
    = ( ^ [R3: set_Pr4305388979797279737on_val,A2: produc9003781588162623004on_val] :
          ( collec7475294502966882545on_val
          @ ^ [B4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B4 @ A2 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1205_under__def,axiom,
    ( order_under_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A2 ) @ R3 ) ) ) ) ).

% under_def
thf(fact_1206_wf__mlex,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( wf_nat @ R2 )
     => ( wf_nat @ ( mlex_prod_nat @ F @ R2 ) ) ) ).

% wf_mlex
thf(fact_1207_irreflp__on__irrefl__on__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ( irrefl7423879665237053122on_val @ A5
        @ ^ [A2: produc9003781588162623004on_val,B4: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ A2 @ B4 ) @ R ) )
      = ( irrefl4606354804476895398on_val @ A5 @ R ) ) ).

% irreflp_on_irrefl_on_eq
thf(fact_1208_irreflp__on__irrefl__on__eq,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( irreflp_on_nat @ A5
        @ ^ [A2: nat,B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B4 ) @ R ) )
      = ( irrefl_on_nat @ A5 @ R ) ) ).

% irreflp_on_irrefl_on_eq
thf(fact_1209_mlex__less,axiom,
    ! [F: produc9003781588162623004on_val > nat,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( mlex_p4921486899325709005on_val @ F @ R2 ) ) ) ).

% mlex_less
thf(fact_1210_mlex__less,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R2 ) ) ) ).

% mlex_less
thf(fact_1211_mlex__iff,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,F: produc9003781588162623004on_val > nat,R2: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( mlex_p4921486899325709005on_val @ F @ R2 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R2 ) ) ) ) ).

% mlex_iff
thf(fact_1212_mlex__iff,axiom,
    ! [X: nat,Y: nat,F: nat > nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R2 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).

% mlex_iff
thf(fact_1213_mlex__leq,axiom,
    ! [F: produc9003781588162623004on_val > nat,X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ R2 )
       => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( mlex_p4921486899325709005on_val @ F @ R2 ) ) ) ) ).

% mlex_leq
thf(fact_1214_mlex__leq,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R2 ) ) ) ) ).

% mlex_leq
thf(fact_1215_wf__pred__nat,axiom,
    wf_nat @ pred_nat ).

% wf_pred_nat
thf(fact_1216_wf__less__than,axiom,
    wf_nat @ less_than ).

% wf_less_than
thf(fact_1217_antisymp__antisym__eq,axiom,
    ! [R: set_Pr4305388979797279737on_val] :
      ( ( antisy7396856984585285295on_val @ top_to1561438921946393474on_val
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( antisy5014944754269107577on_val @ top_to1561438921946393474on_val @ R ) ) ).

% antisymp_antisym_eq
thf(fact_1218_antisymp__antisym__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( antisymp_on_nat @ top_top_set_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( antisym_on_nat @ top_top_set_nat @ R ) ) ).

% antisymp_antisym_eq
thf(fact_1219_trans__less__than,axiom,
    trans_on_nat @ top_top_set_nat @ less_than ).

% trans_less_than
thf(fact_1220_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_1221_irrefl__less__than,axiom,
    irrefl_on_nat @ top_top_set_nat @ less_than ).

% irrefl_less_than
thf(fact_1222_total__on__less__than,axiom,
    ! [A5: set_nat] : ( total_on_nat @ A5 @ less_than ) ).

% total_on_less_than
thf(fact_1223_total__less__than,axiom,
    total_on_nat @ top_top_set_nat @ less_than ).

% total_less_than
thf(fact_1224_asym__less__than,axiom,
    asym_on_nat @ top_top_set_nat @ less_than ).

% asym_less_than
thf(fact_1225_antisymp__on__antisym__on__eq,axiom,
    ! [A5: set_Pr7649655576376376530on_val,R: set_Pr4305388979797279737on_val] :
      ( ( antisy7396856984585285295on_val @ A5
        @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] : ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ Y5 ) @ R ) )
      = ( antisy5014944754269107577on_val @ A5 @ R ) ) ).

% antisymp_on_antisym_on_eq
thf(fact_1226_antisymp__on__antisym__on__eq,axiom,
    ! [A5: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( antisymp_on_nat @ A5
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) )
      = ( antisym_on_nat @ A5 @ R ) ) ).

% antisymp_on_antisym_on_eq
thf(fact_1227_mlex__prod__def,axiom,
    ( mlex_prod_nat
    = ( ^ [F2: nat > nat,R4: set_Pr1261947904930325089at_nat] :
          ( inv_im8721129524585065343at_nat @ ( lex_prod_nat_nat @ less_than @ R4 )
          @ ^ [X2: nat] : ( product_Pair_nat_nat @ ( F2 @ X2 ) @ X2 ) ) ) ) ).

% mlex_prod_def
thf(fact_1228_trancl__insert2,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( transi3605591327991077947on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( sup_su3890347010539237325on_val @ ( transi3605591327991077947on_val @ R )
        @ ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] :
                ( ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ A ) @ ( transi3605591327991077947on_val @ R ) )
                  | ( X2 = A ) )
                & ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ Y5 ) @ ( transi3605591327991077947on_val @ R ) )
                  | ( Y5 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_1229_trancl__insert2,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( transi6264000038957366511cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( sup_su6327502436637775413at_nat @ ( transi6264000038957366511cl_nat @ R )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] :
                ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ A ) @ ( transi6264000038957366511cl_nat @ R ) )
                  | ( X2 = A ) )
                & ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ Y5 ) @ ( transi6264000038957366511cl_nat @ R ) )
                  | ( Y5 = B ) ) ) ) ) ) ) ).

% trancl_insert2
thf(fact_1230_rtrancl__insert,axiom,
    ! [A: produc9003781588162623004on_val,B: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( transi3365842532184619097on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ A @ B ) @ R ) )
      = ( sup_su3890347010539237325on_val @ ( transi3365842532184619097on_val @ R )
        @ ( collec5166231387257825412on_val
          @ ( produc1845727980445322802_val_o
            @ ^ [X2: produc9003781588162623004on_val,Y5: produc9003781588162623004on_val] :
                ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X2 @ A ) @ ( transi3365842532184619097on_val @ R ) )
                & ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ B @ Y5 ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ) ) ) ).

% rtrancl_insert
thf(fact_1231_rtrancl__insert,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( transi2905341329935302413cl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R ) )
      = ( sup_su6327502436637775413at_nat @ ( transi2905341329935302413cl_nat @ R )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X2: nat,Y5: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ A ) @ ( transi2905341329935302413cl_nat @ R ) )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ Y5 ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ) ) ) ).

% rtrancl_insert
thf(fact_1232_wf__insert,axiom,
    ! [Y: produc9003781588162623004on_val,X: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( wf_Pro3300669915054815219on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R ) )
      = ( ( wf_Pro3300669915054815219on_val @ R )
        & ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% wf_insert
thf(fact_1233_wf__insert,axiom,
    ! [Y: nat,X: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( wf_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) )
      = ( ( wf_nat @ R )
        & ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).

% wf_insert
thf(fact_1234_acyclic__insert,axiom,
    ! [Y: produc9003781588162623004on_val,X: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( transi5235682513840182669on_val @ ( insert7062705985672971881on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ R ) )
      = ( ( transi5235682513840182669on_val @ R )
        & ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ).

% acyclic_insert
thf(fact_1235_acyclic__insert,axiom,
    ! [Y: nat,X: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( transi3139058337892597825ic_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R ) )
      = ( ( transi3139058337892597825ic_nat @ R )
        & ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).

% acyclic_insert
thf(fact_1236_converse__rtrancl__induct2,axiom,
    ! [Ax: produc9003781588162623004on_val,Ay: produc9003781588162623004on_val,Bx: produc9003781588162623004on_val,By: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val,P2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi725908587811456796on_val @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,Aa2: produc9003781588162623004on_val,Ba: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A4 @ B3 ) @ ( produc4958730763592889809on_val @ Aa2 @ Ba ) ) @ R )
             => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Aa2 @ Ba ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi725908587811456796on_val @ R ) )
               => ( ( P2 @ Aa2 @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1237_converse__rtrancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: nat,B3: nat,Aa2: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa2 @ Ba ) ) @ R )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa2 @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
               => ( ( P2 @ Aa2 @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1238_converse__rtrancl__induct2,axiom,
    ! [Ax: produc1457211279475724562t_char > prat,Ay: produc1457211279475724562t_char > option_val,Bx: produc1457211279475724562t_char > prat,By: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,Aa2: produc1457211279475724562t_char > prat,Ba: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A4 @ B3 ) @ ( produc74620467203698198on_val @ Aa2 @ Ba ) ) @ R )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Aa2 @ Ba ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3365842532184619097on_val @ R ) )
               => ( ( P2 @ Aa2 @ Ba )
                 => ( P2 @ A4 @ B3 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1239_converse__rtranclE2,axiom,
    ! [Xa2: produc9003781588162623004on_val,Xb: produc9003781588162623004on_val,Za: produc9003781588162623004on_val,Zb: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Xa2 @ Xb ) @ ( produc4958730763592889809on_val @ Za @ Zb ) ) @ ( transi725908587811456796on_val @ R ) )
     => ( ( ( produc4958730763592889809on_val @ Xa2 @ Xb )
         != ( produc4958730763592889809on_val @ Za @ Zb ) )
       => ~ ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Xa2 @ Xb ) @ ( produc4958730763592889809on_val @ A4 @ B3 ) ) @ R )
             => ~ ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A4 @ B3 ) @ ( produc4958730763592889809on_val @ Za @ Zb ) ) @ ( transi725908587811456796on_val @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1240_converse__rtranclE2,axiom,
    ! [Xa2: nat,Xb: nat,Za: nat,Zb: nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( ( product_Pair_nat_nat @ Xa2 @ Xb )
         != ( product_Pair_nat_nat @ Za @ Zb ) )
       => ~ ! [A4: nat,B3: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xb ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1241_converse__rtranclE2,axiom,
    ! [Xa2: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char > option_val,Za: produc1457211279475724562t_char > prat,Zb: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Xa2 @ Xb ) @ ( produc74620467203698198on_val @ Za @ Zb ) ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( ( produc74620467203698198on_val @ Xa2 @ Xb )
         != ( produc74620467203698198on_val @ Za @ Zb ) )
       => ~ ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Xa2 @ Xb ) @ ( produc74620467203698198on_val @ A4 @ B3 ) ) @ R )
             => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A4 @ B3 ) @ ( produc74620467203698198on_val @ Za @ Zb ) ) @ ( transi3365842532184619097on_val @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1242_rtrancl__induct2,axiom,
    ! [Ax: produc9003781588162623004on_val,Ay: produc9003781588162623004on_val,Bx: produc9003781588162623004on_val,By: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val,P2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi725908587811456796on_val @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,Aa2: produc9003781588162623004on_val,Ba: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ A4 @ B3 ) ) @ ( transi725908587811456796on_val @ R ) )
             => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A4 @ B3 ) @ ( produc4958730763592889809on_val @ Aa2 @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1243_rtrancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: nat,B3: nat,Aa2: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi8609417484261754244at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa2 @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1244_rtrancl__induct2,axiom,
    ! [Ax: produc1457211279475724562t_char > prat,Ay: produc1457211279475724562t_char > option_val,Bx: produc1457211279475724562t_char > prat,By: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3365842532184619097on_val @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,Aa2: produc1457211279475724562t_char > prat,Ba: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ A4 @ B3 ) ) @ ( transi3365842532184619097on_val @ R ) )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A4 @ B3 ) @ ( produc74620467203698198on_val @ Aa2 @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1245_trancl__induct2,axiom,
    ! [Ax: produc9003781588162623004on_val,Ay: produc9003781588162623004on_val,Bx: produc9003781588162623004on_val,By: produc9003781588162623004on_val,R: set_Pr4590028993149660487on_val,P2: produc9003781588162623004on_val > produc9003781588162623004on_val > $o] :
      ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ Bx @ By ) ) @ ( transi1250686601697532602on_val @ R ) )
     => ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val] :
            ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ A4 @ B3 ) ) @ R )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: produc9003781588162623004on_val,B3: produc9003781588162623004on_val,Aa2: produc9003781588162623004on_val,Ba: produc9003781588162623004on_val] :
              ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ Ax @ Ay ) @ ( produc4958730763592889809on_val @ A4 @ B3 ) ) @ ( transi1250686601697532602on_val @ R ) )
             => ( ( member2023868432420333456on_val @ ( produc1713567484784205015on_val @ ( produc4958730763592889809on_val @ A4 @ B3 ) @ ( produc4958730763592889809on_val @ Aa2 @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1246_trancl__induct2,axiom,
    ! [Ax: nat,Ay: nat,Bx: nat,By: nat,R: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi243908449541399842at_nat @ R ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ R )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: nat,B3: nat,Aa2: nat,Ba: nat] :
              ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( transi243908449541399842at_nat @ R ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ ( product_Pair_nat_nat @ Aa2 @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1247_trancl__induct2,axiom,
    ! [Ax: produc1457211279475724562t_char > prat,Ay: produc1457211279475724562t_char > option_val,Bx: produc1457211279475724562t_char > prat,By: produc1457211279475724562t_char > option_val,R: set_Pr4305388979797279737on_val,P2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > option_val ) > $o] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ Bx @ By ) ) @ ( transi3605591327991077947on_val @ R ) )
     => ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val] :
            ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ A4 @ B3 ) ) @ R )
           => ( P2 @ A4 @ B3 ) )
       => ( ! [A4: produc1457211279475724562t_char > prat,B3: produc1457211279475724562t_char > option_val,Aa2: produc1457211279475724562t_char > prat,Ba: produc1457211279475724562t_char > option_val] :
              ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ Ax @ Ay ) @ ( produc74620467203698198on_val @ A4 @ B3 ) ) @ ( transi3605591327991077947on_val @ R ) )
             => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ ( produc74620467203698198on_val @ A4 @ B3 ) @ ( produc74620467203698198on_val @ Aa2 @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B3 )
                 => ( P2 @ Aa2 @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% trancl_induct2
thf(fact_1248_irrefl__tranclI,axiom,
    ! [R: set_Pr4305388979797279737on_val,X: produc9003781588162623004on_val] :
      ( ( ( inf_in4072335026415195879on_val @ ( conver3382110228903287541on_val @ R ) @ ( transi3365842532184619097on_val @ R ) )
        = bot_bo402267660617762917on_val )
     => ~ ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ X ) @ ( transi3605591327991077947on_val @ R ) ) ) ).

% irrefl_tranclI
thf(fact_1249_irrefl__tranclI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,X: nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( converse_nat_nat @ R ) @ ( transi2905341329935302413cl_nat @ R ) )
        = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transi6264000038957366511cl_nat @ R ) ) ) ).

% irrefl_tranclI
thf(fact_1250_trancl__converseD,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3605591327991077947on_val @ ( conver3382110228903287541on_val @ R ) ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( conver3382110228903287541on_val @ ( transi3605591327991077947on_val @ R ) ) ) ) ).

% trancl_converseD
thf(fact_1251_trancl__converseD,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R ) ) ) ) ).

% trancl_converseD
thf(fact_1252_trancl__converseI,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( conver3382110228903287541on_val @ ( transi3605591327991077947on_val @ R ) ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3605591327991077947on_val @ ( conver3382110228903287541on_val @ R ) ) ) ) ).

% trancl_converseI
thf(fact_1253_trancl__converseI,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( converse_nat_nat @ ( transi6264000038957366511cl_nat @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi6264000038957366511cl_nat @ ( converse_nat_nat @ R ) ) ) ) ).

% trancl_converseI
thf(fact_1254_rtrancl__converseD,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3365842532184619097on_val @ ( conver3382110228903287541on_val @ R ) ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ ( transi3365842532184619097on_val @ R ) ) ) ).

% rtrancl_converseD
thf(fact_1255_rtrancl__converseD,axiom,
    ! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ).

% rtrancl_converseD
thf(fact_1256_rtrancl__converseI,axiom,
    ! [Y: produc9003781588162623004on_val,X: produc9003781588162623004on_val,R: set_Pr4305388979797279737on_val] :
      ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ Y @ X ) @ ( transi3365842532184619097on_val @ R ) )
     => ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3365842532184619097on_val @ ( conver3382110228903287541on_val @ R ) ) ) ) ).

% rtrancl_converseI
thf(fact_1257_rtrancl__converseI,axiom,
    ! [Y: nat,X: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transi2905341329935302413cl_nat @ R ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ ( converse_nat_nat @ R ) ) ) ) ).

% rtrancl_converseI
thf(fact_1258_Not__Domain__rtrancl,axiom,
    ! [X: produc9003781588162623004on_val,R2: set_Pr4305388979797279737on_val,Y: produc9003781588162623004on_val] :
      ( ~ ( member2879030624002546611on_val @ X @ ( domain6215983837468562630on_val @ R2 ) )
     => ( ( member6505152944228695362on_val @ ( produc4958730763592889809on_val @ X @ Y ) @ ( transi3365842532184619097on_val @ R2 ) )
        = ( X = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1259_Not__Domain__rtrancl,axiom,
    ! [X: nat,R2: set_Pr1261947904930325089at_nat,Y: nat] :
      ( ~ ( member_nat @ X @ ( domain_nat_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
        = ( X = Y ) ) ) ).

% Not_Domain_rtrancl
thf(fact_1260_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_1261_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_1262_less__than__def,axiom,
    ( less_than
    = ( transi6264000038957366511cl_nat @ pred_nat ) ) ).

% less_than_def
thf(fact_1263_Abs__state__inverse,axiom,
    ! [Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi2: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi2 )
              & ( valid_state @ Phi2 ) ) ) )
     => ( ( rep_state @ ( abs_state @ Y ) )
        = Y ) ) ).

% Abs_state_inverse
thf(fact_1264_Rep__state,axiom,
    ! [X: state] :
      ( member2879030624002546611on_val @ ( rep_state @ X )
      @ ( collec7475294502966882545on_val
        @ ^ [Uu3: produc9003781588162623004on_val] :
          ? [Phi2: produc9003781588162623004on_val] :
            ( ( Uu3 = Phi2 )
            & ( valid_state @ Phi2 ) ) ) ) ).

% Rep_state
thf(fact_1265_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_1266_Rep__state__induct,axiom,
    ! [Y: produc9003781588162623004on_val,P2: produc9003781588162623004on_val > $o] :
      ( ( member2879030624002546611on_val @ Y
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi2: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi2 )
              & ( valid_state @ Phi2 ) ) ) )
     => ( ! [X3: state] : ( P2 @ ( rep_state @ X3 ) )
       => ( P2 @ Y ) ) ) ).

% Rep_state_induct
thf(fact_1267_Abs__state__inject,axiom,
    ! [X: produc9003781588162623004on_val,Y: produc9003781588162623004on_val] :
      ( ( member2879030624002546611on_val @ X
        @ ( collec7475294502966882545on_val
          @ ^ [Uu3: produc9003781588162623004on_val] :
            ? [Phi2: produc9003781588162623004on_val] :
              ( ( Uu3 = Phi2 )
              & ( valid_state @ Phi2 ) ) ) )
     => ( ( member2879030624002546611on_val @ Y
          @ ( collec7475294502966882545on_val
            @ ^ [Uu3: produc9003781588162623004on_val] :
              ? [Phi2: produc9003781588162623004on_val] :
                ( ( Uu3 = Phi2 )
                & ( valid_state @ Phi2 ) ) ) )
       => ( ( ( abs_state @ X )
            = ( abs_state @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_state_inject
thf(fact_1268_Abs__state__induct,axiom,
    ! [P2: state > $o,X: state] :
      ( ! [Y3: produc9003781588162623004on_val] :
          ( ( member2879030624002546611on_val @ Y3
            @ ( collec7475294502966882545on_val
              @ ^ [Uu3: produc9003781588162623004on_val] :
                ? [Phi2: produc9003781588162623004on_val] :
                  ( ( Uu3 = Phi2 )
                  & ( valid_state @ Phi2 ) ) ) )
         => ( P2 @ ( abs_state @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% Abs_state_induct
thf(fact_1269_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_1270_type__definition__state,axiom,
    ( type_d7385716227342696708on_val @ rep_state @ abs_state
    @ ( collec7475294502966882545on_val
      @ ^ [Uu3: produc9003781588162623004on_val] :
        ? [Phi2: produc9003781588162623004on_val] :
          ( ( Uu3 = Phi2 )
          & ( valid_state @ Phi2 ) ) ) ) ).

% type_definition_state

% Helper facts (7)
thf(help_If_2_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
    ! [X: option_val,Y: option_val] :
      ( ( if_option_val @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Mask__Oval_J_T,axiom,
    ! [X: option_val,Y: option_val] :
      ( ( if_option_val @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X: option_state,Y: option_state] :
      ( ( if_option_state @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__PartialHeapSA__Ostate_J_T,axiom,
    ! [X: option_state,Y: option_state] :
      ( ( if_option_state @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_T,axiom,
    ! [X: set_Pr7649655576376376530on_val,Y: set_Pr7649655576376376530on_val] :
      ( ( if_set1193336150724294796on_val @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Mask__Oval_J_J_J_J_T,axiom,
    ! [X: set_Pr7649655576376376530on_val,Y: set_Pr7649655576376376530on_val] :
      ( ( if_set1193336150724294796on_val @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( get_h @ unit )
    = empty_7512026568161686585ar_val ) ).

%------------------------------------------------------------------------------