TPTP Problem File: SLH0572^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Combinable_Wands/0001_Mask/prob_00474_015798__7195576_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2176 ( 646 unt; 918 typ;   0 def)
%            Number of atoms       : 3877 (1209 equ;   0 cnn)
%            Maximal formula atoms :   10 (   3 avg)
%            Number of connectives : 13334 ( 347   ~;  47   |; 312   &;11106   @)
%                                         (   0 <=>;1522  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   7 avg)
%            Number of types       :  100 (  99 usr)
%            Number of type conns  : 3463 (3463   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  822 ( 819 usr;  45 con; 0-4 aty)
%            Number of variables   : 4536 ( 882   ^;3528   !; 126   ?;4536   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:08:29.484
%------------------------------------------------------------------------------
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% Explicit typings (819)
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    bNF_Gr3750019412944930853t_char: set_Pr2507339680178222706t_char > ( produc1457211279475724562t_char > list_char ) > produc1457211279475724562t_char > list_char > $o ).

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

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

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
    bNF_Gr2539438056706512010t_char: set_Pr3996677194857090567t_char > ( produc6071853487833700903t_char > produc6859378861088059299t_char ) > produc6071853487833700903t_char > produc6859378861088059299t_char > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr7887112059459224296at_nat: set_Pr3996677194857090567t_char > ( produc6071853487833700903t_char > product_prod_nat_nat ) > produc6071853487833700903t_char > product_prod_nat_nat > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    bNF_Gr472834428334945934od_o_o: set_se491199571785101031od_o_o > ( set_Product_prod_o_o > set_Product_prod_o_o ) > set_Product_prod_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Def_OGrp_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Gr1962207778594255720at_nat: set_se491199571785101031od_o_o > ( set_Product_prod_o_o > set_Pr1261947904930325089at_nat ) > set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o ).

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

thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_fs4859913984995905862t_char: ( nat > list_char > $o ) > ( list_char > list_char > $o ) > produc1457211279475724562t_char > produc1457211279475724562t_char ).

thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_fs2355234947312196021t_char: ( nat > nat > $o ) > ( nat > list_char > $o ) > produc1457211279475724562t_char > product_prod_nat_nat ).

thf(sy_c_BNF__Def_Opick__middlep_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
    bNF_pi2586896789752333176t_char: ( nat > list_char > $o ) > ( list_char > list_char > $o ) > nat > list_char > list_char ).

thf(sy_c_BNF__Def_Opick__middlep_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_pi5668428018083610343t_char: ( nat > nat > $o ) > ( nat > list_char > $o ) > nat > list_char > nat ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    bNF_re7444546633290812577t_char: ( list_char > list_char > $o ) > ( produc1457211279475724562t_char > produc1457211279475724562t_char > $o ) > ( list_char > produc1457211279475724562t_char ) > ( list_char > produc1457211279475724562t_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__List__Olist_It__String__Ochar_J_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re4857190184819116159at_nat: ( list_char > nat > $o ) > ( produc1457211279475724562t_char > product_prod_nat_nat > $o ) > ( list_char > produc1457211279475724562t_char ) > ( nat > product_prod_nat_nat ) > $o ).

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re1173106714776151178_nat_o: ( nat > list_char > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( nat > product_prod_nat_nat > $o ) > ( list_char > product_prod_nat_nat > $o ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    bNF_re7671636171547762529t_char: ( nat > list_char > $o ) > ( product_prod_nat_nat > produc1457211279475724562t_char > $o ) > ( nat > product_prod_nat_nat ) > ( list_char > produc1457211279475724562t_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__List__Olist_It__String__Ochar_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    bNF_re1247391961183351807t_char: ( nat > nat > $o ) > ( ( list_char > produc1457211279475724562t_char ) > ( list_char > produc1457211279475724562t_char ) > $o ) > ( nat > list_char > produc1457211279475724562t_char ) > ( nat > list_char > produc1457211279475724562t_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__List__Olist_It__String__Ochar_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re453200506278319679at_nat: ( nat > nat > $o ) > ( ( list_char > produc1457211279475724562t_char ) > ( nat > product_prod_nat_nat ) > $o ) > ( nat > list_char > produc1457211279475724562t_char ) > ( nat > nat > product_prod_nat_nat ) > $o ).

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    bNF_re596687900092489407t_char: ( nat > nat > $o ) > ( ( nat > product_prod_nat_nat ) > ( list_char > produc1457211279475724562t_char ) > $o ) > ( nat > nat > product_prod_nat_nat ) > ( nat > list_char > produc1457211279475724562t_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    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 ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_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__List__Olist_It__String__Ochar_J,type,
    bNF_re2076907181175516082t_char: ( nat > nat > $o ) > ( nat > list_char > $o ) > ( nat > nat ) > ( nat > list_char ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    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 ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bNF_re4534060058427857109at_nat: ( nat > set_Product_prod_o_o > $o ) > ( ( nat > product_prod_nat_nat ) > ( set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > $o ) > ( nat > nat > product_prod_nat_nat ) > ( set_Product_prod_o_o > set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bNF_re4920675765442204577at_nat: ( nat > set_Pr1261947904930325089at_nat > $o ) > ( ( nat > product_prod_nat_nat ) > ( set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ) > $o ) > ( nat > nat > product_prod_nat_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bNF_re1126598138771558770at_nat: ( nat > set_Pr1261947904930325089at_nat > $o ) > ( product_prod_nat_nat > produc732395585841259969at_nat > $o ) > ( nat > product_prod_nat_nat ) > ( set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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,
    bNF_re1738110731795791768at_nat: ( nat > set_Pr1261947904930325089at_nat > $o ) > ( product_prod_nat_nat > produc3843707927480180839at_nat > $o ) > ( nat > product_prod_nat_nat ) > ( set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__List__Olist_It__String__Ochar_J_M_Eo_J,type,
    bNF_re1503381852333877516char_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( nat > $o ) > ( list_char > $o ) > $o ) > ( product_prod_nat_nat > nat > $o ) > ( product_prod_nat_nat > list_char > $o ) > $o ).

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    bNF_re1392313733760654173_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( product_prod_nat_nat > nat > $o ) > ( product_prod_nat_nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Nat__Onat_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
    bNF_re4549960102941810855t_char: ( set_Product_prod_o_o > nat > $o ) > ( ( set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > ( list_char > produc1457211279475724562t_char ) > $o ) > ( set_Product_prod_o_o > set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > ( nat > list_char > produc1457211279475724562t_char ) > $o ).

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    bNF_re5841918410865130215at_nat: ( set_Product_prod_o_o > nat > $o ) > ( ( set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > ( nat > product_prod_nat_nat ) > $o ) > ( set_Product_prod_o_o > set_Pr1261947904930325089at_nat > produc732395585841259969at_nat ) > ( nat > nat > product_prod_nat_nat ) > $o ).

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thf(sy_c_PosRat_Opmax,type,
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thf(sy_c_PosRat_Opmin,type,
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thf(sy_c_PosRat_Opnone,type,
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thf(sy_c_PosRat_Oppos,type,
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thf(sy_c_PosRat_Opwrite,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__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,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Nat__Onat_J_J,type,
    member3072328306138578198_o_nat: produc1677149977551171693_o_nat > set_Pr7303405868105718221_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4915364185892706192at_nat: produc6164359238935219303at_nat > set_Pr2060405980302660935at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8767205482156871978at_nat: produc7372027153569465473at_nat > set_Pr7390626335576417249at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_J,type,
    member5560047953027865360od_o_o: produc2245965095621756647od_o_o > set_Pr1013884327308616903od_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member3762726516173965008at_nat: produc4432270315841807015at_nat > set_Pr3844393010656751751at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member208136763105818512at_nat: produc266773502759220327at_nat > set_Pr4427867301352232263at_nat > $o ).

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: produc732395585841259969at_nat > set_Pr457366540195662369at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member2037436044582932534od_o_o: produc7898102162437433997od_o_o > set_Pr8055703017043224813od_o_o > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o ).

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

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

thf(sy_v_a,type,
    a: produc1457211279475724562t_char > prat ).

thf(sy_v_b,type,
    b: produc1457211279475724562t_char > prat ).

thf(sy_v_hl,type,
    hl: produc1457211279475724562t_char ).

thf(sy_v_ma,type,
    ma: prat ).

thf(sy_v_mb,type,
    mb: prat ).

% Relevant facts (1253)
thf(fact_0_assms_I2_J,axiom,
    valid_mask @ b ).

% assms(2)
thf(fact_1_assms_I1_J,axiom,
    valid_mask @ a ).

% assms(1)
thf(fact_2_assms_I4_J,axiom,
    upper_bounded @ b @ mb ).

% assms(4)
thf(fact_3_assms_I3_J,axiom,
    upper_bounded @ a @ ma ).

% assms(3)
thf(fact_4_add__masks__asso,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( add_ma6586698345353345811t_char @ ( add_ma6586698345353345811t_char @ A @ B ) @ C )
      = ( add_ma6586698345353345811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) ) ) ).

% add_masks_asso
thf(fact_5_add__masks__comm,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [A2: produc1457211279475724562t_char > prat,B2: produc1457211279475724562t_char > prat] : ( add_ma6586698345353345811t_char @ B2 @ A2 ) ) ) ).

% add_masks_comm
thf(fact_6_add__masks__cancellative,axiom,
    ! [B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ B @ C )
        = ( add_ma6586698345353345811t_char @ B @ D ) )
     => ( C = D ) ) ).

% add_masks_cancellative
thf(fact_7_assms_I5_J,axiom,
    pgte @ pwrite @ ( padd @ ma @ mb ) ).

% assms(5)
thf(fact_8_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_9_pgte__antisym,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pgte @ B @ A )
       => ( A = B ) ) ) ).

% pgte_antisym
thf(fact_10_minus__empty,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( Pi
      = ( add_ma6586698345353345811t_char @ Pi @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_11_mult__distr__masks,axiom,
    ! [A: prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ B ) @ ( multiply_mask @ A @ C ) ) ) ).

% mult_distr_masks
thf(fact_12_upper__boundedI,axiom,
    ! [P: prat,Pi: produc1457211279475724562t_char > prat] :
      ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ P @ ( Pi @ Hl ) )
     => ( upper_bounded @ Pi @ P ) ) ).

% upper_boundedI
thf(fact_13_upper__bounded__def,axiom,
    ( upper_bounded
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,P2: prat] :
        ! [Hl2: produc1457211279475724562t_char] : ( pgte @ P2 @ ( Pi2 @ Hl2 ) ) ) ) ).

% upper_bounded_def
thf(fact_14_upper__bounded__bigger,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,A: prat,B: prat] :
      ( ( upper_bounded @ Pi @ A )
     => ( ( pgte @ B @ A )
       => ( upper_bounded @ Pi @ B ) ) ) ).

% upper_bounded_bigger
thf(fact_15_upper__valid,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( ( valid_mask @ B )
          & ( valid_mask @ C ) ) ) ) ).

% upper_valid
thf(fact_16_upper__valid__aux,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ A )
     => ( ( A
          = ( add_ma6586698345353345811t_char @ B @ C ) )
       => ( valid_mask @ B ) ) ) ).

% upper_valid_aux
thf(fact_17_valid__empty,axiom,
    valid_mask @ empty_3446695950879338768t_char ).

% valid_empty
thf(fact_18_padd__asso,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( padd @ ( padd @ A @ B ) @ C )
      = ( padd @ A @ ( padd @ B @ C ) ) ) ).

% padd_asso
thf(fact_19_padd__comm,axiom,
    ( padd
    = ( ^ [A2: prat,B2: prat] : ( padd @ B2 @ A2 ) ) ) ).

% padd_comm
thf(fact_20_padd__cancellative,axiom,
    ! [A: prat,X: prat,B: prat,Y: prat] :
      ( ( A
        = ( padd @ X @ B ) )
     => ( ( A
          = ( padd @ Y @ B ) )
       => ( X = Y ) ) ) ).

% padd_cancellative
thf(fact_21_p__greater__exists,axiom,
    ( pgte
    = ( ^ [A2: prat,B2: prat] :
        ? [R: prat] :
          ( A2
          = ( padd @ B2 @ R ) ) ) ) ).

% p_greater_exists
thf(fact_22_greater__sum__both,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( pgte @ A @ ( padd @ B @ C ) )
     => ? [A1: prat,A22: prat] :
          ( ( A
            = ( padd @ A1 @ A22 ) )
          & ( pgte @ A1 @ B )
          & ( pgte @ A22 @ C ) ) ) ).

% greater_sum_both
thf(fact_23_mult__add__states,axiom,
    ! [A: prat,B: prat,Pi: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ ( padd @ A @ B ) @ Pi )
      = ( add_ma6586698345353345811t_char @ ( multiply_mask @ A @ Pi ) @ ( multiply_mask @ B @ Pi ) ) ) ).

% mult_add_states
thf(fact_24_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( padd @ ( Pi3 @ Hl2 ) @ ( Pi2 @ Hl2 ) ) ) ) ).

% add_masks.simps
thf(fact_25_add__masks_Oelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_26_valid__mult,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,P: prat] :
      ( ( valid_mask @ Pi )
     => ( ( pgte @ pwrite @ P )
       => ( valid_mask @ ( multiply_mask @ P @ Pi ) ) ) ) ).

% valid_mult
thf(fact_27_mult__write__mask,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( ( multiply_mask @ pwrite @ Pi )
      = Pi ) ).

% mult_write_mask
thf(fact_28_padd__comp__one,axiom,
    ! [X: prat] :
      ( ( pgte @ pwrite @ X )
     => ( ( padd @ X @ ( comp_one @ X ) )
        = pwrite ) ) ).

% padd_comp_one
thf(fact_29_half__plus__half,axiom,
    ( ( padd @ half @ half )
    = pwrite ) ).

% half_plus_half
thf(fact_30_valid__mask__full__mask,axiom,
    valid_mask @ full_mask ).

% valid_mask_full_mask
thf(fact_31_padd__one__ineq__sum,axiom,
    ! [A: prat,B: prat,X: prat,Aa: prat,Bb: prat] :
      ( ( ( padd @ A @ B )
        = pwrite )
     => ( ( pgte @ X @ Aa )
       => ( ( pgte @ X @ Bb )
         => ( pgte @ X @ ( padd @ ( pmult @ A @ Aa ) @ ( pmult @ B @ Bb ) ) ) ) ) ) ).

% padd_one_ineq_sum
thf(fact_32_pgte__pgt,axiom,
    ! [A: prat,B: prat,C: prat,D: prat] :
      ( ( pgt @ A @ B )
     => ( ( pgte @ C @ D )
       => ( pgt @ ( padd @ A @ C ) @ ( padd @ B @ D ) ) ) ) ).

% pgte_pgt
thf(fact_33_upper__bounded__smaller,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,A: prat,P: prat] :
      ( ( upper_bounded @ Pi @ A )
     => ( upper_bounded @ ( multiply_mask @ P @ Pi ) @ ( pmult @ P @ A ) ) ) ).

% upper_bounded_smaller
thf(fact_34_multiply__smaller__pwrite,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ pwrite @ A )
     => ( ( pgte @ pwrite @ B )
       => ( pgte @ pwrite @ ( pmult @ A @ B ) ) ) ) ).

% multiply_smaller_pwrite
thf(fact_35_pinv__pwrite,axiom,
    ( ( pinv @ pwrite )
    = pwrite ) ).

% pinv_pwrite
thf(fact_36_pgt__implies__pgte,axiom,
    ! [A: prat,B: prat] :
      ( ( pgt @ A @ B )
     => ( pgte @ A @ B ) ) ).

% pgt_implies_pgte
thf(fact_37_pmult__comm,axiom,
    ( pmult
    = ( ^ [A2: prat,B2: prat] : ( pmult @ B2 @ A2 ) ) ) ).

% pmult_comm
thf(fact_38_mem__Collect__eq,axiom,
    ! [A: produc2934264451710624999od_o_o,P3: produc2934264451710624999od_o_o > $o] :
      ( ( member444158400953824016od_o_o @ A @ ( collec3302784863738620242od_o_o @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_39_mem__Collect__eq,axiom,
    ! [A: produc732395585841259969at_nat,P3: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( collec1729919521702017196at_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P3: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P3: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o] :
      ( ( collec3302784863738620242od_o_o
        @ ^ [X2: produc2934264451710624999od_o_o] : ( member444158400953824016od_o_o @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
    ! [A3: set_Pr457366540195662369at_nat] :
      ( ( collec1729919521702017196at_nat
        @ ^ [X2: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_48_pmult__order,axiom,
    ! [A: prat,B: prat,P: prat] :
      ( ( pgte @ A @ B )
     => ( pgte @ ( pmult @ P @ A ) @ ( pmult @ B @ P ) ) ) ).

% pmult_order
thf(fact_49_pmult__distr,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( pmult @ A @ ( padd @ B @ C ) )
      = ( padd @ ( pmult @ A @ B ) @ ( pmult @ A @ C ) ) ) ).

% pmult_distr
thf(fact_50_pmult__padd,axiom,
    ! [A: prat,B: prat,X: prat,C: prat,Y: prat] :
      ( ( pmult @ A @ ( padd @ ( pmult @ B @ X ) @ ( pmult @ C @ Y ) ) )
      = ( padd @ ( pmult @ ( pmult @ A @ B ) @ X ) @ ( pmult @ ( pmult @ A @ C ) @ Y ) ) ) ).

% pmult_padd
thf(fact_51_pmult__special_I1_J,axiom,
    ! [X: prat] :
      ( ( pmult @ pwrite @ X )
      = X ) ).

% pmult_special(1)
thf(fact_52_multiply__mask__def,axiom,
    ( multiply_mask
    = ( ^ [P2: prat,Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char] : ( pmult @ P2 @ ( Pi2 @ Hl2 ) ) ) ) ).

% multiply_mask_def
thf(fact_53_not__pgte__charact,axiom,
    ! [A: prat,B: prat] :
      ( ( ~ ( pgte @ A @ B ) )
      = ( pgt @ B @ A ) ) ).

% not_pgte_charact
thf(fact_54_pinv__double__half,axiom,
    ! [P: prat] :
      ( ( ppos @ P )
     => ( ( pmult @ half @ ( pinv @ P ) )
        = ( pinv @ ( padd @ P @ P ) ) ) ) ).

% pinv_double_half
thf(fact_55_half__between__0__1,axiom,
    ( ( ppos @ half )
    & ( pgt @ pwrite @ half ) ) ).

% half_between_0_1
thf(fact_56_pinv__pmult__ok,axiom,
    ! [P: prat] :
      ( ( ppos @ P )
     => ( ( pmult @ P @ ( pinv @ P ) )
        = pwrite ) ) ).

% pinv_pmult_ok
thf(fact_57_pinv__def,axiom,
    ( pinv
    = ( pdiv @ pwrite ) ) ).

% pinv_def
thf(fact_58_valid__larger__mask,axiom,
    ( valid_mask
    = ( greate7680731506090375811t_char @ full_mask ) ) ).

% valid_larger_mask
thf(fact_59_pinv__inverts,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ B )
       => ( pgte @ ( pinv @ B ) @ ( pinv @ A ) ) ) ) ).

% pinv_inverts
thf(fact_60_pmin__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmin @ A @ B )
        = B ) ) ).

% pmin_is
thf(fact_61_pmin__greater,axiom,
    ! [A: prat,B: prat] : ( pgte @ A @ ( pmin @ A @ B ) ) ).

% pmin_greater
thf(fact_62_pmax__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmax @ A @ B )
        = A ) ) ).

% pmax_is
thf(fact_63_greater__mask__properties_I3_J,axiom,
    ! [Pi4: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ Pi4 @ Pi )
        & ( greate7680731506090375811t_char @ Pi @ Pi4 ) )
     => ( Pi = Pi4 ) ) ).

% greater_mask_properties(3)
thf(fact_64_greater__mask__properties_I2_J,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ A @ B )
        & ( greate7680731506090375811t_char @ B @ C ) )
     => ( greate7680731506090375811t_char @ A @ C ) ) ).

% greater_mask_properties(2)
thf(fact_65_greater__mask__properties_I1_J,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi @ Pi ) ).

% greater_mask_properties(1)
thf(fact_66_pmax__comm,axiom,
    ( pmax
    = ( ^ [A2: prat,B2: prat] : ( pmax @ B2 @ A2 ) ) ) ).

% pmax_comm
thf(fact_67_pmin__comm,axiom,
    ( pmin
    = ( ^ [A2: prat,B2: prat] : ( pmin @ B2 @ A2 ) ) ) ).

% pmin_comm
thf(fact_68_pmult__pdiv__cancel,axiom,
    ! [A: prat,X: prat] :
      ( ( ppos @ A )
     => ( ( pmult @ A @ ( pdiv @ X @ A ) )
        = X ) ) ).

% pmult_pdiv_cancel
thf(fact_69_pmin__pmax,axiom,
    ! [X: prat,A: prat,B: prat] :
      ( ( pgte @ X @ ( pmin @ A @ B ) )
     => ( X
        = ( pmin @ ( pmax @ X @ A ) @ ( pmax @ X @ B ) ) ) ) ).

% pmin_pmax
thf(fact_70_greater__mask__equiv__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
        ! [Hl2: produc1457211279475724562t_char] : ( pgte @ ( Pi3 @ Hl2 ) @ ( Pi2 @ Hl2 ) ) ) ) ).

% greater_mask_equiv_def
thf(fact_71_greater__maskI,axiom,
    ! [Pi4: produc1457211279475724562t_char > prat,Pi: produc1457211279475724562t_char > prat] :
      ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ ( Pi4 @ Hl ) @ ( Pi @ Hl ) )
     => ( greate7680731506090375811t_char @ Pi4 @ Pi ) ) ).

% greater_maskI
thf(fact_72_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_73_const__sum__mask__greater,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat,D: produc1457211279475724562t_char > prat] :
      ( ( ( add_ma6586698345353345811t_char @ A @ B )
        = ( add_ma6586698345353345811t_char @ C @ D ) )
     => ( ( greate7680731506090375811t_char @ A @ C )
       => ( greate7680731506090375811t_char @ D @ B ) ) ) ).

% const_sum_mask_greater
thf(fact_74_greater__mask__decomp,axiom,
    ! [A: produc1457211279475724562t_char > prat,B: produc1457211279475724562t_char > prat,C: produc1457211279475724562t_char > prat] :
      ( ( greate7680731506090375811t_char @ A @ ( add_ma6586698345353345811t_char @ B @ C ) )
     => ? [A1: produc1457211279475724562t_char > prat,A22: produc1457211279475724562t_char > prat] :
          ( ( A
            = ( add_ma6586698345353345811t_char @ A1 @ A22 ) )
          & ( greate7680731506090375811t_char @ A1 @ B )
          & ( greate7680731506090375811t_char @ A22 @ C ) ) ) ).

% greater_mask_decomp
thf(fact_75_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
        ? [R: produc1457211279475724562t_char > prat] :
          ( Pi3
          = ( add_ma6586698345353345811t_char @ Pi2 @ R ) ) ) ) ).

% greater_mask_def
thf(fact_76_pdiv__smaller,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ A )
       => ( pgte @ pwrite @ ( pdiv @ B @ A ) ) ) ) ).

% pdiv_smaller
thf(fact_77_sum__coeff,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ( padd @ ( pdiv @ A @ ( padd @ A @ B ) ) @ ( pdiv @ B @ ( padd @ A @ B ) ) )
          = pwrite ) ) ) ).

% sum_coeff
thf(fact_78_pmult__ppos,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ppos @ ( pmult @ A @ B ) ) ) ) ).

% pmult_ppos
thf(fact_79_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_80_ppos__inv,axiom,
    ! [P: prat] :
      ( ( ppos @ P )
     => ( ppos @ ( pinv @ P ) ) ) ).

% ppos_inv
thf(fact_81_update__trans,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat,Pi5: produc1457211279475724562t_char > prat,Dm: produc1457211279475724562t_char > $o] :
      ( ( greate7680731506090375811t_char @ Pi @ Pi4 )
     => ( ( greate7680731506090375811t_char @ Pi4 @ Pi5 )
       => ( ( update5099441844846592839t_char @ ( update5099441844846592839t_char @ Dm @ Pi @ Pi4 ) @ Pi4 @ Pi5 )
          = ( update5099441844846592839t_char @ Dm @ Pi @ Pi5 ) ) ) ) ).

% update_trans
thf(fact_82_pmax__is__smaller,axiom,
    ! [X: prat,A: prat,B: prat] :
      ( ( pgte @ X @ A )
     => ( ( pgte @ X @ B )
       => ( pgte @ X @ ( pmax @ A @ B ) ) ) ) ).

% pmax_is_smaller
thf(fact_83_pmax__smaller,axiom,
    ! [A: prat,B: prat] : ( pgte @ ( pmax @ A @ B ) @ A ) ).

% pmax_smaller
thf(fact_84_greater__mask__uni__equiv,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char,R2: prat] :
      ( ( greate7680731506090375811t_char @ Pi @ ( uni_ma6578582744724537101t_char @ Hl3 @ R2 ) )
      = ( pgte @ ( Pi @ Hl3 ) @ R2 ) ) ).

% greater_mask_uni_equiv
thf(fact_85_greater__mask__uniI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Hl3: produc1457211279475724562t_char,R2: prat] :
      ( ( pgte @ ( Pi @ Hl3 ) @ R2 )
     => ( greate7680731506090375811t_char @ Pi @ ( uni_ma6578582744724537101t_char @ Hl3 @ R2 ) ) ) ).

% greater_mask_uniI
thf(fact_86_empty__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_87_empty__mask_Osimps,axiom,
    ( empty_3446695950879338768t_char
    = ( ^ [Hl2: produc1457211279475724562t_char] : pnone ) ) ).

% empty_mask.simps
thf(fact_88_ppos__eq__pnone,axiom,
    ( ppos
    = ( ^ [P2: prat] : ( P2 != pnone ) ) ) ).

% ppos_eq_pnone
thf(fact_89_pmult__special_I2_J,axiom,
    ! [X: prat] :
      ( ( pmult @ pnone @ X )
      = pnone ) ).

% pmult_special(2)
thf(fact_90_padd__zero,axiom,
    ! [A: prat,B: prat] :
      ( ( pnone
        = ( padd @ A @ B ) )
      = ( ( A = pnone )
        & ( B = pnone ) ) ) ).

% padd_zero
thf(fact_91_add__acc__uni__mask,axiom,
    ( add_ac6043276672051819257t_char
    = ( ^ [Pi2: produc1457211279475724562t_char > prat,Hl2: produc1457211279475724562t_char,P2: prat] : ( add_ma6586698345353345811t_char @ Pi2 @ ( uni_ma6578582744724537101t_char @ Hl2 @ P2 ) ) ) ) ).

% add_acc_uni_mask
thf(fact_92_valid__maskI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat] :
      ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi @ Hl ) )
     => ( ! [F: list_char] :
            ( ( Pi @ ( produc120197956887798346t_char @ null_nat @ F ) )
            = pnone )
       => ( valid_mask @ Pi ) ) ) ).

% valid_maskI
thf(fact_93_valid__mask_Osimps,axiom,
    ( valid_mask
    = ( ^ [Pi2: produc1457211279475724562t_char > prat] :
          ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi2 @ Hl2 ) )
          & ! [F2: list_char] :
              ( ( Pi2 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
              = pnone ) ) ) ) ).

% valid_mask.simps
thf(fact_94_valid__null__def,axiom,
    ( valid_null
    = ( ^ [Pi2: produc1457211279475724562t_char > prat] :
        ! [F2: list_char] :
          ( ( Pi2 @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
          = pnone ) ) ) ).

% valid_null_def
thf(fact_95_valid__mask_Oelims_I3_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X )
     => ~ ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl ) )
          & ! [F: list_char] :
              ( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
              = pnone ) ) ) ).

% valid_mask.elims(3)
thf(fact_96_valid__mask_Oelims_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X )
     => ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl4 ) )
        & ! [F3: list_char] :
            ( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
            = pnone ) ) ) ).

% valid_mask.elims(2)
thf(fact_97_valid__mask_Oelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X )
        = Y )
     => ( Y
        = ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
          & ! [F2: list_char] :
              ( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
              = pnone ) ) ) ) ).

% valid_mask.elims(1)
thf(fact_98_old_Oprod_Oinject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ A @ B )
        = ( produc8517790099723286449at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_99_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_100_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_101_old_Oprod_Oinject,axiom,
    ! [A: nat,B: list_char,A4: nat,B3: list_char] :
      ( ( ( produc120197956887798346t_char @ A @ B )
        = ( produc120197956887798346t_char @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_102_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_103_prod_Oinject,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Pr1261947904930325089at_nat,Y1: set_Product_prod_o_o,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ X1 @ X22 )
        = ( produc8517790099723286449at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_104_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ X1 @ X22 )
        = ( produc9060074326276436823at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_105_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X22 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_106_prod_Oinject,axiom,
    ! [X1: nat,X22: list_char,Y1: nat,Y2: list_char] :
      ( ( ( produc120197956887798346t_char @ X1 @ X22 )
        = ( produc120197956887798346t_char @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_107_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_108_full__mask__def,axiom,
    ( full_mask
    = ( ^ [Hl2: produc1457211279475724562t_char] :
          ( if_prat
          @ ( ( produc6204653154488278070t_char @ Hl2 )
            = null_nat )
          @ pnone
          @ pwrite ) ) ) ).

% full_mask_def
thf(fact_109_valid__mask_Opelims_I3_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
           => ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl ) )
              & ! [F: list_char] :
                  ( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
                  = pnone ) ) ) ) ) ).

% valid_mask.pelims(3)
thf(fact_110_valid__mask_Opelims_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
           => ~ ( ! [Hl4: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl4 ) )
                & ! [F3: list_char] :
                    ( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
                    = pnone ) ) ) ) ) ).

% valid_mask.pelims(2)
thf(fact_111_valid__mask_Opelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( Y
              = ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
                & ! [F2: list_char] :
                    ( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
                    = pnone ) ) )
           => ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X ) ) ) ) ).

% valid_mask.pelims(1)
thf(fact_112_uni__mask_Osimps,axiom,
    ( uni_ma6578582744724537101t_char
    = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char ) ) ).

% uni_mask.simps
thf(fact_113_uni__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( uni_ma6578582744724537101t_char @ X @ Xa )
        = Y )
     => ( Y
        = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X @ Xa ) ) ) ).

% uni_mask.elims
thf(fact_114_fst__conv,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_115_fst__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] :
      ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_116_fst__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_117_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_118_fst__conv,axiom,
    ! [X1: nat,X22: list_char] :
      ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_119_fst__eqD,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat,A: set_Product_prod_o_o] :
      ( ( ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_120_fst__eqD,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_121_fst__eqD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_122_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_123_fst__eqD,axiom,
    ! [X: nat,Y: list_char,A: nat] :
      ( ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_124_old_Oprod_Oexhaust,axiom,
    ! [Y: produc732395585841259969at_nat] :
      ~ ! [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc8517790099723286449at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_125_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1319942482725812455at_nat] :
      ~ ! [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
          ( Y
         != ( produc9060074326276436823at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_126_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_127_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1457211279475724562t_char] :
      ~ ! [A5: nat,B4: list_char] :
          ( Y
         != ( produc120197956887798346t_char @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_128_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B4: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_129_surj__pair,axiom,
    ! [P: produc732395585841259969at_nat] :
    ? [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc8517790099723286449at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_130_surj__pair,axiom,
    ! [P: produc1319942482725812455at_nat] :
    ? [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P
      = ( produc9060074326276436823at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_131_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_132_surj__pair,axiom,
    ! [P: produc1457211279475724562t_char] :
    ? [X3: nat,Y3: list_char] :
      ( P
      = ( produc120197956887798346t_char @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_133_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_134_prod__cases,axiom,
    ! [P3: produc732395585841259969at_nat > $o,P: produc732395585841259969at_nat] :
      ( ! [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] : ( P3 @ ( produc8517790099723286449at_nat @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_135_prod__cases,axiom,
    ! [P3: produc1319942482725812455at_nat > $o,P: produc1319942482725812455at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : ( P3 @ ( produc9060074326276436823at_nat @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_136_prod__cases,axiom,
    ! [P3: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( P3 @ ( produc2922128104949294807at_nat @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_137_prod__cases,axiom,
    ! [P3: produc1457211279475724562t_char > $o,P: produc1457211279475724562t_char] :
      ( ! [A5: nat,B4: list_char] : ( P3 @ ( produc120197956887798346t_char @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_138_prod__cases,axiom,
    ! [P3: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A5: nat,B4: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B4 ) )
     => ( P3 @ P ) ) ).

% prod_cases
thf(fact_139_Pair__inject,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A4: set_Product_prod_o_o,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8517790099723286449at_nat @ A @ B )
        = ( produc8517790099723286449at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_140_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B )
        = ( produc9060074326276436823at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_141_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_142_Pair__inject,axiom,
    ! [A: nat,B: list_char,A4: nat,B3: list_char] :
      ( ( ( produc120197956887798346t_char @ A @ B )
        = ( produc120197956887798346t_char @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_143_Pair__inject,axiom,
    ! [A: nat,B: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_144_empty__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P8892882183480857371t_char @ empty_8141283858255108185t_char @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_145_empty__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: prat] :
      ( ( ( empty_6616431780912394577r_prat @ X )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_P4765339447491148764r_prat @ empty_3412554051312619034r_prat @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_146_eq__fst__iff,axiom,
    ! [A: set_Product_prod_o_o,P: produc732395585841259969at_nat] :
      ( ( A
        = ( produc7275247657300178845at_nat @ P ) )
      = ( ? [B2: set_Pr1261947904930325089at_nat] :
            ( P
            = ( produc8517790099723286449at_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_147_eq__fst__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
      ( ( A
        = ( produc4486455499028916547at_nat @ P ) )
      = ( ? [B2: set_Pr4329608150637261639at_nat] :
            ( P
            = ( produc9060074326276436823at_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_148_eq__fst__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( A
        = ( produc6011863565407846595at_nat @ P ) )
      = ( ? [B2: set_Pr1261947904930325089at_nat] :
            ( P
            = ( produc2922128104949294807at_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_149_eq__fst__iff,axiom,
    ! [A: nat,P: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P ) )
      = ( ? [B2: nat] :
            ( P
            = ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_150_eq__fst__iff,axiom,
    ! [A: nat,P: produc1457211279475724562t_char] :
      ( ( A
        = ( produc6204653154488278070t_char @ P ) )
      = ( ? [B2: list_char] :
            ( P
            = ( produc120197956887798346t_char @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_151_fstI,axiom,
    ! [X: produc732395585841259969at_nat,Y: set_Product_prod_o_o,Z: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc8517790099723286449at_nat @ Y @ Z ) )
     => ( ( produc7275247657300178845at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_152_fstI,axiom,
    ! [X: produc1319942482725812455at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( X
        = ( produc9060074326276436823at_nat @ Y @ Z ) )
     => ( ( produc4486455499028916547at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_153_fstI,axiom,
    ! [X: produc3843707927480180839at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc2922128104949294807at_nat @ Y @ Z ) )
     => ( ( produc6011863565407846595at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_154_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_155_fstI,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat,Z: list_char] :
      ( ( X
        = ( produc120197956887798346t_char @ Y @ Z ) )
     => ( ( produc6204653154488278070t_char @ X )
        = Y ) ) ).

% fstI
thf(fact_156_uni__mask_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Xa: prat,Y: produc1457211279475724562t_char > prat] :
      ( ( ( uni_ma6578582744724537101t_char @ X @ Xa )
        = Y )
     => ( ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X @ Xa ) )
       => ~ ( ( Y
              = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char @ X @ Xa ) )
           => ~ ( accp_P8956218777217113800r_prat @ uni_ma1788645471123007132t_char @ ( produc2920179824973321483r_prat @ X @ Xa ) ) ) ) ) ).

% uni_mask.pelims
thf(fact_157_add__masks_Opelims,axiom,
    ! [X: produc1457211279475724562t_char > prat,Xa: produc1457211279475724562t_char > prat,Xb: produc1457211279475724562t_char,Y: prat] :
      ( ( ( add_ma6586698345353345811t_char @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P224314924008452035t_char @ add_ma2412268806269088278t_char @ ( produc2957999048406202538t_char @ X @ ( produc2591414526589101846t_char @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_158_pre__get__m_Osimps,axiom,
    pre_ge2740421702909910421t_char = produc6204653154488278070t_char ).

% pre_get_m.simps
thf(fact_159_pre__get__m_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X )
        = Y )
     => ( Y
        = ( produc6204653154488278070t_char @ X ) ) ) ).

% pre_get_m.elims
thf(fact_160_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc8146610533011921512t_char] :
      ~ ! [F: nat > list_char > list_char,A5: nat,B4: nat,Acc: list_char] :
          ( X
         != ( produc7013538429079112736t_char @ F @ ( produc8776329868854908137t_char @ A5 @ ( produc120197956887798346t_char @ B4 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_161_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F: nat > nat > nat,A5: nat,B4: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_162_accp__induct__rule,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P3: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X3: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X3 )
           => ( ! [Y4: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y4 @ X3 )
                 => ( P3 @ Y4 ) )
             => ( P3 @ X3 ) ) )
       => ( P3 @ A ) ) ) ).

% accp_induct_rule
thf(fact_163_not__accp__down,axiom,
    ! [R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
      ( ~ ( accp_P4765339447491148764r_prat @ R3 @ X )
     => ~ ! [Z2: produc1457211279475724562t_char > prat] :
            ( ( R3 @ Z2 @ X )
           => ( accp_P4765339447491148764r_prat @ R3 @ Z2 ) ) ) ).

% not_accp_down
thf(fact_164_accp_Ocases,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ! [Y4: produc1457211279475724562t_char > prat] :
          ( ( R2 @ Y4 @ A )
         => ( accp_P4765339447491148764r_prat @ R2 @ Y4 ) ) ) ).

% accp.cases
thf(fact_165_accp_Osimps,axiom,
    ( accp_P4765339447491148764r_prat
    = ( ^ [R: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A2: produc1457211279475724562t_char > prat] :
        ? [X2: produc1457211279475724562t_char > prat] :
          ( ( A2 = X2 )
          & ! [Y5: produc1457211279475724562t_char > prat] :
              ( ( R @ Y5 @ X2 )
             => ( accp_P4765339447491148764r_prat @ R @ Y5 ) ) ) ) ) ).

% accp.simps
thf(fact_166_accpI,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
      ( ! [Y3: produc1457211279475724562t_char > prat] :
          ( ( R2 @ Y3 @ X )
         => ( accp_P4765339447491148764r_prat @ R2 @ Y3 ) )
     => ( accp_P4765339447491148764r_prat @ R2 @ X ) ) ).

% accpI
thf(fact_167_accp__induct,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P3: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X3: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X3 )
           => ( ! [Y4: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y4 @ X3 )
                 => ( P3 @ Y4 ) )
             => ( P3 @ X3 ) ) )
       => ( P3 @ A ) ) ) ).

% accp_induct
thf(fact_168_accp__downward,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,B: produc1457211279475724562t_char > prat,A: produc1457211279475724562t_char > prat] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P4765339447491148764r_prat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_169_pre__get__m_Opelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat] :
      ( ( ( pre_ge2740421702909910421t_char @ X )
        = Y )
     => ( ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X )
       => ~ ( ( Y
              = ( produc6204653154488278070t_char @ X ) )
           => ~ ( accp_P8892882183480857371t_char @ pre_ge8748858970926381898t_char @ X ) ) ) ) ).

% pre_get_m.pelims
thf(fact_170_ssubst__Pair__rhs,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o,S2: set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_171_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_172_ssubst__Pair__rhs,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Pr1261947904930325089at_nat,R3: set_Pr457366540195662369at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_173_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_174_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_175_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: list_char,R3: set_Pr2507339680178222706t_char,S2: list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_176_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R3: set_Pr1261947904930325089at_nat,S2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_177_curryI,axiom,
    ! [F4: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( F4 @ ( produc8517790099723286449at_nat @ A @ B ) )
     => ( produc5440495499350717427_nat_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_178_curryI,axiom,
    ! [F4: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( F4 @ ( produc9060074326276436823at_nat @ A @ B ) )
     => ( produc447603880154497357_nat_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_179_curryI,axiom,
    ! [F4: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F4 @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( produc7820828192191229133_nat_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_180_curryI,axiom,
    ! [F4: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( F4 @ ( produc120197956887798346t_char @ A @ B ) )
     => ( produc80901228256969370char_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_181_curryI,axiom,
    ! [F4: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( F4 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( produc1310100445399344235_nat_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_182_in__lex__prod,axiom,
    ! [A: nat,B: nat,A4: nat,B3: nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( lex_prod_nat_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_183_in__lex__prod,axiom,
    ! [A: nat,B: list_char,A4: nat,B3: list_char,R2: set_Pr1261947904930325089at_nat,S: set_Pr4480099295215233283t_char] :
      ( ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc120197956887798346t_char @ A4 @ B3 ) ) @ ( lex_pr1681056561245511266t_char @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_184_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: nat,A4: product_prod_nat_nat,B3: nat,R2: set_Pr8693737435421807431at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A4 @ B3 ) ) @ ( lex_pr4669217892513370978at_nat @ R2 @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_185_in__lex__prod,axiom,
    ! [A: nat,B: product_prod_nat_nat,A4: nat,B3: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A4 @ B3 ) ) @ ( lex_pr8029265285556086080at_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_186_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: nat,A4: set_Product_prod_o_o,B3: nat,R2: set_Pr1932065953672099015od_o_o,S: set_Pr1261947904930325089at_nat] :
      ( ( member3072328306138578198_o_nat @ ( produc9983997497903781_o_nat @ ( produc6963446911922823808_o_nat @ A @ B ) @ ( produc6963446911922823808_o_nat @ A4 @ B3 ) ) @ ( lex_pr7492273792749525096_o_nat @ R2 @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_187_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: nat,A4: set_Pr1261947904930325089at_nat,B3: nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member3480088626692287306at_nat @ ( produc3585720979125124569at_nat @ ( produc3149970401713881818at_nat @ A @ B ) @ ( produc3149970401713881818at_nat @ A4 @ B3 ) ) @ ( lex_pr1951919194703979714at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_188_in__lex__prod,axiom,
    ! [A: nat,B: set_Product_prod_o_o,A4: nat,B3: set_Product_prod_o_o,R2: set_Pr1261947904930325089at_nat,S: set_Pr1932065953672099015od_o_o] :
      ( ( member1875862901105104226od_o_o @ ( produc5336978198856898033od_o_o @ ( produc8424169554027496414od_o_o @ A @ B ) @ ( produc8424169554027496414od_o_o @ A4 @ B3 ) ) @ ( lex_pr8952996434854197702od_o_o @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_189_in__lex__prod,axiom,
    ! [A: nat,B: set_Pr1261947904930325089at_nat,A4: nat,B3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4107301959146914710at_nat @ ( produc8824474337961625893at_nat @ ( produc5463602008962177208at_nat @ A @ B ) @ ( produc5463602008962177208at_nat @ A4 @ B3 ) ) @ ( lex_pr4265550801952275104at_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_190_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A4 @ B3 ) ) @ ( lex_pr8801849515957261039at_nat @ R2 @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_191_in__lex__prod,axiom,
    ! [A: set_Product_prod_o_o,B: product_prod_nat_nat,A4: set_Product_prod_o_o,B3: product_prod_nat_nat,R2: set_Pr1932065953672099015od_o_o,S: set_Pr8693737435421807431at_nat] :
      ( ( member4915364185892706192at_nat @ ( produc6215473421702830295at_nat @ ( produc4266715247171069521at_nat @ A @ B ) @ ( produc4266715247171069521at_nat @ A4 @ B3 ) ) @ ( lex_pr2931540531659632233at_nat @ R2 @ S ) )
      = ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_192_swap__simp,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Product_prod_o_o] :
      ( ( produc8291025974198835997od_o_o @ ( produc2070086378307464573od_o_o @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_193_swap__simp,axiom,
    ! [X: list_char,Y: nat] :
      ( ( produc4410195085741832554ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) )
      = ( produc120197956887798346t_char @ Y @ X ) ) ).

% swap_simp
thf(fact_194_swap__simp,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc5515357658759882065at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc2070086378307464573od_o_o @ Y @ X ) ) ).

% swap_simp
thf(fact_195_swap__simp,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc5396952528824381687at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_196_swap__simp,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc3946833401584108663at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_197_swap__simp,axiom,
    ! [X: nat,Y: list_char] :
      ( ( produc5051419219066781674t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc8702345860417625034ar_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_198_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_199_prod_Ocollapse,axiom,
    ! [Prod: produc732395585841259969at_nat] :
      ( ( produc8517790099723286449at_nat @ ( produc7275247657300178845at_nat @ Prod ) @ ( produc4551702098526911967at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_200_prod_Ocollapse,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_201_prod_Ocollapse,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_202_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_203_prod_Ocollapse,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_204_snd__swap,axiom,
    ! [X: produc1457211279475724562t_char] :
      ( ( produc1259554233722553080ar_nat @ ( produc5051419219066781674t_char @ X ) )
      = ( produc6204653154488278070t_char @ X ) ) ).

% snd_swap
thf(fact_205_fst__swap,axiom,
    ! [X: produc7606469099237612818ar_nat] :
      ( ( produc6204653154488278070t_char @ ( produc4410195085741832554ar_nat @ X ) )
      = ( produc1259554233722553080ar_nat @ X ) ) ).

% fst_swap
thf(fact_206_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P: produc732395585841259969at_nat] :
      ( ( B
        = ( produc4551702098526911967at_nat @ P ) )
      = ( ? [A2: set_Product_prod_o_o] :
            ( P
            = ( produc8517790099723286449at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_207_eq__snd__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,P: produc1319942482725812455at_nat] :
      ( ( B
        = ( produc826222791318859909at_nat @ P ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat] :
            ( P
            = ( produc9060074326276436823at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_208_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( B
        = ( produc2937545259423399429at_nat @ P ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat] :
            ( P
            = ( produc2922128104949294807at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_209_eq__snd__iff,axiom,
    ! [B: list_char,P: produc1457211279475724562t_char] :
      ( ( B
        = ( produc1900778367047502200t_char @ P ) )
      = ( ? [A2: nat] :
            ( P
            = ( produc120197956887798346t_char @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_210_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_211_sndI,axiom,
    ! [X: produc732395585841259969at_nat,Y: set_Product_prod_o_o,Z: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc8517790099723286449at_nat @ Y @ Z ) )
     => ( ( produc4551702098526911967at_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_212_sndI,axiom,
    ! [X: produc1319942482725812455at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( X
        = ( produc9060074326276436823at_nat @ Y @ Z ) )
     => ( ( produc826222791318859909at_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_213_sndI,axiom,
    ! [X: produc3843707927480180839at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc2922128104949294807at_nat @ Y @ Z ) )
     => ( ( produc2937545259423399429at_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_214_sndI,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat,Z: list_char] :
      ( ( X
        = ( produc120197956887798346t_char @ Y @ Z ) )
     => ( ( produc1900778367047502200t_char @ X )
        = Z ) ) ).

% sndI
thf(fact_215_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_216_snd__eqD,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_217_snd__eqD,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_218_snd__eqD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_219_snd__eqD,axiom,
    ! [X: nat,Y: list_char,A: list_char] :
      ( ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_220_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_221_snd__conv,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_222_snd__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] :
      ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_223_snd__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_224_snd__conv,axiom,
    ! [X1: nat,X22: list_char] :
      ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_225_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_226_prod__eq__iff,axiom,
    ( ( ^ [Y6: produc1457211279475724562t_char,Z3: produc1457211279475724562t_char] : ( Y6 = Z3 ) )
    = ( ^ [S3: produc1457211279475724562t_char,T: produc1457211279475724562t_char] :
          ( ( ( produc6204653154488278070t_char @ S3 )
            = ( produc6204653154488278070t_char @ T ) )
          & ( ( produc1900778367047502200t_char @ S3 )
            = ( produc1900778367047502200t_char @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_227_prod__eqI,axiom,
    ! [P: produc1457211279475724562t_char,Q: produc1457211279475724562t_char] :
      ( ( ( produc6204653154488278070t_char @ P )
        = ( produc6204653154488278070t_char @ Q ) )
     => ( ( ( produc1900778367047502200t_char @ P )
          = ( produc1900778367047502200t_char @ Q ) )
       => ( P = Q ) ) ) ).

% prod_eqI
thf(fact_228_prod_Oexpand,axiom,
    ! [Prod: produc1457211279475724562t_char,Prod2: produc1457211279475724562t_char] :
      ( ( ( ( produc6204653154488278070t_char @ Prod )
          = ( produc6204653154488278070t_char @ Prod2 ) )
        & ( ( produc1900778367047502200t_char @ Prod )
          = ( produc1900778367047502200t_char @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_229_prod_Oswap__def,axiom,
    ( produc8291025974198835997od_o_o
    = ( ^ [P2: produc7898102162437433997od_o_o] : ( produc8517790099723286449at_nat @ ( produc7327370413965865899od_o_o @ P2 ) @ ( produc827543935884356969od_o_o @ P2 ) ) ) ) ).

% prod.swap_def
thf(fact_230_prod_Oswap__def,axiom,
    ( produc5396952528824381687at_nat
    = ( ^ [P2: produc1319942482725812455at_nat] : ( produc9060074326276436823at_nat @ ( produc826222791318859909at_nat @ P2 ) @ ( produc4486455499028916547at_nat @ P2 ) ) ) ) ).

% prod.swap_def
thf(fact_231_prod_Oswap__def,axiom,
    ( produc3946833401584108663at_nat
    = ( ^ [P2: produc3843707927480180839at_nat] : ( produc2922128104949294807at_nat @ ( produc2937545259423399429at_nat @ P2 ) @ ( produc6011863565407846595at_nat @ P2 ) ) ) ) ).

% prod.swap_def
thf(fact_232_prod_Oswap__def,axiom,
    ( produc4410195085741832554ar_nat
    = ( ^ [P2: produc7606469099237612818ar_nat] : ( produc120197956887798346t_char @ ( produc1259554233722553080ar_nat @ P2 ) @ ( produc5563429021163328950ar_nat @ P2 ) ) ) ) ).

% prod.swap_def
thf(fact_233_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P2: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) ) ) ) ).

% prod.swap_def
thf(fact_234_prod_Oswap__def,axiom,
    ( produc5051419219066781674t_char
    = ( ^ [P2: produc1457211279475724562t_char] : ( produc8702345860417625034ar_nat @ ( produc1900778367047502200t_char @ P2 ) @ ( produc6204653154488278070t_char @ P2 ) ) ) ) ).

% prod.swap_def
thf(fact_235_curryE,axiom,
    ! [F4: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc5440495499350717427_nat_o @ F4 @ A @ B )
     => ( F4 @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_236_curryE,axiom,
    ! [F4: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc447603880154497357_nat_o @ F4 @ A @ B )
     => ( F4 @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_237_curryE,axiom,
    ! [F4: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F4 @ A @ B )
     => ( F4 @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryE
thf(fact_238_curryE,axiom,
    ! [F4: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( produc80901228256969370char_o @ F4 @ A @ B )
     => ( F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% curryE
thf(fact_239_curryE,axiom,
    ! [F4: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryE
thf(fact_240_curryD,axiom,
    ! [F4: produc732395585841259969at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc5440495499350717427_nat_o @ F4 @ A @ B )
     => ( F4 @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_241_curryD,axiom,
    ! [F4: produc1319942482725812455at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc447603880154497357_nat_o @ F4 @ A @ B )
     => ( F4 @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_242_curryD,axiom,
    ! [F4: produc3843707927480180839at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7820828192191229133_nat_o @ F4 @ A @ B )
     => ( F4 @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% curryD
thf(fact_243_curryD,axiom,
    ! [F4: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( produc80901228256969370char_o @ F4 @ A @ B )
     => ( F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% curryD
thf(fact_244_curryD,axiom,
    ! [F4: product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc1310100445399344235_nat_o @ F4 @ A @ B )
     => ( F4 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% curryD
thf(fact_245_prod_Oexhaust__sel,axiom,
    ! [Prod: produc732395585841259969at_nat] :
      ( Prod
      = ( produc8517790099723286449at_nat @ ( produc7275247657300178845at_nat @ Prod ) @ ( produc4551702098526911967at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_246_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( Prod
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_247_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( Prod
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_248_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_249_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( Prod
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_250_surjective__pairing,axiom,
    ! [T2: produc732395585841259969at_nat] :
      ( T2
      = ( produc8517790099723286449at_nat @ ( produc7275247657300178845at_nat @ T2 ) @ ( produc4551702098526911967at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_251_surjective__pairing,axiom,
    ! [T2: produc1319942482725812455at_nat] :
      ( T2
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ T2 ) @ ( produc826222791318859909at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_252_surjective__pairing,axiom,
    ! [T2: produc3843707927480180839at_nat] :
      ( T2
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ T2 ) @ ( produc2937545259423399429at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_253_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_254_surjective__pairing,axiom,
    ! [T2: produc1457211279475724562t_char] :
      ( T2
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ T2 ) @ ( produc1900778367047502200t_char @ T2 ) ) ) ).

% surjective_pairing
thf(fact_255_exI__realizer,axiom,
    ! [P3: set_Pr1261947904930325089at_nat > set_Product_prod_o_o > $o,Y: set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o] :
      ( ( P3 @ Y @ X )
     => ( P3 @ ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) ) @ ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_256_exI__realizer,axiom,
    ! [P3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( P3 @ Y @ X )
     => ( P3 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_257_exI__realizer,axiom,
    ! [P3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( P3 @ Y @ X )
     => ( P3 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_258_exI__realizer,axiom,
    ! [P3: nat > nat > $o,Y: nat,X: nat] :
      ( ( P3 @ Y @ X )
     => ( P3 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_259_exI__realizer,axiom,
    ! [P3: list_char > nat > $o,Y: list_char,X: nat] :
      ( ( P3 @ Y @ X )
     => ( P3 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_260_conjI__realizer,axiom,
    ! [P3: set_Product_prod_o_o > $o,P: set_Product_prod_o_o,Q2: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat] :
      ( ( P3 @ P )
     => ( ( Q2 @ Q )
       => ( ( P3 @ ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_261_conjI__realizer,axiom,
    ! [P3: set_Pr4329608150637261639at_nat > $o,P: set_Pr4329608150637261639at_nat,Q2: set_Pr4329608150637261639at_nat > $o,Q: set_Pr4329608150637261639at_nat] :
      ( ( P3 @ P )
     => ( ( Q2 @ Q )
       => ( ( P3 @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_262_conjI__realizer,axiom,
    ! [P3: set_Pr1261947904930325089at_nat > $o,P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat] :
      ( ( P3 @ P )
     => ( ( Q2 @ Q )
       => ( ( P3 @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ P @ Q ) ) )
          & ( Q2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_263_conjI__realizer,axiom,
    ! [P3: nat > $o,P: nat,Q2: nat > $o,Q: nat] :
      ( ( P3 @ P )
     => ( ( Q2 @ Q )
       => ( ( P3 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_264_conjI__realizer,axiom,
    ! [P3: nat > $o,P: nat,Q2: list_char > $o,Q: list_char] :
      ( ( P3 @ P )
     => ( ( Q2 @ Q )
       => ( ( P3 @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ P @ Q ) ) )
          & ( Q2 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ P @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_265_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat,A: produc732395585841259969at_nat] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc8517790099723286449at_nat @ X @ Y ) )
       => ( P3 @ ( produc7275247657300178845at_nat @ A ) @ ( produc4551702098526911967at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_266_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc9060074326276436823at_nat @ X @ Y ) )
       => ( P3 @ ( produc4486455499028916547at_nat @ A ) @ ( produc826222791318859909at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_267_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc2922128104949294807at_nat @ X @ Y ) )
       => ( P3 @ ( produc6011863565407846595at_nat @ A ) @ ( produc2937545259423399429at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_268_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( product_Pair_nat_nat @ X @ Y ) )
       => ( P3 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_269_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P3: nat > list_char > $o,X: nat,Y: list_char,A: produc1457211279475724562t_char] :
      ( ( P3 @ X @ Y )
     => ( ( A
          = ( produc120197956887798346t_char @ X @ Y ) )
       => ( P3 @ ( produc6204653154488278070t_char @ A ) @ ( produc1900778367047502200t_char @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_270_exE__realizer_H,axiom,
    ! [P3: list_char > nat > $o,P: produc1457211279475724562t_char] :
      ( ( P3 @ ( produc1900778367047502200t_char @ P ) @ ( produc6204653154488278070t_char @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ~ ( P3 @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_271_same__fstI,axiom,
    ! [P3: nat > $o,X: nat,Y7: list_char,Y: list_char,R3: nat > set_Pr4480099295215233283t_char] :
      ( ( P3 @ X )
     => ( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ X @ Y7 ) @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( same_f652031872680891423t_char @ P3 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_272_same__fstI,axiom,
    ! [P3: set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat > set_Pr7459493094073627847at_nat] :
      ( ( P3 @ X )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member8215962448328497936at_nat @ ( produc2220582066590712663at_nat @ ( produc9060074326276436823at_nat @ X @ Y7 ) @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( same_f4504843259822650924at_nat @ P3 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_273_same__fstI,axiom,
    ! [P3: set_Product_prod_o_o > $o,X: set_Product_prod_o_o,Y7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Product_prod_o_o > set_Pr4329608150637261639at_nat] :
      ( ( P3 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ X @ Y7 ) @ ( produc8517790099723286449at_nat @ X @ Y ) ) @ ( same_f6937299433074352134at_nat @ P3 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_274_same__fstI,axiom,
    ! [P3: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P3 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X @ Y7 ) @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( same_f1460074399991292076at_nat @ P3 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_275_same__fstI,axiom,
    ! [P3: nat > $o,X: nat,Y7: nat,Y: nat,R3: nat > set_Pr1261947904930325089at_nat] :
      ( ( P3 @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y ) @ ( R3 @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y7 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P3 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_276_fstOp__def,axiom,
    ( bNF_fs4859913984995905862t_char
    = ( ^ [P4: nat > list_char > $o,Q3: list_char > list_char > $o,Ac: produc1457211279475724562t_char] : ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Ac ) @ ( bNF_pi2586896789752333176t_char @ P4 @ Q3 @ ( produc6204653154488278070t_char @ Ac ) @ ( produc1900778367047502200t_char @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_277_fstOp__def,axiom,
    ( bNF_fs2355234947312196021t_char
    = ( ^ [P4: nat > nat > $o,Q3: nat > list_char > $o,Ac: produc1457211279475724562t_char] : ( product_Pair_nat_nat @ ( produc6204653154488278070t_char @ Ac ) @ ( bNF_pi5668428018083610343t_char @ P4 @ Q3 @ ( produc6204653154488278070t_char @ Ac ) @ ( produc1900778367047502200t_char @ Ac ) ) ) ) ) ).

% fstOp_def
thf(fact_278_sndOp__def,axiom,
    ( bNF_sn5442490161743904247t_char
    = ( ^ [P4: nat > nat > $o,Q3: nat > list_char > $o,Ac: produc1457211279475724562t_char] : ( produc120197956887798346t_char @ ( bNF_pi5668428018083610343t_char @ P4 @ Q3 @ ( produc6204653154488278070t_char @ Ac ) @ ( produc1900778367047502200t_char @ Ac ) ) @ ( produc1900778367047502200t_char @ Ac ) ) ) ) ).

% sndOp_def
thf(fact_279_image2__eqI,axiom,
    ! [B: nat,F4: product_prod_nat_nat > nat,X: product_prod_nat_nat,C: nat,G: product_prod_nat_nat > nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr8954129895687480649at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_280_image2__eqI,axiom,
    ! [B: nat,F4: product_prod_nat_nat > nat,X: product_prod_nat_nat,C: list_char,G: product_prod_nat_nat > list_char,A3: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr3738443398878377306t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_281_image2__eqI,axiom,
    ! [B: product_prod_nat_nat,F4: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,C: product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A3 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C ) @ ( bNF_Gr8808630244184418791at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_282_image2__eqI,axiom,
    ! [B: nat,F4: produc859450856879609959at_nat > nat,X: produc859450856879609959at_nat,C: nat,G: produc859450856879609959at_nat > nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr97278709799963567at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_283_image2__eqI,axiom,
    ! [B: nat,F4: produc859450856879609959at_nat > nat,X: produc859450856879609959at_nat,C: list_char,G: produc859450856879609959at_nat > list_char,A3: set_Pr8693737435421807431at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8206827879206165904at_nat @ X @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ B @ C ) @ ( bNF_Gr197193474242742976t_char @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_284_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F4: product_prod_nat_nat > set_Product_prod_o_o,X: product_prod_nat_nat,C: set_Product_prod_o_o,G: product_prod_nat_nat > set_Product_prod_o_o,A3: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ C ) @ ( bNF_Gr6834105381924935271od_o_o @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_285_image2__eqI,axiom,
    ! [B: set_Product_prod_o_o,F4: product_prod_nat_nat > set_Product_prod_o_o,X: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A3 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ B @ C ) @ ( bNF_Gr6227219538689528513at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_286_image2__eqI,axiom,
    ! [B: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8440522571783428010at_nat @ X @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ C ) @ ( bNF_Gr2618164000252033511at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_287_image2__eqI,axiom,
    ! [B: nat,F4: produc2934264451710624999od_o_o > nat,X: produc2934264451710624999od_o_o,C: nat,G: produc2934264451710624999od_o_o > nat,A3: set_Pr1932065953672099015od_o_o] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member444158400953824016od_o_o @ X @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr3709360158638718511at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_288_image2__eqI,axiom,
    ! [B: nat,F4: produc732395585841259969at_nat > nat,X: produc732395585841259969at_nat,C: nat,G: produc732395585841259969at_nat > nat,A3: set_Pr457366540195662369at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member4095101504841534314at_nat @ X @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr8825950589361439753at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_289_pair__in__swap__image,axiom,
    ! [Y: set_Product_prod_o_o,X: set_Product_prod_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ ( image_5377701252353234885od_o_o @ produc4310480016201085175od_o_o @ A3 ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_290_pair__in__swap__image,axiom,
    ! [Y: product_prod_nat_nat,X: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( image_6321802852832981957at_nat @ produc7225117575323628663at_nat @ A3 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_291_pair__in__swap__image,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o,A3: set_Pr457366540195662369at_nat] :
      ( ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ Y @ X ) @ ( image_8843695349120394257od_o_o @ produc5515357658759882065at_nat @ A3 ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_292_pair__in__swap__image,axiom,
    ! [Y: list_char,X: nat,A3: set_Pr2507339680178222706t_char] :
      ( ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ Y @ X ) @ ( image_652198164289537669ar_nat @ produc5051419219066781674t_char @ A3 ) )
      = ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_293_pair__in__swap__image,axiom,
    ! [Y: set_Product_prod_o_o,X: set_Pr1261947904930325089at_nat,A3: set_Pr8055703017043224813od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y @ X ) @ ( image_3720774369771669369at_nat @ produc8291025974198835997od_o_o @ A3 ) )
      = ( member2037436044582932534od_o_o @ ( produc2070086378307464573od_o_o @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_294_pair__in__swap__image,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,A3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ ( image_3313167189156628421at_nat @ produc5396952528824381687at_nat @ A3 ) )
      = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_295_pair__in__swap__image,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ ( image_960689587819713477at_nat @ produc3946833401584108663at_nat @ A3 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_296_pair__in__swap__image,axiom,
    ! [Y: nat,X: list_char,A3: set_Pr8803067691650843250ar_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y @ X ) @ ( image_688197443715608197t_char @ produc4410195085741832554ar_nat @ A3 ) )
      = ( member6633493567087309371ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_297_pair__in__swap__image,axiom,
    ! [Y: nat,X: nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( image_5168914502847457605at_nat @ product_swap_nat_nat @ A3 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A3 ) ) ).

% pair_in_swap_image
thf(fact_298_fst__fstOp,axiom,
    ! [P3: nat > list_char > $o,Q2: list_char > list_char > $o] :
      ( produc6204653154488278070t_char
      = ( comp_P6261748720030321425t_char @ produc6204653154488278070t_char @ ( bNF_fs4859913984995905862t_char @ P3 @ Q2 ) ) ) ).

% fst_fstOp
thf(fact_299_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_300_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat,P: produc1457211279475724562t_char] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc3665563422017311042at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_301_mem__case__prodE,axiom,
    ! [Z: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,P: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc2423807293333670521at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_302_mem__case__prodE,axiom,
    ! [Z: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat,P: produc1457211279475724562t_char] :
      ( ( member8206827879206165904at_nat @ Z @ ( produc3763337624047832232at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_303_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P: produc732395585841259969at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc4192656640593693979at_nat @ C @ P ) )
     => ~ ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc8517790099723286449at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_304_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P: produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc1433696742876942453at_nat @ C @ P ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_305_mem__case__prodE,axiom,
    ! [Z: produc2934264451710624999od_o_o,C: nat > nat > set_Pr1932065953672099015od_o_o,P: product_prod_nat_nat] :
      ( ( member444158400953824016od_o_o @ Z @ ( produc773911639751745017od_o_o @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member444158400953824016od_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_306_mem__case__prodE,axiom,
    ! [Z: produc732395585841259969at_nat,C: nat > nat > set_Pr457366540195662369at_nat,P: product_prod_nat_nat] :
      ( ( member4095101504841534314at_nat @ Z @ ( produc1156700346287174227at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member4095101504841534314at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_307_mem__case__prodE,axiom,
    ! [Z: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,P: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( produc3705604114177432185at_nat @ C @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_308_mem__case__prodE,axiom,
    ! [Z: produc2934264451710624999od_o_o,C: nat > list_char > set_Pr1932065953672099015od_o_o,P: produc1457211279475724562t_char] :
      ( ( member444158400953824016od_o_o @ Z @ ( produc1201903168620932904od_o_o @ C @ P ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member444158400953824016od_o_o @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_309_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc2934264451710624999od_o_o,A3: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( member444158400953824016od_o_o @ X @ ( collec3302784863738620242od_o_o @ ( produc9128050996712119596_o_o_o @ A3 ) ) )
     => ( A3 @ ( produc3308063957627853123od_o_o @ X ) @ ( produc954223133649922181od_o_o @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_310_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc732395585841259969at_nat,A3: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ( member4095101504841534314at_nat @ X @ ( collec1729919521702017196at_nat @ ( produc8045849345425481810_nat_o @ A3 ) ) )
     => ( A3 @ ( produc7275247657300178845at_nat @ X ) @ ( produc4551702098526911967at_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_311_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ A3 ) ) )
     => ( A3 @ ( produc6011863565407846595at_nat @ X ) @ ( produc2937545259423399429at_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_312_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc859450856879609959at_nat,A3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ X @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ A3 ) ) )
     => ( A3 @ ( produc3213797794245857475at_nat @ X ) @ ( produc6408287024330202629at_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_313_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_nat_nat,A3: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A3 ) ) )
     => ( A3 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_314_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc1457211279475724562t_char,A3: nat > list_char > $o] :
      ( ( member484235747325421115t_char @ X @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ A3 ) ) )
     => ( A3 @ ( produc6204653154488278070t_char @ X ) @ ( produc1900778367047502200t_char @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_315_fst__comp__apfst,axiom,
    ! [F4: nat > nat] :
      ( ( comp_P6261748720030321425t_char @ produc6204653154488278070t_char @ ( produc2835005367637779135t_char @ F4 ) )
      = ( comp_n7998183327787346125t_char @ F4 @ produc6204653154488278070t_char ) ) ).

% fst_comp_apfst
thf(fact_316_fst__comp__map__prod,axiom,
    ! [F4: nat > nat,G: list_char > list_char] :
      ( ( comp_P6261748720030321425t_char @ produc6204653154488278070t_char @ ( produc5890707981334396508t_char @ F4 @ G ) )
      = ( comp_n7998183327787346125t_char @ F4 @ produc6204653154488278070t_char ) ) ).

% fst_comp_map_prod
thf(fact_317_fst__comp__apsnd,axiom,
    ! [F4: list_char > list_char] :
      ( ( comp_P6261748720030321425t_char @ produc6204653154488278070t_char @ ( produc5571979602231644306ar_nat @ F4 ) )
      = produc6204653154488278070t_char ) ).

% fst_comp_apsnd
thf(fact_318_map__prod__simp,axiom,
    ! [F4: nat > nat,G: nat > nat,A: nat,B: nat] :
      ( ( produc6977886695330630970at_nat @ F4 @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_319_map__prod__simp,axiom,
    ! [F4: nat > nat,G: list_char > nat,A: nat,B: list_char] :
      ( ( produc3271292527955122507ar_nat @ F4 @ G @ ( produc120197956887798346t_char @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_320_map__prod__simp,axiom,
    ! [F4: nat > nat,G: nat > list_char,A: nat,B: nat] :
      ( ( produc3912516661280071627t_char @ F4 @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_321_map__prod__simp,axiom,
    ! [F4: nat > nat,G: list_char > list_char,A: nat,B: list_char] :
      ( ( produc5890707981334396508t_char @ F4 @ G @ ( produc120197956887798346t_char @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_322_map__prod__simp,axiom,
    ! [F4: set_Product_prod_o_o > nat,G: set_Pr1261947904930325089at_nat > nat,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc6028337107241370000at_nat @ F4 @ G @ ( produc8517790099723286449at_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_323_map__prod__simp,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc7374836315209622838at_nat @ F4 @ G @ ( produc2922128104949294807at_nat @ A @ B ) )
      = ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_324_map__prod__simp,axiom,
    ! [F4: nat > set_Product_prod_o_o,G: nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( produc8030503241784373392at_nat @ F4 @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc8517790099723286449at_nat @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_325_map__prod__simp,axiom,
    ! [F4: nat > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( produc7508474501381341110at_nat @ F4 @ G @ ( product_Pair_nat_nat @ A @ B ) )
      = ( produc2922128104949294807at_nat @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_326_map__prod__simp,axiom,
    ! [F4: set_Product_prod_o_o > nat,G: set_Pr1261947904930325089at_nat > list_char,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc2482972258791676705t_char @ F4 @ G @ ( produc8517790099723286449at_nat @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_327_map__prod__simp,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > nat,G: set_Pr1261947904930325089at_nat > list_char,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc4151949830376967623t_char @ F4 @ G @ ( produc2922128104949294807at_nat @ A @ B ) )
      = ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) ) ).

% map_prod_simp
thf(fact_328_fst__map__prod,axiom,
    ! [F4: nat > nat,G: list_char > list_char,X: produc1457211279475724562t_char] :
      ( ( produc6204653154488278070t_char @ ( produc5890707981334396508t_char @ F4 @ G @ X ) )
      = ( F4 @ ( produc6204653154488278070t_char @ X ) ) ) ).

% fst_map_prod
thf(fact_329_apfst__conv,axiom,
    ! [F4: set_Product_prod_o_o > set_Product_prod_o_o,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc3496622756277034469at_nat @ F4 @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_330_apfst__conv,axiom,
    ! [F4: set_Product_prod_o_o > set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc1719095019421672203at_nat @ F4 @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_331_apfst__conv,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc6529295065807390923at_nat @ F4 @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_332_apfst__conv,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Product_prod_o_o,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc6414627024158678591at_nat @ F4 @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_333_apfst__conv,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc5380969780118646373at_nat @ F4 @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_334_apfst__conv,axiom,
    ! [F4: nat > nat,X: nat,Y: list_char] :
      ( ( produc2835005367637779135t_char @ F4 @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc120197956887798346t_char @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_335_apfst__conv,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( produc986720760941809198at_nat @ F4 @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ ( F4 @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_336_apsnd__conv,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc4927343412728267593od_o_o @ F4 @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( produc8517790099723286449at_nat @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_337_apsnd__conv,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( produc8057050257526108681at_nat @ F4 @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( produc9060074326276436823at_nat @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_338_apsnd__conv,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( produc6468969725218530979at_nat @ F4 @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( produc2922128104949294807at_nat @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_339_apsnd__conv,axiom,
    ! [F4: list_char > list_char,X: nat,Y: list_char] :
      ( ( produc5571979602231644306ar_nat @ F4 @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc120197956887798346t_char @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_340_apsnd__conv,axiom,
    ! [F4: list_char > nat,X: nat,Y: list_char] :
      ( ( produc8286888221270194945at_nat @ F4 @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( product_Pair_nat_nat @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_341_apsnd__conv,axiom,
    ! [F4: nat > list_char,X: nat,Y: nat] :
      ( ( produc561246565970313601ar_nat @ F4 @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( produc120197956887798346t_char @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_342_apsnd__conv,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( produc3094765310956027504at_nat @ F4 @ ( product_Pair_nat_nat @ X @ Y ) )
      = ( product_Pair_nat_nat @ X @ ( F4 @ Y ) ) ) ).

% apsnd_conv
thf(fact_343_fst__apfst,axiom,
    ! [F4: nat > nat,X: produc1457211279475724562t_char] :
      ( ( produc6204653154488278070t_char @ ( produc2835005367637779135t_char @ F4 @ X ) )
      = ( F4 @ ( produc6204653154488278070t_char @ X ) ) ) ).

% fst_apfst
thf(fact_344_fst__apsnd,axiom,
    ! [F4: list_char > list_char,X: produc1457211279475724562t_char] :
      ( ( produc6204653154488278070t_char @ ( produc5571979602231644306ar_nat @ F4 @ X ) )
      = ( produc6204653154488278070t_char @ X ) ) ).

% fst_apsnd
thf(fact_345_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F4: nat > nat,G: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_346_map__prod__imageI,axiom,
    ! [A: nat,B: list_char,R3: set_Pr2507339680178222706t_char,F4: nat > nat,G: list_char > nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_2110972657701675764at_nat @ ( produc3271292527955122507ar_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_347_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F4: nat > nat,G: nat > list_char] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_728565836875346774t_char @ ( produc3912516661280071627t_char @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_348_map__prod__imageI,axiom,
    ! [A: nat,B: list_char,R3: set_Pr2507339680178222706t_char,F4: nat > nat,G: list_char > list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R3 )
     => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_3726312381382425221t_char @ ( produc5890707981334396508t_char @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_349_map__prod__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,F4: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_350_map__prod__imageI,axiom,
    ! [A: nat,B: nat,R3: set_Pr1261947904930325089at_nat,F4: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_351_map__prod__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,F4: product_prod_nat_nat > nat,G: product_prod_nat_nat > list_char] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R3 )
     => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_4209661836242193520t_char @ ( produc8778638755205509575t_char @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_352_map__prod__imageI,axiom,
    ! [A: nat,B: list_char,R3: set_Pr2507339680178222706t_char,F4: nat > product_prod_nat_nat,G: list_char > product_prod_nat_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_5263967778105348570at_nat @ ( produc6494513283846573029at_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_353_map__prod__imageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o,F4: set_Product_prod_o_o > nat,G: set_Product_prod_o_o > nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_2029717158088452831at_nat @ ( produc9048355686397359158_o_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_354_map__prod__imageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R3: set_Pr457366540195662369at_nat,F4: set_Product_prod_o_o > nat,G: set_Pr1261947904930325089at_nat > nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ A ) @ ( G @ B ) ) @ ( image_4401859676472782469at_nat @ ( produc6028337107241370000at_nat @ F4 @ G ) @ R3 ) ) ) ).

% map_prod_imageI
thf(fact_355_apsnd__apfst,axiom,
    ! [F4: list_char > set_Pr1261947904930325089at_nat,G: nat > set_Product_prod_o_o,X: produc1457211279475724562t_char] :
      ( ( produc8409766556590220869od_o_o @ F4 @ ( produc7700126872046322396t_char @ G @ X ) )
      = ( produc8517790099723286449at_nat @ ( G @ ( produc6204653154488278070t_char @ X ) ) @ ( F4 @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_356_apsnd__apfst,axiom,
    ! [F4: list_char > set_Pr4329608150637261639at_nat,G: nat > set_Pr4329608150637261639at_nat,X: produc1457211279475724562t_char] :
      ( ( produc6263020832738037791at_nat @ F4 @ ( produc1495070761899153948t_char @ G @ X ) )
      = ( produc9060074326276436823at_nat @ ( G @ ( produc6204653154488278070t_char @ X ) ) @ ( F4 @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_357_apsnd__apfst,axiom,
    ! [F4: list_char > set_Pr1261947904930325089at_nat,G: nat > set_Pr1261947904930325089at_nat,X: produc1457211279475724562t_char] :
      ( ( produc2344097319275338655at_nat @ F4 @ ( produc8767593173587849910t_char @ G @ X ) )
      = ( produc2922128104949294807at_nat @ ( G @ ( produc6204653154488278070t_char @ X ) ) @ ( F4 @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_358_apsnd__apfst,axiom,
    ! [F4: list_char > list_char,G: nat > nat,X: produc1457211279475724562t_char] :
      ( ( produc5571979602231644306ar_nat @ F4 @ ( produc2835005367637779135t_char @ G @ X ) )
      = ( produc120197956887798346t_char @ ( G @ ( produc6204653154488278070t_char @ X ) ) @ ( F4 @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_359_apsnd__apfst,axiom,
    ! [F4: list_char > nat,G: nat > nat,X: produc1457211279475724562t_char] :
      ( ( produc8286888221270194945at_nat @ F4 @ ( produc2835005367637779135t_char @ G @ X ) )
      = ( product_Pair_nat_nat @ ( G @ ( produc6204653154488278070t_char @ X ) ) @ ( F4 @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_360_apfst__apsnd,axiom,
    ! [F4: nat > set_Product_prod_o_o,G: list_char > set_Pr1261947904930325089at_nat,X: produc1457211279475724562t_char] :
      ( ( produc7380983214354644646at_nat @ F4 @ ( produc3483201489921181714at_nat @ G @ X ) )
      = ( produc8517790099723286449at_nat @ ( F4 @ ( produc6204653154488278070t_char @ X ) ) @ ( G @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_361_apfst__apsnd,axiom,
    ! [F4: nat > set_Pr4329608150637261639at_nat,G: list_char > set_Pr4329608150637261639at_nat,X: produc1457211279475724562t_char] :
      ( ( produc5654963204781008076at_nat @ F4 @ ( produc5930691501305929976at_nat @ G @ X ) )
      = ( produc9060074326276436823at_nat @ ( F4 @ ( produc6204653154488278070t_char @ X ) ) @ ( G @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_362_apfst__apsnd,axiom,
    ! [F4: nat > set_Pr1261947904930325089at_nat,G: list_char > set_Pr1261947904930325089at_nat,X: produc1457211279475724562t_char] :
      ( ( produc919334375020031820at_nat @ F4 @ ( produc3483201489921181714at_nat @ G @ X ) )
      = ( produc2922128104949294807at_nat @ ( F4 @ ( produc6204653154488278070t_char @ X ) ) @ ( G @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_363_apfst__apsnd,axiom,
    ! [F4: nat > nat,G: list_char > list_char,X: produc1457211279475724562t_char] :
      ( ( produc2835005367637779135t_char @ F4 @ ( produc5571979602231644306ar_nat @ G @ X ) )
      = ( produc120197956887798346t_char @ ( F4 @ ( produc6204653154488278070t_char @ X ) ) @ ( G @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_364_apfst__apsnd,axiom,
    ! [F4: nat > nat,G: list_char > nat,X: produc1457211279475724562t_char] :
      ( ( produc986720760941809198at_nat @ F4 @ ( produc8286888221270194945at_nat @ G @ X ) )
      = ( product_Pair_nat_nat @ ( F4 @ ( produc6204653154488278070t_char @ X ) ) @ ( G @ ( produc1900778367047502200t_char @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_365_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F4: nat > nat,G: nat > nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( image_5168914502847457605at_nat @ ( produc6977886695330630970at_nat @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_366_prod__fun__imageE,axiom,
    ! [C: produc1457211279475724562t_char,F4: nat > nat,G: nat > list_char,R3: set_Pr1261947904930325089at_nat] :
      ( ( member484235747325421115t_char @ C @ ( image_728565836875346774t_char @ ( produc3912516661280071627t_char @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc120197956887798346t_char @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_367_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F4: nat > nat,G: list_char > nat,R3: set_Pr2507339680178222706t_char] :
      ( ( member8440522571783428010at_nat @ C @ ( image_2110972657701675764at_nat @ ( produc3271292527955122507ar_nat @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_368_prod__fun__imageE,axiom,
    ! [C: produc1457211279475724562t_char,F4: nat > nat,G: list_char > list_char,R3: set_Pr2507339680178222706t_char] :
      ( ( member484235747325421115t_char @ C @ ( image_3726312381382425221t_char @ ( produc5890707981334396508t_char @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( C
              = ( produc120197956887798346t_char @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_369_prod__fun__imageE,axiom,
    ! [C: produc859450856879609959at_nat,F4: nat > product_prod_nat_nat,G: nat > product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( image_1119892238107900587at_nat @ ( produc1153188318382123958at_nat @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc6161850002892822231at_nat @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_370_prod__fun__imageE,axiom,
    ! [C: product_prod_nat_nat,F4: product_prod_nat_nat > nat,G: product_prod_nat_nat > nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( image_6706873702294942047at_nat @ ( produc5156754000680454454at_nat @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( C
              = ( product_Pair_nat_nat @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_371_prod__fun__imageE,axiom,
    ! [C: produc859450856879609959at_nat,F4: nat > product_prod_nat_nat,G: list_char > product_prod_nat_nat,R3: set_Pr2507339680178222706t_char] :
      ( ( member8206827879206165904at_nat @ C @ ( image_5263967778105348570at_nat @ ( produc6494513283846573029at_nat @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( C
              = ( produc6161850002892822231at_nat @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_372_prod__fun__imageE,axiom,
    ! [C: produc1457211279475724562t_char,F4: product_prod_nat_nat > nat,G: product_prod_nat_nat > list_char,R3: set_Pr8693737435421807431at_nat] :
      ( ( member484235747325421115t_char @ C @ ( image_4209661836242193520t_char @ ( produc8778638755205509575t_char @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( C
              = ( produc120197956887798346t_char @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_373_prod__fun__imageE,axiom,
    ! [C: produc2934264451710624999od_o_o,F4: nat > set_Product_prod_o_o,G: nat > set_Product_prod_o_o,R3: set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ C @ ( image_8095068765341748011od_o_o @ ( produc5809847009553229750od_o_o @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc763777882069021527od_o_o @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_374_prod__fun__imageE,axiom,
    ! [C: produc732395585841259969at_nat,F4: nat > set_Product_prod_o_o,G: nat > set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ C @ ( image_1912541218521307653at_nat @ ( produc8030503241784373392at_nat @ F4 @ G ) @ R3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( C
              = ( produc8517790099723286449at_nat @ ( F4 @ X3 ) @ ( G @ Y3 ) ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 ) ) ) ).

% prod_fun_imageE
thf(fact_375_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc2934264451710624999od_o_o,A3: set_se491199571785101031od_o_o,F4: set_Product_prod_o_o > set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ Z @ ( collec3302784863738620242od_o_o @ ( produc9128050996712119596_o_o_o @ ( bNF_Gr472834428334945934od_o_o @ A3 @ F4 ) ) ) )
     => ( ( comp_s2259134718109610372od_o_o @ F4 @ produc3308063957627853123od_o_o @ Z )
        = ( produc954223133649922181od_o_o @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_376_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc732395585841259969at_nat,A3: set_se491199571785101031od_o_o,F4: set_Product_prod_o_o > set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ Z @ ( collec1729919521702017196at_nat @ ( produc8045849345425481810_nat_o @ ( bNF_Gr1962207778594255720at_nat @ A3 @ F4 ) ) ) )
     => ( ( comp_s3616968522556393092at_nat @ F4 @ produc7275247657300178845at_nat @ Z )
        = ( produc4551702098526911967at_nat @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_377_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A3 @ F4 ) ) ) )
     => ( ( comp_s7355055891894327172at_nat @ F4 @ produc6011863565407846595at_nat @ Z )
        = ( produc2937545259423399429at_nat @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_378_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A3 @ F4 ) ) ) )
     => ( ( comp_P3189659738270478212at_nat @ F4 @ produc3213797794245857475at_nat @ Z )
        = ( produc6408287024330202629at_nat @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_379_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: product_prod_nat_nat,A3: set_nat,F4: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A3 @ F4 ) ) ) )
     => ( ( comp_n2711240574518091068at_nat @ F4 @ product_fst_nat_nat @ Z )
        = ( product_snd_nat_nat @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_380_Collect__case__prod__Grp__eqD,axiom,
    ! [Z: produc1457211279475724562t_char,A3: set_nat,F4: nat > list_char] :
      ( ( member484235747325421115t_char @ Z @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ ( bNF_Gr2905446856833220225t_char @ A3 @ F4 ) ) ) )
     => ( ( comp_n8836744016134526332t_char @ F4 @ produc6204653154488278070t_char @ Z )
        = ( produc1900778367047502200t_char @ Z ) ) ) ).

% Collect_case_prod_Grp_eqD
thf(fact_381_Collect__split__mono__strong,axiom,
    ! [X4: set_nat,A3: set_Pr2507339680178222706t_char,Y8: set_list_char,P3: nat > list_char > $o,Q2: nat > list_char > $o] :
      ( ( X4
        = ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ A3 ) )
     => ( ( Y8
          = ( image_3034139768025083822t_char @ produc1900778367047502200t_char @ A3 ) )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ X4 )
             => ! [Xa2: list_char] :
                  ( ( member_list_char @ Xa2 @ Y8 )
                 => ( ( P3 @ X3 @ Xa2 )
                   => ( Q2 @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P3 ) ) )
           => ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_382_ssubsetI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,B5: set_Pr1932065953672099015od_o_o] :
      ( ! [Pi6: set_Product_prod_o_o,H: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Pi6 @ H ) @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Pi6 @ H ) @ B5 ) )
     => ( ord_le7586318239878589543od_o_o @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_383_ssubsetI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [Pi6: product_prod_nat_nat,H: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Pi6 @ H ) @ A3 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Pi6 @ H ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_384_ssubsetI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,B5: set_Pr457366540195662369at_nat] :
      ( ! [Pi6: set_Product_prod_o_o,H: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Pi6 @ H ) @ A3 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Pi6 @ H ) @ B5 ) )
     => ( ord_le4841768313324958145at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_385_ssubsetI,axiom,
    ! [A3: set_Pr7459493094073627847at_nat,B5: set_Pr7459493094073627847at_nat] :
      ( ! [Pi6: set_Pr4329608150637261639at_nat,H: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Pi6 @ H ) @ A3 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Pi6 @ H ) @ B5 ) )
     => ( ord_le5997549366648089703at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_386_ssubsetI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [Pi6: set_Pr1261947904930325089at_nat,H: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi6 @ H ) @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi6 @ H ) @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_387_ssubsetI,axiom,
    ! [A3: set_Pr2507339680178222706t_char,B5: set_Pr2507339680178222706t_char] :
      ( ! [Pi6: nat,H: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi6 @ H ) @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi6 @ H ) @ B5 ) )
     => ( ord_le6315511645215477266t_char @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_388_ssubsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [Pi6: nat,H: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi6 @ H ) @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi6 @ H ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_389_map__prod__surj,axiom,
    ! [F4: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat @ F4 @ 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 @ F4 @ G ) @ top_to4669805908274784177at_nat )
          = top_to4669805908274784177at_nat ) ) ) ).

% map_prod_surj
thf(fact_390_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc2934264451710624999od_o_o,A3: set_se491199571785101031od_o_o,F4: set_Product_prod_o_o > set_Product_prod_o_o] :
      ( ( member444158400953824016od_o_o @ Z @ ( collec3302784863738620242od_o_o @ ( produc9128050996712119596_o_o_o @ ( bNF_Gr472834428334945934od_o_o @ A3 @ F4 ) ) ) )
     => ( member3157015407008874800od_o_o @ ( produc3308063957627853123od_o_o @ Z ) @ A3 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_391_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc732395585841259969at_nat,A3: set_se491199571785101031od_o_o,F4: set_Product_prod_o_o > set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ Z @ ( collec1729919521702017196at_nat @ ( produc8045849345425481810_nat_o @ ( bNF_Gr1962207778594255720at_nat @ A3 @ F4 ) ) ) )
     => ( member3157015407008874800od_o_o @ ( produc7275247657300178845at_nat @ Z ) @ A3 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_392_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( bNF_Gr6677838525845560078at_nat @ A3 @ F4 ) ) ) )
     => ( member2643936169264416010at_nat @ ( produc6011863565407846595at_nat @ Z ) @ A3 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_393_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ ( bNF_Gr7275288873855648526at_nat @ A3 @ F4 ) ) ) )
     => ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ Z ) @ A3 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_394_Collect__case__prod__Grp__in,axiom,
    ! [Z: product_prod_nat_nat,A3: set_nat,F4: nat > nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( bNF_Grp_nat_nat @ A3 @ F4 ) ) ) )
     => ( member_nat @ ( product_fst_nat_nat @ Z ) @ A3 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_395_Collect__case__prod__Grp__in,axiom,
    ! [Z: produc1457211279475724562t_char,A3: set_nat,F4: nat > list_char] :
      ( ( member484235747325421115t_char @ Z @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ ( bNF_Gr2905446856833220225t_char @ A3 @ F4 ) ) ) )
     => ( member_nat @ ( produc6204653154488278070t_char @ Z ) @ A3 ) ) ).

% Collect_case_prod_Grp_in
thf(fact_396_range__fst,axiom,
    ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ top_to1250346684925019074t_char )
    = top_top_set_nat ) ).

% range_fst
thf(fact_397_subrelI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,S: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ S ) )
     => ( ord_le7586318239878589543od_o_o @ R2 @ S ) ) ).

% subrelI
thf(fact_398_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_399_subrelI,axiom,
    ! [R2: set_Pr457366540195662369at_nat,S: set_Pr457366540195662369at_nat] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X3 @ Y3 ) @ R2 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le4841768313324958145at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_400_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_401_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_402_subrelI,axiom,
    ! [R2: set_Pr2507339680178222706t_char,S: set_Pr2507339680178222706t_char] :
      ( ! [X3: nat,Y3: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ R2 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ Y3 ) @ S ) )
     => ( ord_le6315511645215477266t_char @ R2 @ S ) ) ).

% subrelI
thf(fact_403_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_404_relChain__def,axiom,
    ( bNF_Ca3809476920824728371_o_nat
    = ( ^ [R: set_Pr1932065953672099015od_o_o,As: set_Product_prod_o_o > nat] :
        ! [I: set_Product_prod_o_o,J: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_405_relChain__def,axiom,
    ( bNF_Ca8308629720386654381at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,As: product_prod_nat_nat > nat] :
        ! [I: product_prod_nat_nat,J: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_406_relChain__def,axiom,
    ( bNF_Ca8906692637563564659at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,As: set_Pr4329608150637261639at_nat > nat] :
        ! [I: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_407_relChain__def,axiom,
    ( bNF_Ca4949091444211332621at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,As: set_Pr1261947904930325089at_nat > nat] :
        ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_408_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_409_snd__diag__fst,axiom,
    ( ( comp_P2122579405136797538t_char @ product_snd_nat_nat
      @ ( comp_n1645060710405774596t_char
        @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ X2 )
        @ produc6204653154488278070t_char ) )
    = produc6204653154488278070t_char ) ).

% snd_diag_fst
thf(fact_410_case__prodI,axiom,
    ! [F4: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( F4 @ A @ B )
     => ( produc8045849345425481810_nat_o @ F4 @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_411_case__prodI,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( F4 @ A @ B )
     => ( produc9020218426428501292_nat_o @ F4 @ ( produc9060074326276436823at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_412_case__prodI,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F4 @ A @ B )
     => ( produc410239310623530412_nat_o @ F4 @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_413_case__prodI,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( F4 @ A @ B )
     => ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% case_prodI
thf(fact_414_case__prodI,axiom,
    ! [F4: nat > nat > $o,A: nat,B: nat] :
      ( ( F4 @ A @ B )
     => ( produc6081775807080527818_nat_o @ F4 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_415_case__prodI2,axiom,
    ! [P: produc732395585841259969at_nat,C: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ! [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc8517790099723286449at_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc8045849345425481810_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_416_case__prodI2,axiom,
    ! [P: produc1319942482725812455at_nat,C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
          ( ( P
            = ( produc9060074326276436823at_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc9020218426428501292_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_417_case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc2922128104949294807at_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc410239310623530412_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_418_case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,C: nat > list_char > $o] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc8526122873445342073char_o @ C @ P ) ) ).

% case_prodI2
thf(fact_419_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A5: nat,B4: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( C @ A5 @ B4 ) )
     => ( produc6081775807080527818_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_420_case__prod__Pair,axiom,
    ( ( produc9214616623306952315at_nat @ produc8517790099723286449at_nat )
    = id_Pro7183678340284408754at_nat ) ).

% case_prod_Pair
thf(fact_421_case__prod__Pair,axiom,
    ( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat )
    = id_Pro6043886709289754456at_nat ) ).

% case_prod_Pair
thf(fact_422_case__prod__Pair,axiom,
    ( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat )
    = id_Pro8722402698821300696at_nat ) ).

% case_prod_Pair
thf(fact_423_case__prod__Pair,axiom,
    ( ( produc4071494739252518515t_char @ produc120197956887798346t_char )
    = id_Pro6264179822262827651t_char ) ).

% case_prod_Pair
thf(fact_424_case__prod__Pair,axiom,
    ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat )
    = id_Pro7559728917991913458at_nat ) ).

% case_prod_Pair
thf(fact_425_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_426_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat,A: nat,B: list_char] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc3665563422017311042at_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_427_mem__case__prodI,axiom,
    ! [Z: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,A: nat,B: nat] :
      ( ( member8206827879206165904at_nat @ Z @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2423807293333670521at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_428_mem__case__prodI,axiom,
    ! [Z: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat,A: nat,B: list_char] :
      ( ( member8206827879206165904at_nat @ Z @ ( C @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc3763337624047832232at_nat @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_429_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc4192656640593693979at_nat @ C @ ( produc8517790099723286449at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_430_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc1433696742876942453at_nat @ C @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_431_mem__case__prodI,axiom,
    ! [Z: produc2934264451710624999od_o_o,C: nat > nat > set_Pr1932065953672099015od_o_o,A: nat,B: nat] :
      ( ( member444158400953824016od_o_o @ Z @ ( C @ A @ B ) )
     => ( member444158400953824016od_o_o @ Z @ ( produc773911639751745017od_o_o @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_432_mem__case__prodI,axiom,
    ! [Z: produc732395585841259969at_nat,C: nat > nat > set_Pr457366540195662369at_nat,A: nat,B: nat] :
      ( ( member4095101504841534314at_nat @ Z @ ( C @ A @ B ) )
     => ( member4095101504841534314at_nat @ Z @ ( produc1156700346287174227at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_433_mem__case__prodI,axiom,
    ! [Z: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,A: nat,B: nat] :
      ( ( member8757157785044589968at_nat @ Z @ ( C @ A @ B ) )
     => ( member8757157785044589968at_nat @ Z @ ( produc3705604114177432185at_nat @ C @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_434_mem__case__prodI,axiom,
    ! [Z: produc2934264451710624999od_o_o,C: nat > list_char > set_Pr1932065953672099015od_o_o,A: nat,B: list_char] :
      ( ( member444158400953824016od_o_o @ Z @ ( C @ A @ B ) )
     => ( member444158400953824016od_o_o @ Z @ ( produc1201903168620932904od_o_o @ C @ ( produc120197956887798346t_char @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_435_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc8197505143624133779at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_436_mem__case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,Z: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc3665563422017311042at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_437_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc2423807293333670521at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_438_mem__case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,Z: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( member8206827879206165904at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8206827879206165904at_nat @ Z @ ( produc3763337624047832232at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_439_mem__case__prodI2,axiom,
    ! [P: produc732395585841259969at_nat,Z: product_prod_nat_nat,C: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc8517790099723286449at_nat @ A5 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc4192656640593693979at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_440_mem__case__prodI2,axiom,
    ! [P: produc3843707927480180839at_nat,Z: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( P
            = ( produc2922128104949294807at_nat @ A5 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc1433696742876942453at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_441_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: produc2934264451710624999od_o_o,C: nat > nat > set_Pr1932065953672099015od_o_o] :
      ( ! [A5: nat,B4: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member444158400953824016od_o_o @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member444158400953824016od_o_o @ Z @ ( produc773911639751745017od_o_o @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_442_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: produc732395585841259969at_nat,C: nat > nat > set_Pr457366540195662369at_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member4095101504841534314at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member4095101504841534314at_nat @ Z @ ( produc1156700346287174227at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_443_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member8757157785044589968at_nat @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member8757157785044589968at_nat @ Z @ ( produc3705604114177432185at_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_444_mem__case__prodI2,axiom,
    ! [P: produc1457211279475724562t_char,Z: produc2934264451710624999od_o_o,C: nat > list_char > set_Pr1932065953672099015od_o_o] :
      ( ! [A5: nat,B4: list_char] :
          ( ( P
            = ( produc120197956887798346t_char @ A5 @ B4 ) )
         => ( member444158400953824016od_o_o @ Z @ ( C @ A5 @ B4 ) ) )
     => ( member444158400953824016od_o_o @ Z @ ( produc1201903168620932904od_o_o @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_445_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F4: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F4 @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_446_pair__imageI,axiom,
    ! [A: nat,B: list_char,A3: set_Pr2507339680178222706t_char,F4: nat > list_char > product_prod_nat_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F4 @ A @ B ) @ ( image_2110972657701675764at_nat @ ( produc6775168648774419426at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_447_pair__imageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,F4: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F4 @ A @ B ) @ ( image_6706873702294942047at_nat @ ( produc2982569246487150357at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_448_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F4: nat > nat > produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member8206827879206165904at_nat @ ( F4 @ A @ B ) @ ( image_1119892238107900587at_nat @ ( produc2382474379627796377at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_449_pair__imageI,axiom,
    ! [A: nat,B: list_char,A3: set_Pr2507339680178222706t_char,F4: nat > list_char > produc859450856879609959at_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ A3 )
     => ( member8206827879206165904at_nat @ ( F4 @ A @ B ) @ ( image_5263967778105348570at_nat @ ( produc849435112462063048at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_450_pair__imageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_Pr1932065953672099015od_o_o,F4: set_Product_prod_o_o > set_Product_prod_o_o > product_prod_nat_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F4 @ A @ B ) @ ( image_2029717158088452831at_nat @ ( produc9066432642217594517at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_451_pair__imageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A3: set_Pr457366540195662369at_nat,F4: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F4 @ A @ B ) @ ( image_4401859676472782469at_nat @ ( produc764144141013033915at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_452_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_Pr4329608150637261639at_nat,F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ A3 )
     => ( member8440522571783428010at_nat @ ( F4 @ A @ B ) @ ( image_7325330684212377951at_nat @ ( produc2376066862417776405at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_453_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F4: nat > nat > produc2934264451710624999od_o_o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member444158400953824016od_o_o @ ( F4 @ A @ B ) @ ( image_8095068765341748011od_o_o @ ( produc1490135129000203801od_o_o @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_454_pair__imageI,axiom,
    ! [A: nat,B: nat,A3: set_Pr1261947904930325089at_nat,F4: nat > nat > produc732395585841259969at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A3 )
     => ( member4095101504841534314at_nat @ ( F4 @ A @ B ) @ ( image_1912541218521307653at_nat @ ( produc1215715022925026803at_nat @ F4 ) @ A3 ) ) ) ).

% pair_imageI
thf(fact_455_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,P3: nat > nat > $o,Q2: nat > nat > $o,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re578469030762574527_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P3 @ ( product_fst_nat_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_456_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: nat > list_char > $o,P3: nat > nat > $o,Q2: nat > list_char > $o,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc1457211279475724562t_char,B5: set_Pr2507339680178222706t_char] :
      ( ( bNF_re4495586573714376046char_o @ R1
        @ ( bNF_re1135979186550299152ar_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( member484235747325421115t_char @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le6315511645215477266t_char @ B5 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R22 ) ) )
             => ( ( P3 @ ( product_fst_nat_nat @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_457_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: nat > nat > $o,P3: nat > nat > $o,Q2: list_char > nat > $o,A: produc1457211279475724562t_char,A3: set_Pr2507339680178222706t_char,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re1113502957280211792_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member484235747325421115t_char @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P3 @ ( produc6204653154488278070t_char @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc1900778367047502200t_char @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_458_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: nat > list_char > $o,P3: nat > nat > $o,Q2: list_char > list_char > $o,A: produc1457211279475724562t_char,A3: set_Pr2507339680178222706t_char,B: produc1457211279475724562t_char,B5: set_Pr2507339680178222706t_char] :
      ( ( bNF_re2942033592892840063char_o @ R1
        @ ( bNF_re1135979186550299152ar_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member484235747325421115t_char @ A @ A3 )
       => ( ( member484235747325421115t_char @ B @ B5 )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R1 ) ) )
           => ( ( ord_le6315511645215477266t_char @ B5 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R22 ) ) )
             => ( ( P3 @ ( produc6204653154488278070t_char @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q2 @ ( produc1900778367047502200t_char @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_459_predicate2__transferD,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: nat > nat > $o,P3: product_prod_nat_nat > nat > $o,Q2: product_prod_nat_nat > nat > $o,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re1392313733760654173_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le3000389064537975527at_nat @ A3 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P3 @ ( produc3213797794245857475at_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc6408287024330202629at_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_460_predicate2__transferD,axiom,
    ! [R1: nat > nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o,P3: nat > product_prod_nat_nat > $o,Q2: nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( bNF_re3272885123108610425_nat_o @ R1
        @ ( bNF_re3666534408544137501at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( member8206827879206165904at_nat @ B @ B5 )
         => ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R1 ) ) )
           => ( ( ord_le3000389064537975527at_nat @ B5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R22 ) ) )
             => ( ( P3 @ ( product_fst_nat_nat @ A ) @ ( produc3213797794245857475at_nat @ B ) )
                = ( Q2 @ ( product_snd_nat_nat @ A ) @ ( produc6408287024330202629at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_461_predicate2__transferD,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: nat > list_char > $o,P3: product_prod_nat_nat > nat > $o,Q2: product_prod_nat_nat > list_char > $o,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B: produc1457211279475724562t_char,B5: set_Pr2507339680178222706t_char] :
      ( ( bNF_re1503381852333877516char_o @ R1
        @ ( bNF_re1135979186550299152ar_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( ( member484235747325421115t_char @ B @ B5 )
         => ( ( ord_le3000389064537975527at_nat @ A3 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R1 ) ) )
           => ( ( ord_le6315511645215477266t_char @ B5 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R22 ) ) )
             => ( ( P3 @ ( produc3213797794245857475at_nat @ A ) @ ( produc6204653154488278070t_char @ B ) )
                = ( Q2 @ ( produc6408287024330202629at_nat @ A ) @ ( produc1900778367047502200t_char @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_462_predicate2__transferD,axiom,
    ! [R1: nat > list_char > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o,P3: nat > product_prod_nat_nat > $o,Q2: list_char > product_prod_nat_nat > $o,A: produc1457211279475724562t_char,A3: set_Pr2507339680178222706t_char,B: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( bNF_re1173106714776151178_nat_o @ R1
        @ ( bNF_re3666534408544137501at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member484235747325421115t_char @ A @ A3 )
       => ( ( member8206827879206165904at_nat @ B @ B5 )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R1 ) ) )
           => ( ( ord_le3000389064537975527at_nat @ B5 @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R22 ) ) )
             => ( ( P3 @ ( produc6204653154488278070t_char @ A ) @ ( produc3213797794245857475at_nat @ B ) )
                = ( Q2 @ ( produc1900778367047502200t_char @ A ) @ ( produc6408287024330202629at_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_463_predicate2__transferD,axiom,
    ! [R1: set_Product_prod_o_o > set_Product_prod_o_o > $o,R22: nat > nat > $o,P3: set_Product_prod_o_o > nat > $o,Q2: set_Product_prod_o_o > nat > $o,A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re2789440281400720861_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le7586318239878589543od_o_o @ A3 @ ( collec3302784863738620242od_o_o @ ( produc9128050996712119596_o_o_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P3 @ ( produc3308063957627853123od_o_o @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc954223133649922181od_o_o @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_464_predicate2__transferD,axiom,
    ! [R1: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o,R22: nat > nat > $o,P3: set_Product_prod_o_o > nat > $o,Q2: set_Pr1261947904930325089at_nat > nat > $o,A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( bNF_re929499510339767863_nat_o @ R1
        @ ( bNF_re4705727531993890431at_o_o @ R22
          @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
        @ P3
        @ Q2 )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( ( member8440522571783428010at_nat @ B @ B5 )
         => ( ( ord_le4841768313324958145at_nat @ A3 @ ( collec1729919521702017196at_nat @ ( produc8045849345425481810_nat_o @ R1 ) ) )
           => ( ( ord_le3146513528884898305at_nat @ B5 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R22 ) ) )
             => ( ( P3 @ ( produc7275247657300178845at_nat @ A ) @ ( product_fst_nat_nat @ B ) )
                = ( Q2 @ ( produc4551702098526911967at_nat @ A ) @ ( product_snd_nat_nat @ B ) ) ) ) ) ) ) ) ).

% predicate2_transferD
thf(fact_465_case__prod__Pair__iden,axiom,
    ! [P: produc732395585841259969at_nat] :
      ( ( produc9214616623306952315at_nat @ produc8517790099723286449at_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_466_case__prod__Pair__iden,axiom,
    ! [P: produc1319942482725812455at_nat] :
      ( ( produc5456455677105495163at_nat @ produc9060074326276436823at_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_467_case__prod__Pair__iden,axiom,
    ! [P: produc3843707927480180839at_nat] :
      ( ( produc7548845685434208379at_nat @ produc2922128104949294807at_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_468_case__prod__Pair__iden,axiom,
    ! [P: produc1457211279475724562t_char] :
      ( ( produc4071494739252518515t_char @ produc120197956887798346t_char @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_469_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_470_accp__subset,axiom,
    ! [R1: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,R22: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le6723339807950303054prat_o @ R1 @ R22 )
     => ( ord_le2830794348966088778prat_o @ ( accp_P4765339447491148764r_prat @ R22 ) @ ( accp_P4765339447491148764r_prat @ R1 ) ) ) ).

% accp_subset
thf(fact_471_pred__subset__eq2,axiom,
    ! [R3: set_Pr1932065953672099015od_o_o,S4: set_Pr1932065953672099015od_o_o] :
      ( ( ord_le4789732138934968014_o_o_o
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R3 )
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le7586318239878589543od_o_o @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_472_pred__subset__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R3 )
        @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_473_pred__subset__eq2,axiom,
    ! [R3: set_Pr457366540195662369at_nat,S4: set_Pr457366540195662369at_nat] :
      ( ( ord_le8804154948208483060_nat_o
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ R3 )
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le4841768313324958145at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_474_pred__subset__eq2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R3 )
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le5997549366648089703at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_475_pred__subset__eq2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R3 )
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le1268244103169919719at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_476_pred__subset__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
      ( ( ord_le2712619360209801397char_o
        @ ^ [X2: nat,Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ R3 )
        @ ^ [X2: nat,Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ S4 ) )
      = ( ord_le6315511645215477266t_char @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_477_pred__subset__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [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 ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_478_pred__equals__eq2,axiom,
    ! [R3: set_Pr1932065953672099015od_o_o,S4: set_Pr1932065953672099015od_o_o] :
      ( ( ( ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R3 ) )
        = ( ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_479_pred__equals__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S4: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R3 ) )
        = ( ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_480_pred__equals__eq2,axiom,
    ! [R3: set_Pr457366540195662369at_nat,S4: set_Pr457366540195662369at_nat] :
      ( ( ( ^ [X2: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ R3 ) )
        = ( ^ [X2: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_481_pred__equals__eq2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S4: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R3 ) )
        = ( ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_482_pred__equals__eq2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R3 ) )
        = ( ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_483_pred__equals__eq2,axiom,
    ! [R3: set_Pr2507339680178222706t_char,S4: set_Pr2507339680178222706t_char] :
      ( ( ( ^ [X2: nat,Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ R3 ) )
        = ( ^ [X2: nat,Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_484_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_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 ) @ S4 ) ) )
      = ( R3 = S4 ) ) ).

% pred_equals_eq2
thf(fact_485_fst__diag__id,axiom,
    ! [Z: set_Pr4329608150637261639at_nat] :
      ( ( comp_P8235817900121468292at_nat @ produc4486455499028916547at_nat
        @ ^ [X2: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X2 @ X2 )
        @ Z )
      = ( id_set7849613212452405432at_nat @ Z ) ) ).

% fst_diag_id
thf(fact_486_fst__diag__id,axiom,
    ! [Z: set_Pr1261947904930325089at_nat] :
      ( ( comp_P4528037747288561284at_nat @ produc6011863565407846595at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X2 @ X2 )
        @ Z )
      = ( id_set2334057505560298834at_nat @ Z ) ) ).

% fst_diag_id
thf(fact_487_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_488_snd__diag__id,axiom,
    ! [Z: set_Pr4329608150637261639at_nat] :
      ( ( comp_P8235817900121468292at_nat @ produc826222791318859909at_nat
        @ ^ [X2: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X2 @ X2 )
        @ Z )
      = ( id_set7849613212452405432at_nat @ Z ) ) ).

% snd_diag_id
thf(fact_489_snd__diag__id,axiom,
    ! [Z: set_Pr1261947904930325089at_nat] :
      ( ( comp_P4528037747288561284at_nat @ produc2937545259423399429at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X2 @ X2 )
        @ Z )
      = ( id_set2334057505560298834at_nat @ Z ) ) ).

% snd_diag_id
thf(fact_490_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_491_case__prodD,axiom,
    ! [F4: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat] :
      ( ( produc8045849345425481810_nat_o @ F4 @ ( produc8517790099723286449at_nat @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_492_case__prodD,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( produc9020218426428501292_nat_o @ F4 @ ( produc9060074326276436823at_nat @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_493_case__prodD,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc410239310623530412_nat_o @ F4 @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_494_case__prodD,axiom,
    ! [F4: nat > list_char > $o,A: nat,B: list_char] :
      ( ( produc8526122873445342073char_o @ F4 @ ( produc120197956887798346t_char @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_495_case__prodD,axiom,
    ! [F4: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F4 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F4 @ A @ B ) ) ).

% case_prodD
thf(fact_496_case__prodE,axiom,
    ! [C: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o,P: produc732395585841259969at_nat] :
      ( ( produc8045849345425481810_nat_o @ C @ P )
     => ~ ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc8517790099723286449at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_497_case__prodE,axiom,
    ! [C: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,P: produc1319942482725812455at_nat] :
      ( ( produc9020218426428501292_nat_o @ C @ P )
     => ~ ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
            ( ( P
              = ( produc9060074326276436823at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_498_case__prodE,axiom,
    ! [C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ( produc410239310623530412_nat_o @ C @ P )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_499_case__prodE,axiom,
    ! [C: nat > list_char > $o,P: produc1457211279475724562t_char] :
      ( ( produc8526122873445342073char_o @ C @ P )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_500_case__prodE,axiom,
    ! [C: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_501_fst__def,axiom,
    ( produc6204653154488278070t_char
    = ( produc9163108120003956271ar_nat
      @ ^ [X12: nat,X23: list_char] : X12 ) ) ).

% fst_def
thf(fact_502_accp__subset__induct,axiom,
    ! [D2: ( produc1457211279475724562t_char > prat ) > $o,R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,P3: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le2830794348966088778prat_o @ D2 @ ( accp_P4765339447491148764r_prat @ R3 ) )
     => ( ! [X3: produc1457211279475724562t_char > prat,Z2: produc1457211279475724562t_char > prat] :
            ( ( D2 @ X3 )
           => ( ( R3 @ Z2 @ X3 )
             => ( D2 @ Z2 ) ) )
       => ( ( D2 @ X )
         => ( ! [X3: produc1457211279475724562t_char > prat] :
                ( ( D2 @ X3 )
               => ( ! [Z4: produc1457211279475724562t_char > prat] :
                      ( ( R3 @ Z4 @ X3 )
                     => ( P3 @ Z4 ) )
                 => ( P3 @ X3 ) ) )
           => ( P3 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_503_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A2: nat,B2: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A6: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_504_lex__prod__def,axiom,
    ( lex_pr4669217892513370978at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2200730763347676620at_nat
          @ ( produc3600351907215576810_nat_o
            @ ( produc6198819664464039784_nat_o
              @ ^ [A2: product_prod_nat_nat,B2: nat] :
                  ( produc8758269395934548017_nat_o
                  @ ^ [A6: product_prod_nat_nat,B6: nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_505_lex__prod__def,axiom,
    ( lex_pr8029265285556086080at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec3253044014973990424at_nat
          @ ( produc2869316064735525022_nat_o
            @ ( produc7480869177107369060_nat_o
              @ ^ [A2: nat,B2: product_prod_nat_nat] :
                  ( produc5864757623865647827_nat_o
                  @ ^ [A6: nat,B6: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_506_lex__prod__def,axiom,
    ( lex_pr7492273792749525096_o_nat
    = ( ^ [Ra: set_Pr1932065953672099015od_o_o,Rb: set_Pr1261947904930325089at_nat] :
          ( collec5679390776719607896_o_nat
          @ ( produc4348875432616242270_nat_o
            @ ( produc7792188603650235612_nat_o
              @ ^ [A2: set_Product_prod_o_o,B2: nat] :
                  ( produc1844271551222047659_nat_o
                  @ ^ [A6: set_Product_prod_o_o,B6: nat] :
                      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_507_lex__prod__def,axiom,
    ( lex_pr1951919194703979714at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec1461227866750739596at_nat
          @ ( produc8701595406162815018_nat_o
            @ ( produc7110675705651593384_nat_o
              @ ^ [A2: set_Pr1261947904930325089at_nat,B2: nat] :
                  ( produc3595121349294487249_nat_o
                  @ ^ [A6: set_Pr1261947904930325089at_nat,B6: nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_508_lex__prod__def,axiom,
    ( lex_pr8952996434854197702od_o_o
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1932065953672099015od_o_o] :
          ( collec4482925371686133924od_o_o
          @ ( produc3597285735490004498_o_o_o
            @ ( produc5816153797853224152_o_o_o
              @ ^ [A2: nat,B2: set_Product_prod_o_o] :
                  ( produc327537277274621645_o_o_o
                  @ ^ [A6: nat,B6: set_Product_prod_o_o] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_509_lex__prod__def,axiom,
    ( lex_pr4265550801952275104at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec2088441199205367000at_nat
          @ ( produc7830147906768200158_nat_o
            @ ( produc1908958340173245860_nat_o
              @ ^ [A2: nat,B2: set_Pr1261947904930325089at_nat] :
                  ( produc7997411485876045171_nat_o
                  @ ^ [A6: nat,B6: set_Pr1261947904930325089at_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_510_lex__prod__def,axiom,
    ( lex_pr8801849515957261039at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7334067512558549330at_nat
          @ ( produc3603361296207914284_nat_o
            @ ( produc5077327844045112642_nat_o
              @ ^ [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
                  ( produc6590410687421337004_nat_o
                  @ ^ [A6: product_prod_nat_nat,B6: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_511_lex__prod__def,axiom,
    ( lex_pr2931540531659632233at_nat
    = ( ^ [Ra: set_Pr1932065953672099015od_o_o,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7722287194462926802at_nat
          @ ( produc926537989614425004_nat_o
            @ ( produc4838932303645241282_nat_o
              @ ^ [A2: set_Product_prod_o_o,B2: product_prod_nat_nat] :
                  ( produc120290491662694834_nat_o
                  @ ^ [A6: set_Product_prod_o_o,B6: product_prod_nat_nat] :
                      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_512_lex__prod__def,axiom,
    ( lex_pr2425111879933873909od_o_o
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1932065953672099015od_o_o] :
          ( collec5067964091810496210od_o_o
          @ ( produc7101619085578143404_o_o_o
            @ ( produc1659243639860068802_o_o_o
              @ ^ [A2: product_prod_nat_nat,B2: set_Product_prod_o_o] :
                  ( produc6482769799778684454_o_o_o
                  @ ^ [A6: product_prod_nat_nat,B6: set_Product_prod_o_o] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ A6 ) @ Ra )
                      | ( ( A2 = A6 )
                        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_513_snd__fst__flip,axiom,
    ( produc7327370413965865899od_o_o
    = ( comp_P1188652384226297060od_o_o @ produc7275247657300178845at_nat
      @ ( produc4152889022482007471at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Product_prod_o_o] : ( produc8517790099723286449at_nat @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_514_snd__fst__flip,axiom,
    ( produc826222791318859909at_nat
    = ( comp_P3728553842126764068at_nat @ produc4486455499028916547at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_515_snd__fst__flip,axiom,
    ( produc2937545259423399429at_nat
    = ( comp_P4974406650942564362at_nat @ produc6011863565407846595at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_516_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_517_snd__fst__flip,axiom,
    ( produc1259554233722553080ar_nat
    = ( comp_P3187634502937433873ar_nat @ produc6204653154488278070t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X2: list_char,Y5: nat] : ( produc120197956887798346t_char @ Y5 @ X2 ) ) ) ) ).

% snd_fst_flip
thf(fact_518_fst__snd__flip,axiom,
    ( produc827543935884356969od_o_o
    = ( comp_P2909091037308380810od_o_o @ produc4551702098526911967at_nat
      @ ( produc4152889022482007471at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Product_prod_o_o] : ( produc8517790099723286449at_nat @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_519_fst__snd__flip,axiom,
    ( produc4486455499028916547at_nat
    = ( comp_P3728553842126764068at_nat @ produc826222791318859909at_nat
      @ ( produc5456455677105495163at_nat
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_520_fst__snd__flip,axiom,
    ( produc6011863565407846595at_nat
    = ( comp_P4974406650942564362at_nat @ produc2937545259423399429at_nat
      @ ( produc7548845685434208379at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_521_fst__snd__flip,axiom,
    ( produc5563429021163328950ar_nat
    = ( comp_P26140635296239040ar_nat @ produc1900778367047502200t_char
      @ ( produc6166083679720228083t_char
        @ ^ [X2: list_char,Y5: nat] : ( produc120197956887798346t_char @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_522_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_523_fst__snd__flip,axiom,
    ( produc6204653154488278070t_char
    = ( comp_P680174135349460753t_char @ produc1259554233722553080ar_nat
      @ ( produc997380522159630963ar_nat
        @ ^ [X2: nat,Y5: list_char] : ( produc8702345860417625034ar_nat @ Y5 @ X2 ) ) ) ) ).

% fst_snd_flip
thf(fact_524_fst__diag__fst,axiom,
    ( ( comp_P2122579405136797538t_char @ product_fst_nat_nat
      @ ( comp_n1645060710405774596t_char
        @ ^ [X2: nat] : ( product_Pair_nat_nat @ X2 @ X2 )
        @ produc6204653154488278070t_char ) )
    = produc6204653154488278070t_char ) ).

% fst_diag_fst
thf(fact_525_fun_Oin__rel,axiom,
    ! [R3: nat > list_char > $o,A: nat > nat,B: nat > list_char] :
      ( ( bNF_re2076907181175516082t_char
        @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
        @ R3
        @ A
        @ B )
      = ( ? [Z5: nat > produc1457211279475724562t_char] :
            ( ( member3158901901048444578t_char @ Z5
              @ ( collec4820088019371311076t_char
                @ ^ [X2: nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_1003422490162143483t_char @ X2 @ top_top_set_nat ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) ) )
            & ( ( comp_P8410057226579278353at_nat @ produc6204653154488278070t_char @ Z5 )
              = A )
            & ( ( comp_P4974907838745982882ar_nat @ produc1900778367047502200t_char @ Z5 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_526_fun_Orel__compp__Grp,axiom,
    ! [R3: nat > list_char > $o] :
      ( ( bNF_re2076907181175516082t_char
        @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
        @ R3 )
      = ( relcom2152729800865970265t_char
        @ ( conver457542469546801155at_nat
          @ ( bNF_Gr7887393120999648746at_nat
            @ ( collec4820088019371311076t_char
              @ ^ [X2: nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_1003422490162143483t_char @ X2 @ top_top_set_nat ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) )
            @ ( comp_P8410057226579278353at_nat @ produc6204653154488278070t_char ) ) )
        @ ( bNF_Gr962715995036458107t_char
          @ ( collec4820088019371311076t_char
            @ ^ [X2: nat > produc1457211279475724562t_char] : ( ord_le6315511645215477266t_char @ ( image_1003422490162143483t_char @ X2 @ top_top_set_nat ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) )
          @ ( comp_P4974907838745982882ar_nat @ produc1900778367047502200t_char ) ) ) ) ).

% fun.rel_compp_Grp
thf(fact_527_comp__one__def,axiom,
    ( comp_one
    = ( ^ [P2: prat] :
          ( fChoice_prat
          @ ^ [R: prat] :
              ( ( padd @ P2 @ R )
              = pwrite ) ) ) ) ).

% comp_one_def
thf(fact_528_subset__fst__snd,axiom,
    ! [A3: set_Pr2507339680178222706t_char] :
      ( ord_le6315511645215477266t_char @ A3
      @ ( produc6030673636645808891t_char @ ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ A3 )
        @ ^ [Uu: nat] : ( image_3034139768025083822t_char @ produc1900778367047502200t_char @ A3 ) ) ) ).

% subset_fst_snd
thf(fact_529_flip__pred,axiom,
    ! [A3: set_Pr8055703017043224813od_o_o,R3: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o] :
      ( ( ord_le3216732753317744781od_o_o @ A3 @ ( collec8895626098298191224od_o_o @ ( produc4907931132024049542_o_o_o @ ( conver100682298699653391at_nat @ R3 ) ) ) )
     => ( ord_le4841768313324958145at_nat
        @ ( image_3720774369771669369at_nat
          @ ( produc4152889022482007471at_nat
            @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Product_prod_o_o] : ( produc8517790099723286449at_nat @ Y5 @ X2 ) )
          @ A3 )
        @ ( collec1729919521702017196at_nat @ ( produc8045849345425481810_nat_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_530_flip__pred,axiom,
    ! [A3: set_Pr7459493094073627847at_nat,R3: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ( ord_le5997549366648089703at_nat @ A3 @ ( collec7872792499061297490at_nat @ ( produc9020218426428501292_nat_o @ ( conver9095369822871743413at_nat @ R3 ) ) ) )
     => ( ord_le5997549366648089703at_nat
        @ ( image_3313167189156628421at_nat
          @ ( produc5456455677105495163at_nat
            @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ Y5 @ X2 ) )
          @ A3 )
        @ ( collec7872792499061297490at_nat @ ( produc9020218426428501292_nat_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_531_flip__pred,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R3: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ A3 @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ ( conver5243090411617913141at_nat @ R3 ) ) ) )
     => ( ord_le1268244103169919719at_nat
        @ ( image_960689587819713477at_nat
          @ ( produc7548845685434208379at_nat
            @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ Y5 @ X2 ) )
          @ A3 )
        @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_532_flip__pred,axiom,
    ! [A3: set_Pr8803067691650843250ar_nat,R3: nat > list_char > $o] :
      ( ( ord_le3387867619833322002ar_nat @ A3 @ ( collec8608152508653127293ar_nat @ ( produc2492928673396900345_nat_o @ ( conver4209408378539545768t_char @ R3 ) ) ) )
     => ( ord_le6315511645215477266t_char
        @ ( image_688197443715608197t_char
          @ ( produc6166083679720228083t_char
            @ ^ [X2: list_char,Y5: nat] : ( produc120197956887798346t_char @ Y5 @ X2 ) )
          @ A3 )
        @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_533_flip__pred,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R3: nat > nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ( conversep_nat_nat @ R3 ) ) ) )
     => ( ord_le3146513528884898305at_nat
        @ ( image_5168914502847457605at_nat
          @ ( produc2626176000494625587at_nat
            @ ^ [X2: nat,Y5: nat] : ( product_Pair_nat_nat @ Y5 @ X2 ) )
          @ A3 )
        @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R3 ) ) ) ) ).

% flip_pred
thf(fact_534_The__case__prod,axiom,
    ! [P3: nat > list_char > $o] :
      ( ( the_Pr202245514940600089t_char @ ( produc8526122873445342073char_o @ P3 ) )
      = ( the_Pr202245514940600089t_char
        @ ^ [Xy: produc1457211279475724562t_char] : ( P3 @ ( produc6204653154488278070t_char @ Xy ) @ ( produc1900778367047502200t_char @ Xy ) ) ) ) ).

% The_case_prod
thf(fact_535_mem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A3: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A3 @ B5 ) )
      = ( ( member_nat @ A @ A3 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_536_mem__Sigma__iff,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B5 ) )
      = ( ( member_nat @ A @ A3 )
        & ( member_list_char @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_537_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A3 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A3 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_538_mem__Sigma__iff,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc514367270615792904od_o_o @ A3 @ B5 ) )
      = ( ( member3157015407008874800od_o_o @ A @ A3 )
        & ( member3157015407008874800od_o_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_539_mem__Sigma__iff,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc7365643037244224738at_nat @ A3 @ B5 ) )
      = ( ( member3157015407008874800od_o_o @ A @ A3 )
        & ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_540_mem__Sigma__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A3 @ B5 ) )
      = ( ( member2643936169264416010at_nat @ A @ A3 )
        & ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_541_mem__Sigma__iff,axiom,
    ! [A: produc859450856879609959at_nat,B: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A3 @ B5 ) )
      = ( ( member8206827879206165904at_nat @ A @ A3 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_542_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A3 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A3 )
        & ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_543_mem__Sigma__iff,axiom,
    ! [A: produc2934264451710624999od_o_o,B: product_prod_nat_nat,A3: set_Pr1932065953672099015od_o_o,B5: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat] :
      ( ( member8767205482156871978at_nat @ ( produc6591419942417647473at_nat @ A @ B ) @ ( produc726318440123720610at_nat @ A3 @ B5 ) )
      = ( ( member444158400953824016od_o_o @ A @ A3 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_544_mem__Sigma__iff,axiom,
    ! [A: produc732395585841259969at_nat,B: product_prod_nat_nat,A3: set_Pr457366540195662369at_nat,B5: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3762726516173965008at_nat @ ( produc5685408946904390167at_nat @ A @ B ) @ ( produc4339135674454976968at_nat @ A3 @ B5 ) )
      = ( ( member4095101504841534314at_nat @ A @ A3 )
        & ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_545_SigmaI,axiom,
    ! [A: nat,A3: set_nat,B: nat,B5: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_546_SigmaI,axiom,
    ! [A: nat,A3: set_nat,B: list_char,B5: nat > set_list_char] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_list_char @ B @ ( B5 @ A ) )
       => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_547_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_548_SigmaI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Product_prod_o_o,B5: set_Product_prod_o_o > set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( ( member3157015407008874800od_o_o @ B @ ( B5 @ A ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc514367270615792904od_o_o @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_549_SigmaI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Pr1261947904930325089at_nat,B5: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( ( member2643936169264416010at_nat @ B @ ( B5 @ A ) )
       => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc7365643037244224738at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_550_SigmaI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( ( member2643936169264416010at_nat @ B @ ( B5 @ A ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_551_SigmaI,axiom,
    ! [A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_552_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc859450856879609959at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member8206827879206165904at_nat @ B @ ( B5 @ A ) )
       => ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_553_SigmaI,axiom,
    ! [A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,B: product_prod_nat_nat,B5: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat] :
      ( ( member444158400953824016od_o_o @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member8767205482156871978at_nat @ ( produc6591419942417647473at_nat @ A @ B ) @ ( produc726318440123720610at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_554_SigmaI,axiom,
    ! [A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,B: product_prod_nat_nat,B5: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member4095101504841534314at_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ B @ ( B5 @ A ) )
       => ( member3762726516173965008at_nat @ ( produc5685408946904390167at_nat @ A @ B ) @ ( produc4339135674454976968at_nat @ A3 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_555_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F4: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F4 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_556_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im2857804880773004637at_nat @ R2 @ F4 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_557_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr8693737435421807431at_nat,F4: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im8721129524585065343at_nat @ R2 @ F4 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_558_in__inv__image,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,R2: set_Pr1261947904930325089at_nat,F4: set_Product_prod_o_o > nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( inv_im1273697872756756579od_o_o @ R2 @ F4 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_559_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,F4: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im6750873321330482877at_nat @ R2 @ F4 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_560_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1932065953672099015od_o_o,F4: nat > set_Product_prod_o_o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im9036347267506859781_o_nat @ R2 @ F4 ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_561_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr4329608150637261639at_nat,F4: nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im4437241714082187487at_nat @ R2 @ F4 ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_562_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,F4: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im5896302226278515346at_nat @ R2 @ F4 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_563_in__inv__image,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,R2: set_Pr8693737435421807431at_nat,F4: set_Product_prod_o_o > product_prod_nat_nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( inv_im6294380050766602264od_o_o @ R2 @ F4 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_564_in__inv__image,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr1932065953672099015od_o_o,F4: product_prod_nat_nat > set_Product_prod_o_o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( inv_im6800808702492360588at_nat @ R2 @ F4 ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_565_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_566_The__split__eq,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( the_Pr31138996699769672at_nat
        @ ( produc8045849345425481810_nat_o
          @ ^ [X5: set_Product_prod_o_o,Y9: set_Pr1261947904930325089at_nat] :
              ( ( X = X5 )
              & ( Y = Y9 ) ) ) )
      = ( produc8517790099723286449at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_567_The__split__eq,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( the_Pr4877703158196064750at_nat
        @ ( produc9020218426428501292_nat_o
          @ ^ [X5: set_Pr4329608150637261639at_nat,Y9: set_Pr4329608150637261639at_nat] :
              ( ( X = X5 )
              & ( Y = Y9 ) ) ) )
      = ( produc9060074326276436823at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_568_The__split__eq,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( the_Pr4604535484834969198at_nat
        @ ( produc410239310623530412_nat_o
          @ ^ [X5: set_Pr1261947904930325089at_nat,Y9: set_Pr1261947904930325089at_nat] :
              ( ( X = X5 )
              & ( Y = Y9 ) ) ) )
      = ( produc2922128104949294807at_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_569_The__split__eq,axiom,
    ! [X: nat,Y: list_char] :
      ( ( the_Pr202245514940600089t_char
        @ ( produc8526122873445342073char_o
          @ ^ [X5: nat,Y9: list_char] :
              ( ( X = X5 )
              & ( Y = Y9 ) ) ) )
      = ( produc120197956887798346t_char @ X @ Y ) ) ).

% The_split_eq
thf(fact_570_The__split__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X5: nat,Y9: nat] :
              ( ( X = X5 )
              & ( Y = Y9 ) ) ) )
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_571_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A3: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A3 @ B5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_572_SigmaE,axiom,
    ! [C: produc1457211279475724562t_char,A3: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ C @ ( produc6030673636645808891t_char @ A3 @ B5 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ! [Y3: list_char] :
                ( ( member_list_char @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc120197956887798346t_char @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_573_SigmaE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( produc2761391749766926216at_nat @ A3 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_574_SigmaE,axiom,
    ! [C: produc2934264451710624999od_o_o,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ C @ ( produc514367270615792904od_o_o @ A3 @ B5 ) )
     => ~ ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A3 )
           => ! [Y3: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc763777882069021527od_o_o @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_575_SigmaE,axiom,
    ! [C: produc732395585841259969at_nat,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ C @ ( produc7365643037244224738at_nat @ A3 @ B5 ) )
     => ~ ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A3 )
           => ! [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc8517790099723286449at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_576_SigmaE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( produc7726050920695093128at_nat @ A3 @ B5 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A3 )
           => ! [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_577_SigmaE,axiom,
    ! [C: produc7148259590854449153at_nat,A3: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ C @ ( produc3829605335166519074at_nat @ A3 @ B5 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc1860397064299428849at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_578_SigmaE,axiom,
    ! [C: produc39072733474941005at_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ C @ ( produc7465995907834253422at_nat @ A3 @ B5 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ! [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc5496787636967163197at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_579_SigmaE,axiom,
    ! [C: produc7372027153569465473at_nat,A3: set_Pr1932065953672099015od_o_o,B5: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat] :
      ( ( member8767205482156871978at_nat @ C @ ( produc726318440123720610at_nat @ A3 @ B5 ) )
     => ~ ! [X3: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc6591419942417647473at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_580_SigmaE,axiom,
    ! [C: produc4432270315841807015at_nat,A3: set_Pr457366540195662369at_nat,B5: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3762726516173965008at_nat @ C @ ( produc4339135674454976968at_nat @ A3 @ B5 ) )
     => ~ ! [X3: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( B5 @ X3 ) )
               => ( C
                 != ( produc5685408946904390167at_nat @ X3 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_581_SigmaD1,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc514367270615792904od_o_o @ A3 @ B5 ) )
     => ( member3157015407008874800od_o_o @ A @ A3 ) ) ).

% SigmaD1
thf(fact_582_SigmaD1,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A3 @ B5 ) )
     => ( member8440522571783428010at_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_583_SigmaD1,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc7365643037244224738at_nat @ A3 @ B5 ) )
     => ( member3157015407008874800od_o_o @ A @ A3 ) ) ).

% SigmaD1
thf(fact_584_SigmaD1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat,B5: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc6258361011560675592at_nat @ A3 @ B5 ) )
     => ( member6273874620674182256at_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_585_SigmaD1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A3 @ B5 ) )
     => ( member2643936169264416010at_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_586_SigmaD1,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B5 ) )
     => ( member_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_587_SigmaD1,axiom,
    ! [A: nat,B: nat,A3: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A3 @ B5 ) )
     => ( member_nat @ A @ A3 ) ) ).

% SigmaD1
thf(fact_588_SigmaD2,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc514367270615792904od_o_o @ A3 @ B5 ) )
     => ( member3157015407008874800od_o_o @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_589_SigmaD2,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A3 @ B5 ) )
     => ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_590_SigmaD2,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc7365643037244224738at_nat @ A3 @ B5 ) )
     => ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_591_SigmaD2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat,B5: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( produc6258361011560675592at_nat @ A3 @ B5 ) )
     => ( member6273874620674182256at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_592_SigmaD2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A3 @ B5 ) )
     => ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_593_SigmaD2,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B5 ) )
     => ( member_list_char @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_594_SigmaD2,axiom,
    ! [A: nat,B: nat,A3: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A3 @ B5 ) )
     => ( member_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_595_SigmaE2,axiom,
    ! [A: nat,B: nat,A3: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A3 @ B5 ) )
     => ~ ( ( member_nat @ A @ A3 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_596_SigmaE2,axiom,
    ! [A: nat,B: list_char,A3: set_nat,B5: nat > set_list_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ ( produc6030673636645808891t_char @ A3 @ B5 ) )
     => ~ ( ( member_nat @ A @ A3 )
         => ~ ( member_list_char @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_597_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc2761391749766926216at_nat @ A3 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_598_SigmaE2,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( produc514367270615792904od_o_o @ A3 @ B5 ) )
     => ~ ( ( member3157015407008874800od_o_o @ A @ A3 )
         => ~ ( member3157015407008874800od_o_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_599_SigmaE2,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,A3: set_se491199571785101031od_o_o,B5: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc7365643037244224738at_nat @ A3 @ B5 ) )
     => ~ ( ( member3157015407008874800od_o_o @ A @ A3 )
         => ~ ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_600_SigmaE2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A3 @ B5 ) )
     => ~ ( ( member2643936169264416010at_nat @ A @ A3 )
         => ~ ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_601_SigmaE2,axiom,
    ! [A: produc859450856879609959at_nat,B: product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ ( produc3829605335166519074at_nat @ A3 @ B5 ) )
     => ~ ( ( member8206827879206165904at_nat @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_602_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ ( produc7465995907834253422at_nat @ A3 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A3 )
         => ~ ( member8206827879206165904at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_603_SigmaE2,axiom,
    ! [A: produc2934264451710624999od_o_o,B: product_prod_nat_nat,A3: set_Pr1932065953672099015od_o_o,B5: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat] :
      ( ( member8767205482156871978at_nat @ ( produc6591419942417647473at_nat @ A @ B ) @ ( produc726318440123720610at_nat @ A3 @ B5 ) )
     => ~ ( ( member444158400953824016od_o_o @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_604_SigmaE2,axiom,
    ! [A: produc732395585841259969at_nat,B: product_prod_nat_nat,A3: set_Pr457366540195662369at_nat,B5: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member3762726516173965008at_nat @ ( produc5685408946904390167at_nat @ A @ B ) @ ( produc4339135674454976968at_nat @ A3 @ B5 ) )
     => ~ ( ( member4095101504841534314at_nat @ A @ A3 )
         => ~ ( member8440522571783428010at_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_605_mem__Times__iff,axiom,
    ! [X: product_prod_nat_nat,A3: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B5 ) )
      = ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A3 )
        & ( member_nat @ ( product_snd_nat_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_606_mem__Times__iff,axiom,
    ! [X: produc1457211279475724562t_char,A3: set_nat,B5: set_list_char] :
      ( ( member484235747325421115t_char @ X
        @ ( produc6030673636645808891t_char @ A3
          @ ^ [Uu: nat] : B5 ) )
      = ( ( member_nat @ ( produc6204653154488278070t_char @ X ) @ A3 )
        & ( member_list_char @ ( produc1900778367047502200t_char @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_607_mem__Times__iff,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc3213797794245857475at_nat @ X ) @ A3 )
        & ( member8440522571783428010at_nat @ ( produc6408287024330202629at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_608_mem__Times__iff,axiom,
    ! [X: produc2934264451710624999od_o_o,A3: set_se491199571785101031od_o_o,B5: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ X
        @ ( produc514367270615792904od_o_o @ A3
          @ ^ [Uu: set_Product_prod_o_o] : B5 ) )
      = ( ( member3157015407008874800od_o_o @ ( produc3308063957627853123od_o_o @ X ) @ A3 )
        & ( member3157015407008874800od_o_o @ ( produc954223133649922181od_o_o @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_609_mem__Times__iff,axiom,
    ! [X: produc732395585841259969at_nat,A3: set_se491199571785101031od_o_o,B5: set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ X
        @ ( produc7365643037244224738at_nat @ A3
          @ ^ [Uu: set_Product_prod_o_o] : B5 ) )
      = ( ( member3157015407008874800od_o_o @ ( produc7275247657300178845at_nat @ X ) @ A3 )
        & ( member2643936169264416010at_nat @ ( produc4551702098526911967at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_610_mem__Times__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ X
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) )
      = ( ( member2643936169264416010at_nat @ ( produc6011863565407846595at_nat @ X ) @ A3 )
        & ( member2643936169264416010at_nat @ ( produc2937545259423399429at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_611_mem__Times__iff,axiom,
    ! [X: produc7148259590854449153at_nat,A3: set_Pr8693737435421807431at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member3919319858682911658at_nat @ X
        @ ( produc3829605335166519074at_nat @ A3
          @ ^ [Uu: produc859450856879609959at_nat] : B5 ) )
      = ( ( member8206827879206165904at_nat @ ( produc4362022507300774877at_nat @ X ) @ A3 )
        & ( member8440522571783428010at_nat @ ( produc1191852045998802463at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_612_mem__Times__iff,axiom,
    ! [X: produc39072733474941005at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( member6033505038158179318at_nat @ X
        @ ( produc7465995907834253422at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 ) )
      = ( ( member8440522571783428010at_nat @ ( produc7998413079968509225at_nat @ X ) @ A3 )
        & ( member8206827879206165904at_nat @ ( produc4828242618666536811at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_613_mem__Times__iff,axiom,
    ! [X: produc7372027153569465473at_nat,A3: set_Pr1932065953672099015od_o_o,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8767205482156871978at_nat @ X
        @ ( produc726318440123720610at_nat @ A3
          @ ^ [Uu: produc2934264451710624999od_o_o] : B5 ) )
      = ( ( member444158400953824016od_o_o @ ( produc1201395269668887389at_nat @ X ) @ A3 )
        & ( member8440522571783428010at_nat @ ( produc175674587098333087at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_614_mem__Times__iff,axiom,
    ! [X: produc4432270315841807015at_nat,A3: set_Pr457366540195662369at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member3762726516173965008at_nat @ X
        @ ( produc4339135674454976968at_nat @ A3
          @ ^ [Uu: produc732395585841259969at_nat] : B5 ) )
      = ( ( member4095101504841534314at_nat @ ( produc8509921526640248835at_nat @ X ) @ A3 )
        & ( member8440522571783428010at_nat @ ( produc9111005830278538053at_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_615_swap__product,axiom,
    ! [A3: set_se7855581050983116737at_nat,B5: set_se491199571785101031od_o_o] :
      ( ( image_3720774369771669369at_nat
        @ ( produc4152889022482007471at_nat
          @ ^ [I: set_Pr1261947904930325089at_nat,J: set_Product_prod_o_o] : ( produc8517790099723286449at_nat @ J @ I ) )
        @ ( produc917939315828402862od_o_o @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) )
      = ( produc7365643037244224738at_nat @ B5
        @ ^ [Uu: set_Product_prod_o_o] : A3 ) ) ).

% swap_product
thf(fact_616_swap__product,axiom,
    ! [A3: set_se5479201105039950375at_nat,B5: set_se5479201105039950375at_nat] :
      ( ( image_3313167189156628421at_nat
        @ ( produc5456455677105495163at_nat
          @ ^ [I: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ J @ I ) )
        @ ( produc6258361011560675592at_nat @ A3
          @ ^ [Uu: set_Pr4329608150637261639at_nat] : B5 ) )
      = ( produc6258361011560675592at_nat @ B5
        @ ^ [Uu: set_Pr4329608150637261639at_nat] : A3 ) ) ).

% swap_product
thf(fact_617_swap__product,axiom,
    ! [A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( image_960689587819713477at_nat
        @ ( produc7548845685434208379at_nat
          @ ^ [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ J @ I ) )
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) )
      = ( produc7726050920695093128at_nat @ B5
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : A3 ) ) ).

% swap_product
thf(fact_618_swap__product,axiom,
    ! [A3: set_list_char,B5: set_nat] :
      ( ( image_688197443715608197t_char
        @ ( produc6166083679720228083t_char
          @ ^ [I: list_char,J: nat] : ( produc120197956887798346t_char @ J @ I ) )
        @ ( produc5389449503320859771ar_nat @ A3
          @ ^ [Uu: list_char] : B5 ) )
      = ( produc6030673636645808891t_char @ B5
        @ ^ [Uu: nat] : A3 ) ) ).

% swap_product
thf(fact_619_swap__product,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [I: nat,J: nat] : ( product_Pair_nat_nat @ J @ I ) )
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ B5
        @ ^ [Uu: nat] : A3 ) ) ).

% swap_product
thf(fact_620_Grp__fst__snd,axiom,
    ! [R3: nat > list_char > $o] :
      ( ( relcom6922018513477707698t_char @ ( conver1523793232969761197ar_nat @ ( bNF_Gr3073014387538510996ar_nat @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) @ produc6204653154488278070t_char ) ) @ ( bNF_Gr3750019412944930853t_char @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R3 ) ) @ produc1900778367047502200t_char ) )
      = R3 ) ).

% Grp_fst_snd
thf(fact_621_member__product,axiom,
    ! [X: produc2934264451710624999od_o_o,A3: set_se491199571785101031od_o_o,B5: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ X @ ( produc240105137174993938od_o_o @ A3 @ B5 ) )
      = ( member444158400953824016od_o_o @ X
        @ ( produc514367270615792904od_o_o @ A3
          @ ^ [Uu: set_Product_prod_o_o] : B5 ) ) ) ).

% member_product
thf(fact_622_member__product,axiom,
    ! [X: produc732395585841259969at_nat,A3: set_se491199571785101031od_o_o,B5: set_se7855581050983116737at_nat] :
      ( ( member4095101504841534314at_nat @ X @ ( produc5117767642622527212at_nat @ A3 @ B5 ) )
      = ( member4095101504841534314at_nat @ X
        @ ( produc7365643037244224738at_nat @ A3
          @ ^ [Uu: set_Product_prod_o_o] : B5 ) ) ) ).

% member_product
thf(fact_623_member__product,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( produc3241716407206042770at_nat @ A3 @ B5 ) )
      = ( member8757157785044589968at_nat @ X
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) ) ) ).

% member_product
thf(fact_624_member__product,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ X @ ( produc6773009516019950738at_nat @ A3 @ B5 ) )
      = ( member8206827879206165904at_nat @ X
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ).

% member_product
thf(fact_625_member__product,axiom,
    ! [X: product_prod_nat_nat,A3: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( produc929334515565554804at_nat @ A3 @ B5 ) )
      = ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B5 ) ) ) ).

% member_product
thf(fact_626_snd__image__times,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( A3 = bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc6408287024330202629at_nat
            @ ( produc2761391749766926216at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A3 != bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc6408287024330202629at_nat
            @ ( produc2761391749766926216at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_627_snd__image__times,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( A3 = bot_bo2099793752762293965at_nat )
       => ( ( image_2020710134525389919at_nat @ produc9109627233917564779at_nat
            @ ( produc3090783324798197870at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A3 != bot_bo2099793752762293965at_nat )
       => ( ( image_2020710134525389919at_nat @ produc9109627233917564779at_nat
            @ ( produc3090783324798197870at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_628_snd__image__times,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( A3 = bot_bo228742789529271731at_nat )
       => ( ( image_1780646082475352133at_nat @ produc5538728265880653343at_nat
            @ ( produc8743256393616062242at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A3 != bot_bo228742789529271731at_nat )
       => ( ( image_1780646082475352133at_nat @ produc5538728265880653343at_nat
            @ ( produc8743256393616062242at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_629_snd__image__times,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( A3 = bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6345472114899077253at_nat
            @ ( produc4441300235185154312at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A3 != bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6345472114899077253at_nat
            @ ( produc4441300235185154312at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_630_Sigma__empty1,axiom,
    ! [B5: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B5 )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty1
thf(fact_631_Sigma__empty1,axiom,
    ! [B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ bot_bo3083307316010499117at_nat @ B5 )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty1
thf(fact_632_Sigma__empty2,axiom,
    ! [A3: set_nat] :
      ( ( produc457027306803732586at_nat @ A3
        @ ^ [Uu: nat] : bot_bot_set_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty2
thf(fact_633_Sigma__empty2,axiom,
    ! [A3: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ A3
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : bot_bo3083307316010499117at_nat )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty2
thf(fact_634_Times__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = bot_bo5327735625951526323at_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_635_Times__empty,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( produc3090783324798197870at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = bot_bo3433114129128967449at_nat )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( B5 = bot_bo228742789529271731at_nat ) ) ) ).

% Times_empty
thf(fact_636_Times__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8743256393616062242at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = bot_bo7908653319349560013at_nat )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( B5 = bot_bo2099793752762293965at_nat ) ) ) ).

% Times_empty
thf(fact_637_Times__empty,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = bot_bo7193250159860175667at_nat )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( B5 = bot_bo228742789529271731at_nat ) ) ) ).

% Times_empty
thf(fact_638_Times__empty,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_639_Times__empty,axiom,
    ! [A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 )
        = bot_bo228742789529271731at_nat )
      = ( ( A3 = bot_bo3083307316010499117at_nat )
        | ( B5 = bot_bo3083307316010499117at_nat ) ) ) ).

% Times_empty
thf(fact_640_Collect__const__case__prod,axiom,
    ! [P3: $o] :
      ( ( P3
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B2: nat] : P3 ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P3
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A2: nat,B2: nat] : P3 ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_641_Collect__const__case__prod,axiom,
    ! [P3: $o] :
      ( ( P3
       => ( ( collec6321179662152712658at_nat
            @ ( produc410239310623530412_nat_o
              @ ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : P3 ) )
          = top_to6833984726390702231at_nat ) )
      & ( ~ P3
       => ( ( collec6321179662152712658at_nat
            @ ( produc410239310623530412_nat_o
              @ ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : P3 ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_642_fst__image__times,axiom,
    ! [B5: set_list_char,A3: set_nat] :
      ( ( ( B5 = bot_bo8503570744482963627t_char )
       => ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char
            @ ( produc6030673636645808891t_char @ A3
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( B5 != bot_bo8503570744482963627t_char )
       => ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char
            @ ( produc6030673636645808891t_char @ A3
              @ ^ [Uu: nat] : B5 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_643_fst__image__times,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( B5 = bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc3213797794245857475at_nat
            @ ( produc2761391749766926216at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( B5 != bot_bo2099793752762293965at_nat )
       => ( ( image_6706873702294942047at_nat @ produc3213797794245857475at_nat
            @ ( produc2761391749766926216at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_644_fst__image__times,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( B5 = bot_bo2099793752762293965at_nat )
       => ( ( image_8065348968207030699at_nat @ produc9061435184492477405at_nat
            @ ( produc8743256393616062242at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( B5 != bot_bo2099793752762293965at_nat )
       => ( ( image_8065348968207030699at_nat @ produc9061435184492477405at_nat
            @ ( produc8743256393616062242at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_645_fst__image__times,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( B5 = bot_bo228742789529271731at_nat )
       => ( ( image_7444983900720083193at_nat @ produc3408962115674613033at_nat
            @ ( produc3090783324798197870at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( B5 != bot_bo228742789529271731at_nat )
       => ( ( image_7444983900720083193at_nat @ produc3408962115674613033at_nat
            @ ( produc3090783324798197870at_nat @ A3
              @ ^ [Uu: product_prod_nat_nat] : B5 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_646_fst__image__times,axiom,
    ! [B5: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( B5 = bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6021103657482432835at_nat
            @ ( produc4441300235185154312at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( B5 != bot_bo228742789529271731at_nat )
       => ( ( image_4030181298759670085at_nat @ produc6021103657482432835at_nat
            @ ( produc4441300235185154312at_nat @ A3
              @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) )
          = A3 ) ) ) ).

% fst_image_times
thf(fact_647_Sigma__empty__iff,axiom,
    ! [I2: set_nat,X4: nat > set_nat] :
      ( ( ( produc457027306803732586at_nat @ I2 @ X4 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ I2 )
           => ( ( X4 @ X2 )
              = bot_bot_set_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_648_Sigma__empty__iff,axiom,
    ! [I2: set_se7855581050983116737at_nat,X4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ I2 @ X4 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ I2 )
           => ( ( X4 @ X2 )
              = bot_bo3083307316010499117at_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_649_times__eq__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = ( produc2761391749766926216at_nat @ C2
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( ( A3 = C2 )
          & ( B5 = D2 ) )
        | ( ( ( A3 = bot_bo2099793752762293965at_nat )
            | ( B5 = bot_bo2099793752762293965at_nat ) )
          & ( ( C2 = bot_bo2099793752762293965at_nat )
            | ( D2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_650_times__eq__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat,C2: set_Pr1261947904930325089at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc3090783324798197870at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = ( produc3090783324798197870at_nat @ C2
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( ( A3 = C2 )
          & ( B5 = D2 ) )
        | ( ( ( A3 = bot_bo2099793752762293965at_nat )
            | ( B5 = bot_bo228742789529271731at_nat ) )
          & ( ( C2 = bot_bo2099793752762293965at_nat )
            | ( D2 = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_651_times__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,C2: set_Pr4329608150637261639at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc8743256393616062242at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = ( produc8743256393616062242at_nat @ C2
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( ( A3 = C2 )
          & ( B5 = D2 ) )
        | ( ( ( A3 = bot_bo228742789529271731at_nat )
            | ( B5 = bot_bo2099793752762293965at_nat ) )
          & ( ( C2 = bot_bo228742789529271731at_nat )
            | ( D2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_652_times__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : B5 )
        = ( produc4441300235185154312at_nat @ C2
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( ( A3 = C2 )
          & ( B5 = D2 ) )
        | ( ( ( A3 = bot_bo228742789529271731at_nat )
            | ( B5 = bot_bo228742789529271731at_nat ) )
          & ( ( C2 = bot_bo228742789529271731at_nat )
            | ( D2 = bot_bo228742789529271731at_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_653_times__subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : C2 )
        @ ( produc2761391749766926216at_nat @ B5
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( C2 = bot_bo2099793752762293965at_nat )
        | ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
          & ( ord_le3146513528884898305at_nat @ C2 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_654_times__subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le7005211468645301325at_nat
        @ ( produc3090783324798197870at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : C2 )
        @ ( produc3090783324798197870at_nat @ B5
          @ ^ [Uu: product_prod_nat_nat] : D2 ) )
      = ( ( A3 = bot_bo2099793752762293965at_nat )
        | ( C2 = bot_bo228742789529271731at_nat )
        | ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
          & ( ord_le1268244103169919719at_nat @ C2 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_655_times__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2257378622011118081at_nat
        @ ( produc8743256393616062242at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : C2 )
        @ ( produc8743256393616062242at_nat @ B5
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( C2 = bot_bo2099793752762293965at_nat )
        | ( ( ord_le1268244103169919719at_nat @ A3 @ B5 )
          & ( ord_le3146513528884898305at_nat @ C2 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_656_times__subset__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat
        @ ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : C2 )
        @ ( produc4441300235185154312at_nat @ B5
          @ ^ [Uu: produc3843707927480180839at_nat] : D2 ) )
      = ( ( A3 = bot_bo228742789529271731at_nat )
        | ( C2 = bot_bo228742789529271731at_nat )
        | ( ( ord_le1268244103169919719at_nat @ A3 @ B5 )
          & ( ord_le1268244103169919719at_nat @ C2 @ D2 ) ) ) ) ).

% times_subset_iff
thf(fact_657_fst__image__Sigma,axiom,
    ! [A3: set_nat,B5: nat > set_list_char] :
      ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ ( produc6030673636645808891t_char @ A3 @ B5 ) )
      = ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo8503570744482963627t_char ) ) ) ) ).

% fst_image_Sigma
thf(fact_658_fst__image__Sigma,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_6706873702294942047at_nat @ produc3213797794245857475at_nat @ ( produc2761391749766926216at_nat @ A3 @ B5 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_659_fst__image__Sigma,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_229310333994078635at_nat @ produc4362022507300774877at_nat @ ( produc3829605335166519074at_nat @ A3 @ B5 ) )
      = ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_660_fst__image__Sigma,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,B5: produc2934264451710624999od_o_o > set_Pr1261947904930325089at_nat] :
      ( ( image_2414159597220281259od_o_o @ produc1201395269668887389at_nat @ ( produc726318440123720610at_nat @ A3 @ B5 ) )
      = ( collec3302784863738620242od_o_o
        @ ^ [X2: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_661_fst__image__Sigma,axiom,
    ! [A3: set_Pr457366540195662369at_nat,B5: produc732395585841259969at_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_1472044358370609119at_nat @ produc8509921526640248835at_nat @ ( produc4339135674454976968at_nat @ A3 @ B5 ) )
      = ( collec1729919521702017196at_nat
        @ ^ [X2: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_662_fst__image__Sigma,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( image_8065348968207030699at_nat @ produc9061435184492477405at_nat @ ( produc8743256393616062242at_nat @ A3 @ B5 ) )
      = ( collec6321179662152712658at_nat
        @ ^ [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo2099793752762293965at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_663_fst__image__Sigma,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( image_7444983900720083193at_nat @ produc3408962115674613033at_nat @ ( produc3090783324798197870at_nat @ A3 @ B5 ) )
      = ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_664_fst__image__Sigma,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B5: produc859450856879609959at_nat > set_Pr4329608150637261639at_nat] :
      ( ( image_2925403533633446213at_nat @ produc7486072714475936067at_nat @ ( produc8155347593902061832at_nat @ A3 @ B5 ) )
      = ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_665_fst__image__Sigma,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,B5: produc2934264451710624999od_o_o > set_Pr4329608150637261639at_nat] :
      ( ( image_1244762028972942661od_o_o @ produc2135782952764625091at_nat @ ( produc5548153629540029832at_nat @ A3 @ B5 ) )
      = ( collec3302784863738620242od_o_o
        @ ^ [X2: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_666_fst__image__Sigma,axiom,
    ! [A3: set_Pr457366540195662369at_nat,B5: produc732395585841259969at_nat > set_Pr4329608150637261639at_nat] :
      ( ( image_7224720686980701561at_nat @ produc6776774009670138985at_nat @ ( produc1691665436710640302at_nat @ A3 @ B5 ) )
      = ( collec1729919521702017196at_nat
        @ ^ [X2: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X2 @ A3 )
            & ( ( B5 @ X2 )
             != bot_bo228742789529271731at_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_667_refl__onI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( ord_le7586318239878589543od_o_o @ R2
        @ ( produc514367270615792904od_o_o @ A3
          @ ^ [Uu: set_Product_prod_o_o] : A3 ) )
     => ( ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A3 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ X3 ) @ R2 ) )
       => ( refl_o935183656833700870od_o_o @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_668_refl__onI,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( ord_le8053465387879095399od_o_o @ R2
        @ ( produc3997963044239729416od_o_o @ A3
          @ ^ [Uu: produc2934264451710624999od_o_o] : A3 ) )
     => ( ! [X3: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ A3 )
           => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ X3 ) @ R2 ) )
       => ( refl_o8365827894042769382od_o_o @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_669_refl__onI,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( ord_le1067877976815774951at_nat @ R2
        @ ( produc8020639878719958920at_nat @ A3
          @ ^ [Uu: produc732395585841259969at_nat] : A3 ) )
     => ( ! [X3: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ A3 )
           => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_o3078925143892389184at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_670_refl__onI,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3091953645718357095at_nat @ R2
        @ ( produc4441300235185154312at_nat @ A3
          @ ^ [Uu: produc3843707927480180839at_nat] : A3 ) )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A3 )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_o7446464703481863270at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_671_refl__onI,axiom,
    ! [R2: set_Pr553994874890374343at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le2286497330737841255at_nat @ R2
        @ ( produc18557253517588744at_nat @ A3
          @ ^ [Uu: produc859450856879609959at_nat] : A3 ) )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A3 )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_o6679620997798956134at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_672_refl__onI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3000389064537975527at_nat @ R2
        @ ( produc2761391749766926216at_nat @ A3
          @ ^ [Uu: product_prod_nat_nat] : A3 ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_o7122697186039111552at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_673_refl__onI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( ord_le5997549366648089703at_nat @ R2
        @ ( produc6258361011560675592at_nat @ A3
          @ ^ [Uu: set_Pr4329608150637261639at_nat] : A3 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A3 )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_o6272981002401706822at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_674_refl__onI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( ord_le1268244103169919719at_nat @ R2
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : A3 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A3 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_o6955724340487615712at_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_675_refl__onI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( ord_le3146513528884898305at_nat @ R2
        @ ( produc457027306803732586at_nat @ A3
          @ ^ [Uu: nat] : A3 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 ) )
       => ( refl_on_nat @ A3 @ R2 ) ) ) ).

% refl_onI
thf(fact_676_refl__on__def,axiom,
    ( refl_o935183656833700870od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
          ( ( ord_le7586318239878589543od_o_o @ R
            @ ( produc514367270615792904od_o_o @ A7
              @ ^ [Uu: set_Product_prod_o_o] : A7 ) )
          & ! [X2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X2 @ A7 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_677_refl__on__def,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
          ( ( ord_le3000389064537975527at_nat @ R
            @ ( produc2761391749766926216at_nat @ A7
              @ ^ [Uu: product_prod_nat_nat] : A7 ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A7 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_678_refl__on__def,axiom,
    ( refl_o6272981002401706822at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
          ( ( ord_le5997549366648089703at_nat @ R
            @ ( produc6258361011560675592at_nat @ A7
              @ ^ [Uu: set_Pr4329608150637261639at_nat] : A7 ) )
          & ! [X2: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X2 @ A7 )
             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_679_refl__on__def,axiom,
    ( refl_o6955724340487615712at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
          ( ( ord_le1268244103169919719at_nat @ R
            @ ( produc7726050920695093128at_nat @ A7
              @ ^ [Uu: set_Pr1261947904930325089at_nat] : A7 ) )
          & ! [X2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X2 @ A7 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_680_refl__on__def,axiom,
    ( refl_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
          ( ( ord_le3146513528884898305at_nat @ R
            @ ( produc457027306803732586at_nat @ A7
              @ ^ [Uu: nat] : A7 ) )
          & ! [X2: nat] :
              ( ( member_nat @ X2 @ A7 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def
thf(fact_681_relation__of__def,axiom,
    ( order_9102575157408253013od_o_o
    = ( ^ [P4: set_Product_prod_o_o > set_Product_prod_o_o > $o,A7: set_se491199571785101031od_o_o] :
          ( collec3302784863738620242od_o_o
          @ ( produc9128050996712119596_o_o_o
            @ ^ [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 )
                  @ ( produc514367270615792904od_o_o @ A7
                    @ ^ [Uu: set_Product_prod_o_o] : A7 ) )
                & ( P4 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_682_relation__of__def,axiom,
    ( order_1381308103030626383at_nat
    = ( ^ [P4: product_prod_nat_nat > product_prod_nat_nat > $o,A7: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 )
                  @ ( produc2761391749766926216at_nat @ A7
                    @ ^ [Uu: product_prod_nat_nat] : A7 ) )
                & ( P4 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_683_relation__of__def,axiom,
    ( order_8308734813930759573at_nat
    = ( ^ [P4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,A7: set_se5479201105039950375at_nat] :
          ( collec7872792499061297490at_nat
          @ ( produc9020218426428501292_nat_o
            @ ^ [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
                ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B2 )
                  @ ( produc6258361011560675592at_nat @ A7
                    @ ^ [Uu: set_Pr4329608150637261639at_nat] : A7 ) )
                & ( P4 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_684_relation__of__def,axiom,
    ( order_8809246210994205103at_nat
    = ( ^ [P4: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A7: set_se7855581050983116737at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 )
                  @ ( produc7726050920695093128at_nat @ A7
                    @ ^ [Uu: set_Pr1261947904930325089at_nat] : A7 ) )
                & ( P4 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_685_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P4: nat > nat > $o,A7: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A2: nat,B2: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 )
                  @ ( produc457027306803732586at_nat @ A7
                    @ ^ [Uu: nat] : A7 ) )
                & ( P4 @ A2 @ B2 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_686_refl__on__domain,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
       => ( ( member444158400953824016od_o_o @ A @ A3 )
          & ( member444158400953824016od_o_o @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_687_refl__on__domain,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
       => ( ( member4095101504841534314at_nat @ A @ A3 )
          & ( member4095101504841534314at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_688_refl__on__domain,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( ( member8757157785044589968at_nat @ A @ A3 )
          & ( member8757157785044589968at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_689_refl__on__domain,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 )
       => ( ( member8206827879206165904at_nat @ A @ A3 )
          & ( member8206827879206165904at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_690_refl__on__domain,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ( member3157015407008874800od_o_o @ A @ A3 )
          & ( member3157015407008874800od_o_o @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_691_refl__on__domain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
          & ( member8440522571783428010at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_692_refl__on__domain,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
       => ( ( member6273874620674182256at_nat @ A @ A3 )
          & ( member6273874620674182256at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_693_refl__on__domain,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ( member2643936169264416010at_nat @ A @ A3 )
          & ( member2643936169264416010at_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_694_refl__on__domain,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( refl_on_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( ( member_nat @ A @ A3 )
          & ( member_nat @ B @ A3 ) ) ) ) ).

% refl_on_domain
thf(fact_695_refl__onD,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_696_refl__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_697_refl__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_698_refl__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ A3 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_699_refl__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_700_refl__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_701_refl__onD,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ A @ A3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_702_refl__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_703_refl__onD,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A3 @ R2 )
     => ( ( member_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R2 ) ) ) ).

% refl_onD
thf(fact_704_refl__onD1,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
       => ( member444158400953824016od_o_o @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_705_refl__onD1,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
       => ( member4095101504841534314at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_706_refl__onD1,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
       => ( member8757157785044589968at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_707_refl__onD1,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
       => ( member8206827879206165904at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_708_refl__onD1,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
       => ( member3157015407008874800od_o_o @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_709_refl__onD1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( member8440522571783428010at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_710_refl__onD1,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( member6273874620674182256at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_711_refl__onD1,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( member2643936169264416010at_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_712_refl__onD1,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( member_nat @ X @ A3 ) ) ) ).

% refl_onD1
thf(fact_713_refl__onD2,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( refl_o8365827894042769382od_o_o @ A3 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
       => ( member444158400953824016od_o_o @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_714_refl__onD2,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( refl_o3078925143892389184at_nat @ A3 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
       => ( member4095101504841534314at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_715_refl__onD2,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( refl_o7446464703481863270at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
       => ( member8757157785044589968at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_716_refl__onD2,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( refl_o6679620997798956134at_nat @ A3 @ R2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
       => ( member8206827879206165904at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_717_refl__onD2,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
       => ( member3157015407008874800od_o_o @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_718_refl__onD2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( member8440522571783428010at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_719_refl__onD2,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( member6273874620674182256at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_720_refl__onD2,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( member2643936169264416010at_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_721_refl__onD2,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( refl_on_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( member_nat @ Y @ A3 ) ) ) ).

% refl_onD2
thf(fact_722_reflD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( refl_o935183656833700870od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_723_reflD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_724_reflD,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( refl_o6272981002401706822at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_725_reflD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_726_reflD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R2 ) ) ).

% reflD
thf(fact_727_reflI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ X3 ) @ R2 )
     => ( refl_o935183656833700870od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% reflI
thf(fact_728_reflI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R2 )
     => ( refl_o7122697186039111552at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% reflI
thf(fact_729_reflI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ R2 )
     => ( refl_o6272981002401706822at_nat @ top_to1754375979021635447at_nat @ R2 ) ) ).

% reflI
thf(fact_730_reflI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ R2 )
     => ( refl_o6955724340487615712at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% reflI
thf(fact_731_reflI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
     => ( refl_on_nat @ top_top_set_nat @ R2 ) ) ).

% reflI
thf(fact_732_aboveS__def,axiom,
    ( order_780859930024226034od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A2: set_Product_prod_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] :
              ( ( B2 != A2 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_733_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] :
              ( ( B2 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_734_aboveS__def,axiom,
    ( order_2442354982230848562at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A2: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] :
              ( ( B2 != A2 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_735_aboveS__def,axiom,
    ( order_4440095836999870412at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] :
              ( ( B2 != A2 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_736_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B2: nat] :
              ( ( B2 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) ) ) ) ) ).

% aboveS_def
thf(fact_737_above__def,axiom,
    ( order_6680364264651538695od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A2: set_Product_prod_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_738_above__def,axiom,
    ( order_793026671032835073at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_739_above__def,axiom,
    ( order_3445360974473395783at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A2: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_740_above__def,axiom,
    ( order_4986753419052871009at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_741_above__def,axiom,
    ( order_above_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) ) ) ) ).

% above_def
thf(fact_742_Sigma__def,axiom,
    ( produc7365643037244224738at_nat
    = ( ^ [A7: set_se491199571785101031od_o_o,B7: set_Product_prod_o_o > set_se7855581050983116737at_nat] :
          ( comple4626563411906277390at_nat
          @ ( image_3475158639969665759at_nat
            @ ^ [X2: set_Product_prod_o_o] :
                ( comple4626563411906277390at_nat
                @ ( image_1314253420279835909at_nat
                  @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ bot_bo1676454896636820109at_nat )
                  @ ( B7 @ X2 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_743_Sigma__def,axiom,
    ( produc6258361011560675592at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,B7: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X2: set_Pr4329608150637261639at_nat] :
                ( comple7342073961973170484at_nat
                @ ( image_1124969593608450885at_nat
                  @ ^ [Y5: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ bot_bo4948859079157340979at_nat )
                  @ ( B7 @ X2 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_744_Sigma__def,axiom,
    ( produc6030673636645808891t_char
    = ( ^ [A7: set_nat,B7: nat > set_list_char] :
          ( comple9178978156834306527t_char
          @ ( image_7918911184113263195t_char
            @ ^ [X2: nat] :
                ( comple9178978156834306527t_char
                @ ( image_366428816823531018t_char
                  @ ^ [Y5: list_char] : ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ bot_bo1414094081888755934t_char )
                  @ ( B7 @ X2 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_745_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A7: 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 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_746_Sigma__def,axiom,
    ( produc7726050920695093128at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,B7: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X2: set_Pr1261947904930325089at_nat] :
                ( comple4364388969068068532at_nat
                @ ( image_6898982319499645995at_nat
                  @ ^ [Y5: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ bot_bo228742789529271731at_nat )
                  @ ( B7 @ X2 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_747_asymI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
         => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R2 ) )
     => ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% asymI
thf(fact_748_asymI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% asymI
thf(fact_749_asymI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R2 ) ) ).

% asymI
thf(fact_750_asymI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% asymI
thf(fact_751_asymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 ) )
     => ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymI
thf(fact_752_asym__onI,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A3 )
         => ( ( member3157015407008874800od_o_o @ Y3 @ A3 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
             => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o8381145231034218809od_o_o @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_753_asym__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X3 @ A3 )
         => ( ( member444158400953824016od_o_o @ Y3 @ A3 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R2 )
             => ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o7797749731265203737od_o_o @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_754_asym__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X3 @ A3 )
         => ( ( member4095101504841534314at_nat @ Y3 @ A3 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o194845430079802867at_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_755_asym__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A3 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o9101232109321078681at_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_756_asym__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A3 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A3 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o9200482492015546265at_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_757_asym__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o8546584678760569651at_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_758_asym__onI,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A3 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A3 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o8654330542403097209at_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_759_asym__onI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A3 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_o5033339280856310931at_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_760_asym__onI,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( member_nat @ Y3 @ A3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_nat @ A3 @ R2 ) ) ).

% asym_onI
thf(fact_761_asym__on__def,axiom,
    ( asym_o8381145231034218809od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ A7 )
         => ! [Y5: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y5 @ A7 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R )
               => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_762_asym__on__def,axiom,
    ( asym_o8546584678760569651at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A7 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A7 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_763_asym__on__def,axiom,
    ( asym_o8654330542403097209at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ A7 )
         => ! [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A7 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R )
               => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_764_asym__on__def,axiom,
    ( asym_o5033339280856310931at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A7 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A7 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R )
               => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_765_asym__on__def,axiom,
    ( asym_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A7 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A7 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_766_asym__onD,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( asym_o8381145231034218809od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y @ A3 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_767_asym__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( asym_o7797749731265203737od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( member444158400953824016od_o_o @ Y @ A3 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
           => ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_768_asym__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( asym_o194845430079802867at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( member4095101504841534314at_nat @ Y @ A3 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
           => ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_769_asym__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( asym_o9101232109321078681at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( member8757157785044589968at_nat @ Y @ A3 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
           => ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_770_asym__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( asym_o9200482492015546265at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( member8206827879206165904at_nat @ Y @ A3 )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
           => ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_771_asym__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( member8440522571783428010at_nat @ Y @ A3 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_772_asym__onD,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( asym_o8654330542403097209at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ A3 )
       => ( ( member6273874620674182256at_nat @ Y @ A3 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
           => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_773_asym__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( asym_o5033339280856310931at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( member2643936169264416010at_nat @ Y @ A3 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_774_asym__onD,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_775_refl__on__singleton,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( refl_o6272981002401706822at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ bot_bo4948859079157340979at_nat ) ) ).

% refl_on_singleton
thf(fact_776_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_777_refl__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( refl_o6955724340487615712at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% refl_on_singleton
thf(fact_778_refl__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( refl_o7122697186039111552at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% refl_on_singleton
thf(fact_779_refl__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( refl_o7446464703481863270at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% refl_on_singleton
thf(fact_780_asym__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 )
      = ( ! [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_781_asym__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 )
      = ( ! [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_782_asym__iff,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R2 )
      = ( ! [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] :
            ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R2 )
           => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_783_asym__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 )
      = ( ! [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_784_asym__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X2: nat,Y5: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_785_asymD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
       => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_786_asymD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_787_asymD,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_788_asymD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_789_asymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) ) ) ).

% asymD
thf(fact_790_Id__on__def,axiom,
    ( id_on_3275838214495744996at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X2: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ bot_bo4948859079157340979at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_791_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A7: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X2: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ bot_bo2099793752762293965at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_792_Id__on__def,axiom,
    ( id_on_6713504163309144702at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X2: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ bot_bo228742789529271731at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_793_linear__order__on__singleton,axiom,
    ! [X: set_Pr4329608150637261639at_nat] : ( order_1878278401035043452at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ bot_bo4948859079157340979at_nat ) ) ).

% linear_order_on_singleton
thf(fact_794_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_795_linear__order__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( order_5942792088423767318at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% linear_order_on_singleton
thf(fact_796_linear__order__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( order_7480980593162297270at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% linear_order_on_singleton
thf(fact_797_linear__order__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( order_2711500226769404828at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% linear_order_on_singleton
thf(fact_798_insert__Times__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( produc4441300235185154312at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
        @ ^ [Uu: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
      = ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B )
        @ ( sup_su6664651903789602459at_nat
          @ ( produc4441300235185154312at_nat @ A3
            @ ^ [Uu: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
          @ ( produc4441300235185154312at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
            @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_799_insert__Times__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( produc8743256393616062242at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
        @ ^ [Uu: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
      = ( insert373714669040926161at_nat @ ( produc5023393015413421041at_nat @ A @ B )
        @ ( sup_su6071048768109290037at_nat
          @ ( produc8743256393616062242at_nat @ A3
            @ ^ [Uu: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
          @ ( produc8743256393616062242at_nat @ ( insert9069300056098147895at_nat @ A @ A3 )
            @ ^ [Uu: produc3843707927480180839at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_800_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( produc3090783324798197870at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
        @ ^ [Uu: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
      = ( insert8243672688981495325at_nat @ ( produc8594291983450332477at_nat @ A @ B )
        @ ( sup_su1595509577888697473at_nat
          @ ( produc3090783324798197870at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ B @ B5 ) )
          @ ( produc3090783324798197870at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
            @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_801_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( produc2761391749766926216at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
        @ ^ [Uu: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
      = ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B )
        @ ( sup_su718114333110466843at_nat
          @ ( produc2761391749766926216at_nat @ A3
            @ ^ [Uu: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ B @ B5 ) )
          @ ( produc2761391749766926216at_nat @ ( insert8211810215607154385at_nat @ A @ A3 )
            @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_802_insert__Times__insert,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( produc7365643037244224738at_nat @ ( insert3817420688681518039od_o_o @ A @ A3 )
        @ ^ [Uu: set_Product_prod_o_o] : ( insert9200635055090092081at_nat @ B @ B5 ) )
      = ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B )
        @ ( sup_su7309111544468589045at_nat
          @ ( produc7365643037244224738at_nat @ A3
            @ ^ [Uu: set_Product_prod_o_o] : ( insert9200635055090092081at_nat @ B @ B5 ) )
          @ ( produc7365643037244224738at_nat @ ( insert3817420688681518039od_o_o @ A @ A3 )
            @ ^ [Uu: set_Product_prod_o_o] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_803_insert__Times__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat,B: set_Pr4329608150637261639at_nat,B5: set_se5479201105039950375at_nat] :
      ( ( produc6258361011560675592at_nat @ ( insert2290795007606087959at_nat @ A @ A3 )
        @ ^ [Uu: set_Pr4329608150637261639at_nat] : ( insert2290795007606087959at_nat @ B @ B5 ) )
      = ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B )
        @ ( sup_su5395584912770621083at_nat
          @ ( produc6258361011560675592at_nat @ A3
            @ ^ [Uu: set_Pr4329608150637261639at_nat] : ( insert2290795007606087959at_nat @ B @ B5 ) )
          @ ( produc6258361011560675592at_nat @ ( insert2290795007606087959at_nat @ A @ A3 )
            @ ^ [Uu: set_Pr4329608150637261639at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_804_insert__Times__insert,axiom,
    ! [A: nat,A3: set_nat,B: list_char,B5: set_list_char] :
      ( ( produc6030673636645808891t_char @ ( insert_nat @ A @ A3 )
        @ ^ [Uu: nat] : ( insert_list_char @ B @ B5 ) )
      = ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B )
        @ ( sup_su7899656352754981958t_char
          @ ( produc6030673636645808891t_char @ A3
            @ ^ [Uu: nat] : ( insert_list_char @ B @ B5 ) )
          @ ( produc6030673636645808891t_char @ ( insert_nat @ A @ A3 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_805_insert__Times__insert,axiom,
    ! [A: nat,A3: set_nat,B: nat,B5: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A3 )
        @ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A3
            @ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A3 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_806_insert__Times__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( insert9200635055090092081at_nat @ A @ A3 )
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : ( insert9200635055090092081at_nat @ B @ B5 ) )
      = ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B )
        @ ( sup_su5525570899277871387at_nat
          @ ( produc7726050920695093128at_nat @ A3
            @ ^ [Uu: set_Pr1261947904930325089at_nat] : ( insert9200635055090092081at_nat @ B @ B5 ) )
          @ ( produc7726050920695093128at_nat @ ( insert9200635055090092081at_nat @ A @ A3 )
            @ ^ [Uu: set_Pr1261947904930325089at_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_807_asymp__asym__eq,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( asymp_1374842860160563877od_o_o @ top_to1478448522960722999od_o_o
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o8381145231034218809od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% asymp_asym_eq
thf(fact_808_asymp__asym__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( asymp_3201933403969103519at_nat @ top_to4669805908274784177at_nat
        @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_809_asymp__asym__eq,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( asymp_7147290840880511973at_nat @ top_to1754375979021635447at_nat
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o8654330542403097209at_nat @ top_to1754375979021635447at_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_810_asymp__asym__eq,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( asymp_8721585889738326015at_nat @ top_to7629004291339433233at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o5033339280856310931at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_811_asymp__asym__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ top_top_set_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_812_prod__set__simps_I2_J,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_7607750056555722463at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_813_prod__set__simps_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( basic_5713354656523905413at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) ).

% prod_set_simps(2)
thf(fact_814_prod__set__simps_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_484852565060450053at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(2)
thf(fact_815_prod__set__simps_I2_J,axiom,
    ! [X: nat,Y: list_char] :
      ( ( basic_2224790070303523448t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( insert_list_char @ Y @ bot_bo8503570744482963627t_char ) ) ).

% prod_set_simps(2)
thf(fact_816_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_817_Id__onI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ A ) @ ( id_on_3944400147138547876od_o_o @ A3 ) ) ) ).

% Id_onI
thf(fact_818_Id__onI,axiom,
    ! [A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ A @ A3 )
     => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ A ) @ ( id_on_3920255446293593732od_o_o @ A3 ) ) ) ).

% Id_onI
thf(fact_819_Id__onI,axiom,
    ! [A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ A @ A3 )
     => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ A ) @ ( id_on_7507360449079020766at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_820_Id__onI,axiom,
    ! [A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A3 )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( id_on_7869290023980668164at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_821_Id__onI,axiom,
    ! [A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A3 )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( id_on_7917584626360399108at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_822_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_823_Id__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ A @ A3 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( id_on_3275838214495744996at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_824_Id__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( id_on_6713504163309144702at_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_825_Id__onI,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A3 ) ) ) ).

% Id_onI
thf(fact_826_Sigma__Un__distrib1,axiom,
    ! [I2: set_se7855581050983116737at_nat,J2: set_se7855581050983116737at_nat,C2: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ I2 @ J2 ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I2 @ C2 ) @ ( produc7726050920695093128at_nat @ J2 @ C2 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_827_Sigma__Un__distrib2,axiom,
    ! [I2: set_se7855581050983116737at_nat,A3: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ I2
        @ ^ [I: set_Pr1261947904930325089at_nat] : ( sup_su3642409539654194069at_nat @ ( A3 @ I ) @ ( B5 @ I ) ) )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I2 @ A3 ) @ ( produc7726050920695093128at_nat @ I2 @ B5 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_828_Times__Un__distrib1,axiom,
    ! [A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat,C2: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ A3 @ B5 )
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : C2 )
      = ( sup_su5525570899277871387at_nat
        @ ( produc7726050920695093128at_nat @ A3
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : C2 )
        @ ( produc7726050920695093128at_nat @ B5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : C2 ) ) ) ).

% Times_Un_distrib1
thf(fact_829_Id__onE,axiom,
    ! [C: produc2245965095621756647od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ C @ ( id_on_3920255446293593732od_o_o @ A3 ) )
     => ~ ! [X3: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ A3 )
           => ( C
             != ( produc6658621592305443671od_o_o @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_830_Id__onE,axiom,
    ! [C: produc266773502759220327at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( member208136763105818512at_nat @ C @ ( id_on_7507360449079020766at_nat @ A3 ) )
     => ~ ! [X3: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ A3 )
           => ( C
             != ( produc4785852428192286935at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_831_Id__onE,axiom,
    ! [C: produc1906917888121415911at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ C @ ( id_on_7869290023980668164at_nat @ A3 ) )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A3 )
           => ( C
             != ( produc2335810615602371415at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_832_Id__onE,axiom,
    ! [C: produc349518998152878311at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C @ ( id_on_7917584626360399108at_nat @ A3 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A3 )
           => ( C
             != ( produc4662710985925991255at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_833_Id__onE,axiom,
    ! [C: produc2934264451710624999od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ C @ ( id_on_3944400147138547876od_o_o @ A3 ) )
     => ~ ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ A3 )
           => ( C
             != ( produc763777882069021527od_o_o @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_834_Id__onE,axiom,
    ! [C: produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( id_on_2554058798563519774at_nat @ A3 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ( C
             != ( produc6161850002892822231at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_835_Id__onE,axiom,
    ! [C: produc1319942482725812455at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ C @ ( id_on_3275838214495744996at_nat @ A3 ) )
     => ~ ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A3 )
           => ( C
             != ( produc9060074326276436823at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_836_Id__onE,axiom,
    ! [C: produc3843707927480180839at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( id_on_6713504163309144702at_nat @ A3 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A3 )
           => ( C
             != ( produc2922128104949294807at_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_837_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( C
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_838_Id__on__eqI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( A = B )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ ( id_on_3944400147138547876od_o_o @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_839_Id__on__eqI,axiom,
    ! [A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( A = B )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ ( id_on_3920255446293593732od_o_o @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_840_Id__on__eqI,axiom,
    ! [A: produc732395585841259969at_nat,B: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( A = B )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ ( id_on_7507360449079020766at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_841_Id__on__eqI,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( A = B )
     => ( ( member8757157785044589968at_nat @ A @ A3 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ ( id_on_7869290023980668164at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_842_Id__on__eqI,axiom,
    ! [A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( A = B )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ ( id_on_7917584626360399108at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_843_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( A = B )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_844_Id__on__eqI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( A = B )
     => ( ( member6273874620674182256at_nat @ A @ A3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ ( id_on_3275838214495744996at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_845_Id__on__eqI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( A = B )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( id_on_6713504163309144702at_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_846_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A3: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A3 ) ) ) ) ).

% Id_on_eqI
thf(fact_847_Id__on__iff,axiom,
    ! [X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ ( id_on_3920255446293593732od_o_o @ A3 ) )
      = ( ( X = Y )
        & ( member444158400953824016od_o_o @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_848_Id__on__iff,axiom,
    ! [X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ ( id_on_7507360449079020766at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member4095101504841534314at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_849_Id__on__iff,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ ( id_on_7869290023980668164at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member8757157785044589968at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_850_Id__on__iff,axiom,
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ ( id_on_7917584626360399108at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member8206827879206165904at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_851_Id__on__iff,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( id_on_3944400147138547876od_o_o @ A3 ) )
      = ( ( X = Y )
        & ( member3157015407008874800od_o_o @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_852_Id__on__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( id_on_2554058798563519774at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member8440522571783428010at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_853_Id__on__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( id_on_3275838214495744996at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member6273874620674182256at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_854_Id__on__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( id_on_6713504163309144702at_nat @ A3 ) )
      = ( ( X = Y )
        & ( member2643936169264416010at_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_855_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A3 ) )
      = ( ( X = Y )
        & ( member_nat @ X @ A3 ) ) ) ).

% Id_on_iff
thf(fact_856_asymp__on__asym__on__eq,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( asymp_1374842860160563877od_o_o @ A3
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o8381145231034218809od_o_o @ A3 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_857_asymp__on__asym__on__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( asymp_3201933403969103519at_nat @ A3
        @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o8546584678760569651at_nat @ A3 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_858_asymp__on__asym__on__eq,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( asymp_7147290840880511973at_nat @ A3
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o8654330542403097209at_nat @ A3 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_859_asymp__on__asym__on__eq,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( asymp_8721585889738326015at_nat @ A3
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_o5033339280856310931at_nat @ A3 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_860_asymp__on__asym__on__eq,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ A3
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) )
      = ( asym_on_nat @ A3 @ R2 ) ) ).

% asymp_on_asym_on_eq
thf(fact_861_Field__insert,axiom,
    ! [A: $o,B: $o,R2: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 ) )
      = ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).

% Field_insert
thf(fact_862_Field__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( field_1428494348145443988at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) )
      = ( sup_su474284620922681339at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_863_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_864_Field__insert,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_865_Field__insert,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_866_Field__insert,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) )
      = ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_867_prod__set__simps_I1_J,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_3533134110227475617at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) )
      = ( insert3817420688681518039od_o_o @ X @ bot_bo8006819982598807891od_o_o ) ) ).

% prod_set_simps(1)
thf(fact_868_prod__set__simps_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( basic_1274512288150691911at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) )
      = ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ).

% prod_set_simps(1)
thf(fact_869_prod__set__simps_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( basic_3661821009137050567at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) )
      = ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) ).

% prod_set_simps(1)
thf(fact_870_prod__set__simps_I1_J,axiom,
    ! [X: nat,Y: list_char] :
      ( ( basic_4688923273072336698t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% prod_set_simps(1)
thf(fact_871_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_872_Range__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( range_8412904782085253007at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_8412904782085253007at_nat @ R2 ) ) ) ).

% Range_insert
thf(fact_873_Range__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( range_7218127889571902005at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) )
      = ( insert2290795007606087959at_nat @ B @ ( range_7218127889571902005at_nat @ R2 ) ) ) ).

% Range_insert
thf(fact_874_Range__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( range_880292732526890933at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( insert9200635055090092081at_nat @ B @ ( range_880292732526890933at_nat @ R2 ) ) ) ).

% Range_insert
thf(fact_875_Range__insert,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( range_nat_list_char @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 ) )
      = ( insert_list_char @ B @ ( range_nat_list_char @ R2 ) ) ) ).

% Range_insert
thf(fact_876_Range__insert,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( range_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( insert_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).

% Range_insert
thf(fact_877_Domain__insert,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( domain8494901049294654374at_nat @ ( insert8109617542356073361at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 ) )
      = ( insert3817420688681518039od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) ) ) ).

% Domain_insert
thf(fact_878_Domain__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( domain6426273929576919500at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) )
      = ( insert2290795007606087959at_nat @ A @ ( domain6426273929576919500at_nat @ R2 ) ) ) ).

% Domain_insert
thf(fact_879_Domain__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( domain8802589944165416012at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( insert9200635055090092081at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) ) ) ).

% Domain_insert
thf(fact_880_Domain__insert,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( domain_nat_list_char @ ( insert3372640069787401954t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 ) )
      = ( insert_nat @ A @ ( domain_nat_list_char @ R2 ) ) ) ).

% Domain_insert
thf(fact_881_Domain__insert,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( domain_nat_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( insert_nat @ A @ ( domain_nat_nat @ R2 ) ) ) ).

% Domain_insert
thf(fact_882_trans__singleton,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ bot_bo4948859079157340979at_nat ) ) ).

% trans_singleton
thf(fact_883_trans__singleton,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ bot_bo228742789529271731at_nat ) ) ).

% trans_singleton
thf(fact_884_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_885_FieldI2,axiom,
    ! [I3: produc2934264451710624999od_o_o,J3: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I3 @ J3 ) @ R3 )
     => ( member444158400953824016od_o_o @ J3 @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% FieldI2
thf(fact_886_FieldI2,axiom,
    ! [I3: produc732395585841259969at_nat,J3: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I3 @ J3 ) @ R3 )
     => ( member4095101504841534314at_nat @ J3 @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_887_FieldI2,axiom,
    ! [I3: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J3 ) @ R3 )
     => ( member8757157785044589968at_nat @ J3 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_888_FieldI2,axiom,
    ! [I3: produc859450856879609959at_nat,J3: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J3 ) @ R3 )
     => ( member8206827879206165904at_nat @ J3 @ ( field_3044959742847878580at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_889_FieldI2,axiom,
    ! [I3: set_Product_prod_o_o,J3: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I3 @ J3 ) @ R3 )
     => ( member3157015407008874800od_o_o @ J3 @ ( field_6101175518749917524od_o_o @ R3 ) ) ) ).

% FieldI2
thf(fact_890_FieldI2,axiom,
    ! [I3: product_prod_nat_nat,J3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J3 ) @ R3 )
     => ( member8440522571783428010at_nat @ J3 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_891_FieldI2,axiom,
    ! [I3: $o,J3: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I3 @ J3 ) @ R3 )
     => ( member_o @ J3 @ ( field_o @ R3 ) ) ) ).

% FieldI2
thf(fact_892_FieldI2,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J3: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J3 ) @ R3 )
     => ( member6273874620674182256at_nat @ J3 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_893_FieldI2,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J3 ) @ R3 )
     => ( member2643936169264416010at_nat @ J3 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_894_FieldI2,axiom,
    ! [I3: nat,J3: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J3 ) @ R3 )
     => ( member_nat @ J3 @ ( field_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_895_FieldI1,axiom,
    ! [I3: produc2934264451710624999od_o_o,J3: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I3 @ J3 ) @ R3 )
     => ( member444158400953824016od_o_o @ I3 @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% FieldI1
thf(fact_896_FieldI1,axiom,
    ! [I3: produc732395585841259969at_nat,J3: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I3 @ J3 ) @ R3 )
     => ( member4095101504841534314at_nat @ I3 @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_897_FieldI1,axiom,
    ! [I3: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J3 ) @ R3 )
     => ( member8757157785044589968at_nat @ I3 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_898_FieldI1,axiom,
    ! [I3: produc859450856879609959at_nat,J3: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J3 ) @ R3 )
     => ( member8206827879206165904at_nat @ I3 @ ( field_3044959742847878580at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_899_FieldI1,axiom,
    ! [I3: set_Product_prod_o_o,J3: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I3 @ J3 ) @ R3 )
     => ( member3157015407008874800od_o_o @ I3 @ ( field_6101175518749917524od_o_o @ R3 ) ) ) ).

% FieldI1
thf(fact_900_FieldI1,axiom,
    ! [I3: product_prod_nat_nat,J3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J3 ) @ R3 )
     => ( member8440522571783428010at_nat @ I3 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_901_FieldI1,axiom,
    ! [I3: $o,J3: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I3 @ J3 ) @ R3 )
     => ( member_o @ I3 @ ( field_o @ R3 ) ) ) ).

% FieldI1
thf(fact_902_FieldI1,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J3: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J3 ) @ R3 )
     => ( member6273874620674182256at_nat @ I3 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_903_FieldI1,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J3 ) @ R3 )
     => ( member2643936169264416010at_nat @ I3 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_904_FieldI1,axiom,
    ! [I3: nat,J3: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J3 ) @ R3 )
     => ( member_nat @ I3 @ ( field_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_905_trans__on__def,axiom,
    ( trans_7029024365671003047od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ A7 )
         => ! [Y5: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y5 @ A7 )
             => ! [Z5: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Z5 @ A7 )
                 => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R )
                   => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Z5 ) @ R )
                     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Z5 ) @ R ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_906_trans__on__def,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A7 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A7 )
             => ! [Z5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z5 @ A7 )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R )
                   => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ Z5 ) @ R )
                     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Z5 ) @ R ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_907_trans__on__def,axiom,
    ( trans_707035017233543399at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ A7 )
         => ! [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A7 )
             => ! [Z5: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ Z5 @ A7 )
                 => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R )
                   => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ Z5 ) @ R )
                     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Z5 ) @ R ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_908_trans__on__def,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A7 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A7 )
             => ! [Z5: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Z5 @ A7 )
                 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R )
                   => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Z5 ) @ R )
                     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Z5 ) @ R ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_909_trans__on__def,axiom,
    ( trans_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A7 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A7 )
             => ! [Z5: nat] :
                  ( ( member_nat @ Z5 @ A7 )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
                   => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z5 ) @ R )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z5 ) @ R ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_910_trans__onI,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A3 )
         => ( ( member3157015407008874800od_o_o @ Y3 @ A3 )
           => ( ( member3157015407008874800od_o_o @ Z2 @ A3 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z2 ) @ R2 )
                 => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_7029024365671003047od_o_o @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_911_trans__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o,Z2: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X3 @ A3 )
         => ( ( member444158400953824016od_o_o @ Y3 @ A3 )
           => ( ( member444158400953824016od_o_o @ Z2 @ A3 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R2 )
               => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ Z2 ) @ R2 )
                 => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_2549647969203962503od_o_o @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_912_trans__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat,Z2: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X3 @ A3 )
         => ( ( member4095101504841534314at_nat @ Y3 @ A3 )
           => ( ( member4095101504841534314at_nat @ Z2 @ A3 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_3798871764278489953at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_913_trans__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,Z2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A3 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
           => ( ( member8757157785044589968at_nat @ Z2 @ A3 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_4802246114490097159at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_914_trans__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat,Z2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A3 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A3 )
           => ( ( member8206827879206165904at_nat @ Z2 @ A3 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_613433433080944135at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_915_trans__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
           => ( ( member8440522571783428010at_nat @ Z2 @ A3 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_3241703717017304737at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_916_trans__onI,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A3 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A3 )
           => ( ( member6273874620674182256at_nat @ Z2 @ A3 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_707035017233543399at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_917_trans__onI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A3 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
           => ( ( member2643936169264416010at_nat @ Z2 @ A3 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_3732288485672914945at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_918_trans__onI,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( member_nat @ Y3 @ A3 )
           => ( ( member_nat @ Z2 @ A3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R2 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R2 ) ) ) ) ) )
     => ( trans_on_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_919_trans__onD,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y @ A3 )
         => ( ( member3157015407008874800od_o_o @ Z @ A3 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_920_trans__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o,Z: produc2934264451710624999od_o_o] :
      ( ( trans_2549647969203962503od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( member444158400953824016od_o_o @ Y @ A3 )
         => ( ( member444158400953824016od_o_o @ Z @ A3 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ Z ) @ R2 )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_921_trans__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat,Z: produc732395585841259969at_nat] :
      ( ( trans_3798871764278489953at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( member4095101504841534314at_nat @ Y @ A3 )
         => ( ( member4095101504841534314at_nat @ Z @ A3 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ Z ) @ R2 )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_922_trans__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] :
      ( ( trans_4802246114490097159at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( member8757157785044589968at_nat @ Y @ A3 )
         => ( ( member8757157785044589968at_nat @ Z @ A3 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ Z ) @ R2 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_923_trans__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
      ( ( trans_613433433080944135at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( member8206827879206165904at_nat @ Y @ A3 )
         => ( ( member8206827879206165904at_nat @ Z @ A3 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ Z ) @ R2 )
               => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_924_trans__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( member8440522571783428010at_nat @ Y @ A3 )
         => ( ( member8440522571783428010at_nat @ Z @ A3 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z ) @ R2 )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_925_trans__onD,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ A3 )
       => ( ( member6273874620674182256at_nat @ Y @ A3 )
         => ( ( member6273874620674182256at_nat @ Z @ A3 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z ) @ R2 )
               => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_926_trans__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( member2643936169264416010at_nat @ Y @ A3 )
         => ( ( member2643936169264416010at_nat @ Z @ A3 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_927_trans__onD,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( ( trans_on_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( member_nat @ Z @ A3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_928_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
     => ~ ! [B4: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_929_Domain_Ocases,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R2 ) )
     => ~ ! [B4: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_930_Domain_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
     => ~ ! [B4: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_931_Domain_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R2 ) )
     => ~ ! [B4: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_932_Domain_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
     => ~ ! [B4: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_933_Domain_Ocases,axiom,
    ! [A: nat,R2: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R2 ) )
     => ~ ! [B4: list_char] :
            ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_934_Domain_Ocases,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R2 ) )
     => ~ ! [B4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ R2 ) ) ).

% Domain.cases
thf(fact_935_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
      = ( ? [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
            ( ( A = A2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_936_Domain_Osimps,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R2 ) )
      = ( ? [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
            ( ( A = A2 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_937_Domain_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
      = ( ? [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat] :
            ( ( A = A2 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_938_Domain_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R2 ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
            ( ( A = A2 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_939_Domain_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
            ( ( A = A2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_940_Domain_Osimps,axiom,
    ! [A: nat,R2: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R2 ) )
      = ( ? [A2: nat,B2: list_char] :
            ( ( A = A2 )
            & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_941_Domain_Osimps,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R2 ) )
      = ( ? [A2: nat,B2: nat] :
            ( ( A = A2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Domain.simps
thf(fact_942_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_943_Domain_ODomainI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_944_Domain_ODomainI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
     => ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_945_Domain_ODomainI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
     => ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_946_Domain_ODomainI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_947_Domain_ODomainI,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
     => ( member_nat @ A @ ( domain_nat_list_char @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_948_Domain_ODomainI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member_nat @ A @ ( domain_nat_nat @ R2 ) ) ) ).

% Domain.DomainI
thf(fact_949_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
     => ~ ! [B4: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_950_DomainE,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R2 ) )
     => ~ ! [B4: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_951_DomainE,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
     => ~ ! [B4: set_Pr1261947904930325089at_nat] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_952_DomainE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R2 ) )
     => ~ ! [B4: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_953_DomainE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
     => ~ ! [B4: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_954_DomainE,axiom,
    ! [A: nat,R2: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R2 ) )
     => ~ ! [B4: list_char] :
            ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_955_DomainE,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R2 ) )
     => ~ ! [B4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ R2 ) ) ).

% DomainE
thf(fact_956_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8908270702770587084od_o_o @ R2 ) )
      = ( ? [Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_957_Domain__iff,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( domain2386162569883866188at_nat @ R2 ) )
      = ( ? [Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_958_Domain__iff,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ ( domain8494901049294654374at_nat @ R2 ) )
      = ( ? [Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_959_Domain__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( domain6426273929576919500at_nat @ R2 ) )
      = ( ? [Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_960_Domain__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( domain8802589944165416012at_nat @ R2 ) )
      = ( ? [Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_961_Domain__iff,axiom,
    ! [A: nat,R2: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ ( domain_nat_list_char @ R2 ) )
      = ( ? [Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_962_Domain__iff,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( domain_nat_nat @ R2 ) )
      = ( ? [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y5 ) @ R2 ) ) ) ).

% Domain_iff
thf(fact_963_Range_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R2 ) )
     => ~ ! [A5: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_964_Range_Ocases,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( range_4455291928337903541at_nat @ R2 ) )
     => ~ ! [A5: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_965_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R2 ) )
     => ~ ! [A5: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_966_Range_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( range_7218127889571902005at_nat @ R2 ) )
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_967_Range_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R2 ) )
     => ~ ! [A5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_968_Range_Ocases,axiom,
    ! [A: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member_list_char @ A @ ( range_nat_list_char @ R2 ) )
     => ~ ! [A5: nat] :
            ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_969_Range_Ocases,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R2 ) )
     => ~ ! [A5: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A ) @ R2 ) ) ).

% Range.cases
thf(fact_970_Range_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R2 ) )
      = ( ? [A2: set_Product_prod_o_o,B2: set_Product_prod_o_o] :
            ( ( A = B2 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_971_Range_Osimps,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( range_4455291928337903541at_nat @ R2 ) )
      = ( ? [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
            ( ( A = B2 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_972_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R2 ) )
      = ( ? [A2: set_Product_prod_o_o,B2: set_Pr1261947904930325089at_nat] :
            ( ( A = B2 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_973_Range_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( range_7218127889571902005at_nat @ R2 ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
            ( ( A = B2 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_974_Range_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R2 ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
            ( ( A = B2 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_975_Range_Osimps,axiom,
    ! [A: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member_list_char @ A @ ( range_nat_list_char @ R2 ) )
      = ( ? [A2: nat,B2: list_char] :
            ( ( A = B2 )
            & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_976_Range_Osimps,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R2 ) )
      = ( ? [A2: nat,B2: nat] :
            ( ( A = B2 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) ) ) ) ).

% Range.simps
thf(fact_977_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R2 ) ) ) ).

% Range.intros
thf(fact_978_Range_Ointros,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( member8440522571783428010at_nat @ B @ ( range_4455291928337903541at_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_979_Range_Ointros,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
     => ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_980_Range_Ointros,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
     => ( member6273874620674182256at_nat @ B @ ( range_7218127889571902005at_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_981_Range_Ointros,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_982_Range_Ointros,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
     => ( member_list_char @ B @ ( range_nat_list_char @ R2 ) ) ) ).

% Range.intros
thf(fact_983_Range_Ointros,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( member_nat @ B @ ( range_nat_nat @ R2 ) ) ) ).

% Range.intros
thf(fact_984_RangeE,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( range_1318165146934585909od_o_o @ R2 ) )
     => ~ ! [A5: set_Product_prod_o_o] :
            ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_985_RangeE,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( range_4455291928337903541at_nat @ R2 ) )
     => ~ ! [A5: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_986_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_8412904782085253007at_nat @ R2 ) )
     => ~ ! [A5: set_Product_prod_o_o] :
            ~ ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_987_RangeE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ B @ ( range_7218127889571902005at_nat @ R2 ) )
     => ~ ! [A5: set_Pr4329608150637261639at_nat] :
            ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_988_RangeE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( range_880292732526890933at_nat @ R2 ) )
     => ~ ! [A5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_989_RangeE,axiom,
    ! [B: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member_list_char @ B @ ( range_nat_list_char @ R2 ) )
     => ~ ! [A5: nat] :
            ~ ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_990_RangeE,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ B @ ( range_nat_nat @ R2 ) )
     => ~ ! [A5: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B ) @ R2 ) ) ).

% RangeE
thf(fact_991_Range__iff,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( range_1318165146934585909od_o_o @ R2 ) )
      = ( ? [Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_992_Range__iff,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( range_4455291928337903541at_nat @ R2 ) )
      = ( ? [Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_993_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_8412904782085253007at_nat @ R2 ) )
      = ( ? [Y5: set_Product_prod_o_o] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_994_Range__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( range_7218127889571902005at_nat @ R2 ) )
      = ( ? [Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_995_Range__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( range_880292732526890933at_nat @ R2 ) )
      = ( ? [Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_996_Range__iff,axiom,
    ! [A: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member_list_char @ A @ ( range_nat_list_char @ R2 ) )
      = ( ? [Y5: nat] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_997_Range__iff,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( range_nat_nat @ R2 ) )
      = ( ? [Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ A ) @ R2 ) ) ) ).

% Range_iff
thf(fact_998_fsts_Ointros,axiom,
    ! [P: produc1457211279475724562t_char] : ( member_nat @ ( produc6204653154488278070t_char @ P ) @ ( basic_4688923273072336698t_char @ P ) ) ).

% fsts.intros
thf(fact_999_fsts_Osimps,axiom,
    ! [A: nat,P: produc1457211279475724562t_char] :
      ( ( member_nat @ A @ ( basic_4688923273072336698t_char @ P ) )
      = ( A
        = ( produc6204653154488278070t_char @ P ) ) ) ).

% fsts.simps
thf(fact_1000_fsts_Ocases,axiom,
    ! [A: nat,P: produc1457211279475724562t_char] :
      ( ( member_nat @ A @ ( basic_4688923273072336698t_char @ P ) )
     => ( A
        = ( produc6204653154488278070t_char @ P ) ) ) ).

% fsts.cases
thf(fact_1001_transI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o,Z2: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z2 ) @ R2 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z2 ) @ R2 ) ) )
     => ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% transI
thf(fact_1002_transI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z2 ) @ R2 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z2 ) @ R2 ) ) )
     => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% transI
thf(fact_1003_transI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z2 ) @ R2 )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Z2 ) @ R2 ) ) )
     => ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 ) ) ).

% transI
thf(fact_1004_transI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z2 ) @ R2 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z2 ) @ R2 ) ) )
     => ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% transI
thf(fact_1005_transI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R2 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z2 ) @ R2 ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R2 ) ) ).

% transI
thf(fact_1006_transE,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_1007_transE,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_1008_transE,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_1009_transE,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_1010_transE,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transE
thf(fact_1011_transD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_1012_transD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_1013_transD,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_1014_transD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_1015_transD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z ) @ R2 ) ) ) ) ).

% transD
thf(fact_1016_Domain__fst,axiom,
    ( domain_nat_list_char
    = ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char ) ) ).

% Domain_fst
thf(fact_1017_fst__eq__Domain,axiom,
    ! [R3: set_Pr2507339680178222706t_char] :
      ( ( image_3548405822317538717ar_nat @ produc6204653154488278070t_char @ R3 )
      = ( domain_nat_list_char @ R3 ) ) ).

% fst_eq_Domain
thf(fact_1018_prod__set__defs_I1_J,axiom,
    ( basic_4688923273072336698t_char
    = ( ^ [P2: produc1457211279475724562t_char] : ( insert_nat @ ( produc6204653154488278070t_char @ P2 ) @ bot_bot_set_nat ) ) ) ).

% prod_set_defs(1)
thf(fact_1019_cofinal__def,axiom,
    ( bNF_Ca3407573850950408152od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ ( field_6101175518749917524od_o_o @ R ) )
         => ? [Y5: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y5 @ A7 )
              & ( X2 != Y5 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_1020_cofinal__def,axiom,
    ( bNF_Ca1979673041776187986at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ ( field_4706345496283612622at_nat @ R ) )
         => ? [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A7 )
              & ( X2 != Y5 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_1021_cofinal__def,axiom,
    ( bNF_Ca4363140458462388713inal_o
    = ( ^ [A7: set_o,R: set_Product_prod_o_o] :
        ! [X2: $o] :
          ( ( member_o @ X2 @ ( field_o @ R ) )
         => ? [Y5: $o] :
              ( ( member_o @ Y5 @ A7 )
              & ( X2 = (~ Y5) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_1022_cofinal__def,axiom,
    ( bNF_Ca6037544080215331608at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ ( field_1428494348145443988at_nat @ R ) )
         => ? [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A7 )
              & ( X2 != Y5 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_1023_cofinal__def,axiom,
    ( bNF_Ca8448034394003515314at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ ( field_3145993188331922222at_nat @ R ) )
         => ? [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A7 )
              & ( X2 != Y5 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_1024_cofinal__def,axiom,
    ( bNF_Ca5159621140400407487al_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ ( field_nat @ R ) )
         => ? [Y5: nat] :
              ( ( member_nat @ Y5 @ A7 )
              & ( X2 != Y5 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_1025_Rangep__Range__eq,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( rangep4714679473018996347od_o_o
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: set_Product_prod_o_o] : ( member3157015407008874800od_o_o @ X2 @ ( range_1318165146934585909od_o_o @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1026_Rangep__Range__eq,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( rangep8386759446810740219at_nat
        @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ ( range_4455291928337903541at_nat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1027_Rangep__Range__eq,axiom,
    ! [R2: set_Pr457366540195662369at_nat] :
      ( ( rangep7742664902157527253at_nat
        @ ^ [X2: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X2 @ ( range_8412904782085253007at_nat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1028_Rangep__Range__eq,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( rangep5814831589058262651at_nat
        @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ X2 @ ( range_7218127889571902005at_nat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1029_Rangep__Range__eq,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( rangep5542131597093287419at_nat
        @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X2 @ ( range_880292732526890933at_nat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1030_Rangep__Range__eq,axiom,
    ! [R2: set_Pr2507339680178222706t_char] :
      ( ( rangep_nat_list_char
        @ ^ [X2: nat,Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: list_char] : ( member_list_char @ X2 @ ( range_nat_list_char @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1031_Rangep__Range__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( rangep_nat_nat
        @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) )
      = ( ^ [X2: nat] : ( member_nat @ X2 @ ( range_nat_nat @ R2 ) ) ) ) ).

% Rangep_Range_eq
thf(fact_1032_Range__def,axiom,
    ( range_1318165146934585909od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o] :
          ( collec2441215771612176754od_o_o
          @ ( rangep4714679473018996347od_o_o
            @ ^ [X2: set_Product_prod_o_o,Y5: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1033_Range__def,axiom,
    ( range_4455291928337903541at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat] :
          ( collec3392354462482085612at_nat
          @ ( rangep8386759446810740219at_nat
            @ ^ [X2: product_prod_nat_nat,Y5: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1034_Range__def,axiom,
    ( range_8412904782085253007at_nat
    = ( ^ [R: set_Pr457366540195662369at_nat] :
          ( collec5514110066124741708at_nat
          @ ( rangep7742664902157527253at_nat
            @ ^ [X2: set_Product_prod_o_o,Y5: set_Pr1261947904930325089at_nat] : ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1035_Range__def,axiom,
    ( range_7218127889571902005at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat] :
          ( collec4573900263915890866at_nat
          @ ( rangep5814831589058262651at_nat
            @ ^ [X2: set_Pr4329608150637261639at_nat,Y5: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1036_Range__def,axiom,
    ( range_880292732526890933at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat] :
          ( collec5514110066124741708at_nat
          @ ( rangep5542131597093287419at_nat
            @ ^ [X2: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1037_Range__def,axiom,
    ( range_nat_list_char
    = ( ^ [R: set_Pr2507339680178222706t_char] :
          ( collect_list_char
          @ ( rangep_nat_list_char
            @ ^ [X2: nat,Y5: list_char] : ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1038_Range__def,axiom,
    ( range_nat_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat] :
          ( collect_nat
          @ ( rangep_nat_nat
            @ ^ [X2: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% Range_def
thf(fact_1039_under__incr,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ord_le2654036014095304327od_o_o @ ( order_8825441150112336748od_o_o @ R2 @ A ) @ ( order_8825441150112336748od_o_o @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_1040_under__incr,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( ord_le3146513528884898305at_nat @ ( order_6448319830299018214at_nat @ R2 @ A ) @ ( order_6448319830299018214at_nat @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_1041_under__incr,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
       => ( ord_le7966958417815640519at_nat @ ( order_5971310966627442348at_nat @ R2 @ A ) @ ( order_5971310966627442348at_nat @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_1042_under__incr,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ord_le2077887516847798113at_nat @ ( order_8557040927991152966at_nat @ R2 @ A ) @ ( order_8557040927991152966at_nat @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_1043_under__incr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( ord_less_eq_set_nat @ ( order_under_nat @ R2 @ A ) @ ( order_under_nat @ R2 @ B ) ) ) ) ).

% under_incr
thf(fact_1044_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( order_3767514495197212837od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
        = ( ! [A7: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A7 @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( ( A7 != bot_bo8006819982598807891od_o_o )
               => ? [X2: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X2 @ A7 )
                    & ! [Y5: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y5 @ A7 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1045_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( order_2615775362358406454r_on_o @ ( field_o @ R2 ) @ R2 )
        = ( ! [A7: set_o] :
              ( ( ord_less_eq_set_o @ A7 @ ( field_o @ R2 ) )
             => ( ( A7 != bot_bot_set_o )
               => ? [X2: $o] :
                    ( ( member_o @ X2 @ A7 )
                    & ! [Y5: $o] :
                        ( ( member_o @ Y5 @ A7 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1046_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R2 ) @ R2 )
     => ( ( order_423636163458052581at_nat @ ( field_1428494348145443988at_nat @ R2 ) @ R2 )
        = ( ! [A7: set_se5479201105039950375at_nat] :
              ( ( ord_le7966958417815640519at_nat @ A7 @ ( field_1428494348145443988at_nat @ R2 ) )
             => ( ( A7 != bot_bo1534825418459847827at_nat )
               => ? [X2: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X2 @ A7 )
                    & ! [Y5: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Y5 @ A7 )
                       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1047_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( order_1968183844437341695at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
        = ( ! [A7: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A7 @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( ( A7 != bot_bo3083307316010499117at_nat )
               => ? [X2: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X2 @ A7 )
                    & ! [Y5: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y5 @ A7 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1048_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        = ( ! [A7: set_nat] :
              ( ( ord_less_eq_set_nat @ A7 @ ( field_nat @ R2 ) )
             => ( ( A7 != bot_bot_set_nat )
               => ? [X2: nat] :
                    ( ( member_nat @ X2 @ A7 )
                    & ! [Y5: nat] :
                        ( ( member_nat @ Y5 @ A7 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1049_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R2 ) @ R2 )
     => ( ( order_8331247224601271455at_nat @ ( field_4706345496283612622at_nat @ R2 ) @ R2 )
        = ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ A7 @ ( field_4706345496283612622at_nat @ R2 ) )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ? [X2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X2 @ A7 )
                    & ! [Y5: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Y5 @ A7 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1050_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr5564308138774400199at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( order_2339240034117818117at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
        = ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ A7 @ ( field_2862844103579723188at_nat @ R2 ) )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ? [X2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X2 @ A7 )
                    & ! [Y5: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Y5 @ A7 )
                       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1051_well__order__on__domain,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_2783320356396525957od_o_o @ A3 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
       => ( ( member444158400953824016od_o_o @ A @ A3 )
          & ( member444158400953824016od_o_o @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1052_well__order__on__domain,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_1951249980326167391at_nat @ A3 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
       => ( ( member4095101504841534314at_nat @ A @ A3 )
          & ( member4095101504841534314at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1053_well__order__on__domain,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2339240034117818117at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( ( member8757157785044589968at_nat @ A @ A3 )
          & ( member8757157785044589968at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1054_well__order__on__domain,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( order_8549180852824817413at_nat @ A3 @ R2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 )
       => ( ( member8206827879206165904at_nat @ A @ A3 )
          & ( member8206827879206165904at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1055_well__order__on__domain,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_3767514495197212837od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ( member3157015407008874800od_o_o @ A @ A3 )
          & ( member3157015407008874800od_o_o @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1056_well__order__on__domain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( order_8331247224601271455at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( ( member8440522571783428010at_nat @ A @ A3 )
          & ( member8440522571783428010at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1057_well__order__on__domain,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( order_423636163458052581at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
       => ( ( member6273874620674182256at_nat @ A @ A3 )
          & ( member6273874620674182256at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1058_well__order__on__domain,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_1968183844437341695at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ( member2643936169264416010at_nat @ A @ A3 )
          & ( member2643936169264416010at_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1059_well__order__on__domain,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_2888998067076097458on_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( ( member_nat @ A @ A3 )
          & ( member_nat @ B @ A3 ) ) ) ) ).

% well_order_on_domain
thf(fact_1060_under__def,axiom,
    ( order_8825441150112336748od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A2: set_Product_prod_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] : ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ A2 ) @ R ) ) ) ) ).

% under_def
thf(fact_1061_under__def,axiom,
    ( order_6448319830299018214at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ A2 ) @ R ) ) ) ) ).

% under_def
thf(fact_1062_under__def,axiom,
    ( order_5971310966627442348at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A2: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ A2 ) @ R ) ) ) ) ).

% under_def
thf(fact_1063_under__def,axiom,
    ( order_8557040927991152966at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A2 ) @ R ) ) ) ) ).

% under_def
thf(fact_1064_under__def,axiom,
    ( order_under_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A2 ) @ R ) ) ) ) ).

% under_def
thf(fact_1065_underS__incl__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_2625247082195685180od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ B @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R2 @ A ) @ ( order_3915675016455977933od_o_o @ R2 @ B ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1066_underS__incl__iff,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_91209724698019100od_o_o @ ( field_5121862139100741428od_o_o @ R2 ) @ R2 )
     => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
       => ( ( member444158400953824016od_o_o @ B @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( ord_le7586318239878589543od_o_o @ ( order_1929855754104155309od_o_o @ R2 @ A ) @ ( order_1929855754104155309od_o_o @ R2 @ B ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1067_underS__incl__iff,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_9086815480479278966at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( ord_le4841768313324958145at_nat @ ( order_6745690092771398279at_nat @ R2 @ A ) @ ( order_6745690092771398279at_nat @ R2 @ B ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1068_underS__incl__iff,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_2711500226769404828at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( ord_le1268244103169919719at_nat @ ( order_3736018788886245933at_nat @ R2 @ A ) @ ( order_3736018788886245933at_nat @ R2 @ B ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1069_underS__incl__iff,axiom,
    ! [R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( order_7200634732473852828at_nat @ ( field_3044959742847878580at_nat @ R2 ) @ R2 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R2 ) )
       => ( ( member8206827879206165904at_nat @ B @ ( field_3044959742847878580at_nat @ R2 ) )
         => ( ( ord_le3000389064537975527at_nat @ ( order_4712251856133020205at_nat @ R2 @ A ) @ ( order_4712251856133020205at_nat @ R2 @ B ) )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1070_underS__incl__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( order_7480980593162297270at_nat @ ( field_4706345496283612622at_nat @ R2 ) @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R2 ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R2 ) )
         => ( ( ord_le3146513528884898305at_nat @ ( order_4046151456452760519at_nat @ R2 @ A ) @ ( order_4046151456452760519at_nat @ R2 @ B ) )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1071_underS__incl__iff,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_6238756238976269133r_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( ord_less_eq_set_o @ ( order_underS_o @ R2 @ A ) @ ( order_underS_o @ R2 @ B ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1072_underS__incl__iff,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( order_1878278401035043452at_nat @ ( field_1428494348145443988at_nat @ R2 ) @ R2 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R2 ) )
       => ( ( member6273874620674182256at_nat @ B @ ( field_1428494348145443988at_nat @ R2 ) )
         => ( ( ord_le7966958417815640519at_nat @ ( order_4357248199484075277at_nat @ R2 @ A ) @ ( order_4357248199484075277at_nat @ R2 @ B ) )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1073_underS__incl__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_5942792088423767318at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R2 @ A ) @ ( order_2644262826035875111at_nat @ R2 @ B ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1074_underS__incl__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A ) @ ( order_underS_nat @ R2 @ B ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1075_prod_Orel__compp__Grp,axiom,
    ( basic_6618480896482986139t_char
    = ( ^ [R12: nat > list_char > $o,R23: nat > list_char > $o] :
          ( relcom7513465730990557260t_char
          @ ( conver2024995914067322255at_nat
            @ ( bNF_Gr7887112059459224296at_nat
              @ ( collec1162968953386343058t_char
                @ ^ [X2: produc6071853487833700903t_char] :
                    ( ( ord_le6315511645215477266t_char @ ( basic_2214135432706510599t_char @ X2 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R12 ) ) )
                    & ( ord_le6315511645215477266t_char @ ( basic_5792570394382837829t_char @ X2 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R23 ) ) ) ) )
              @ ( produc881455346247319798ar_nat @ produc6204653154488278070t_char @ produc6204653154488278070t_char ) ) )
          @ ( bNF_Gr2539438056706512010t_char
            @ ( collec1162968953386343058t_char
              @ ^ [X2: produc6071853487833700903t_char] :
                  ( ( ord_le6315511645215477266t_char @ ( basic_2214135432706510599t_char @ X2 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R12 ) ) )
                  & ( ord_le6315511645215477266t_char @ ( basic_5792570394382837829t_char @ X2 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R23 ) ) ) ) )
            @ ( produc4656792822516519862t_char @ produc1900778367047502200t_char @ produc1900778367047502200t_char ) ) ) ) ) ).

% prod.rel_compp_Grp
thf(fact_1076_prod_Oin__rel,axiom,
    ( basic_5328504652464829177at_nat
    = ( ^ [R12: nat > nat > $o,R23: nat > nat > $o,A2: product_prod_nat_nat,B2: 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 )
            = B2 ) ) ) ) ).

% prod.in_rel
thf(fact_1077_prod_Oin__rel,axiom,
    ( basic_6618480896482986139t_char
    = ( ^ [R12: nat > list_char > $o,R23: nat > list_char > $o,A2: product_prod_nat_nat,B2: produc6859378861088059299t_char] :
        ? [Z5: produc6071853487833700903t_char] :
          ( ( member151853725258021968t_char @ Z5
            @ ( collec1162968953386343058t_char
              @ ^ [X2: produc6071853487833700903t_char] :
                  ( ( ord_le6315511645215477266t_char @ ( basic_2214135432706510599t_char @ X2 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R12 ) ) )
                  & ( ord_le6315511645215477266t_char @ ( basic_5792570394382837829t_char @ X2 ) @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ R23 ) ) ) ) ) )
          & ( ( produc881455346247319798ar_nat @ produc6204653154488278070t_char @ produc6204653154488278070t_char @ Z5 )
            = A2 )
          & ( ( produc4656792822516519862t_char @ produc1900778367047502200t_char @ produc1900778367047502200t_char @ Z5 )
            = B2 ) ) ) ) ).

% prod.in_rel
thf(fact_1078_trans__join,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
      = ( ! [X2: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X2 @ R2 )
           => ( produc9128050996712119596_o_o_o
              @ ^ [Y5: set_Product_prod_o_o,Y12: set_Product_prod_o_o] :
                ! [Z5: produc2934264451710624999od_o_o] :
                  ( ( member444158400953824016od_o_o @ Z5 @ R2 )
                 => ( produc9128050996712119596_o_o_o
                    @ ^ [Y22: set_Product_prod_o_o,Aa2: set_Product_prod_o_o] :
                        ( ( Y12 = Y22 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Aa2 ) @ R2 ) )
                    @ Z5 ) )
              @ X2 ) ) ) ) ).

% trans_join
thf(fact_1079_trans__join,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
      = ( ! [X2: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X2 @ R2 )
           => ( produc6590410687421337004_nat_o
              @ ^ [Y5: product_prod_nat_nat,Y12: product_prod_nat_nat] :
                ! [Z5: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ Z5 @ R2 )
                 => ( produc6590410687421337004_nat_o
                    @ ^ [Y22: product_prod_nat_nat,Aa2: product_prod_nat_nat] :
                        ( ( Y12 = Y22 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ Aa2 ) @ R2 ) )
                    @ Z5 ) )
              @ X2 ) ) ) ) ).

% trans_join
thf(fact_1080_trans__join,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
      = ( ! [X2: produc1319942482725812455at_nat] :
            ( ( member1466754251312161552at_nat @ X2 @ R2 )
           => ( produc9020218426428501292_nat_o
              @ ^ [Y5: set_Pr4329608150637261639at_nat,Y12: set_Pr4329608150637261639at_nat] :
                ! [Z5: produc1319942482725812455at_nat] :
                  ( ( member1466754251312161552at_nat @ Z5 @ R2 )
                 => ( produc9020218426428501292_nat_o
                    @ ^ [Y22: set_Pr4329608150637261639at_nat,Aa2: set_Pr4329608150637261639at_nat] :
                        ( ( Y12 = Y22 )
                       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ Aa2 ) @ R2 ) )
                    @ Z5 ) )
              @ X2 ) ) ) ) ).

% trans_join
thf(fact_1081_trans__join,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
      = ( ! [X2: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X2 @ R2 )
           => ( produc410239310623530412_nat_o
              @ ^ [Y5: set_Pr1261947904930325089at_nat,Y12: set_Pr1261947904930325089at_nat] :
                ! [Z5: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ Z5 @ R2 )
                 => ( produc410239310623530412_nat_o
                    @ ^ [Y22: set_Pr1261947904930325089at_nat,Aa2: set_Pr1261947904930325089at_nat] :
                        ( ( Y12 = Y22 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Aa2 ) @ R2 ) )
                    @ Z5 ) )
              @ X2 ) ) ) ) ).

% trans_join
thf(fact_1082_trans__join,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ R2 )
           => ( produc6081775807080527818_nat_o
              @ ^ [Y5: nat,Y12: nat] :
                ! [Z5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z5 @ R2 )
                 => ( produc6081775807080527818_nat_o
                    @ ^ [Y22: nat,Aa2: nat] :
                        ( ( Y12 = Y22 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Aa2 ) @ R2 ) )
                    @ Z5 ) )
              @ X2 ) ) ) ) ).

% trans_join
thf(fact_1083_rel__prod__inject,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o,A: nat,B: nat,C: nat,D: nat] :
      ( ( basic_5328504652464829177at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1084_rel__prod__inject,axiom,
    ! [R1: nat > nat > $o,R22: list_char > nat > $o,A: nat,B: list_char,C: nat,D: nat] :
      ( ( basic_5421180659935147146ar_nat @ R1 @ R22 @ ( produc120197956887798346t_char @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1085_rel__prod__inject,axiom,
    ! [R1: nat > nat > $o,R22: nat > list_char > $o,A: nat,B: nat,C: nat,D: list_char] :
      ( ( basic_6062404793260096266t_char @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( produc120197956887798346t_char @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1086_rel__prod__inject,axiom,
    ! [R1: nat > nat > $o,R22: list_char > list_char > $o,A: nat,B: list_char,C: nat,D: list_char] :
      ( ( basic_2137218353159982875t_char @ R1 @ R22 @ ( produc120197956887798346t_char @ A @ B ) @ ( produc120197956887798346t_char @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1087_rel__prod__inject,axiom,
    ! [R1: set_Product_prod_o_o > nat > $o,R22: set_Pr1261947904930325089at_nat > nat > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,C: nat,D: nat] :
      ( ( basic_8774556366321213519at_nat @ R1 @ R22 @ ( produc8517790099723286449at_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1088_rel__prod__inject,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > nat > $o,R22: set_Pr1261947904930325089at_nat > nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: nat,D: nat] :
      ( ( basic_332869774602983541at_nat @ R1 @ R22 @ ( produc2922128104949294807at_nat @ A @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1089_rel__prod__inject,axiom,
    ! [R1: nat > set_Product_prod_o_o > $o,R22: nat > set_Pr1261947904930325089at_nat > $o,A: nat,B: nat,C: set_Product_prod_o_o,D: set_Pr1261947904930325089at_nat] :
      ( ( basic_1553350464009441103at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( produc8517790099723286449at_nat @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1090_rel__prod__inject,axiom,
    ! [R1: nat > set_Pr1261947904930325089at_nat > $o,R22: nat > set_Pr1261947904930325089at_nat > $o,A: nat,B: nat,C: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( basic_466507960774701813at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1091_rel__prod__inject,axiom,
    ! [R1: set_Product_prod_o_o > nat > $o,R22: set_Pr1261947904930325089at_nat > list_char > $o,A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,C: nat,D: list_char] :
      ( ( basic_2953984095916011360t_char @ R1 @ R22 @ ( produc8517790099723286449at_nat @ A @ B ) @ ( produc120197956887798346t_char @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1092_rel__prod__inject,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > nat > $o,R22: set_Pr1261947904930325089at_nat > list_char > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: nat,D: list_char] :
      ( ( basic_3952808556802799238t_char @ R1 @ R22 @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc120197956887798346t_char @ C @ D ) )
      = ( ( R1 @ A @ C )
        & ( R22 @ B @ D ) ) ) ).

% rel_prod_inject
thf(fact_1093_underS__I,axiom,
    ! [I3: produc2934264451710624999od_o_o,J3: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( I3 != J3 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I3 @ J3 ) @ R3 )
       => ( member444158400953824016od_o_o @ I3 @ ( order_1929855754104155309od_o_o @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1094_underS__I,axiom,
    ! [I3: produc732395585841259969at_nat,J3: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( I3 != J3 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I3 @ J3 ) @ R3 )
       => ( member4095101504841534314at_nat @ I3 @ ( order_6745690092771398279at_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1095_underS__I,axiom,
    ! [I3: produc3843707927480180839at_nat,J3: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( I3 != J3 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J3 ) @ R3 )
       => ( member8757157785044589968at_nat @ I3 @ ( order_3736018788886245933at_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1096_underS__I,axiom,
    ! [I3: produc859450856879609959at_nat,J3: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( I3 != J3 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J3 ) @ R3 )
       => ( member8206827879206165904at_nat @ I3 @ ( order_4712251856133020205at_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1097_underS__I,axiom,
    ! [I3: set_Product_prod_o_o,J3: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( I3 != J3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I3 @ J3 ) @ R3 )
       => ( member3157015407008874800od_o_o @ I3 @ ( order_3915675016455977933od_o_o @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1098_underS__I,axiom,
    ! [I3: product_prod_nat_nat,J3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( I3 != J3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J3 ) @ R3 )
       => ( member8440522571783428010at_nat @ I3 @ ( order_4046151456452760519at_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1099_underS__I,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,J3: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( I3 != J3 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J3 ) @ R3 )
       => ( member6273874620674182256at_nat @ I3 @ ( order_4357248199484075277at_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1100_underS__I,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,J3: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( I3 != J3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J3 ) @ R3 )
       => ( member2643936169264416010at_nat @ I3 @ ( order_2644262826035875111at_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1101_underS__I,axiom,
    ! [I3: nat,J3: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( I3 != J3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J3 ) @ R3 )
       => ( member_nat @ I3 @ ( order_underS_nat @ R3 @ J3 ) ) ) ) ).

% underS_I
thf(fact_1102_underS__E,axiom,
    ! [I3: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o,J3: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ I3 @ ( order_3915675016455977933od_o_o @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1103_underS__E,axiom,
    ! [I3: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o,J3: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I3 @ ( order_1929855754104155309od_o_o @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1104_underS__E,axiom,
    ! [I3: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat,J3: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I3 @ ( order_6745690092771398279at_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1105_underS__E,axiom,
    ! [I3: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat,J3: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I3 @ ( order_3736018788886245933at_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1106_underS__E,axiom,
    ! [I3: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat,J3: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ I3 @ ( order_4712251856133020205at_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1107_underS__E,axiom,
    ! [I3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,J3: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ I3 @ ( order_4046151456452760519at_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1108_underS__E,axiom,
    ! [I3: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,J3: set_Pr4329608150637261639at_nat] :
      ( ( member6273874620674182256at_nat @ I3 @ ( order_4357248199484075277at_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1109_underS__E,axiom,
    ! [I3: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,J3: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I3 @ ( order_2644262826035875111at_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1110_underS__E,axiom,
    ! [I3: nat,R3: set_Pr1261947904930325089at_nat,J3: nat] :
      ( ( member_nat @ I3 @ ( order_underS_nat @ R3 @ J3 ) )
     => ( ( I3 != J3 )
        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I3 @ J3 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1111_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 )
     => ~ ! [A5: nat,B4: nat,C3: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A5 @ C3 ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1112_rel__prod_Ocases,axiom,
    ! [R1: nat > nat > $o,R22: list_char > nat > $o,A12: produc1457211279475724562t_char,A23: product_prod_nat_nat] :
      ( ( basic_5421180659935147146ar_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: nat,B4: nat,C3: list_char] :
            ( ( A12
              = ( produc120197956887798346t_char @ A5 @ C3 ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1113_rel__prod_Ocases,axiom,
    ! [R1: nat > nat > $o,R22: nat > list_char > $o,A12: product_prod_nat_nat,A23: produc1457211279475724562t_char] :
      ( ( basic_6062404793260096266t_char @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: nat,B4: nat,C3: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A5 @ C3 ) )
           => ! [D3: list_char] :
                ( ( A23
                  = ( produc120197956887798346t_char @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1114_rel__prod_Ocases,axiom,
    ! [R1: nat > nat > $o,R22: list_char > list_char > $o,A12: produc1457211279475724562t_char,A23: produc1457211279475724562t_char] :
      ( ( basic_2137218353159982875t_char @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: nat,B4: nat,C3: list_char] :
            ( ( A12
              = ( produc120197956887798346t_char @ A5 @ C3 ) )
           => ! [D3: list_char] :
                ( ( A23
                  = ( produc120197956887798346t_char @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1115_rel__prod_Ocases,axiom,
    ! [R1: set_Product_prod_o_o > nat > $o,R22: set_Pr1261947904930325089at_nat > nat > $o,A12: produc732395585841259969at_nat,A23: product_prod_nat_nat] :
      ( ( basic_8774556366321213519at_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: set_Product_prod_o_o,B4: nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12
              = ( produc8517790099723286449at_nat @ A5 @ C3 ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1116_rel__prod_Ocases,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > nat > $o,R22: set_Pr1261947904930325089at_nat > nat > $o,A12: produc3843707927480180839at_nat,A23: product_prod_nat_nat] :
      ( ( basic_332869774602983541at_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: set_Pr1261947904930325089at_nat,B4: nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12
              = ( produc2922128104949294807at_nat @ A5 @ C3 ) )
           => ! [D3: nat] :
                ( ( A23
                  = ( product_Pair_nat_nat @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1117_rel__prod_Ocases,axiom,
    ! [R1: nat > set_Product_prod_o_o > $o,R22: nat > set_Pr1261947904930325089at_nat > $o,A12: product_prod_nat_nat,A23: produc732395585841259969at_nat] :
      ( ( basic_1553350464009441103at_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: nat,B4: set_Product_prod_o_o,C3: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A5 @ C3 ) )
           => ! [D3: set_Pr1261947904930325089at_nat] :
                ( ( A23
                  = ( produc8517790099723286449at_nat @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1118_rel__prod_Ocases,axiom,
    ! [R1: nat > set_Pr1261947904930325089at_nat > $o,R22: nat > set_Pr1261947904930325089at_nat > $o,A12: product_prod_nat_nat,A23: produc3843707927480180839at_nat] :
      ( ( basic_466507960774701813at_nat @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: nat,B4: set_Pr1261947904930325089at_nat,C3: nat] :
            ( ( A12
              = ( product_Pair_nat_nat @ A5 @ C3 ) )
           => ! [D3: set_Pr1261947904930325089at_nat] :
                ( ( A23
                  = ( produc2922128104949294807at_nat @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1119_rel__prod_Ocases,axiom,
    ! [R1: set_Product_prod_o_o > nat > $o,R22: set_Pr1261947904930325089at_nat > list_char > $o,A12: produc732395585841259969at_nat,A23: produc1457211279475724562t_char] :
      ( ( basic_2953984095916011360t_char @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: set_Product_prod_o_o,B4: nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12
              = ( produc8517790099723286449at_nat @ A5 @ C3 ) )
           => ! [D3: list_char] :
                ( ( A23
                  = ( produc120197956887798346t_char @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1120_rel__prod_Ocases,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > nat > $o,R22: set_Pr1261947904930325089at_nat > list_char > $o,A12: produc3843707927480180839at_nat,A23: produc1457211279475724562t_char] :
      ( ( basic_3952808556802799238t_char @ R1 @ R22 @ A12 @ A23 )
     => ~ ! [A5: set_Pr1261947904930325089at_nat,B4: nat,C3: set_Pr1261947904930325089at_nat] :
            ( ( A12
              = ( produc2922128104949294807at_nat @ A5 @ C3 ) )
           => ! [D3: list_char] :
                ( ( A23
                  = ( produc120197956887798346t_char @ B4 @ D3 ) )
               => ( ( R1 @ A5 @ B4 )
                 => ~ ( R22 @ C3 @ D3 ) ) ) ) ) ).

% rel_prod.cases
thf(fact_1121_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,B2: nat,C4: nat,D4: nat] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C4 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1122_rel__prod_Osimps,axiom,
    ( basic_5421180659935147146ar_nat
    = ( ^ [R12: nat > nat > $o,R23: list_char > nat > $o,A13: produc1457211279475724562t_char,A24: product_prod_nat_nat] :
        ? [A2: nat,B2: nat,C4: list_char,D4: nat] :
          ( ( A13
            = ( produc120197956887798346t_char @ A2 @ C4 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1123_rel__prod_Osimps,axiom,
    ( basic_6062404793260096266t_char
    = ( ^ [R12: nat > nat > $o,R23: nat > list_char > $o,A13: product_prod_nat_nat,A24: produc1457211279475724562t_char] :
        ? [A2: nat,B2: nat,C4: nat,D4: list_char] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C4 ) )
          & ( A24
            = ( produc120197956887798346t_char @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1124_rel__prod_Osimps,axiom,
    ( basic_2137218353159982875t_char
    = ( ^ [R12: nat > nat > $o,R23: list_char > list_char > $o,A13: produc1457211279475724562t_char,A24: produc1457211279475724562t_char] :
        ? [A2: nat,B2: nat,C4: list_char,D4: list_char] :
          ( ( A13
            = ( produc120197956887798346t_char @ A2 @ C4 ) )
          & ( A24
            = ( produc120197956887798346t_char @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1125_rel__prod_Osimps,axiom,
    ( basic_8774556366321213519at_nat
    = ( ^ [R12: set_Product_prod_o_o > nat > $o,R23: set_Pr1261947904930325089at_nat > nat > $o,A13: produc732395585841259969at_nat,A24: product_prod_nat_nat] :
        ? [A2: set_Product_prod_o_o,B2: nat,C4: set_Pr1261947904930325089at_nat,D4: nat] :
          ( ( A13
            = ( produc8517790099723286449at_nat @ A2 @ C4 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1126_rel__prod_Osimps,axiom,
    ( basic_332869774602983541at_nat
    = ( ^ [R12: set_Pr1261947904930325089at_nat > nat > $o,R23: set_Pr1261947904930325089at_nat > nat > $o,A13: produc3843707927480180839at_nat,A24: product_prod_nat_nat] :
        ? [A2: set_Pr1261947904930325089at_nat,B2: nat,C4: set_Pr1261947904930325089at_nat,D4: nat] :
          ( ( A13
            = ( produc2922128104949294807at_nat @ A2 @ C4 ) )
          & ( A24
            = ( product_Pair_nat_nat @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1127_rel__prod_Osimps,axiom,
    ( basic_1553350464009441103at_nat
    = ( ^ [R12: nat > set_Product_prod_o_o > $o,R23: nat > set_Pr1261947904930325089at_nat > $o,A13: product_prod_nat_nat,A24: produc732395585841259969at_nat] :
        ? [A2: nat,B2: set_Product_prod_o_o,C4: nat,D4: set_Pr1261947904930325089at_nat] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C4 ) )
          & ( A24
            = ( produc8517790099723286449at_nat @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1128_rel__prod_Osimps,axiom,
    ( basic_466507960774701813at_nat
    = ( ^ [R12: nat > set_Pr1261947904930325089at_nat > $o,R23: nat > set_Pr1261947904930325089at_nat > $o,A13: product_prod_nat_nat,A24: produc3843707927480180839at_nat] :
        ? [A2: nat,B2: set_Pr1261947904930325089at_nat,C4: nat,D4: set_Pr1261947904930325089at_nat] :
          ( ( A13
            = ( product_Pair_nat_nat @ A2 @ C4 ) )
          & ( A24
            = ( produc2922128104949294807at_nat @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1129_rel__prod_Osimps,axiom,
    ( basic_2953984095916011360t_char
    = ( ^ [R12: set_Product_prod_o_o > nat > $o,R23: set_Pr1261947904930325089at_nat > list_char > $o,A13: produc732395585841259969at_nat,A24: produc1457211279475724562t_char] :
        ? [A2: set_Product_prod_o_o,B2: nat,C4: set_Pr1261947904930325089at_nat,D4: list_char] :
          ( ( A13
            = ( produc8517790099723286449at_nat @ A2 @ C4 ) )
          & ( A24
            = ( produc120197956887798346t_char @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1130_rel__prod_Osimps,axiom,
    ( basic_3952808556802799238t_char
    = ( ^ [R12: set_Pr1261947904930325089at_nat > nat > $o,R23: set_Pr1261947904930325089at_nat > list_char > $o,A13: produc3843707927480180839at_nat,A24: produc1457211279475724562t_char] :
        ? [A2: set_Pr1261947904930325089at_nat,B2: nat,C4: set_Pr1261947904930325089at_nat,D4: list_char] :
          ( ( A13
            = ( produc2922128104949294807at_nat @ A2 @ C4 ) )
          & ( A24
            = ( produc120197956887798346t_char @ B2 @ D4 ) )
          & ( R12 @ A2 @ B2 )
          & ( R23 @ C4 @ D4 ) ) ) ) ).

% rel_prod.simps
thf(fact_1131_rel__prod_Ointros,axiom,
    ! [R1: nat > nat > $o,A: nat,B: nat,R22: nat > nat > $o,C: nat,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_5328504652464829177at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1132_rel__prod_Ointros,axiom,
    ! [R1: nat > nat > $o,A: nat,B: nat,R22: list_char > nat > $o,C: list_char,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_5421180659935147146ar_nat @ R1 @ R22 @ ( produc120197956887798346t_char @ A @ C ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1133_rel__prod_Ointros,axiom,
    ! [R1: nat > nat > $o,A: nat,B: nat,R22: nat > list_char > $o,C: nat,D: list_char] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_6062404793260096266t_char @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C ) @ ( produc120197956887798346t_char @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1134_rel__prod_Ointros,axiom,
    ! [R1: nat > nat > $o,A: nat,B: nat,R22: list_char > list_char > $o,C: list_char,D: list_char] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_2137218353159982875t_char @ R1 @ R22 @ ( produc120197956887798346t_char @ A @ C ) @ ( produc120197956887798346t_char @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1135_rel__prod_Ointros,axiom,
    ! [R1: set_Product_prod_o_o > nat > $o,A: set_Product_prod_o_o,B: nat,R22: set_Pr1261947904930325089at_nat > nat > $o,C: set_Pr1261947904930325089at_nat,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_8774556366321213519at_nat @ R1 @ R22 @ ( produc8517790099723286449at_nat @ A @ C ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1136_rel__prod_Ointros,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > nat > $o,A: set_Pr1261947904930325089at_nat,B: nat,R22: set_Pr1261947904930325089at_nat > nat > $o,C: set_Pr1261947904930325089at_nat,D: nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_332869774602983541at_nat @ R1 @ R22 @ ( produc2922128104949294807at_nat @ A @ C ) @ ( product_Pair_nat_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1137_rel__prod_Ointros,axiom,
    ! [R1: nat > set_Product_prod_o_o > $o,A: nat,B: set_Product_prod_o_o,R22: nat > set_Pr1261947904930325089at_nat > $o,C: nat,D: set_Pr1261947904930325089at_nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_1553350464009441103at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C ) @ ( produc8517790099723286449at_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1138_rel__prod_Ointros,axiom,
    ! [R1: nat > set_Pr1261947904930325089at_nat > $o,A: nat,B: set_Pr1261947904930325089at_nat,R22: nat > set_Pr1261947904930325089at_nat > $o,C: nat,D: set_Pr1261947904930325089at_nat] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_466507960774701813at_nat @ R1 @ R22 @ ( product_Pair_nat_nat @ A @ C ) @ ( produc2922128104949294807at_nat @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1139_rel__prod_Ointros,axiom,
    ! [R1: set_Product_prod_o_o > nat > $o,A: set_Product_prod_o_o,B: nat,R22: set_Pr1261947904930325089at_nat > list_char > $o,C: set_Pr1261947904930325089at_nat,D: list_char] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_2953984095916011360t_char @ R1 @ R22 @ ( produc8517790099723286449at_nat @ A @ C ) @ ( produc120197956887798346t_char @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1140_rel__prod_Ointros,axiom,
    ! [R1: set_Pr1261947904930325089at_nat > nat > $o,A: set_Pr1261947904930325089at_nat,B: nat,R22: set_Pr1261947904930325089at_nat > list_char > $o,C: set_Pr1261947904930325089at_nat,D: list_char] :
      ( ( R1 @ A @ B )
     => ( ( R22 @ C @ D )
       => ( basic_3952808556802799238t_char @ R1 @ R22 @ ( produc2922128104949294807at_nat @ A @ C ) @ ( produc120197956887798346t_char @ B @ D ) ) ) ) ).

% rel_prod.intros
thf(fact_1141_Pair__transfer,axiom,
    ! [A3: nat > nat > $o,B5: nat > nat > $o] : ( bNF_re3696488243098009855at_nat @ A3 @ ( bNF_re901410630616637503at_nat @ B5 @ ( basic_5328504652464829177at_nat @ A3 @ B5 ) ) @ product_Pair_nat_nat @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_1142_Pair__transfer,axiom,
    ! [A3: nat > nat > $o,B5: list_char > nat > $o] : ( bNF_re453200506278319679at_nat @ A3 @ ( bNF_re4857190184819116159at_nat @ B5 @ ( basic_5421180659935147146ar_nat @ A3 @ B5 ) ) @ produc120197956887798346t_char @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_1143_Pair__transfer,axiom,
    ! [A3: nat > nat > $o,B5: nat > list_char > $o] : ( bNF_re596687900092489407t_char @ A3 @ ( bNF_re7671636171547762529t_char @ B5 @ ( basic_6062404793260096266t_char @ A3 @ B5 ) ) @ product_Pair_nat_nat @ produc120197956887798346t_char ) ).

% Pair_transfer
thf(fact_1144_Pair__transfer,axiom,
    ! [A3: nat > nat > $o,B5: list_char > list_char > $o] : ( bNF_re1247391961183351807t_char @ A3 @ ( bNF_re7444546633290812577t_char @ B5 @ ( basic_2137218353159982875t_char @ A3 @ B5 ) ) @ produc120197956887798346t_char @ produc120197956887798346t_char ) ).

% Pair_transfer
thf(fact_1145_Pair__transfer,axiom,
    ! [A3: set_Product_prod_o_o > nat > $o,B5: set_Pr1261947904930325089at_nat > nat > $o] : ( bNF_re5841918410865130215at_nat @ A3 @ ( bNF_re1234328061921555220at_nat @ B5 @ ( basic_8774556366321213519at_nat @ A3 @ B5 ) ) @ produc8517790099723286449at_nat @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_1146_Pair__transfer,axiom,
    ! [A3: set_Pr1261947904930325089at_nat > nat > $o,B5: set_Pr1261947904930325089at_nat > nat > $o] : ( bNF_re547192559215464167at_nat @ A3 @ ( bNF_re3575669401655913070at_nat @ B5 @ ( basic_332869774602983541at_nat @ A3 @ B5 ) ) @ produc2922128104949294807at_nat @ product_Pair_nat_nat ) ).

% Pair_transfer
thf(fact_1147_Pair__transfer,axiom,
    ! [A3: nat > set_Product_prod_o_o > $o,B5: nat > set_Pr1261947904930325089at_nat > $o] : ( bNF_re4534060058427857109at_nat @ A3 @ ( bNF_re1126598138771558770at_nat @ B5 @ ( basic_1553350464009441103at_nat @ A3 @ B5 ) ) @ product_Pair_nat_nat @ produc8517790099723286449at_nat ) ).

% Pair_transfer
thf(fact_1148_Pair__transfer,axiom,
    ! [A3: nat > set_Pr1261947904930325089at_nat > $o,B5: nat > set_Pr1261947904930325089at_nat > $o] : ( bNF_re4920675765442204577at_nat @ A3 @ ( bNF_re1738110731795791768at_nat @ B5 @ ( basic_466507960774701813at_nat @ A3 @ B5 ) ) @ product_Pair_nat_nat @ produc2922128104949294807at_nat ) ).

% Pair_transfer
thf(fact_1149_Pair__transfer,axiom,
    ! [A3: set_Product_prod_o_o > nat > $o,B5: set_Pr1261947904930325089at_nat > list_char > $o] : ( bNF_re4549960102941810855t_char @ A3 @ ( bNF_re6785196917787558454t_char @ B5 @ ( basic_2953984095916011360t_char @ A3 @ B5 ) ) @ produc8517790099723286449at_nat @ produc120197956887798346t_char ) ).

% Pair_transfer
thf(fact_1150_Pair__transfer,axiom,
    ! [A3: set_Pr1261947904930325089at_nat > nat > $o,B5: set_Pr1261947904930325089at_nat > list_char > $o] : ( bNF_re7367706772112929959t_char @ A3 @ ( bNF_re2575847962171715088t_char @ B5 @ ( basic_3952808556802799238t_char @ A3 @ B5 ) ) @ produc2922128104949294807at_nat @ produc120197956887798346t_char ) ).

% Pair_transfer
thf(fact_1151_fst__transfer,axiom,
    ! [A3: nat > nat > $o,B5: list_char > list_char > $o] : ( bNF_re5269148346131715199at_nat @ ( basic_2137218353159982875t_char @ A3 @ B5 ) @ A3 @ produc6204653154488278070t_char @ produc6204653154488278070t_char ) ).

% fst_transfer
thf(fact_1152_underS__def,axiom,
    ( order_3915675016455977933od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A2: set_Product_prod_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] :
              ( ( B2 != A2 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1153_underS__def,axiom,
    ( order_4046151456452760519at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A2: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] :
              ( ( B2 != A2 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1154_underS__def,axiom,
    ( order_4357248199484075277at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A2: set_Pr4329608150637261639at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] :
              ( ( B2 != A2 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1155_underS__def,axiom,
    ( order_2644262826035875111at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] :
              ( ( B2 != A2 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1156_underS__def,axiom,
    ( order_underS_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A2: nat] :
          ( collect_nat
          @ ^ [B2: nat] :
              ( ( B2 != A2 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ A2 ) @ R ) ) ) ) ) ).

% underS_def
thf(fact_1157_rel__prod__sel,axiom,
    ( basic_2137218353159982875t_char
    = ( ^ [R12: nat > nat > $o,R23: list_char > list_char > $o,P2: produc1457211279475724562t_char,Q4: produc1457211279475724562t_char] :
          ( ( R12 @ ( produc6204653154488278070t_char @ P2 ) @ ( produc6204653154488278070t_char @ Q4 ) )
          & ( R23 @ ( produc1900778367047502200t_char @ P2 ) @ ( produc1900778367047502200t_char @ Q4 ) ) ) ) ) ).

% rel_prod_sel
thf(fact_1158_trans__on__join,axiom,
    ( trans_7029024365671003047od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X2 @ R )
         => ( produc9128050996712119596_o_o_o
            @ ^ [Y5: set_Product_prod_o_o,Y12: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ Y5 @ A7 )
               => ( ( member3157015407008874800od_o_o @ Y12 @ A7 )
                 => ! [Z5: produc2934264451710624999od_o_o] :
                      ( ( member444158400953824016od_o_o @ Z5 @ R )
                     => ( produc9128050996712119596_o_o_o
                        @ ^ [Y22: set_Product_prod_o_o,Aa2: set_Product_prod_o_o] :
                            ( ( Y12 = Y22 )
                           => ( ( member3157015407008874800od_o_o @ Aa2 @ A7 )
                             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1159_trans__on__join,axiom,
    ( trans_2549647969203962503od_o_o
    = ( ^ [A7: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o] :
        ! [X2: produc2245965095621756647od_o_o] :
          ( ( member5560047953027865360od_o_o @ X2 @ R )
         => ( produc758850913379988780_o_o_o
            @ ^ [Y5: produc2934264451710624999od_o_o,Y12: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ Y5 @ A7 )
               => ( ( member444158400953824016od_o_o @ Y12 @ A7 )
                 => ! [Z5: produc2245965095621756647od_o_o] :
                      ( ( member5560047953027865360od_o_o @ Z5 @ R )
                     => ( produc758850913379988780_o_o_o
                        @ ^ [Y22: produc2934264451710624999od_o_o,Aa2: produc2934264451710624999od_o_o] :
                            ( ( Y12 = Y22 )
                           => ( ( member444158400953824016od_o_o @ Aa2 @ A7 )
                             => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1160_trans__on__join,axiom,
    ( trans_3798871764278489953at_nat
    = ( ^ [A7: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat] :
        ! [X2: produc266773502759220327at_nat] :
          ( ( member208136763105818512at_nat @ X2 @ R )
         => ( produc5191426448209871788_nat_o
            @ ^ [Y5: produc732395585841259969at_nat,Y12: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ Y5 @ A7 )
               => ( ( member4095101504841534314at_nat @ Y12 @ A7 )
                 => ! [Z5: produc266773502759220327at_nat] :
                      ( ( member208136763105818512at_nat @ Z5 @ R )
                     => ( produc5191426448209871788_nat_o
                        @ ^ [Y22: produc732395585841259969at_nat,Aa2: produc732395585841259969at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member4095101504841534314at_nat @ Aa2 @ A7 )
                             => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1161_trans__on__join,axiom,
    ( trans_4802246114490097159at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
        ! [X2: produc1906917888121415911at_nat] :
          ( ( member6009668390874684176at_nat @ X2 @ R )
         => ( produc1561160019221259564_nat_o
            @ ^ [Y5: produc3843707927480180839at_nat,Y12: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y5 @ A7 )
               => ( ( member8757157785044589968at_nat @ Y12 @ A7 )
                 => ! [Z5: produc1906917888121415911at_nat] :
                      ( ( member6009668390874684176at_nat @ Z5 @ R )
                     => ( produc1561160019221259564_nat_o
                        @ ^ [Y22: produc3843707927480180839at_nat,Aa2: produc3843707927480180839at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8757157785044589968at_nat @ Aa2 @ A7 )
                             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1162_trans__on__join,axiom,
    ( trans_613433433080944135at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
        ! [X2: produc349518998152878311at_nat] :
          ( ( member8062223511168850704at_nat @ X2 @ R )
         => ( produc3603361296207914284_nat_o
            @ ^ [Y5: produc859450856879609959at_nat,Y12: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y5 @ A7 )
               => ( ( member8206827879206165904at_nat @ Y12 @ A7 )
                 => ! [Z5: produc349518998152878311at_nat] :
                      ( ( member8062223511168850704at_nat @ Z5 @ R )
                     => ( produc3603361296207914284_nat_o
                        @ ^ [Y22: produc859450856879609959at_nat,Aa2: produc859450856879609959at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8206827879206165904at_nat @ Aa2 @ A7 )
                             => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1163_trans__on__join,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X2 @ R )
         => ( produc6590410687421337004_nat_o
            @ ^ [Y5: product_prod_nat_nat,Y12: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y5 @ A7 )
               => ( ( member8440522571783428010at_nat @ Y12 @ A7 )
                 => ! [Z5: produc859450856879609959at_nat] :
                      ( ( member8206827879206165904at_nat @ Z5 @ R )
                     => ( produc6590410687421337004_nat_o
                        @ ^ [Y22: product_prod_nat_nat,Aa2: product_prod_nat_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member8440522571783428010at_nat @ Aa2 @ A7 )
                             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1164_trans__on__join,axiom,
    ( trans_707035017233543399at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: produc1319942482725812455at_nat] :
          ( ( member1466754251312161552at_nat @ X2 @ R )
         => ( produc9020218426428501292_nat_o
            @ ^ [Y5: set_Pr4329608150637261639at_nat,Y12: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Y5 @ A7 )
               => ( ( member6273874620674182256at_nat @ Y12 @ A7 )
                 => ! [Z5: produc1319942482725812455at_nat] :
                      ( ( member1466754251312161552at_nat @ Z5 @ R )
                     => ( produc9020218426428501292_nat_o
                        @ ^ [Y22: set_Pr4329608150637261639at_nat,Aa2: set_Pr4329608150637261639at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member6273874620674182256at_nat @ Aa2 @ A7 )
                             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1165_trans__on__join,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X2 @ R )
         => ( produc410239310623530412_nat_o
            @ ^ [Y5: set_Pr1261947904930325089at_nat,Y12: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y5 @ A7 )
               => ( ( member2643936169264416010at_nat @ Y12 @ A7 )
                 => ! [Z5: produc3843707927480180839at_nat] :
                      ( ( member8757157785044589968at_nat @ Z5 @ R )
                     => ( produc410239310623530412_nat_o
                        @ ^ [Y22: set_Pr1261947904930325089at_nat,Aa2: set_Pr1261947904930325089at_nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member2643936169264416010at_nat @ Aa2 @ A7 )
                             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1166_trans__on__join,axiom,
    ( trans_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ R )
         => ( produc6081775807080527818_nat_o
            @ ^ [Y5: nat,Y12: nat] :
                ( ( member_nat @ Y5 @ A7 )
               => ( ( member_nat @ Y12 @ A7 )
                 => ! [Z5: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ Z5 @ R )
                     => ( produc6081775807080527818_nat_o
                        @ ^ [Y22: nat,Aa2: nat] :
                            ( ( Y12 = Y22 )
                           => ( ( member_nat @ Aa2 @ A7 )
                             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Aa2 ) @ R ) ) )
                        @ Z5 ) ) ) )
            @ X2 ) ) ) ) ).

% trans_on_join
thf(fact_1167_refl__on__def_H,axiom,
    ( refl_o935183656833700870od_o_o
    = ( ^ [A7: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
          ( ! [X2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X2 @ R )
             => ( produc9128050996712119596_o_o_o
                @ ^ [Y5: set_Product_prod_o_o,Z5: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ Y5 @ A7 )
                    & ( member3157015407008874800od_o_o @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ X2 @ A7 )
             => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1168_refl__on__def_H,axiom,
    ( refl_o8365827894042769382od_o_o
    = ( ^ [A7: set_Pr1932065953672099015od_o_o,R: set_Pr1013884327308616903od_o_o] :
          ( ! [X2: produc2245965095621756647od_o_o] :
              ( ( member5560047953027865360od_o_o @ X2 @ R )
             => ( produc758850913379988780_o_o_o
                @ ^ [Y5: produc2934264451710624999od_o_o,Z5: produc2934264451710624999od_o_o] :
                    ( ( member444158400953824016od_o_o @ Y5 @ A7 )
                    & ( member444158400953824016od_o_o @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X2 @ A7 )
             => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1169_refl__on__def_H,axiom,
    ( refl_o3078925143892389184at_nat
    = ( ^ [A7: set_Pr457366540195662369at_nat,R: set_Pr4427867301352232263at_nat] :
          ( ! [X2: produc266773502759220327at_nat] :
              ( ( member208136763105818512at_nat @ X2 @ R )
             => ( produc5191426448209871788_nat_o
                @ ^ [Y5: produc732395585841259969at_nat,Z5: produc732395585841259969at_nat] :
                    ( ( member4095101504841534314at_nat @ Y5 @ A7 )
                    & ( member4095101504841534314at_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X2 @ A7 )
             => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1170_refl__on__def_H,axiom,
    ( refl_o7446464703481863270at_nat
    = ( ^ [A7: set_Pr4329608150637261639at_nat,R: set_Pr5564308138774400199at_nat] :
          ( ! [X2: produc1906917888121415911at_nat] :
              ( ( member6009668390874684176at_nat @ X2 @ R )
             => ( produc1561160019221259564_nat_o
                @ ^ [Y5: produc3843707927480180839at_nat,Z5: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Y5 @ A7 )
                    & ( member8757157785044589968at_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X2 @ A7 )
             => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1171_refl__on__def_H,axiom,
    ( refl_o6679620997798956134at_nat
    = ( ^ [A7: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
          ( ! [X2: produc349518998152878311at_nat] :
              ( ( member8062223511168850704at_nat @ X2 @ R )
             => ( produc3603361296207914284_nat_o
                @ ^ [Y5: produc859450856879609959at_nat,Z5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Y5 @ A7 )
                    & ( member8206827879206165904at_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ A7 )
             => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1172_refl__on__def_H,axiom,
    ( refl_o7122697186039111552at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
          ( ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ R )
             => ( produc6590410687421337004_nat_o
                @ ^ [Y5: product_prod_nat_nat,Z5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ A7 )
                    & ( member8440522571783428010at_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A7 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1173_refl__on__def_H,axiom,
    ( refl_o6272981002401706822at_nat
    = ( ^ [A7: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
          ( ! [X2: produc1319942482725812455at_nat] :
              ( ( member1466754251312161552at_nat @ X2 @ R )
             => ( produc9020218426428501292_nat_o
                @ ^ [Y5: set_Pr4329608150637261639at_nat,Z5: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ Y5 @ A7 )
                    & ( member6273874620674182256at_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X2 @ A7 )
             => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1174_refl__on__def_H,axiom,
    ( refl_o6955724340487615712at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
          ( ! [X2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X2 @ R )
             => ( produc410239310623530412_nat_o
                @ ^ [Y5: set_Pr1261947904930325089at_nat,Z5: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Y5 @ A7 )
                    & ( member2643936169264416010at_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ X2 @ A7 )
             => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1175_refl__on__def_H,axiom,
    ( refl_on_nat
    = ( ^ [A7: set_nat,R: set_Pr1261947904930325089at_nat] :
          ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ R )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y5: nat,Z5: nat] :
                    ( ( member_nat @ Y5 @ A7 )
                    & ( member_nat @ Z5 @ A7 ) )
                @ X2 ) )
          & ! [X2: nat] :
              ( ( member_nat @ X2 @ A7 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R ) ) ) ) ) ).

% refl_on_def'
thf(fact_1176_UnderS__def,axiom,
    ( order_8790096249059994541od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A7: set_se491199571785101031od_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R ) )
              & ! [X2: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1177_UnderS__def,axiom,
    ( order_5230482973827120269od_o_o
    = ( ^ [R: set_Pr1013884327308616903od_o_o,A7: set_Pr1932065953672099015od_o_o] :
          ( collec3302784863738620242od_o_o
          @ ^ [B2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R ) )
              & ! [X2: produc2934264451710624999od_o_o] :
                  ( ( member444158400953824016od_o_o @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1178_UnderS__def,axiom,
    ( order_4440812209384976999at_nat
    = ( ^ [R: set_Pr4427867301352232263at_nat,A7: set_Pr457366540195662369at_nat] :
          ( collec1729919521702017196at_nat
          @ ^ [B2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ B2 @ ( field_7361900449976055182at_nat @ R ) )
              & ! [X2: produc732395585841259969at_nat] :
                  ( ( member4095101504841534314at_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1179_UnderS__def,axiom,
    ( order_18415343535366669at_nat
    = ( ^ [R: set_Pr5564308138774400199at_nat,A7: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [B2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R ) )
              & ! [X2: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1180_UnderS__def,axiom,
    ( order_2814193433121557005at_nat
    = ( ^ [R: set_Pr553994874890374343at_nat,A7: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [B2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R ) )
              & ! [X2: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1181_UnderS__def,axiom,
    ( order_5965028037015491495at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R ) )
              & ! [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1182_UnderS__def,axiom,
    ( order_UnderS_o
    = ( ^ [R: set_Product_prod_o_o,A7: set_o] :
          ( collect_o
          @ ^ [B2: $o] :
              ( ( member_o @ B2 @ ( field_o @ R ) )
              & ! [X2: $o] :
                  ( ( member_o @ X2 @ A7 )
                 => ( ( B2 = (~ X2) )
                    & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1183_UnderS__def,axiom,
    ( order_6705503380902354157at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A7: set_se5479201105039950375at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R ) )
              & ! [X2: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1184_UnderS__def,axiom,
    ( order_7299894921538178311at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A7: set_se7855581050983116737at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R ) )
              & ! [X2: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1185_UnderS__def,axiom,
    ( order_UnderS_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A7: set_nat] :
          ( collect_nat
          @ ^ [B2: nat] :
              ( ( member_nat @ B2 @ ( field_nat @ R ) )
              & ! [X2: nat] :
                  ( ( member_nat @ X2 @ A7 )
                 => ( ( B2 != X2 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ) ).

% UnderS_def
thf(fact_1186_Under__def,axiom,
    ( order_6002923745599660940od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A7: set_se491199571785101031od_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R ) )
              & ! [X2: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ X2 @ A7 )
                 => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1187_Under__def,axiom,
    ( order_6472461514257025388od_o_o
    = ( ^ [R: set_Pr1013884327308616903od_o_o,A7: set_Pr1932065953672099015od_o_o] :
          ( collec3302784863738620242od_o_o
          @ ^ [B2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R ) )
              & ! [X2: produc2934264451710624999od_o_o] :
                  ( ( member444158400953824016od_o_o @ X2 @ A7 )
                 => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1188_Under__def,axiom,
    ( order_5208262014832739270at_nat
    = ( ^ [R: set_Pr4427867301352232263at_nat,A7: set_Pr457366540195662369at_nat] :
          ( collec1729919521702017196at_nat
          @ ^ [B2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ B2 @ ( field_7361900449976055182at_nat @ R ) )
              & ! [X2: produc732395585841259969at_nat] :
                  ( ( member4095101504841534314at_nat @ X2 @ A7 )
                 => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1189_Under__def,axiom,
    ( order_7144270531945035756at_nat
    = ( ^ [R: set_Pr5564308138774400199at_nat,A7: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [B2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R ) )
              & ! [X2: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ X2 @ A7 )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1190_Under__def,axiom,
    ( order_1452147409139849196at_nat
    = ( ^ [R: set_Pr553994874890374343at_nat,A7: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [B2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R ) )
              & ! [X2: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X2 @ A7 )
                 => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1191_Under__def,axiom,
    ( order_7744983739816774662at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R ) )
              & ! [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ A7 )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1192_Under__def,axiom,
    ( order_Under_o
    = ( ^ [R: set_Product_prod_o_o,A7: set_o] :
          ( collect_o
          @ ^ [B2: $o] :
              ( ( member_o @ B2 @ ( field_o @ R ) )
              & ! [X2: $o] :
                  ( ( member_o @ X2 @ A7 )
                 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1193_Under__def,axiom,
    ( order_3471437392943445708at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A7: set_se5479201105039950375at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R ) )
              & ! [X2: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ X2 @ A7 )
                 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1194_Under__def,axiom,
    ( order_7546462373915380070at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A7: set_se7855581050983116737at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R ) )
              & ! [X2: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ X2 @ A7 )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1195_Under__def,axiom,
    ( order_Under_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A7: set_nat] :
          ( collect_nat
          @ ^ [B2: nat] :
              ( ( member_nat @ B2 @ ( field_nat @ R ) )
              & ! [X2: nat] :
                  ( ( member_nat @ X2 @ A7 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B2 @ X2 ) @ R ) ) ) ) ) ) ).

% Under_def
thf(fact_1196_Above__def,axiom,
    ( order_3857846860138862887od_o_o
    = ( ^ [R: set_Pr1932065953672099015od_o_o,A7: set_se491199571785101031od_o_o] :
          ( collec2441215771612176754od_o_o
          @ ^ [B2: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ B2 @ ( field_6101175518749917524od_o_o @ R ) )
              & ! [X2: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ X2 @ A7 )
                 => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1197_Above__def,axiom,
    ( order_78083019343882247od_o_o
    = ( ^ [R: set_Pr1013884327308616903od_o_o,A7: set_Pr1932065953672099015od_o_o] :
          ( collec3302784863738620242od_o_o
          @ ^ [B2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ B2 @ ( field_5121862139100741428od_o_o @ R ) )
              & ! [X2: produc2934264451710624999od_o_o] :
                  ( ( member444158400953824016od_o_o @ X2 @ A7 )
                 => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1198_Above__def,axiom,
    ( order_4078086773371869409at_nat
    = ( ^ [R: set_Pr4427867301352232263at_nat,A7: set_Pr457366540195662369at_nat] :
          ( collec1729919521702017196at_nat
          @ ^ [B2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ B2 @ ( field_7361900449976055182at_nat @ R ) )
              & ! [X2: produc732395585841259969at_nat] :
                  ( ( member4095101504841534314at_nat @ X2 @ A7 )
                 => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1199_Above__def,axiom,
    ( order_1832958076520874887at_nat
    = ( ^ [R: set_Pr5564308138774400199at_nat,A7: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [B2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ R ) )
              & ! [X2: produc3843707927480180839at_nat] :
                  ( ( member8757157785044589968at_nat @ X2 @ A7 )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1200_Above__def,axiom,
    ( order_4829706783164889991at_nat
    = ( ^ [R: set_Pr553994874890374343at_nat,A7: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ^ [B2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ R ) )
              & ! [X2: produc859450856879609959at_nat] :
                  ( ( member8206827879206165904at_nat @ X2 @ A7 )
                 => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1201_Above__def,axiom,
    ( order_2089690580550591521at_nat
    = ( ^ [R: set_Pr8693737435421807431at_nat,A7: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ R ) )
              & ! [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ A7 )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1202_Above__def,axiom,
    ( order_Above_o
    = ( ^ [R: set_Product_prod_o_o,A7: set_o] :
          ( collect_o
          @ ^ [B2: $o] :
              ( ( member_o @ B2 @ ( field_o @ R ) )
              & ! [X2: $o] :
                  ( ( member_o @ X2 @ A7 )
                 => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1203_Above__def,axiom,
    ( order_945487400789399143at_nat
    = ( ^ [R: set_Pr7459493094073627847at_nat,A7: set_se5479201105039950375at_nat] :
          ( collec4573900263915890866at_nat
          @ ^ [B2: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ R ) )
              & ! [X2: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ X2 @ A7 )
                 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1204_Above__def,axiom,
    ( order_3976174864977098113at_nat
    = ( ^ [R: set_Pr4329608150637261639at_nat,A7: set_se7855581050983116737at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ R ) )
              & ! [X2: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ X2 @ A7 )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1205_Above__def,axiom,
    ( order_Above_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,A7: set_nat] :
          ( collect_nat
          @ ^ [B2: nat] :
              ( ( member_nat @ B2 @ ( field_nat @ R ) )
              & ! [X2: nat] :
                  ( ( member_nat @ X2 @ A7 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ B2 ) @ R ) ) ) ) ) ) ).

% Above_def
thf(fact_1206_underS__incr,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 )
     => ( ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R2 )
       => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
         => ( ord_le2654036014095304327od_o_o @ ( order_3915675016455977933od_o_o @ R2 @ A ) @ ( order_3915675016455977933od_o_o @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1207_underS__incr,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( antisy1245043278253178980at_nat @ top_to4669805908274784177at_nat @ R2 )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
         => ( ord_le3146513528884898305at_nat @ ( order_4046151456452760519at_nat @ R2 @ A ) @ ( order_4046151456452760519at_nat @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1208_underS__incr,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( antisy2402629262940634666at_nat @ top_to1754375979021635447at_nat @ R2 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
         => ( ord_le7966958417815640519at_nat @ ( order_4357248199484075277at_nat @ R2 @ A ) @ ( order_4357248199484075277at_nat @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1209_underS__incr,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
         => ( ord_le2077887516847798113at_nat @ ( order_2644262826035875111at_nat @ R2 @ A ) @ ( order_2644262826035875111at_nat @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1210_underS__incr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( trans_on_nat @ top_top_set_nat @ R2 )
     => ( ( antisym_on_nat @ top_top_set_nat @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
         => ( ord_less_eq_set_nat @ ( order_underS_nat @ R2 @ A ) @ ( order_underS_nat @ R2 @ B ) ) ) ) ) ).

% underS_incr
thf(fact_1211_antisym__onD,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( antisy777991898044409578od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y @ A3 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1212_antisym__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( antisy2259881126919591626od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( member444158400953824016od_o_o @ Y @ A3 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1213_antisym__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( antisy1857962754316978724at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( member4095101504841534314at_nat @ Y @ A3 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1214_antisym__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( antisy6856624793204979530at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( member8757157785044589968at_nat @ Y @ A3 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1215_antisym__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( antisy3264817248818186058at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( member8206827879206165904at_nat @ Y @ A3 )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1216_antisym__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( antisy1245043278253178980at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( member8440522571783428010at_nat @ Y @ A3 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1217_antisym__onD,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( antisy2402629262940634666at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ A3 )
       => ( ( member6273874620674182256at_nat @ Y @ A3 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1218_antisym__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( antisy1130263102092445124at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( member2643936169264416010at_nat @ Y @ A3 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1219_antisym__onD,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( antisym_on_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1220_antisym__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X3 @ A3 )
         => ( ( member444158400953824016od_o_o @ Y3 @ A3 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy2259881126919591626od_o_o @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1221_antisym__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X3 @ A3 )
         => ( ( member4095101504841534314at_nat @ Y3 @ A3 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy1857962754316978724at_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1222_antisym__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A3 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy6856624793204979530at_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1223_antisym__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A3 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A3 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy3264817248818186058at_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1224_antisym__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy1245043278253178980at_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1225_antisym__onI,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A3 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A3 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy2402629262940634666at_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1226_antisym__onI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A3 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy1130263102092445124at_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1227_antisym__onI,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( member_nat @ Y3 @ A3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R2 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisym_on_nat @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1228_ctwo__Cnotzero,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).

% ctwo_Cnotzero
thf(fact_1229_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_1230_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_Field_natLeq
thf(fact_1231_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat
thf(fact_1232_smax__emptyI,axiom,
    ! [Y8: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y8 )
     => ( ( Y8 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y8 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_1233_wmax__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X4 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X4 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_1234_wmin__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1235_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1236_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_1237_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_1238_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1239_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1240_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_1241_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_1242_smin__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] :
      ( ( X4 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_1243_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_1244_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y8: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y8 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y8 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X4 ) @ Y8 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_1245_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1246_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_1247_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1248_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T2 ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1249_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_1250_nat__gcd_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [X3: nat,Y3: nat] :
          ( X
         != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% nat_gcd.cases
thf(fact_1251_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_1252_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

% Helper facts (4)
thf(help_If_3_1_If_001t__PosRat__Oprat_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X: prat,Y: prat] :
      ( ( if_prat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__PosRat__Oprat_T,axiom,
    ! [X: prat,Y: prat] :
      ( ( if_prat @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__PosRat__Oprat_T,axiom,
    ! [P3: prat > $o] :
      ( ( P3 @ ( fChoice_prat @ P3 ) )
      = ( ? [X6: prat] : ( P3 @ X6 ) ) ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    pgte @ pwrite @ ( add_ma6586698345353345811t_char @ a @ b @ hl ) ).

%------------------------------------------------------------------------------