%------------------------------------------------------------------------------
% 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_00182_005030__7104720_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2062 ( 385 unt; 783 typ;   0 def)
%            Number of atoms       : 4558 (1097 equ;   0 cnn)
%            Maximal formula atoms :    9 (   3 avg)
%            Number of connectives : 16884 ( 407   ~;  32   |; 252   &;13711   @)
%                                         (   0 <=>;2482  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   9 avg)
%            Number of types       :  100 (  99 usr)
%            Number of type conns  : 2472 (2472   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  687 ( 684 usr;  40 con; 0-4 aty)
%            Number of variables   : 4550 ( 155   ^;4271   !; 124   ?;4550   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 09:05:12.282
%------------------------------------------------------------------------------
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thf(ty_n_t__Set__Oset_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_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_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_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_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_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(ty_n_t__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,
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thf(ty_n_t__Set__Oset_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_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(ty_n_t__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,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__PosRat__Oprat_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__PosRat__Oprat,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (684)
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
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thf(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001_Eo,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal_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,
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thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
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thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Nat__Onat,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_BNF__Def_OfstOp_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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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,
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thf(sy_c_BNF__Def_Opick__middlep_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_BNF__Def_OsndOp_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
    bNF_Gr3738443398878377306t_char: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > list_char ) > set_Pr2507339680178222706t_char ).

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

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr8808630244184418791at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr8693737435421807431at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_Oimage2_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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_Gr6834105381924935271od_o_o: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > set_Product_prod_o_o ) > ( product_prod_nat_nat > set_Product_prod_o_o ) > set_Pr1932065953672099015od_o_o ).

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    bij_be2725198553504426486at_nat: ( produc732395585841259969at_nat > product_prod_nat_nat ) > set_Pr457366540195662369at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Fun_Obij__betw_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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Fun_Obij__betw_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_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,
    bij_be236100170522119478at_nat: ( produc3843707927480180839at_nat > produc3843707927480180839at_nat ) > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J,type,
    comp_n7998183327787346125t_char: ( nat > nat ) > ( produc1457211279475724562t_char > nat ) > produc1457211279475724562t_char > nat ).

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    comp_P6261748720030321425t_char: ( produc1457211279475724562t_char > nat ) > ( produc1457211279475724562t_char > produc1457211279475724562t_char ) > produc1457211279475724562t_char > nat ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_001t__PosRat__Oprat,type,
    fun_up829850848796697765r_prat: ( produc1457211279475724562t_char > prat ) > produc1457211279475724562t_char > prat > produc1457211279475724562t_char > prat ).

thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_up6858051064228255165at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

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    fun_up7313344445903829795at_nat: ( product_prod_nat_nat > produc3843707927480180839at_nat ) > product_prod_nat_nat > produc3843707927480180839at_nat > product_prod_nat_nat > produc3843707927480180839at_nat ).

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    fun_up105607794567514583at_nat: ( produc859450856879609959at_nat > product_prod_nat_nat ) > produc859450856879609959at_nat > product_prod_nat_nat > produc859450856879609959at_nat > product_prod_nat_nat ).

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    fun_up110622245137070935at_nat: ( produc2934264451710624999od_o_o > product_prod_nat_nat ) > produc2934264451710624999od_o_o > product_prod_nat_nat > produc2934264451710624999od_o_o > product_prod_nat_nat ).

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    fun_up3446698188135383485at_nat: ( produc2934264451710624999od_o_o > produc3843707927480180839at_nat ) > produc2934264451710624999od_o_o > produc3843707927480180839at_nat > produc2934264451710624999od_o_o > produc3843707927480180839at_nat ).

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    fun_up4035006559045952253at_nat: ( produc732395585841259969at_nat > product_prod_nat_nat ) > produc732395585841259969at_nat > product_prod_nat_nat > produc732395585841259969at_nat > product_prod_nat_nat ).

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    fun_up2755145109609588835at_nat: ( produc732395585841259969at_nat > produc3843707927480180839at_nat ) > produc732395585841259969at_nat > produc3843707927480180839at_nat > produc732395585841259969at_nat > produc3843707927480180839at_nat ).

thf(sy_c_Fun_Ofun__upd_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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_up3742445477866918359at_nat: ( produc3843707927480180839at_nat > product_prod_nat_nat ) > produc3843707927480180839at_nat > product_prod_nat_nat > produc3843707927480180839at_nat > product_prod_nat_nat ).

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    fun_up3138462846764619837at_nat: ( produc3843707927480180839at_nat > produc3843707927480180839at_nat ) > produc3843707927480180839at_nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat ).

thf(sy_c_Fun_Ofun__upd_001tf__a_001t__PosRat__Oprat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_PosRat_Ocomp__one,type,
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thf(sy_c_PosRat_Ohalf,type,
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thf(sy_c_PosRat_Opadd,type,
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thf(sy_c_PosRat_Opdiv,type,
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thf(sy_c_PosRat_Opgt,type,
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thf(sy_c_PosRat_Opgte,type,
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thf(sy_c_PosRat_Opinv,type,
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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_Opmult,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_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_Mt__PosRat__Oprat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__String__Ochar_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_Itf__a_Mt__PosRat__Oprat_J_001tf__a,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
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    member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).

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    member_set_nat: set_nat > set_set_nat > $o ).

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    member3157015407008874800od_o_o: set_Product_prod_o_o > set_se491199571785101031od_o_o > $o ).

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    member3250002818903848048at_nat: set_Pr8693737435421807431at_nat > set_se6425085446898223655at_nat > $o ).

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    member6273874620674182256at_nat: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat > $o ).

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    member2239101972932892176od_o_o: set_se491199571785101031od_o_o > set_se5342866143169225671od_o_o > $o ).

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    member8313122109188517482at_nat: set_se7855581050983116737at_nat > set_se5158245632345848097at_nat > $o ).

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    member2479695886705571664at_nat: set_se5479201105039950375at_nat > set_se6387396333462195463at_nat > $o ).

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

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

thf(sy_v_c,type,
    c: a > prat ).

% Relevant facts (1274)
thf(fact_0_assms,axiom,
    greater_mask_a @ a2 @ ( add_masks_a @ b @ c ) ).

% assms
thf(fact_1_greater__mask__properties_I3_J,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
      ( ( ( greate7680731506090375811t_char @ Pi @ Pi2 )
        & ( greate7680731506090375811t_char @ Pi2 @ Pi ) )
     => ( Pi2 = Pi ) ) ).

% greater_mask_properties(3)
thf(fact_2_greater__mask__properties_I3_J,axiom,
    ! [Pi: a > prat,Pi2: a > prat] :
      ( ( ( greater_mask_a @ Pi @ Pi2 )
        & ( greater_mask_a @ Pi2 @ Pi ) )
     => ( Pi2 = Pi ) ) ).

% greater_mask_properties(3)
thf(fact_3_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_4_greater__mask__properties_I2_J,axiom,
    ! [A: a > prat,B: a > prat,C: a > prat] :
      ( ( ( greater_mask_a @ A @ B )
        & ( greater_mask_a @ B @ C ) )
     => ( greater_mask_a @ A @ C ) ) ).

% greater_mask_properties(2)
thf(fact_5_greater__mask__properties_I1_J,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] : ( greate7680731506090375811t_char @ Pi2 @ Pi2 ) ).

% greater_mask_properties(1)
thf(fact_6_greater__mask__properties_I1_J,axiom,
    ! [Pi2: a > prat] : ( greater_mask_a @ Pi2 @ Pi2 ) ).

% greater_mask_properties(1)
thf(fact_7_add__masks__asso,axiom,
    ! [A: a > prat,B: a > prat,C: a > prat] :
      ( ( add_masks_a @ ( add_masks_a @ A @ B ) @ C )
      = ( add_masks_a @ A @ ( add_masks_a @ B @ C ) ) ) ).

% add_masks_asso
thf(fact_8_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_9_add__masks__comm,axiom,
    ( add_masks_a
    = ( ^ [A2: a > prat,B2: a > prat] : ( add_masks_a @ B2 @ A2 ) ) ) ).

% add_masks_comm
thf(fact_10_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_11_greater__mask__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat] :
        ? [R: produc1457211279475724562t_char > prat] :
          ( Pi3
          = ( add_ma6586698345353345811t_char @ Pi4 @ R ) ) ) ) ).

% greater_mask_def
thf(fact_12_greater__mask__def,axiom,
    ( greater_mask_a
    = ( ^ [Pi3: a > prat,Pi4: a > prat] :
        ? [R: a > prat] :
          ( Pi3
          = ( add_masks_a @ Pi4 @ R ) ) ) ) ).

% greater_mask_def
thf(fact_13_minus__empty,axiom,
    ! [Pi2: a > prat] :
      ( Pi2
      = ( add_masks_a @ Pi2 @ empty_mask_a ) ) ).

% minus_empty
thf(fact_14_minus__empty,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] :
      ( Pi2
      = ( add_ma6586698345353345811t_char @ Pi2 @ empty_3446695950879338768t_char ) ) ).

% minus_empty
thf(fact_15_add__acc__uni__mask,axiom,
    ( add_acc_a
    = ( ^ [Pi4: a > prat,Hl: a,P: prat] : ( add_masks_a @ Pi4 @ ( uni_mask_a @ Hl @ P ) ) ) ) ).

% add_acc_uni_mask
thf(fact_16_add__acc__uni__mask,axiom,
    ( add_ac6043276672051819257t_char
    = ( ^ [Pi4: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char,P: prat] : ( add_ma6586698345353345811t_char @ Pi4 @ ( uni_ma6578582744724537101t_char @ Hl @ P ) ) ) ) ).

% add_acc_uni_mask
thf(fact_17_greater__maskI,axiom,
    ! [Pi: a > prat,Pi2: a > prat] :
      ( ! [Hl2: a] : ( pgte @ ( Pi @ Hl2 ) @ ( Pi2 @ Hl2 ) )
     => ( greater_mask_a @ Pi @ Pi2 ) ) ).

% greater_maskI
thf(fact_18_greater__maskI,axiom,
    ! [Pi: produc1457211279475724562t_char > prat,Pi2: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ ( Pi @ Hl2 ) @ ( Pi2 @ Hl2 ) )
     => ( greate7680731506090375811t_char @ Pi @ Pi2 ) ) ).

% greater_maskI
thf(fact_19_greater__mask__equiv__def,axiom,
    ( greater_mask_a
    = ( ^ [Pi3: a > prat,Pi4: a > prat] :
        ! [Hl: a] : ( pgte @ ( Pi3 @ Hl ) @ ( Pi4 @ Hl ) ) ) ) ).

% greater_mask_equiv_def
thf(fact_20_greater__mask__equiv__def,axiom,
    ( greate7680731506090375811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat] :
        ! [Hl: produc1457211279475724562t_char] : ( pgte @ ( Pi3 @ Hl ) @ ( Pi4 @ Hl ) ) ) ) ).

% greater_mask_equiv_def
thf(fact_21_add__masks_Oelims,axiom,
    ! [X: a > prat,Xa: a > prat,Xb: a,Y: prat] :
      ( ( ( add_masks_a @ X @ Xa @ Xb )
        = Y )
     => ( Y
        = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) ) ) ).

% add_masks.elims
thf(fact_22_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_23_add__masks_Osimps,axiom,
    ( add_masks_a
    = ( ^ [Pi3: a > prat,Pi4: a > prat,Hl: a] : ( padd @ ( Pi3 @ Hl ) @ ( Pi4 @ Hl ) ) ) ) ).

% add_masks.simps
thf(fact_24_add__masks_Osimps,axiom,
    ( add_ma6586698345353345811t_char
    = ( ^ [Pi3: produc1457211279475724562t_char > prat,Pi4: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( padd @ ( Pi3 @ Hl ) @ ( Pi4 @ Hl ) ) ) ) ).

% add_masks.simps
thf(fact_25_upper__bounded__def,axiom,
    ( upper_bounded
    = ( ^ [Pi4: produc1457211279475724562t_char > prat,P: prat] :
        ! [Hl: produc1457211279475724562t_char] : ( pgte @ P @ ( Pi4 @ Hl ) ) ) ) ).

% upper_bounded_def
thf(fact_26_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_27_p__greater__exists,axiom,
    ( pgte
    = ( ^ [A2: prat,B2: prat] :
        ? [R: prat] :
          ( A2
          = ( padd @ B2 @ R ) ) ) ) ).

% p_greater_exists
thf(fact_28_padd__asso,axiom,
    ! [A: prat,B: prat,C: prat] :
      ( ( padd @ ( padd @ A @ B ) @ C )
      = ( padd @ A @ ( padd @ B @ C ) ) ) ).

% padd_asso
thf(fact_29_padd__comm,axiom,
    ( padd
    = ( ^ [A2: prat,B2: prat] : ( padd @ B2 @ A2 ) ) ) ).

% padd_comm
thf(fact_30_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_31_pgte__antisym,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pgte @ B @ A )
       => ( A = B ) ) ) ).

% pgte_antisym
thf(fact_32_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_33_uni__mask_Oelims,axiom,
    ! [X: a,Xa: prat,Y: a > prat] :
      ( ( ( uni_mask_a @ X @ Xa )
        = Y )
     => ( Y
        = ( fun_upd_a_prat @ empty_mask_a @ X @ Xa ) ) ) ).

% uni_mask.elims
thf(fact_34_uni__mask_Osimps,axiom,
    ( uni_ma6578582744724537101t_char
    = ( fun_up829850848796697765r_prat @ empty_3446695950879338768t_char ) ) ).

% uni_mask.simps
thf(fact_35_uni__mask_Osimps,axiom,
    ( uni_mask_a
    = ( fun_upd_a_prat @ empty_mask_a ) ) ).

% uni_mask.simps
thf(fact_36_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_37_pgt__implies__pgte,axiom,
    ! [A: prat,B: prat] :
      ( ( pgt @ A @ B )
     => ( pgte @ A @ B ) ) ).

% pgt_implies_pgte
thf(fact_38_not__pgte__charact,axiom,
    ! [A: prat,B: prat] :
      ( ( ~ ( pgte @ A @ B ) )
      = ( pgt @ B @ A ) ) ).

% not_pgte_charact
thf(fact_39_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
    ! [A: produc2934264451710624999od_o_o,P2: produc2934264451710624999od_o_o > $o] :
      ( ( member444158400953824016od_o_o @ A @ ( collec3302784863738620242od_o_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
    ! [A: produc732395585841259969at_nat,P2: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( collec1729919521702017196at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_45_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_46_Collect__mem__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X2: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_47_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_48_Collect__mem__eq,axiom,
    ! [A3: set_Pr457366540195662369at_nat] :
      ( ( collec1729919521702017196at_nat
        @ ^ [X2: produc732395585841259969at_nat] : ( member4095101504841534314at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_49_pmin__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmin @ A @ B )
        = B ) ) ).

% pmin_is
thf(fact_50_pmin__greater,axiom,
    ! [A: prat,B: prat] : ( pgte @ A @ ( pmin @ A @ B ) ) ).

% pmin_greater
thf(fact_51_padd__comp__one,axiom,
    ! [X: prat] :
      ( ( pgte @ pwrite @ X )
     => ( ( padd @ X @ ( comp_one @ X ) )
        = pwrite ) ) ).

% padd_comp_one
thf(fact_52_empty__mask_Oelims,axiom,
    ! [X: produc1457211279475724562t_char,Y: prat] :
      ( ( ( empty_3446695950879338768t_char @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_53_empty__mask_Oelims,axiom,
    ! [X: a,Y: prat] :
      ( ( ( empty_mask_a @ X )
        = Y )
     => ( Y = pnone ) ) ).

% empty_mask.elims
thf(fact_54_empty__mask_Osimps,axiom,
    ( empty_3446695950879338768t_char
    = ( ^ [Hl: produc1457211279475724562t_char] : pnone ) ) ).

% empty_mask.simps
thf(fact_55_empty__mask_Osimps,axiom,
    ( empty_mask_a
    = ( ^ [Hl: a] : pnone ) ) ).

% empty_mask.simps
thf(fact_56_pmax__is,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( pmax @ A @ B )
        = A ) ) ).

% pmax_is
thf(fact_57_pmax__smaller,axiom,
    ! [A: prat,B: prat] : ( pgte @ ( pmax @ A @ B ) @ A ) ).

% pmax_smaller
thf(fact_58_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_59_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_60_uni__mask_Opelims,axiom,
    ! [X: a,Xa: prat,Y: a > prat] :
      ( ( ( uni_mask_a @ X @ Xa )
        = Y )
     => ( ( accp_P8961225421893513396a_prat @ uni_mask_rel_a @ ( product_Pair_a_prat @ X @ Xa ) )
       => ~ ( ( Y
              = ( fun_upd_a_prat @ empty_mask_a @ X @ Xa ) )
           => ~ ( accp_P8961225421893513396a_prat @ uni_mask_rel_a @ ( product_Pair_a_prat @ X @ Xa ) ) ) ) ) ).

% uni_mask.pelims
thf(fact_61_padd__zero,axiom,
    ! [A: prat,B: prat] :
      ( ( pnone
        = ( padd @ A @ B ) )
      = ( ( A = pnone )
        & ( B = pnone ) ) ) ).

% padd_zero
thf(fact_62_ppos__add,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ppos @ ( padd @ A @ B ) ) ) ).

% ppos_add
thf(fact_63_ppos__eq__pnone,axiom,
    ( ppos
    = ( ^ [P: prat] : ( P != pnone ) ) ) ).

% ppos_eq_pnone
thf(fact_64_pmax__comm,axiom,
    ( pmax
    = ( ^ [A2: prat,B2: prat] : ( pmax @ B2 @ A2 ) ) ) ).

% pmax_comm
thf(fact_65_pmin__comm,axiom,
    ( pmin
    = ( ^ [A2: prat,B2: prat] : ( pmin @ B2 @ A2 ) ) ) ).

% pmin_comm
thf(fact_66_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_67_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_68_empty__mask_Opelims,axiom,
    ! [X: a,Y: prat] :
      ( ( ( empty_mask_a @ X )
        = Y )
     => ( ( accp_a @ empty_mask_rel_a @ X )
       => ~ ( ( Y = pnone )
           => ~ ( accp_a @ empty_mask_rel_a @ X ) ) ) ) ).

% empty_mask.pelims
thf(fact_69_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_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_half__between__0__1,axiom,
    ( ( ppos @ half )
    & ( pgt @ pwrite @ half ) ) ).

% half_between_0_1
thf(fact_81_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_82_pdiv__smaller,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ A )
       => ( pgte @ pwrite @ ( pdiv @ B @ A ) ) ) ) ).

% pdiv_smaller
thf(fact_83_empty__bmask_Opelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( empty_7634413185502380953r_prat @ X )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X )
       => ~ ( ~ Y
           => ~ ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X ) ) ) ) ).

% empty_bmask.pelims(1)
thf(fact_84_empty__bmask_Opelims_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ( empty_7634413185502380953r_prat @ X )
     => ~ ( accp_P4765339447491148764r_prat @ empty_2051610590842377682r_prat @ X ) ) ).

% empty_bmask.pelims(2)
thf(fact_85_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_86_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_87_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_88_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1457211279475724562t_char] :
      ~ ! [A5: nat,B4: list_char] :
          ( Y
         != ( produc120197956887798346t_char @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_89_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_90_surj__pair,axiom,
    ! [P3: produc732395585841259969at_nat] :
    ? [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
      ( P3
      = ( produc8517790099723286449at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_91_surj__pair,axiom,
    ! [P3: produc1319942482725812455at_nat] :
    ? [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P3
      = ( produc9060074326276436823at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_92_surj__pair,axiom,
    ! [P3: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P3
      = ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_93_surj__pair,axiom,
    ! [P3: produc1457211279475724562t_char] :
    ? [X3: nat,Y3: list_char] :
      ( P3
      = ( produc120197956887798346t_char @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_94_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_95_prod__cases,axiom,
    ! [P2: produc732395585841259969at_nat > $o,P3: produc732395585841259969at_nat] :
      ( ! [A5: set_Product_prod_o_o,B4: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc8517790099723286449at_nat @ A5 @ B4 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_96_prod__cases,axiom,
    ! [P2: produc1319942482725812455at_nat > $o,P3: produc1319942482725812455at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : ( P2 @ ( produc9060074326276436823at_nat @ A5 @ B4 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_97_prod__cases,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,P3: produc3843707927480180839at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A5 @ B4 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_98_prod__cases,axiom,
    ! [P2: produc1457211279475724562t_char > $o,P3: produc1457211279475724562t_char] :
      ( ! [A5: nat,B4: list_char] : ( P2 @ ( produc120197956887798346t_char @ A5 @ B4 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_99_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A5: nat,B4: nat] : ( P2 @ ( product_Pair_nat_nat @ A5 @ B4 ) )
     => ( P2 @ P3 ) ) ).

% prod_cases
thf(fact_100_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_101_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_102_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_103_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_104_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_105_half__plus__half,axiom,
    ( ( padd @ half @ half )
    = pwrite ) ).

% half_plus_half
thf(fact_106_add__masks_Opelims,axiom,
    ! [X: a > prat,Xa: a > prat,Xb: a,Y: prat] :
      ( ( ( add_masks_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8439977412596685783prat_a @ add_masks_rel_a @ ( produc2856268368059332798prat_a @ X @ ( produc3557287861060643862prat_a @ Xa @ Xb ) ) )
       => ~ ( ( Y
              = ( padd @ ( X @ Xb ) @ ( Xa @ Xb ) ) )
           => ~ ( accp_P8439977412596685783prat_a @ add_masks_rel_a @ ( produc2856268368059332798prat_a @ X @ ( produc3557287861060643862prat_a @ Xa @ Xb ) ) ) ) ) ) ).

% add_masks.pelims
thf(fact_107_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_108_pinv__def,axiom,
    ( pinv
    = ( pdiv @ pwrite ) ) ).

% pinv_def
thf(fact_109_pinv__inverts,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ A @ B )
     => ( ( ppos @ B )
       => ( pgte @ ( pinv @ B ) @ ( pinv @ A ) ) ) ) ).

% pinv_inverts
thf(fact_110_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_111_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_112_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_113_pmult__comm,axiom,
    ( pmult
    = ( ^ [A2: prat,B2: prat] : ( pmult @ B2 @ A2 ) ) ) ).

% pmult_comm
thf(fact_114_pmult__order,axiom,
    ! [A: prat,B: prat,P3: prat] :
      ( ( pgte @ A @ B )
     => ( pgte @ ( pmult @ P3 @ A ) @ ( pmult @ B @ P3 ) ) ) ).

% pmult_order
thf(fact_115_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_116_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_117_pmult__special_I1_J,axiom,
    ! [X: prat] :
      ( ( pmult @ pwrite @ X )
      = X ) ).

% pmult_special(1)
thf(fact_118_pmult__special_I2_J,axiom,
    ! [X: prat] :
      ( ( pmult @ pnone @ X )
      = pnone ) ).

% pmult_special(2)
thf(fact_119_pmult__ppos,axiom,
    ! [A: prat,B: prat] :
      ( ( ppos @ A )
     => ( ( ppos @ B )
       => ( ppos @ ( pmult @ A @ B ) ) ) ) ).

% pmult_ppos
thf(fact_120_pinv__pmult__ok,axiom,
    ! [P3: prat] :
      ( ( ppos @ P3 )
     => ( ( pmult @ P3 @ ( pinv @ P3 ) )
        = pwrite ) ) ).

% pinv_pmult_ok
thf(fact_121_pinv__pwrite,axiom,
    ( ( pinv @ pwrite )
    = pwrite ) ).

% pinv_pwrite
thf(fact_122_ppos__inv,axiom,
    ! [P3: prat] :
      ( ( ppos @ P3 )
     => ( ppos @ ( pinv @ P3 ) ) ) ).

% ppos_inv
thf(fact_123_pinv__double__half,axiom,
    ! [P3: prat] :
      ( ( ppos @ P3 )
     => ( ( pmult @ half @ ( pinv @ P3 ) )
        = ( pinv @ ( padd @ P3 @ P3 ) ) ) ) ).

% pinv_double_half
thf(fact_124_multiply__smaller__pwrite,axiom,
    ! [A: prat,B: prat] :
      ( ( pgte @ pwrite @ A )
     => ( ( pgte @ pwrite @ B )
       => ( pgte @ pwrite @ ( pmult @ A @ B ) ) ) ) ).

% multiply_smaller_pwrite
thf(fact_125_pmult__pdiv__cancel,axiom,
    ! [A: prat,X: prat] :
      ( ( ppos @ A )
     => ( ( pmult @ A @ ( pdiv @ X @ A ) )
        = X ) ) ).

% pmult_pdiv_cancel
thf(fact_126_accp__induct__rule,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X3: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X3 )
           => ( ! [Y4: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct_rule
thf(fact_127_not__accp__down,axiom,
    ! [R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat] :
      ( ~ ( accp_P4765339447491148764r_prat @ R3 @ X )
     => ~ ! [Z: produc1457211279475724562t_char > prat] :
            ( ( R3 @ Z @ X )
           => ( accp_P4765339447491148764r_prat @ R3 @ Z ) ) ) ).

% not_accp_down
thf(fact_128_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_129_accp__induct,axiom,
    ! [R2: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,A: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( accp_P4765339447491148764r_prat @ R2 @ A )
     => ( ! [X3: produc1457211279475724562t_char > prat] :
            ( ( accp_P4765339447491148764r_prat @ R2 @ X3 )
           => ( ! [Y4: produc1457211279475724562t_char > prat] :
                  ( ( R2 @ Y4 @ X3 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% accp_induct
thf(fact_130_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_131_multiply__mask__def,axiom,
    ( multiply_mask
    = ( ^ [P: prat,Pi4: produc1457211279475724562t_char > prat,Hl: produc1457211279475724562t_char] : ( pmult @ P @ ( Pi4 @ Hl ) ) ) ) ).

% multiply_mask_def
thf(fact_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: set_Product_prod_o_o,A4: product_prod_nat_nat,B3: set_Product_prod_o_o,R2: set_Pr8693737435421807431at_nat,S: set_Pr1932065953672099015od_o_o] :
      ( ( member2261041083240275600od_o_o @ ( produc4737999710111638999od_o_o @ ( produc3760286595445311197od_o_o @ A @ B ) @ ( produc3760286595445311197od_o_o @ A4 @ B3 ) ) @ ( lex_pr2425111879933873909od_o_o @ R2 @ S ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_142_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y6: list_char,Y: list_char,R3: nat > set_Pr4480099295215233283t_char] :
      ( ( P2 @ X )
     => ( ( member6023471406134104908t_char @ ( produc1606739095562851931t_char @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member151853725258021968t_char @ ( produc8192741843292627479t_char @ ( produc120197956887798346t_char @ X @ Y6 ) @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( same_f652031872680891423t_char @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_143_same__fstI,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y6: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr4329608150637261639at_nat > set_Pr7459493094073627847at_nat] :
      ( ( P2 @ X )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member8215962448328497936at_nat @ ( produc2220582066590712663at_nat @ ( produc9060074326276436823at_nat @ X @ Y6 ) @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( same_f4504843259822650924at_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_144_same__fstI,axiom,
    ! [P2: set_Product_prod_o_o > $o,X: set_Product_prod_o_o,Y6: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Product_prod_o_o > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ ( produc8517790099723286449at_nat @ X @ Y6 ) @ ( produc8517790099723286449at_nat @ X @ Y ) ) @ ( same_f6937299433074352134at_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_145_same__fstI,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat > set_Pr4329608150637261639at_nat] :
      ( ( P2 @ X )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ X @ Y6 ) @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( same_f1460074399991292076at_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_146_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y6: nat,Y: nat,R3: nat > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R3 @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P2 @ R3 ) ) ) ) ).

% same_fstI
thf(fact_147_valid__mask_Oelims_I3_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ~ ( valid_mask @ X )
     => ~ ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
          & ! [F: list_char] :
              ( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
              = pnone ) ) ) ).

% valid_mask.elims(3)
thf(fact_148_valid__mask_Oelims_I2_J,axiom,
    ! [X: produc1457211279475724562t_char > prat] :
      ( ( valid_mask @ X )
     => ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
        & ! [F2: list_char] :
            ( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
            = pnone ) ) ) ).

% valid_mask.elims(2)
thf(fact_149_valid__mask_Oelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X )
        = Y )
     => ( Y
        = ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl ) )
          & ! [F3: list_char] :
              ( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
              = pnone ) ) ) ) ).

% valid_mask.elims(1)
thf(fact_150_valid__mask_Osimps,axiom,
    ( valid_mask
    = ( ^ [Pi4: produc1457211279475724562t_char > prat] :
          ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi4 @ Hl ) )
          & ! [F3: list_char] :
              ( ( Pi4 @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
              = pnone ) ) ) ) ).

% valid_mask.simps
thf(fact_151_valid__maskI,axiom,
    ! [Pi2: produc1457211279475724562t_char > prat] :
      ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( Pi2 @ Hl2 ) )
     => ( ! [F: list_char] :
            ( ( Pi2 @ ( produc120197956887798346t_char @ null_nat @ F ) )
            = pnone )
       => ( valid_mask @ Pi2 ) ) ) ).

% valid_maskI
thf(fact_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_curryI,axiom,
    ! [F4: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( F4 @ ( produc120197956887798346t_char @ A @ B ) )
     => ( produc80901228256969370char_o @ F4 @ A @ B ) ) ).

% curryI
thf(fact_163_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_164_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_165_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_166_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_167_curryD,axiom,
    ! [F4: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( produc80901228256969370char_o @ F4 @ A @ B )
     => ( F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% curryD
thf(fact_168_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_169_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_170_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_171_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_172_curryE,axiom,
    ! [F4: produc1457211279475724562t_char > $o,A: nat,B: list_char] :
      ( ( produc80901228256969370char_o @ F4 @ A @ B )
     => ( F4 @ ( produc120197956887798346t_char @ A @ B ) ) ) ).

% curryE
thf(fact_173_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_174_valid__mask_Opelims_I1_J,axiom,
    ! [X: produc1457211279475724562t_char > prat,Y: $o] :
      ( ( ( valid_mask @ X )
        = Y )
     => ( ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X )
       => ~ ( ( Y
              = ( ! [Hl: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl ) )
                & ! [F3: list_char] :
                    ( ( X @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
                    = pnone ) ) )
           => ~ ( accp_P4765339447491148764r_prat @ valid_mask_rel @ X ) ) ) ) ).

% valid_mask.pelims(1)
thf(fact_175_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 )
           => ~ ( ! [Hl3: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl3 ) )
                & ! [F2: list_char] :
                    ( ( X @ ( produc120197956887798346t_char @ null_nat @ F2 ) )
                    = pnone ) ) ) ) ) ).

% valid_mask.pelims(2)
thf(fact_176_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 )
           => ( ! [Hl2: produc1457211279475724562t_char] : ( pgte @ pwrite @ ( X @ Hl2 ) )
              & ! [F: list_char] :
                  ( ( X @ ( produc120197956887798346t_char @ null_nat @ F ) )
                  = pnone ) ) ) ) ) ).

% valid_mask.pelims(3)
thf(fact_177_valid__null__def,axiom,
    ( valid_null
    = ( ^ [Pi4: produc1457211279475724562t_char > prat] :
        ! [F3: list_char] :
          ( ( Pi4 @ ( produc120197956887798346t_char @ null_nat @ F3 ) )
          = pnone ) ) ) ).

% valid_null_def
thf(fact_178_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_179_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_180_full__mask__def,axiom,
    ( full_mask
    = ( ^ [Hl: produc1457211279475724562t_char] :
          ( if_prat
          @ ( ( produc6204653154488278070t_char @ Hl )
            = null_nat )
          @ pnone
          @ pwrite ) ) ) ).

% full_mask_def
thf(fact_181_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_182_swap__simp,axiom,
    ! [X: list_char,Y: nat] :
      ( ( produc4410195085741832554ar_nat @ ( produc8702345860417625034ar_nat @ X @ Y ) )
      = ( produc120197956887798346t_char @ Y @ X ) ) ).

% swap_simp
thf(fact_183_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_184_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_185_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_186_swap__simp,axiom,
    ! [X: nat,Y: list_char] :
      ( ( produc5051419219066781674t_char @ ( produc120197956887798346t_char @ X @ Y ) )
      = ( produc8702345860417625034ar_nat @ Y @ X ) ) ).

% swap_simp
thf(fact_187_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_188_pre__get__m_Osimps,axiom,
    pre_ge2740421702909910421t_char = produc6204653154488278070t_char ).

% pre_get_m.simps
thf(fact_189_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_190_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_191_fst__conv,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_192_fst__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] :
      ( ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_193_fst__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_194_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_195_fst__conv,axiom,
    ! [X1: nat,X22: list_char] :
      ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_196_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_197_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_198_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_199_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_200_fst__eqD,axiom,
    ! [X: nat,Y: list_char,A: nat] :
      ( ( ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_201_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_202_eq__fst__iff,axiom,
    ! [A: set_Product_prod_o_o,P3: produc732395585841259969at_nat] :
      ( ( A
        = ( produc7275247657300178845at_nat @ P3 ) )
      = ( ? [B2: set_Pr1261947904930325089at_nat] :
            ( P3
            = ( produc8517790099723286449at_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_203_eq__fst__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,P3: produc1319942482725812455at_nat] :
      ( ( A
        = ( produc4486455499028916547at_nat @ P3 ) )
      = ( ? [B2: set_Pr4329608150637261639at_nat] :
            ( P3
            = ( produc9060074326276436823at_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_204_eq__fst__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P3: produc3843707927480180839at_nat] :
      ( ( A
        = ( produc6011863565407846595at_nat @ P3 ) )
      = ( ? [B2: set_Pr1261947904930325089at_nat] :
            ( P3
            = ( produc2922128104949294807at_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_205_eq__fst__iff,axiom,
    ! [A: nat,P3: product_prod_nat_nat] :
      ( ( A
        = ( product_fst_nat_nat @ P3 ) )
      = ( ? [B2: nat] :
            ( P3
            = ( product_Pair_nat_nat @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_206_eq__fst__iff,axiom,
    ! [A: nat,P3: produc1457211279475724562t_char] :
      ( ( A
        = ( produc6204653154488278070t_char @ P3 ) )
      = ( ? [B2: list_char] :
            ( P3
            = ( produc120197956887798346t_char @ A @ B2 ) ) ) ) ).

% eq_fst_iff
thf(fact_207_fstI,axiom,
    ! [X: produc732395585841259969at_nat,Y: set_Product_prod_o_o,Z2: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc8517790099723286449at_nat @ Y @ Z2 ) )
     => ( ( produc7275247657300178845at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_208_fstI,axiom,
    ! [X: produc1319942482725812455at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( X
        = ( produc9060074326276436823at_nat @ Y @ Z2 ) )
     => ( ( produc4486455499028916547at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_209_fstI,axiom,
    ! [X: produc3843707927480180839at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc2922128104949294807at_nat @ Y @ Z2 ) )
     => ( ( produc6011863565407846595at_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_210_fstI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z2: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z2 ) )
     => ( ( product_fst_nat_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_211_fstI,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat,Z2: list_char] :
      ( ( X
        = ( produc120197956887798346t_char @ Y @ Z2 ) )
     => ( ( produc6204653154488278070t_char @ X )
        = Y ) ) ).

% fstI
thf(fact_212_prod_Oswap__def,axiom,
    ( produc8291025974198835997od_o_o
    = ( ^ [P: produc7898102162437433997od_o_o] : ( produc8517790099723286449at_nat @ ( produc7327370413965865899od_o_o @ P ) @ ( produc827543935884356969od_o_o @ P ) ) ) ) ).

% prod.swap_def
thf(fact_213_prod_Oswap__def,axiom,
    ( produc5396952528824381687at_nat
    = ( ^ [P: produc1319942482725812455at_nat] : ( produc9060074326276436823at_nat @ ( produc826222791318859909at_nat @ P ) @ ( produc4486455499028916547at_nat @ P ) ) ) ) ).

% prod.swap_def
thf(fact_214_prod_Oswap__def,axiom,
    ( produc3946833401584108663at_nat
    = ( ^ [P: produc3843707927480180839at_nat] : ( produc2922128104949294807at_nat @ ( produc2937545259423399429at_nat @ P ) @ ( produc6011863565407846595at_nat @ P ) ) ) ) ).

% prod.swap_def
thf(fact_215_prod_Oswap__def,axiom,
    ( produc4410195085741832554ar_nat
    = ( ^ [P: produc7606469099237612818ar_nat] : ( produc120197956887798346t_char @ ( produc1259554233722553080ar_nat @ P ) @ ( produc5563429021163328950ar_nat @ P ) ) ) ) ).

% prod.swap_def
thf(fact_216_prod_Oswap__def,axiom,
    ( product_swap_nat_nat
    = ( ^ [P: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( product_snd_nat_nat @ P ) @ ( product_fst_nat_nat @ P ) ) ) ) ).

% prod.swap_def
thf(fact_217_prod_Oswap__def,axiom,
    ( produc5051419219066781674t_char
    = ( ^ [P: produc1457211279475724562t_char] : ( produc8702345860417625034ar_nat @ ( produc1900778367047502200t_char @ P ) @ ( produc6204653154488278070t_char @ P ) ) ) ) ).

% prod.swap_def
thf(fact_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_fst__swap,axiom,
    ! [X: produc7606469099237612818ar_nat] :
      ( ( produc6204653154488278070t_char @ ( produc4410195085741832554ar_nat @ X ) )
      = ( produc1259554233722553080ar_nat @ X ) ) ).

% fst_swap
thf(fact_228_snd__swap,axiom,
    ! [X: produc1457211279475724562t_char] :
      ( ( produc1259554233722553080ar_nat @ ( produc5051419219066781674t_char @ X ) )
      = ( produc6204653154488278070t_char @ X ) ) ).

% snd_swap
thf(fact_229_prod_Ocollapse,axiom,
    ! [Prod: produc732395585841259969at_nat] :
      ( ( produc8517790099723286449at_nat @ ( produc7275247657300178845at_nat @ Prod ) @ ( produc4551702098526911967at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_230_prod_Ocollapse,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_231_prod_Ocollapse,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_232_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_233_prod_Ocollapse,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_234_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P3: produc732395585841259969at_nat] :
      ( ( B
        = ( produc4551702098526911967at_nat @ P3 ) )
      = ( ? [A2: set_Product_prod_o_o] :
            ( P3
            = ( produc8517790099723286449at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_235_eq__snd__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,P3: produc1319942482725812455at_nat] :
      ( ( B
        = ( produc826222791318859909at_nat @ P3 ) )
      = ( ? [A2: set_Pr4329608150637261639at_nat] :
            ( P3
            = ( produc9060074326276436823at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_236_eq__snd__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,P3: produc3843707927480180839at_nat] :
      ( ( B
        = ( produc2937545259423399429at_nat @ P3 ) )
      = ( ? [A2: set_Pr1261947904930325089at_nat] :
            ( P3
            = ( produc2922128104949294807at_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_237_eq__snd__iff,axiom,
    ! [B: list_char,P3: produc1457211279475724562t_char] :
      ( ( B
        = ( produc1900778367047502200t_char @ P3 ) )
      = ( ? [A2: nat] :
            ( P3
            = ( produc120197956887798346t_char @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_238_eq__snd__iff,axiom,
    ! [B: nat,P3: product_prod_nat_nat] :
      ( ( B
        = ( product_snd_nat_nat @ P3 ) )
      = ( ? [A2: nat] :
            ( P3
            = ( product_Pair_nat_nat @ A2 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_239_sndI,axiom,
    ! [X: produc732395585841259969at_nat,Y: set_Product_prod_o_o,Z2: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc8517790099723286449at_nat @ Y @ Z2 ) )
     => ( ( produc4551702098526911967at_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_240_sndI,axiom,
    ! [X: produc1319942482725812455at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( X
        = ( produc9060074326276436823at_nat @ Y @ Z2 ) )
     => ( ( produc826222791318859909at_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_241_sndI,axiom,
    ! [X: produc3843707927480180839at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( X
        = ( produc2922128104949294807at_nat @ Y @ Z2 ) )
     => ( ( produc2937545259423399429at_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_242_sndI,axiom,
    ! [X: produc1457211279475724562t_char,Y: nat,Z2: list_char] :
      ( ( X
        = ( produc120197956887798346t_char @ Y @ Z2 ) )
     => ( ( produc1900778367047502200t_char @ X )
        = Z2 ) ) ).

% sndI
thf(fact_243_sndI,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Z2: nat] :
      ( ( X
        = ( product_Pair_nat_nat @ Y @ Z2 ) )
     => ( ( product_snd_nat_nat @ X )
        = Z2 ) ) ).

% sndI
thf(fact_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_snd__eqD,axiom,
    ! [X: nat,Y: list_char,A: list_char] :
      ( ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_254_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_255_snd__conv,axiom,
    ! [X1: set_Product_prod_o_o,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_256_snd__conv,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X22: set_Pr4329608150637261639at_nat] :
      ( ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_257_snd__conv,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X22: set_Pr1261947904930325089at_nat] :
      ( ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_258_snd__conv,axiom,
    ! [X1: nat,X22: list_char] :
      ( ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_259_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_260_prod__eq__iff,axiom,
    ( ( ^ [Y7: produc1457211279475724562t_char,Z3: produc1457211279475724562t_char] : ( Y7 = Z3 ) )
    = ( ^ [S3: produc1457211279475724562t_char,T: produc1457211279475724562t_char] :
          ( ( ( produc6204653154488278070t_char @ S3 )
            = ( produc6204653154488278070t_char @ T ) )
          & ( ( produc1900778367047502200t_char @ S3 )
            = ( produc1900778367047502200t_char @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_261_prod__eqI,axiom,
    ! [P3: produc1457211279475724562t_char,Q: produc1457211279475724562t_char] :
      ( ( ( produc6204653154488278070t_char @ P3 )
        = ( produc6204653154488278070t_char @ Q ) )
     => ( ( ( produc1900778367047502200t_char @ P3 )
          = ( produc1900778367047502200t_char @ Q ) )
       => ( P3 = Q ) ) ) ).

% prod_eqI
thf(fact_262_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_263_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: nat > nat > set_Pr1261947904930325089at_nat,P3: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc8197505143624133779at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_264_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: nat > list_char > set_Pr1261947904930325089at_nat,P3: produc1457211279475724562t_char] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc3665563422017311042at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P3
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_265_mem__case__prodE,axiom,
    ! [Z2: produc859450856879609959at_nat,C: nat > nat > set_Pr8693737435421807431at_nat,P3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( produc2423807293333670521at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_266_mem__case__prodE,axiom,
    ! [Z2: produc859450856879609959at_nat,C: nat > list_char > set_Pr8693737435421807431at_nat,P3: produc1457211279475724562t_char] :
      ( ( member8206827879206165904at_nat @ Z2 @ ( produc3763337624047832232at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P3
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8206827879206165904at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_267_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P3: produc732395585841259969at_nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc4192656640593693979at_nat @ C @ P3 ) )
     => ~ ! [X3: set_Product_prod_o_o,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P3
              = ( produc8517790099723286449at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_268_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,P3: produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc1433696742876942453at_nat @ C @ P3 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
            ( ( P3
              = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_269_mem__case__prodE,axiom,
    ! [Z2: produc3843707927480180839at_nat,C: nat > nat > set_Pr4329608150637261639at_nat,P3: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( produc3705604114177432185at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_270_mem__case__prodE,axiom,
    ! [Z2: produc2934264451710624999od_o_o,C: nat > nat > set_Pr1932065953672099015od_o_o,P3: product_prod_nat_nat] :
      ( ( member444158400953824016od_o_o @ Z2 @ ( produc773911639751745017od_o_o @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member444158400953824016od_o_o @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_271_mem__case__prodE,axiom,
    ! [Z2: produc732395585841259969at_nat,C: nat > nat > set_Pr457366540195662369at_nat,P3: product_prod_nat_nat] :
      ( ( member4095101504841534314at_nat @ Z2 @ ( produc1156700346287174227at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P3
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member4095101504841534314at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_272_mem__case__prodE,axiom,
    ! [Z2: produc3843707927480180839at_nat,C: nat > list_char > set_Pr4329608150637261639at_nat,P3: produc1457211279475724562t_char] :
      ( ( member8757157785044589968at_nat @ Z2 @ ( produc7856237077994235048at_nat @ C @ P3 ) )
     => ~ ! [X3: nat,Y3: list_char] :
            ( ( P3
              = ( produc120197956887798346t_char @ X3 @ Y3 ) )
           => ~ ( member8757157785044589968at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_273_prod_Oexhaust__sel,axiom,
    ! [Prod: produc732395585841259969at_nat] :
      ( Prod
      = ( produc8517790099723286449at_nat @ ( produc7275247657300178845at_nat @ Prod ) @ ( produc4551702098526911967at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_274_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1319942482725812455at_nat] :
      ( Prod
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ Prod ) @ ( produc826222791318859909at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_275_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3843707927480180839at_nat] :
      ( Prod
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ Prod ) @ ( produc2937545259423399429at_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_276_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_277_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1457211279475724562t_char] :
      ( Prod
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ Prod ) @ ( produc1900778367047502200t_char @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_278_surjective__pairing,axiom,
    ! [T2: produc732395585841259969at_nat] :
      ( T2
      = ( produc8517790099723286449at_nat @ ( produc7275247657300178845at_nat @ T2 ) @ ( produc4551702098526911967at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_279_surjective__pairing,axiom,
    ! [T2: produc1319942482725812455at_nat] :
      ( T2
      = ( produc9060074326276436823at_nat @ ( produc4486455499028916547at_nat @ T2 ) @ ( produc826222791318859909at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_280_surjective__pairing,axiom,
    ! [T2: produc3843707927480180839at_nat] :
      ( T2
      = ( produc2922128104949294807at_nat @ ( produc6011863565407846595at_nat @ T2 ) @ ( produc2937545259423399429at_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_281_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_282_surjective__pairing,axiom,
    ! [T2: produc1457211279475724562t_char] :
      ( T2
      = ( produc120197956887798346t_char @ ( produc6204653154488278070t_char @ T2 ) @ ( produc1900778367047502200t_char @ T2 ) ) ) ).

% surjective_pairing
thf(fact_283_exI__realizer,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > set_Product_prod_o_o > $o,Y: set_Pr1261947904930325089at_nat,X: set_Product_prod_o_o] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) ) @ ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_284_exI__realizer,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,Y: set_Pr4329608150637261639at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) ) @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_285_exI__realizer,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) ) @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_286_exI__realizer,axiom,
    ! [P2: nat > nat > $o,Y: nat,X: nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_287_exI__realizer,axiom,
    ! [P2: list_char > nat > $o,Y: list_char,X: nat] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ X @ Y ) ) @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_288_conjI__realizer,axiom,
    ! [P2: set_Product_prod_o_o > $o,P3: set_Product_prod_o_o,Q2: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat] :
      ( ( P2 @ P3 )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc7275247657300178845at_nat @ ( produc8517790099723286449at_nat @ P3 @ Q ) ) )
          & ( Q2 @ ( produc4551702098526911967at_nat @ ( produc8517790099723286449at_nat @ P3 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_289_conjI__realizer,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > $o,P3: set_Pr4329608150637261639at_nat,Q2: set_Pr4329608150637261639at_nat > $o,Q: set_Pr4329608150637261639at_nat] :
      ( ( P2 @ P3 )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc4486455499028916547at_nat @ ( produc9060074326276436823at_nat @ P3 @ Q ) ) )
          & ( Q2 @ ( produc826222791318859909at_nat @ ( produc9060074326276436823at_nat @ P3 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_290_conjI__realizer,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > $o,P3: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat] :
      ( ( P2 @ P3 )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6011863565407846595at_nat @ ( produc2922128104949294807at_nat @ P3 @ Q ) ) )
          & ( Q2 @ ( produc2937545259423399429at_nat @ ( produc2922128104949294807at_nat @ P3 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_291_conjI__realizer,axiom,
    ! [P2: nat > $o,P3: nat,Q2: nat > $o,Q: nat] :
      ( ( P2 @ P3 )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q ) ) )
          & ( Q2 @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P3 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_292_conjI__realizer,axiom,
    ! [P2: nat > $o,P3: nat,Q2: list_char > $o,Q: list_char] :
      ( ( P2 @ P3 )
     => ( ( Q2 @ Q )
       => ( ( P2 @ ( produc6204653154488278070t_char @ ( produc120197956887798346t_char @ P3 @ Q ) ) )
          & ( Q2 @ ( produc1900778367047502200t_char @ ( produc120197956887798346t_char @ P3 @ Q ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_293_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_Product_prod_o_o > set_Pr1261947904930325089at_nat > $o,X: set_Product_prod_o_o,Y: set_Pr1261947904930325089at_nat,A: produc732395585841259969at_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc8517790099723286449at_nat @ X @ Y ) )
       => ( P2 @ ( produc7275247657300178845at_nat @ A ) @ ( produc4551702098526911967at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_294_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A: produc1319942482725812455at_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc9060074326276436823at_nat @ X @ Y ) )
       => ( P2 @ ( produc4486455499028916547at_nat @ A ) @ ( produc826222791318859909at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_295_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc2922128104949294807at_nat @ X @ Y ) )
       => ( P2 @ ( produc6011863565407846595at_nat @ A ) @ ( produc2937545259423399429at_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_296_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > nat > $o,X: nat,Y: nat,A: product_prod_nat_nat] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( product_Pair_nat_nat @ X @ Y ) )
       => ( P2 @ ( product_fst_nat_nat @ A ) @ ( product_snd_nat_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_297_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: nat > list_char > $o,X: nat,Y: list_char,A: produc1457211279475724562t_char] :
      ( ( P2 @ X @ Y )
     => ( ( A
          = ( produc120197956887798346t_char @ X @ Y ) )
       => ( P2 @ ( produc6204653154488278070t_char @ A ) @ ( produc1900778367047502200t_char @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_298_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_299_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_300_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_301_surjD,axiom,
    ! [F4: nat > nat,Y: nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X3: nat] :
          ( Y
          = ( F4 @ X3 ) ) ) ).

% surjD
thf(fact_302_surjE,axiom,
    ! [F4: nat > nat,Y: nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X3: nat] :
            ( Y
           != ( F4 @ X3 ) ) ) ).

% surjE
thf(fact_303_surjI,axiom,
    ! [G: nat > nat,F4: nat > nat] :
      ( ! [X3: nat] :
          ( ( G @ ( F4 @ X3 ) )
          = X3 )
     => ( ( image_nat_nat2 @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_304_surj__def,axiom,
    ! [F4: nat > nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y5: nat] :
          ? [X2: nat] :
            ( Y5
            = ( F4 @ X2 ) ) ) ) ).

% surj_def
thf(fact_305_Collect__split__mono__strong,axiom,
    ! [X4: set_nat,A3: set_Pr2507339680178222706t_char,Y8: set_list_char,P2: 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 )
                 => ( ( P2 @ X3 @ Xa2 )
                   => ( Q2 @ X3 @ Xa2 ) ) ) )
         => ( ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ P2 ) ) )
           => ( ord_le6315511645215477266t_char @ A3 @ ( collec2458894688891239037t_char @ ( produc8526122873445342073char_o @ Q2 ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_306_map__prod__surj,axiom,
    ! [F4: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat2 @ 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_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_image2__eqI,axiom,
    ! [B: nat,F4: produc3843707927480180839at_nat > nat,X: produc3843707927480180839at_nat,C: nat,G: produc3843707927480180839at_nat > nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( B
        = ( F4 @ X ) )
     => ( ( C
          = ( G @ X ) )
       => ( ( member8757157785044589968at_nat @ X @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C ) @ ( bNF_Gr1074675703407240111at_nat @ A3 @ F4 @ G ) ) ) ) ) ).

% image2_eqI
thf(fact_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_surj__Compl__image__subset,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat2 @ F4 @ A3 ) ) @ ( image_nat_nat2 @ F4 @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_339_ssubsetI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ! [Pi5: product_prod_nat_nat,H: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Pi5 @ H ) @ A3 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Pi5 @ H ) @ B5 ) )
     => ( ord_le3000389064537975527at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_340_ssubsetI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,B5: set_Pr1932065953672099015od_o_o] :
      ( ! [Pi5: set_Product_prod_o_o,H: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Pi5 @ H ) @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Pi5 @ H ) @ B5 ) )
     => ( ord_le7586318239878589543od_o_o @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_341_ssubsetI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,B5: set_Pr457366540195662369at_nat] :
      ( ! [Pi5: set_Product_prod_o_o,H: set_Pr1261947904930325089at_nat] :
          ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Pi5 @ H ) @ A3 )
         => ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ Pi5 @ H ) @ B5 ) )
     => ( ord_le4841768313324958145at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_342_ssubsetI,axiom,
    ! [A3: set_Pr7459493094073627847at_nat,B5: set_Pr7459493094073627847at_nat] :
      ( ! [Pi5: set_Pr4329608150637261639at_nat,H: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Pi5 @ H ) @ A3 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Pi5 @ H ) @ B5 ) )
     => ( ord_le5997549366648089703at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_343_ssubsetI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ! [Pi5: set_Pr1261947904930325089at_nat,H: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi5 @ H ) @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Pi5 @ H ) @ B5 ) )
     => ( ord_le1268244103169919719at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_344_ssubsetI,axiom,
    ! [A3: set_Pr2507339680178222706t_char,B5: set_Pr2507339680178222706t_char] :
      ( ! [Pi5: nat,H: list_char] :
          ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi5 @ H ) @ A3 )
         => ( member484235747325421115t_char @ ( produc120197956887798346t_char @ Pi5 @ H ) @ B5 ) )
     => ( ord_le6315511645215477266t_char @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_345_ssubsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [Pi5: nat,H: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi5 @ H ) @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Pi5 @ H ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ A3 @ B5 ) ) ).

% ssubsetI
thf(fact_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_vimage__subsetD,axiom,
    ! [F4: nat > nat,B5: set_nat,A3: set_nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F4 @ B5 ) @ A3 )
       => ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F4 @ A3 ) ) ) ) ).

% vimage_subsetD
thf(fact_369_mono__add,axiom,
    ! [A: nat] : ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( plus_plus_nat @ A ) ) ).

% mono_add
thf(fact_370_acc__subset__induct,axiom,
    ! [D2: set_se491199571785101031od_o_o,R3: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,P2: set_Product_prod_o_o > $o] :
      ( ( ord_le2654036014095304327od_o_o @ D2 @ ( acc_se454823075469241932od_o_o @ R3 ) )
     => ( ! [X3: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ D2 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Z @ X3 ) @ R3 )
             => ( member3157015407008874800od_o_o @ Z @ D2 ) ) )
       => ( ( member3157015407008874800od_o_o @ X @ D2 )
         => ( ! [X3: set_Product_prod_o_o] :
                ( ( member3157015407008874800od_o_o @ X3 @ D2 )
               => ( ! [Z4: set_Product_prod_o_o] :
                      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_371_acc__subset__induct,axiom,
    ! [D2: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ D2 @ ( acc_Pr5580635297509630636at_nat @ R3 ) )
     => ( ! [X3: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ D2 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Z @ X3 ) @ R3 )
             => ( member8206827879206165904at_nat @ Z @ D2 ) ) )
       => ( ( member8206827879206165904at_nat @ X @ D2 )
         => ( ! [X3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X3 @ D2 )
               => ( ! [Z4: produc859450856879609959at_nat] :
                      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_372_acc__subset__induct,axiom,
    ! [D2: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ D2 @ ( acc_Pr2806007504492836550at_nat @ R3 ) )
     => ( ! [X3: product_prod_nat_nat,Z: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ D2 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z @ X3 ) @ R3 )
             => ( member8440522571783428010at_nat @ Z @ D2 ) ) )
       => ( ( member8440522571783428010at_nat @ X @ D2 )
         => ( ! [X3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X3 @ D2 )
               => ( ! [Z4: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_373_acc__subset__induct,axiom,
    ! [D2: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( ord_le1268244103169919719at_nat @ D2 @ ( acc_Pr8930264548482646700at_nat @ R3 ) )
     => ( ! [X3: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ D2 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Z @ X3 ) @ R3 )
             => ( member8757157785044589968at_nat @ Z @ D2 ) ) )
       => ( ( member8757157785044589968at_nat @ X @ D2 )
         => ( ! [X3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X3 @ D2 )
               => ( ! [Z4: produc3843707927480180839at_nat] :
                      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_374_acc__subset__induct,axiom,
    ! [D2: set_Pr1932065953672099015od_o_o,R3: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,P2: produc2934264451710624999od_o_o > $o] :
      ( ( ord_le7586318239878589543od_o_o @ D2 @ ( acc_Pr8465663490306642988od_o_o @ R3 ) )
     => ( ! [X3: produc2934264451710624999od_o_o,Z: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ D2 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Z @ X3 ) @ R3 )
             => ( member444158400953824016od_o_o @ Z @ D2 ) ) )
       => ( ( member444158400953824016od_o_o @ X @ D2 )
         => ( ! [X3: produc2934264451710624999od_o_o] :
                ( ( member444158400953824016od_o_o @ X3 @ D2 )
               => ( ! [Z4: produc2934264451710624999od_o_o] :
                      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_375_acc__subset__induct,axiom,
    ! [D2: set_Pr457366540195662369at_nat,R3: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,P2: produc732395585841259969at_nat > $o] :
      ( ( ord_le4841768313324958145at_nat @ D2 @ ( acc_Pr503387772527767174at_nat @ R3 ) )
     => ( ! [X3: produc732395585841259969at_nat,Z: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ D2 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Z @ X3 ) @ R3 )
             => ( member4095101504841534314at_nat @ Z @ D2 ) ) )
       => ( ( member4095101504841534314at_nat @ X @ D2 )
         => ( ! [X3: produc732395585841259969at_nat] :
                ( ( member4095101504841534314at_nat @ X3 @ D2 )
               => ( ! [Z4: produc732395585841259969at_nat] :
                      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_376_acc__subset__induct,axiom,
    ! [D2: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( ord_le7966958417815640519at_nat @ D2 @ ( acc_se5703560775919894924at_nat @ R3 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ D2 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Z @ X3 ) @ R3 )
             => ( member6273874620674182256at_nat @ Z @ D2 ) ) )
       => ( ( member6273874620674182256at_nat @ X @ D2 )
         => ( ! [X3: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ X3 @ D2 )
               => ( ! [Z4: set_Pr4329608150637261639at_nat] :
                      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_377_acc__subset__induct,axiom,
    ! [D2: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( ord_le2077887516847798113at_nat @ D2 @ ( acc_se3482671534671426598at_nat @ R3 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ D2 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z @ X3 ) @ R3 )
             => ( member2643936169264416010at_nat @ Z @ D2 ) ) )
       => ( ( member2643936169264416010at_nat @ X @ D2 )
         => ( ! [X3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X3 @ D2 )
               => ( ! [Z4: set_Pr1261947904930325089at_nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_378_acc__subset__induct,axiom,
    ! [D2: set_nat,R3: set_Pr1261947904930325089at_nat,X: nat,P2: nat > $o] :
      ( ( ord_less_eq_set_nat @ D2 @ ( acc_nat @ R3 ) )
     => ( ! [X3: nat,Z: nat] :
            ( ( member_nat @ X3 @ D2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ X3 ) @ R3 )
             => ( member_nat @ Z @ D2 ) ) )
       => ( ( member_nat @ X @ D2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ D2 )
               => ( ! [Z4: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z4 @ X3 ) @ R3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% acc_subset_induct
thf(fact_379_bij__is__inj,axiom,
    ! [F4: nat > nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ( inj_on_nat_nat @ F4 @ top_top_set_nat ) ) ).

% bij_is_inj
thf(fact_380_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_381_bij__betw__apply,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( bij_be1923491781212474806at_nat @ F4 @ A3 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_382_bij__betw__apply,axiom,
    ! [F4: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,B5: set_Pr1261947904930325089at_nat,A: produc859450856879609959at_nat] :
      ( ( bij_be5362278798767818832at_nat @ F4 @ A3 @ B5 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_383_bij__betw__apply,axiom,
    ! [F4: product_prod_nat_nat > produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( bij_be8998669371435553180at_nat @ F4 @ A3 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_384_bij__betw__apply,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat,A: product_prod_nat_nat] :
      ( ( bij_be7339570596641255836at_nat @ F4 @ A3 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_385_bij__betw__apply,axiom,
    ! [F4: product_prod_nat_nat > produc2934264451710624999od_o_o,A3: set_Pr1261947904930325089at_nat,B5: set_Pr1932065953672099015od_o_o,A: product_prod_nat_nat] :
      ( ( bij_be6955527471085564700od_o_o @ F4 @ A3 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member444158400953824016od_o_o @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_386_bij__betw__apply,axiom,
    ! [F4: product_prod_nat_nat > produc732395585841259969at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr457366540195662369at_nat,A: product_prod_nat_nat] :
      ( ( bij_be235880095552951670at_nat @ F4 @ A3 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member4095101504841534314at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_387_bij__betw__apply,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat] :
      ( ( bij_be3768671628604344400at_nat @ F4 @ A3 @ B5 )
     => ( ( member8757157785044589968at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_388_bij__betw__apply,axiom,
    ! [F4: produc2934264451710624999od_o_o > product_prod_nat_nat,A3: set_Pr1932065953672099015od_o_o,B5: set_Pr1261947904930325089at_nat,A: produc2934264451710624999od_o_o] :
      ( ( bij_be890175863832269520at_nat @ F4 @ A3 @ B5 )
     => ( ( member444158400953824016od_o_o @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_389_bij__betw__apply,axiom,
    ! [F4: produc732395585841259969at_nat > product_prod_nat_nat,A3: set_Pr457366540195662369at_nat,B5: set_Pr1261947904930325089at_nat,A: produc732395585841259969at_nat] :
      ( ( bij_be2725198553504426486at_nat @ F4 @ A3 @ B5 )
     => ( ( member4095101504841534314at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_390_bij__betw__apply,axiom,
    ! [F4: produc859450856879609959at_nat > produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( bij_be4529855521105804598at_nat @ F4 @ A3 @ B5 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ ( F4 @ A ) @ B5 ) ) ) ).

% bij_betw_apply
thf(fact_391_bij__betw__iff__bijections,axiom,
    ( bij_be1923491781212474806at_nat
    = ( ^ [F3: product_prod_nat_nat > product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ? [G2: product_prod_nat_nat > product_prod_nat_nat] :
          ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B6 )
             => ( ( member8440522571783428010at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_392_bij__betw__iff__bijections,axiom,
    ( bij_be8998669371435553180at_nat
    = ( ^ [F3: product_prod_nat_nat > produc859450856879609959at_nat,A6: set_Pr1261947904930325089at_nat,B6: set_Pr8693737435421807431at_nat] :
        ? [G2: produc859450856879609959at_nat > product_prod_nat_nat] :
          ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A6 )
             => ( ( member8206827879206165904at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ B6 )
             => ( ( member8440522571783428010at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_393_bij__betw__iff__bijections,axiom,
    ( bij_be5362278798767818832at_nat
    = ( ^ [F3: produc859450856879609959at_nat > product_prod_nat_nat,A6: set_Pr8693737435421807431at_nat,B6: set_Pr1261947904930325089at_nat] :
        ? [G2: product_prod_nat_nat > produc859450856879609959at_nat] :
          ( ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B6 )
             => ( ( member8206827879206165904at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_394_bij__betw__iff__bijections,axiom,
    ( bij_be3768671628604344400at_nat
    = ( ^ [F3: produc3843707927480180839at_nat > product_prod_nat_nat,A6: set_Pr4329608150637261639at_nat,B6: set_Pr1261947904930325089at_nat] :
        ? [G2: product_prod_nat_nat > produc3843707927480180839at_nat] :
          ( ! [X2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X2 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B6 )
             => ( ( member8757157785044589968at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_395_bij__betw__iff__bijections,axiom,
    ( bij_be890175863832269520at_nat
    = ( ^ [F3: produc2934264451710624999od_o_o > product_prod_nat_nat,A6: set_Pr1932065953672099015od_o_o,B6: set_Pr1261947904930325089at_nat] :
        ? [G2: product_prod_nat_nat > produc2934264451710624999od_o_o] :
          ( ! [X2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X2 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B6 )
             => ( ( member444158400953824016od_o_o @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_396_bij__betw__iff__bijections,axiom,
    ( bij_be2725198553504426486at_nat
    = ( ^ [F3: produc732395585841259969at_nat > product_prod_nat_nat,A6: set_Pr457366540195662369at_nat,B6: set_Pr1261947904930325089at_nat] :
        ? [G2: product_prod_nat_nat > produc732395585841259969at_nat] :
          ( ! [X2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X2 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B6 )
             => ( ( member4095101504841534314at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_397_bij__betw__iff__bijections,axiom,
    ( bij_be7339570596641255836at_nat
    = ( ^ [F3: product_prod_nat_nat > produc3843707927480180839at_nat,A6: set_Pr1261947904930325089at_nat,B6: set_Pr4329608150637261639at_nat] :
        ? [G2: produc3843707927480180839at_nat > product_prod_nat_nat] :
          ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A6 )
             => ( ( member8757157785044589968at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X2 @ B6 )
             => ( ( member8440522571783428010at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_398_bij__betw__iff__bijections,axiom,
    ( bij_be6955527471085564700od_o_o
    = ( ^ [F3: product_prod_nat_nat > produc2934264451710624999od_o_o,A6: set_Pr1261947904930325089at_nat,B6: set_Pr1932065953672099015od_o_o] :
        ? [G2: produc2934264451710624999od_o_o > product_prod_nat_nat] :
          ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A6 )
             => ( ( member444158400953824016od_o_o @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: produc2934264451710624999od_o_o] :
              ( ( member444158400953824016od_o_o @ X2 @ B6 )
             => ( ( member8440522571783428010at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_399_bij__betw__iff__bijections,axiom,
    ( bij_be235880095552951670at_nat
    = ( ^ [F3: product_prod_nat_nat > produc732395585841259969at_nat,A6: set_Pr1261947904930325089at_nat,B6: set_Pr457366540195662369at_nat] :
        ? [G2: produc732395585841259969at_nat > product_prod_nat_nat] :
          ( ! [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ A6 )
             => ( ( member4095101504841534314at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: produc732395585841259969at_nat] :
              ( ( member4095101504841534314at_nat @ X2 @ B6 )
             => ( ( member8440522571783428010at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_400_bij__betw__iff__bijections,axiom,
    ( bij_be4529855521105804598at_nat
    = ( ^ [F3: produc859450856879609959at_nat > produc859450856879609959at_nat,A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
        ? [G2: produc859450856879609959at_nat > produc859450856879609959at_nat] :
          ( ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ A6 )
             => ( ( member8206827879206165904at_nat @ ( F3 @ X2 ) @ B6 )
                & ( ( G2 @ ( F3 @ X2 ) )
                  = X2 ) ) )
          & ! [X2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X2 @ B6 )
             => ( ( member8206827879206165904at_nat @ ( G2 @ X2 ) @ A6 )
                & ( ( F3 @ ( G2 @ X2 ) )
                  = X2 ) ) ) ) ) ) ).

% bij_betw_iff_bijections
thf(fact_401_bij__def,axiom,
    ! [F4: nat > nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
      = ( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
        & ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% bij_def
thf(fact_402_bijI,axiom,
    ! [F4: nat > nat] :
      ( ( inj_on_nat_nat @ F4 @ top_top_set_nat )
     => ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
          = top_top_set_nat )
       => ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat ) ) ) ).

% bijI
thf(fact_403_ord_Omono__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,Less_eq: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,F4: produc859450856879609959at_nat > nat,R2: produc859450856879609959at_nat,S: produc859450856879609959at_nat] :
      ( ( monoto5881712244326581776at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
     => ( ( member8206827879206165904at_nat @ R2 @ A3 )
       => ( ( member8206827879206165904at_nat @ S @ A3 )
         => ( ( Less_eq @ R2 @ S )
           => ( ord_less_eq_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.mono_onD
thf(fact_404_ord_Omono__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Less_eq: product_prod_nat_nat > product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat,R2: product_prod_nat_nat,S: product_prod_nat_nat] :
      ( ( monoto3238940775400213162at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
     => ( ( member8440522571783428010at_nat @ R2 @ A3 )
       => ( ( member8440522571783428010at_nat @ S @ A3 )
         => ( ( Less_eq @ R2 @ S )
           => ( ord_less_eq_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.mono_onD
thf(fact_405_ord_Omono__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,Less_eq: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,F4: produc3843707927480180839at_nat > nat,R2: produc3843707927480180839at_nat,S: produc3843707927480180839at_nat] :
      ( ( monoto4038163285145374224at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
     => ( ( member8757157785044589968at_nat @ R2 @ A3 )
       => ( ( member8757157785044589968at_nat @ S @ A3 )
         => ( ( Less_eq @ R2 @ S )
           => ( ord_less_eq_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.mono_onD
thf(fact_406_ord_Omono__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,Less_eq: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > $o,F4: produc2934264451710624999od_o_o > nat,R2: produc2934264451710624999od_o_o,S: produc2934264451710624999od_o_o] :
      ( ( monoto3031015839379671696_o_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
     => ( ( member444158400953824016od_o_o @ R2 @ A3 )
       => ( ( member444158400953824016od_o_o @ S @ A3 )
         => ( ( Less_eq @ R2 @ S )
           => ( ord_less_eq_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.mono_onD
thf(fact_407_ord_Omono__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,Less_eq: produc732395585841259969at_nat > produc732395585841259969at_nat > $o,F4: produc732395585841259969at_nat > nat,R2: produc732395585841259969at_nat,S: produc732395585841259969at_nat] :
      ( ( monoto3172675574582585194at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
     => ( ( member4095101504841534314at_nat @ R2 @ A3 )
       => ( ( member4095101504841534314at_nat @ S @ A3 )
         => ( ( Less_eq @ R2 @ S )
           => ( ord_less_eq_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.mono_onD
thf(fact_408_ord_Omono__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,Less_eq: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,F4: produc859450856879609959at_nat > nat] :
      ( ! [R4: produc859450856879609959at_nat,S4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ R4 @ A3 )
         => ( ( member8206827879206165904at_nat @ S4 @ A3 )
           => ( ( Less_eq @ R4 @ S4 )
             => ( ord_less_eq_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto5881712244326581776at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 ) ) ).

% ord.mono_onI
thf(fact_409_ord_Omono__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Less_eq: product_prod_nat_nat > product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat] :
      ( ! [R4: product_prod_nat_nat,S4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ R4 @ A3 )
         => ( ( member8440522571783428010at_nat @ S4 @ A3 )
           => ( ( Less_eq @ R4 @ S4 )
             => ( ord_less_eq_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto3238940775400213162at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 ) ) ).

% ord.mono_onI
thf(fact_410_ord_Omono__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,Less_eq: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,F4: produc3843707927480180839at_nat > nat] :
      ( ! [R4: produc3843707927480180839at_nat,S4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ R4 @ A3 )
         => ( ( member8757157785044589968at_nat @ S4 @ A3 )
           => ( ( Less_eq @ R4 @ S4 )
             => ( ord_less_eq_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto4038163285145374224at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 ) ) ).

% ord.mono_onI
thf(fact_411_ord_Omono__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,Less_eq: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > $o,F4: produc2934264451710624999od_o_o > nat] :
      ( ! [R4: produc2934264451710624999od_o_o,S4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ R4 @ A3 )
         => ( ( member444158400953824016od_o_o @ S4 @ A3 )
           => ( ( Less_eq @ R4 @ S4 )
             => ( ord_less_eq_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto3031015839379671696_o_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 ) ) ).

% ord.mono_onI
thf(fact_412_ord_Omono__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,Less_eq: produc732395585841259969at_nat > produc732395585841259969at_nat > $o,F4: produc732395585841259969at_nat > nat] :
      ( ! [R4: produc732395585841259969at_nat,S4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ R4 @ A3 )
         => ( ( member4095101504841534314at_nat @ S4 @ A3 )
           => ( ( Less_eq @ R4 @ S4 )
             => ( ord_less_eq_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto3172675574582585194at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 ) ) ).

% ord.mono_onI
thf(fact_413_ord_Omono__on__def,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,Less_eq: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,F4: produc859450856879609959at_nat > nat] :
      ( ( monoto5881712244326581776at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
      = ( ! [R: produc859450856879609959at_nat,S3: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ R @ A3 )
              & ( member8206827879206165904at_nat @ S3 @ A3 )
              & ( Less_eq @ R @ S3 ) )
           => ( ord_less_eq_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.mono_on_def
thf(fact_414_ord_Omono__on__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Less_eq: product_prod_nat_nat > product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat] :
      ( ( monoto3238940775400213162at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
      = ( ! [R: product_prod_nat_nat,S3: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ R @ A3 )
              & ( member8440522571783428010at_nat @ S3 @ A3 )
              & ( Less_eq @ R @ S3 ) )
           => ( ord_less_eq_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.mono_on_def
thf(fact_415_ord_Omono__on__def,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,Less_eq: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,F4: produc3843707927480180839at_nat > nat] :
      ( ( monoto4038163285145374224at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
      = ( ! [R: produc3843707927480180839at_nat,S3: produc3843707927480180839at_nat] :
            ( ( ( member8757157785044589968at_nat @ R @ A3 )
              & ( member8757157785044589968at_nat @ S3 @ A3 )
              & ( Less_eq @ R @ S3 ) )
           => ( ord_less_eq_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.mono_on_def
thf(fact_416_ord_Omono__on__def,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,Less_eq: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > $o,F4: produc2934264451710624999od_o_o > nat] :
      ( ( monoto3031015839379671696_o_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
      = ( ! [R: produc2934264451710624999od_o_o,S3: produc2934264451710624999od_o_o] :
            ( ( ( member444158400953824016od_o_o @ R @ A3 )
              & ( member444158400953824016od_o_o @ S3 @ A3 )
              & ( Less_eq @ R @ S3 ) )
           => ( ord_less_eq_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.mono_on_def
thf(fact_417_ord_Omono__on__def,axiom,
    ! [A3: set_Pr457366540195662369at_nat,Less_eq: produc732395585841259969at_nat > produc732395585841259969at_nat > $o,F4: produc732395585841259969at_nat > nat] :
      ( ( monoto3172675574582585194at_nat @ A3 @ Less_eq @ ord_less_eq_nat @ F4 )
      = ( ! [R: produc732395585841259969at_nat,S3: produc732395585841259969at_nat] :
            ( ( ( member4095101504841534314at_nat @ R @ A3 )
              & ( member4095101504841534314at_nat @ S3 @ A3 )
              & ( Less_eq @ R @ S3 ) )
           => ( ord_less_eq_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.mono_on_def
thf(fact_418_mono__onD,axiom,
    ! [A3: set_nat,F4: nat > nat,R2: nat,S: nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( member_nat @ R2 @ A3 )
       => ( ( member_nat @ S @ A3 )
         => ( ( ord_less_eq_nat @ R2 @ S )
           => ( ord_less_eq_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% mono_onD
thf(fact_419_mono__onI,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ! [R4: nat,S4: nat] :
          ( ( member_nat @ R4 @ A3 )
         => ( ( member_nat @ S4 @ A3 )
           => ( ( ord_less_eq_nat @ R4 @ S4 )
             => ( ord_less_eq_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F4 ) ) ).

% mono_onI
thf(fact_420_bij__iff,axiom,
    ! [F4: nat > nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
      = ( ! [X2: nat] :
          ? [Y5: nat] :
            ( ( ( F4 @ Y5 )
              = X2 )
            & ! [Z5: nat] :
                ( ( ( F4 @ Z5 )
                  = X2 )
               => ( Z5 = Y5 ) ) ) ) ) ).

% bij_iff
thf(fact_421_bij__pointE,axiom,
    ! [F4: nat > nat,Y: nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ~ ! [X3: nat] :
            ( ( Y
              = ( F4 @ X3 ) )
           => ~ ! [X5: nat] :
                  ( ( Y
                    = ( F4 @ X5 ) )
                 => ( X5 = X3 ) ) ) ) ).

% bij_pointE
thf(fact_422_involuntory__imp__bij,axiom,
    ! [F4: nat > nat] :
      ( ! [X3: nat] :
          ( ( F4 @ ( F4 @ X3 ) )
          = X3 )
     => ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat ) ) ).

% involuntory_imp_bij
thf(fact_423_accp__subset__induct,axiom,
    ! [D2: ( produc1457211279475724562t_char > prat ) > $o,R3: ( produc1457211279475724562t_char > prat ) > ( produc1457211279475724562t_char > prat ) > $o,X: produc1457211279475724562t_char > prat,P2: ( produc1457211279475724562t_char > prat ) > $o] :
      ( ( ord_le2830794348966088778prat_o @ D2 @ ( accp_P4765339447491148764r_prat @ R3 ) )
     => ( ! [X3: produc1457211279475724562t_char > prat,Z: produc1457211279475724562t_char > prat] :
            ( ( D2 @ X3 )
           => ( ( R3 @ Z @ X3 )
             => ( D2 @ Z ) ) )
       => ( ( D2 @ X )
         => ( ! [X3: produc1457211279475724562t_char > prat] :
                ( ( D2 @ X3 )
               => ( ! [Z4: produc1457211279475724562t_char > prat] :
                      ( ( R3 @ Z4 @ X3 )
                     => ( P2 @ Z4 ) )
                 => ( P2 @ X3 ) ) )
           => ( P2 @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_424_vimage__subset__eq,axiom,
    ! [F4: nat > nat,B5: set_nat,A3: set_nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F4 @ B5 ) @ A3 )
        = ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F4 @ A3 ) ) ) ) ).

% vimage_subset_eq
thf(fact_425_acc__induct__rule,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o] :
      ( ( member3157015407008874800od_o_o @ A @ ( acc_se454823075469241932od_o_o @ R2 ) )
     => ( ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ ( acc_se454823075469241932od_o_o @ R2 ) )
           => ( ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_426_acc__induct__rule,axiom,
    ! [A: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
           => ( ! [Y4: produc859450856879609959at_nat] :
                  ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_427_acc__induct__rule,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
           => ( ! [Y4: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_428_acc__induct__rule,axiom,
    ! [A: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
           => ( ! [Y4: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_429_acc__induct__rule,axiom,
    ! [A: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,P2: produc2934264451710624999od_o_o > $o] :
      ( ( member444158400953824016od_o_o @ A @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
     => ( ! [X3: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
           => ( ! [Y4: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_430_acc__induct__rule,axiom,
    ! [A: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,P2: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( acc_Pr503387772527767174at_nat @ R2 ) )
     => ( ! [X3: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ ( acc_Pr503387772527767174at_nat @ R2 ) )
           => ( ! [Y4: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_431_acc__induct__rule,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( member6273874620674182256at_nat @ A @ ( acc_se5703560775919894924at_nat @ R2 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ ( acc_se5703560775919894924at_nat @ R2 ) )
           => ( ! [Y4: set_Pr4329608150637261639at_nat] :
                  ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_432_acc__induct__rule,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( member2643936169264416010at_nat @ A @ ( acc_se3482671534671426598at_nat @ R2 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ ( acc_se3482671534671426598at_nat @ R2 ) )
           => ( ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_433_acc__induct__rule,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( acc_nat @ R2 ) )
           => ( ! [Y4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct_rule
thf(fact_434_not__acc__down,axiom,
    ! [X: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ~ ( member3157015407008874800od_o_o @ X @ ( acc_se454823075469241932od_o_o @ R3 ) )
     => ~ ! [Z: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Z @ X ) @ R3 )
           => ( member3157015407008874800od_o_o @ Z @ ( acc_se454823075469241932od_o_o @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_435_not__acc__down,axiom,
    ! [X: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X @ ( acc_Pr5580635297509630636at_nat @ R3 ) )
     => ~ ! [Z: produc859450856879609959at_nat] :
            ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Z @ X ) @ R3 )
           => ( member8206827879206165904at_nat @ Z @ ( acc_Pr5580635297509630636at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_436_not__acc__down,axiom,
    ! [X: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( acc_Pr2806007504492836550at_nat @ R3 ) )
     => ~ ! [Z: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z @ X ) @ R3 )
           => ( member8440522571783428010at_nat @ Z @ ( acc_Pr2806007504492836550at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_437_not__acc__down,axiom,
    ! [X: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ ( acc_Pr8930264548482646700at_nat @ R3 ) )
     => ~ ! [Z: produc3843707927480180839at_nat] :
            ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Z @ X ) @ R3 )
           => ( member8757157785044589968at_nat @ Z @ ( acc_Pr8930264548482646700at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_438_not__acc__down,axiom,
    ! [X: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ~ ( member444158400953824016od_o_o @ X @ ( acc_Pr8465663490306642988od_o_o @ R3 ) )
     => ~ ! [Z: produc2934264451710624999od_o_o] :
            ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Z @ X ) @ R3 )
           => ( member444158400953824016od_o_o @ Z @ ( acc_Pr8465663490306642988od_o_o @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_439_not__acc__down,axiom,
    ! [X: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ~ ( member4095101504841534314at_nat @ X @ ( acc_Pr503387772527767174at_nat @ R3 ) )
     => ~ ! [Z: produc732395585841259969at_nat] :
            ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Z @ X ) @ R3 )
           => ( member4095101504841534314at_nat @ Z @ ( acc_Pr503387772527767174at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_440_not__acc__down,axiom,
    ! [X: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ~ ( member6273874620674182256at_nat @ X @ ( acc_se5703560775919894924at_nat @ R3 ) )
     => ~ ! [Z: set_Pr4329608150637261639at_nat] :
            ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Z @ X ) @ R3 )
           => ( member6273874620674182256at_nat @ Z @ ( acc_se5703560775919894924at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_441_not__acc__down,axiom,
    ! [X: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ~ ( member2643936169264416010at_nat @ X @ ( acc_se3482671534671426598at_nat @ R3 ) )
     => ~ ! [Z: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Z @ X ) @ R3 )
           => ( member2643936169264416010at_nat @ Z @ ( acc_se3482671534671426598at_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_442_not__acc__down,axiom,
    ! [X: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member_nat @ X @ ( acc_nat @ R3 ) )
     => ~ ! [Z: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z @ X ) @ R3 )
           => ( member_nat @ Z @ ( acc_nat @ R3 ) ) ) ) ).

% not_acc_down
thf(fact_443_acc__downward,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( acc_se454823075469241932od_o_o @ R2 ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( member3157015407008874800od_o_o @ A @ ( acc_se454823075469241932od_o_o @ R2 ) ) ) ) ).

% acc_downward
thf(fact_444_acc__downward,axiom,
    ! [B: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 )
       => ( member8206827879206165904at_nat @ A @ ( acc_Pr5580635297509630636at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_445_acc__downward,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_446_acc__downward,axiom,
    ! [B: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( member8757157785044589968at_nat @ A @ ( acc_Pr8930264548482646700at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_447_acc__downward,axiom,
    ! [B: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ B @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 )
       => ( member444158400953824016od_o_o @ A @ ( acc_Pr8465663490306642988od_o_o @ R2 ) ) ) ) ).

% acc_downward
thf(fact_448_acc__downward,axiom,
    ! [B: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ B @ ( acc_Pr503387772527767174at_nat @ R2 ) )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 )
       => ( member4095101504841534314at_nat @ A @ ( acc_Pr503387772527767174at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_449_acc__downward,axiom,
    ! [B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( member6273874620674182256at_nat @ B @ ( acc_se5703560775919894924at_nat @ R2 ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
       => ( member6273874620674182256at_nat @ A @ ( acc_se5703560775919894924at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_450_acc__downward,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( acc_se3482671534671426598at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( member2643936169264416010at_nat @ A @ ( acc_se3482671534671426598at_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_451_acc__downward,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( member_nat @ B @ ( acc_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( member_nat @ A @ ( acc_nat @ R2 ) ) ) ) ).

% acc_downward
thf(fact_452_acc__induct,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o] :
      ( ( member3157015407008874800od_o_o @ A @ ( acc_se454823075469241932od_o_o @ R2 ) )
     => ( ! [X3: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X3 @ ( acc_se454823075469241932od_o_o @ R2 ) )
           => ( ! [Y4: set_Product_prod_o_o] :
                  ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_453_acc__induct,axiom,
    ! [A: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat,P2: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
           => ( ! [Y4: produc859450856879609959at_nat] :
                  ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_454_acc__induct,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
           => ( ! [Y4: product_prod_nat_nat] :
                  ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_455_acc__induct,axiom,
    ! [A: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat,P2: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
           => ( ! [Y4: produc3843707927480180839at_nat] :
                  ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_456_acc__induct,axiom,
    ! [A: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o,P2: produc2934264451710624999od_o_o > $o] :
      ( ( member444158400953824016od_o_o @ A @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
     => ( ! [X3: produc2934264451710624999od_o_o] :
            ( ( member444158400953824016od_o_o @ X3 @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
           => ( ! [Y4: produc2934264451710624999od_o_o] :
                  ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_457_acc__induct,axiom,
    ! [A: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat,P2: produc732395585841259969at_nat > $o] :
      ( ( member4095101504841534314at_nat @ A @ ( acc_Pr503387772527767174at_nat @ R2 ) )
     => ( ! [X3: produc732395585841259969at_nat] :
            ( ( member4095101504841534314at_nat @ X3 @ ( acc_Pr503387772527767174at_nat @ R2 ) )
           => ( ! [Y4: produc732395585841259969at_nat] :
                  ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_458_acc__induct,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,P2: set_Pr4329608150637261639at_nat > $o] :
      ( ( member6273874620674182256at_nat @ A @ ( acc_se5703560775919894924at_nat @ R2 ) )
     => ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ ( acc_se5703560775919894924at_nat @ R2 ) )
           => ( ! [Y4: set_Pr4329608150637261639at_nat] :
                  ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_459_acc__induct,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o] :
      ( ( member2643936169264416010at_nat @ A @ ( acc_se3482671534671426598at_nat @ R2 ) )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ ( acc_se3482671534671426598at_nat @ R2 ) )
           => ( ! [Y4: set_Pr1261947904930325089at_nat] :
                  ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_460_acc__induct,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( acc_nat @ R2 ) )
           => ( ! [Y4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 )
                 => ( P2 @ Y4 ) )
             => ( P2 @ X3 ) ) )
       => ( P2 @ A ) ) ) ).

% acc_induct
thf(fact_461_acc_Ointros,axiom,
    ! [X: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [Y3: produc859450856879609959at_nat] :
          ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ X ) @ R2 )
         => ( member8206827879206165904at_nat @ Y3 @ ( acc_Pr5580635297509630636at_nat @ R2 ) ) )
     => ( member8206827879206165904at_nat @ X @ ( acc_Pr5580635297509630636at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_462_acc_Ointros,axiom,
    ! [X: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [Y3: produc3843707927480180839at_nat] :
          ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ X ) @ R2 )
         => ( member8757157785044589968at_nat @ Y3 @ ( acc_Pr8930264548482646700at_nat @ R2 ) ) )
     => ( member8757157785044589968at_nat @ X @ ( acc_Pr8930264548482646700at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_463_acc_Ointros,axiom,
    ! [X: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [Y3: produc2934264451710624999od_o_o] :
          ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ X ) @ R2 )
         => ( member444158400953824016od_o_o @ Y3 @ ( acc_Pr8465663490306642988od_o_o @ R2 ) ) )
     => ( member444158400953824016od_o_o @ X @ ( acc_Pr8465663490306642988od_o_o @ R2 ) ) ) ).

% acc.intros
thf(fact_464_acc_Ointros,axiom,
    ! [X: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [Y3: produc732395585841259969at_nat] :
          ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ X ) @ R2 )
         => ( member4095101504841534314at_nat @ Y3 @ ( acc_Pr503387772527767174at_nat @ R2 ) ) )
     => ( member4095101504841534314at_nat @ X @ ( acc_Pr503387772527767174at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_465_acc_Ointros,axiom,
    ! [X: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X ) @ R2 )
         => ( member8440522571783428010at_nat @ Y3 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) )
     => ( member8440522571783428010at_nat @ X @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_466_acc_Ointros,axiom,
    ! [X: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ! [Y3: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ X ) @ R2 )
         => ( member3157015407008874800od_o_o @ Y3 @ ( acc_se454823075469241932od_o_o @ R2 ) ) )
     => ( member3157015407008874800od_o_o @ X @ ( acc_se454823075469241932od_o_o @ R2 ) ) ) ).

% acc.intros
thf(fact_467_acc_Ointros,axiom,
    ! [X: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ! [Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ X ) @ R2 )
         => ( member6273874620674182256at_nat @ Y3 @ ( acc_se5703560775919894924at_nat @ R2 ) ) )
     => ( member6273874620674182256at_nat @ X @ ( acc_se5703560775919894924at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_468_acc_Ointros,axiom,
    ! [X: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X ) @ R2 )
         => ( member2643936169264416010at_nat @ Y3 @ ( acc_se3482671534671426598at_nat @ R2 ) ) )
     => ( member2643936169264416010at_nat @ X @ ( acc_se3482671534671426598at_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_469_acc_Ointros,axiom,
    ! [X: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R2 )
         => ( member_nat @ Y3 @ ( acc_nat @ R2 ) ) )
     => ( member_nat @ X @ ( acc_nat @ R2 ) ) ) ).

% acc.intros
thf(fact_470_acc_Osimps,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( acc_se454823075469241932od_o_o @ R2 ) )
      = ( ? [X2: set_Product_prod_o_o] :
            ( ( A = X2 )
            & ! [Y5: set_Product_prod_o_o] :
                ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X2 ) @ R2 )
               => ( member3157015407008874800od_o_o @ Y5 @ ( acc_se454823075469241932od_o_o @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_471_acc_Osimps,axiom,
    ! [A: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
      = ( ? [X2: produc859450856879609959at_nat] :
            ( ( A = X2 )
            & ! [Y5: produc859450856879609959at_nat] :
                ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y5 @ X2 ) @ R2 )
               => ( member8206827879206165904at_nat @ Y5 @ ( acc_Pr5580635297509630636at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_472_acc_Osimps,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
      = ( ? [X2: product_prod_nat_nat] :
            ( ( A = X2 )
            & ! [Y5: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R2 )
               => ( member8440522571783428010at_nat @ Y5 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_473_acc_Osimps,axiom,
    ! [A: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
      = ( ? [X2: produc3843707927480180839at_nat] :
            ( ( A = X2 )
            & ! [Y5: produc3843707927480180839at_nat] :
                ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ X2 ) @ R2 )
               => ( member8757157785044589968at_nat @ Y5 @ ( acc_Pr8930264548482646700at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_474_acc_Osimps,axiom,
    ! [A: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member444158400953824016od_o_o @ A @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
      = ( ? [X2: produc2934264451710624999od_o_o] :
            ( ( A = X2 )
            & ! [Y5: produc2934264451710624999od_o_o] :
                ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y5 @ X2 ) @ R2 )
               => ( member444158400953824016od_o_o @ Y5 @ ( acc_Pr8465663490306642988od_o_o @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_475_acc_Osimps,axiom,
    ! [A: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member4095101504841534314at_nat @ A @ ( acc_Pr503387772527767174at_nat @ R2 ) )
      = ( ? [X2: produc732395585841259969at_nat] :
            ( ( A = X2 )
            & ! [Y5: produc732395585841259969at_nat] :
                ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y5 @ X2 ) @ R2 )
               => ( member4095101504841534314at_nat @ Y5 @ ( acc_Pr503387772527767174at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_476_acc_Osimps,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( acc_se5703560775919894924at_nat @ R2 ) )
      = ( ? [X2: set_Pr4329608150637261639at_nat] :
            ( ( A = X2 )
            & ! [Y5: set_Pr4329608150637261639at_nat] :
                ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X2 ) @ R2 )
               => ( member6273874620674182256at_nat @ Y5 @ ( acc_se5703560775919894924at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_477_acc_Osimps,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( acc_se3482671534671426598at_nat @ R2 ) )
      = ( ? [X2: set_Pr1261947904930325089at_nat] :
            ( ( A = X2 )
            & ! [Y5: set_Pr1261947904930325089at_nat] :
                ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R2 )
               => ( member2643936169264416010at_nat @ Y5 @ ( acc_se3482671534671426598at_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_478_acc_Osimps,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
      = ( ? [X2: nat] :
            ( ( A = X2 )
            & ! [Y5: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 )
               => ( member_nat @ Y5 @ ( acc_nat @ R2 ) ) ) ) ) ) ).

% acc.simps
thf(fact_479_acc_Ocases,axiom,
    ! [A: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ ( acc_se454823075469241932od_o_o @ R2 ) )
     => ! [Y4: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ A ) @ R2 )
         => ( member3157015407008874800od_o_o @ Y4 @ ( acc_se454823075469241932od_o_o @ R2 ) ) ) ) ).

% acc.cases
thf(fact_480_acc_Ocases,axiom,
    ! [A: produc859450856879609959at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( acc_Pr5580635297509630636at_nat @ R2 ) )
     => ! [Y4: produc859450856879609959at_nat] :
          ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y4 @ A ) @ R2 )
         => ( member8206827879206165904at_nat @ Y4 @ ( acc_Pr5580635297509630636at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_481_acc_Ocases,axiom,
    ! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( acc_Pr2806007504492836550at_nat @ R2 ) )
     => ! [Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ A ) @ R2 )
         => ( member8440522571783428010at_nat @ Y4 @ ( acc_Pr2806007504492836550at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_482_acc_Ocases,axiom,
    ! [A: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( acc_Pr8930264548482646700at_nat @ R2 ) )
     => ! [Y4: produc3843707927480180839at_nat] :
          ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ A ) @ R2 )
         => ( member8757157785044589968at_nat @ Y4 @ ( acc_Pr8930264548482646700at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_483_acc_Ocases,axiom,
    ! [A: produc2934264451710624999od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ( member444158400953824016od_o_o @ A @ ( acc_Pr8465663490306642988od_o_o @ R2 ) )
     => ! [Y4: produc2934264451710624999od_o_o] :
          ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ A ) @ R2 )
         => ( member444158400953824016od_o_o @ Y4 @ ( acc_Pr8465663490306642988od_o_o @ R2 ) ) ) ) ).

% acc.cases
thf(fact_484_acc_Ocases,axiom,
    ! [A: produc732395585841259969at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ( member4095101504841534314at_nat @ A @ ( acc_Pr503387772527767174at_nat @ R2 ) )
     => ! [Y4: produc732395585841259969at_nat] :
          ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ A ) @ R2 )
         => ( member4095101504841534314at_nat @ Y4 @ ( acc_Pr503387772527767174at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_485_acc_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( acc_se5703560775919894924at_nat @ R2 ) )
     => ! [Y4: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y4 @ A ) @ R2 )
         => ( member6273874620674182256at_nat @ Y4 @ ( acc_se5703560775919894924at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_486_acc_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( acc_se3482671534671426598at_nat @ R2 ) )
     => ! [Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ A ) @ R2 )
         => ( member2643936169264416010at_nat @ Y4 @ ( acc_se3482671534671426598at_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_487_acc_Ocases,axiom,
    ! [A: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( acc_nat @ R2 ) )
     => ! [Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ A ) @ R2 )
         => ( member_nat @ Y4 @ ( acc_nat @ R2 ) ) ) ) ).

% acc.cases
thf(fact_488_mono__imp__mono__on,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F4 ) ) ).

% mono_imp_mono_on
thf(fact_489_monoI,axiom,
    ! [F4: nat > nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
         => ( ord_less_eq_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 ) ) ).

% monoI
thf(fact_490_monoE,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( ord_less_eq_nat @ X @ Y )
       => ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ) ).

% monoE
thf(fact_491_monoD,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( ord_less_eq_nat @ X @ Y )
       => ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ) ).

% monoD
thf(fact_492_mono__on__subset,axiom,
    ! [A3: set_nat,F4: nat > nat,B5: set_nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ B5 @ A3 )
       => ( monotone_on_nat_nat @ B5 @ ord_less_eq_nat @ ord_less_eq_nat @ F4 ) ) ) ).

% mono_on_subset
thf(fact_493_bij__betw__imp__surj,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( bij_betw_nat_nat @ F4 @ A3 @ top_top_set_nat )
     => ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% bij_betw_imp_surj
thf(fact_494_bij__is__surj,axiom,
    ! [F4: nat > nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% bij_is_surj
thf(fact_495_bij__image__Compl__eq,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ( ( image_nat_nat2 @ F4 @ ( uminus5710092332889474511et_nat @ A3 ) )
        = ( uminus5710092332889474511et_nat @ ( image_nat_nat2 @ F4 @ A3 ) ) ) ) ).

% bij_image_Compl_eq
thf(fact_496_surj__image__vimage__eq,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( image_nat_nat2 @ F4 @ ( vimage_nat_nat @ F4 @ A3 ) )
        = A3 ) ) ).

% surj_image_vimage_eq
thf(fact_497_inj__image__mem__iff,axiom,
    ! [F4: nat > product_prod_nat_nat,A: nat,A3: set_nat] :
      ( ( inj_on5538052773655684606at_nat @ F4 @ top_top_set_nat )
     => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_5846123807819985514at_nat @ F4 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_498_inj__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F4 @ top_to4669805908274784177at_nat )
     => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_5168914502847457605at_nat @ F4 @ A3 ) )
        = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_499_inj__image__mem__iff,axiom,
    ! [F4: nat > produc859450856879609959at_nat,A: nat,A3: set_nat] :
      ( ( inj_on6829720656803765732at_nat @ F4 @ top_top_set_nat )
     => ( ( member8206827879206165904at_nat @ ( F4 @ A ) @ ( image_3276603626957510736at_nat @ F4 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_500_inj__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > produc859450856879609959at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F4 @ top_to4669805908274784177at_nat )
     => ( ( member8206827879206165904at_nat @ ( F4 @ A ) @ ( image_1119892238107900587at_nat @ F4 @ A3 ) )
        = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_501_inj__image__mem__iff,axiom,
    ! [F4: produc859450856879609959at_nat > product_prod_nat_nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F4 @ top_to8454300755403694231at_nat )
     => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_6706873702294942047at_nat @ F4 @ A3 ) )
        = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_502_inj__image__mem__iff,axiom,
    ! [F4: nat > produc3843707927480180839at_nat,A: nat,A3: set_nat] :
      ( ( inj_on7264072676751113700at_nat @ F4 @ top_top_set_nat )
     => ( ( member8757157785044589968at_nat @ ( F4 @ A ) @ ( image_2499013659892586576at_nat @ F4 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_503_inj__image__mem__iff,axiom,
    ! [F4: nat > produc2934264451710624999od_o_o,A: nat,A3: set_nat] :
      ( ( inj_on2663835283014092644od_o_o @ F4 @ top_top_set_nat )
     => ( ( member444158400953824016od_o_o @ ( F4 @ A ) @ ( image_3802643350302125520od_o_o @ F4 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_504_inj__image__mem__iff,axiom,
    ! [F4: nat > produc732395585841259969at_nat,A: nat,A3: set_nat] :
      ( ( inj_on400939309715711422at_nat @ F4 @ top_top_set_nat )
     => ( ( member4095101504841534314at_nat @ ( F4 @ A ) @ ( image_6796201771609466410at_nat @ F4 @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_505_inj__image__mem__iff,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F4 @ top_to6833984726390702231at_nat )
     => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_7325330684212377951at_nat @ F4 @ A3 ) )
        = ( member8757157785044589968at_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_506_inj__image__mem__iff,axiom,
    ! [F4: produc2934264451710624999od_o_o > product_prod_nat_nat,A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on8923534230957082699at_nat @ F4 @ top_to271382412937794071od_o_o )
     => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_2029717158088452831at_nat @ F4 @ A3 ) )
        = ( member444158400953824016od_o_o @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_507_range__ex1__eq,axiom,
    ! [F4: nat > produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( inj_on6829720656803765732at_nat @ F4 @ top_top_set_nat )
     => ( ( member8206827879206165904at_nat @ B @ ( image_3276603626957510736at_nat @ F4 @ top_top_set_nat ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F4 @ X2 ) )
              & ! [Y5: nat] :
                  ( ( B
                    = ( F4 @ Y5 ) )
                 => ( Y5 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_508_range__ex1__eq,axiom,
    ! [F4: nat > product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( inj_on5538052773655684606at_nat @ F4 @ top_top_set_nat )
     => ( ( member8440522571783428010at_nat @ B @ ( image_5846123807819985514at_nat @ F4 @ top_top_set_nat ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F4 @ X2 ) )
              & ! [Y5: nat] :
                  ( ( B
                    = ( F4 @ Y5 ) )
                 => ( Y5 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_509_range__ex1__eq,axiom,
    ! [F4: nat > produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( inj_on7264072676751113700at_nat @ F4 @ top_top_set_nat )
     => ( ( member8757157785044589968at_nat @ B @ ( image_2499013659892586576at_nat @ F4 @ top_top_set_nat ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F4 @ X2 ) )
              & ! [Y5: nat] :
                  ( ( B
                    = ( F4 @ Y5 ) )
                 => ( Y5 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_510_range__ex1__eq,axiom,
    ! [F4: nat > produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( inj_on2663835283014092644od_o_o @ F4 @ top_top_set_nat )
     => ( ( member444158400953824016od_o_o @ B @ ( image_3802643350302125520od_o_o @ F4 @ top_top_set_nat ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F4 @ X2 ) )
              & ! [Y5: nat] :
                  ( ( B
                    = ( F4 @ Y5 ) )
                 => ( Y5 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_511_range__ex1__eq,axiom,
    ! [F4: nat > produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( inj_on400939309715711422at_nat @ F4 @ top_top_set_nat )
     => ( ( member4095101504841534314at_nat @ B @ ( image_6796201771609466410at_nat @ F4 @ top_top_set_nat ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F4 @ X2 ) )
              & ! [Y5: nat] :
                  ( ( B
                    = ( F4 @ Y5 ) )
                 => ( Y5 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_512_inj__on__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F4 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_5168914502847457605at_nat @ F4 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_513_inj__on__image__mem__iff,axiom,
    ! [F4: produc859450856879609959at_nat > product_prod_nat_nat,B5: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F4 @ B5 )
     => ( ( member8206827879206165904at_nat @ A @ B5 )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B5 )
         => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_6706873702294942047at_nat @ F4 @ A3 ) )
            = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_514_inj__on__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > produc859450856879609959at_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F4 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( ( member8206827879206165904at_nat @ ( F4 @ A ) @ ( image_1119892238107900587at_nat @ F4 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_515_inj__on__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F4 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( ( member8757157785044589968at_nat @ ( F4 @ A ) @ ( image_1672857615394513579at_nat @ F4 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_516_inj__on__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > produc2934264451710624999od_o_o,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on5765513801355602071od_o_o @ F4 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( ( member444158400953824016od_o_o @ ( F4 @ A ) @ ( image_8095068765341748011od_o_o @ F4 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_517_inj__on__image__mem__iff,axiom,
    ! [F4: product_prod_nat_nat > produc732395585841259969at_nat,B5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on7902545142154350961at_nat @ F4 @ B5 )
     => ( ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( ( member4095101504841534314at_nat @ ( F4 @ A ) @ ( image_1912541218521307653at_nat @ F4 @ A3 ) )
            = ( member8440522571783428010at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_518_inj__on__image__mem__iff,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,B5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F4 @ B5 )
     => ( ( member8757157785044589968at_nat @ A @ B5 )
       => ( ( ord_le1268244103169919719at_nat @ A3 @ B5 )
         => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_7325330684212377951at_nat @ F4 @ A3 ) )
            = ( member8757157785044589968at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_519_inj__on__image__mem__iff,axiom,
    ! [F4: produc2934264451710624999od_o_o > product_prod_nat_nat,B5: set_Pr1932065953672099015od_o_o,A: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on8923534230957082699at_nat @ F4 @ B5 )
     => ( ( member444158400953824016od_o_o @ A @ B5 )
       => ( ( ord_le7586318239878589543od_o_o @ A3 @ B5 )
         => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_2029717158088452831at_nat @ F4 @ A3 ) )
            = ( member444158400953824016od_o_o @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_520_inj__on__image__mem__iff,axiom,
    ! [F4: produc732395585841259969at_nat > product_prod_nat_nat,B5: set_Pr457366540195662369at_nat,A: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( inj_on1168491563251049969at_nat @ F4 @ B5 )
     => ( ( member4095101504841534314at_nat @ A @ B5 )
       => ( ( ord_le4841768313324958145at_nat @ A3 @ B5 )
         => ( ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_4401859676472782469at_nat @ F4 @ A3 ) )
            = ( member4095101504841534314at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_521_inj__on__image__mem__iff,axiom,
    ! [F4: produc859450856879609959at_nat > produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( inj_on7797493862353638193at_nat @ F4 @ B5 )
     => ( ( member8206827879206165904at_nat @ A @ B5 )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B5 )
         => ( ( member8206827879206165904at_nat @ ( F4 @ A ) @ ( image_6321802852832981957at_nat @ F4 @ A3 ) )
            = ( member8206827879206165904at_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_522_the__inv__into__into,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F4 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_5168914502847457605at_nat @ F4 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( member8440522571783428010at_nat @ ( the_in1699675864273310451at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_523_the__inv__into__into,axiom,
    ! [F4: product_prod_nat_nat > produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,X: produc859450856879609959at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F4 @ A3 )
     => ( ( member8206827879206165904at_nat @ X @ ( image_1119892238107900587at_nat @ F4 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( member8440522571783428010at_nat @ ( the_in4366950361296922457at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_524_the__inv__into__into,axiom,
    ! [F4: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( inj_on229527429751349963at_nat @ F4 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_6706873702294942047at_nat @ F4 @ A3 ) )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B5 )
         => ( member8206827879206165904at_nat @ ( the_in730559788629188109at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_525_the__inv__into__into,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,X: product_prod_nat_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F4 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_7325330684212377951at_nat @ F4 @ A3 ) )
       => ( ( ord_le1268244103169919719at_nat @ A3 @ B5 )
         => ( member8757157785044589968at_nat @ ( the_in6287914319223354893at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_526_the__inv__into__into,axiom,
    ! [F4: produc2934264451710624999od_o_o > product_prod_nat_nat,A3: set_Pr1932065953672099015od_o_o,X: product_prod_nat_nat,B5: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on8923534230957082699at_nat @ F4 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_2029717158088452831at_nat @ F4 @ A3 ) )
       => ( ( ord_le7586318239878589543od_o_o @ A3 @ B5 )
         => ( member444158400953824016od_o_o @ ( the_in6866092425208432013at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_527_the__inv__into__into,axiom,
    ! [F4: produc732395585841259969at_nat > product_prod_nat_nat,A3: set_Pr457366540195662369at_nat,X: product_prod_nat_nat,B5: set_Pr457366540195662369at_nat] :
      ( ( inj_on1168491563251049969at_nat @ F4 @ A3 )
     => ( ( member8440522571783428010at_nat @ X @ ( image_4401859676472782469at_nat @ F4 @ A3 ) )
       => ( ( ord_le4841768313324958145at_nat @ A3 @ B5 )
         => ( member4095101504841534314at_nat @ ( the_in8584769135873776179at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_528_the__inv__into__into,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat,X: produc3843707927480180839at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F4 @ A3 )
     => ( ( member8757157785044589968at_nat @ X @ ( image_1672857615394513579at_nat @ F4 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( member8440522571783428010at_nat @ ( the_in635441250405490521at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_529_the__inv__into__into,axiom,
    ! [F4: product_prod_nat_nat > produc2934264451710624999od_o_o,A3: set_Pr1261947904930325089at_nat,X: produc2934264451710624999od_o_o,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on5765513801355602071od_o_o @ F4 @ A3 )
     => ( ( member444158400953824016od_o_o @ X @ ( image_8095068765341748011od_o_o @ F4 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( member8440522571783428010at_nat @ ( the_in3708071995606951385od_o_o @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_530_the__inv__into__into,axiom,
    ! [F4: product_prod_nat_nat > produc732395585841259969at_nat,A3: set_Pr1261947904930325089at_nat,X: produc732395585841259969at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on7902545142154350961at_nat @ F4 @ A3 )
     => ( ( member4095101504841534314at_nat @ X @ ( image_1912541218521307653at_nat @ F4 @ A3 ) )
       => ( ( ord_le3146513528884898305at_nat @ A3 @ B5 )
         => ( member8440522571783428010at_nat @ ( the_in6095450677922301363at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_531_the__inv__into__into,axiom,
    ! [F4: produc859450856879609959at_nat > produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( inj_on7797493862353638193at_nat @ F4 @ A3 )
     => ( ( member8206827879206165904at_nat @ X @ ( image_6321802852832981957at_nat @ F4 @ A3 ) )
       => ( ( ord_le3000389064537975527at_nat @ A3 @ B5 )
         => ( member8206827879206165904at_nat @ ( the_in7149738279505699699at_nat @ A3 @ F4 @ X ) @ B5 ) ) ) ) ).

% the_inv_into_into
thf(fact_532_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_533_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_534_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_535_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_536_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_537_surj__vimage__empty,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_5168914502847457605at_nat @ F4 @ top_to4669805908274784177at_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( vimage2449269961533847803at_nat @ F4 @ A3 )
          = bot_bo2099793752762293965at_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_538_surj__vimage__empty,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_1672857615394513579at_nat @ F4 @ top_to4669805908274784177at_nat )
        = top_to6833984726390702231at_nat )
     => ( ( ( vimage1489446588345997665at_nat @ F4 @ A3 )
          = bot_bo2099793752762293965at_nat )
        = ( A3 = bot_bo228742789529271731at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_539_surj__vimage__empty,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_7325330684212377951at_nat @ F4 @ top_to6833984726390702231at_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( vimage7141919657163862037at_nat @ F4 @ A3 )
          = bot_bo228742789529271731at_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_540_surj__vimage__empty,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_960689587819713477at_nat @ F4 @ top_to6833984726390702231at_nat )
        = top_to6833984726390702231at_nat )
     => ( ( ( vimage6896080417876799867at_nat @ F4 @ A3 )
          = bot_bo228742789529271731at_nat )
        = ( A3 = bot_bo228742789529271731at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_541_surj__vimage__empty,axiom,
    ! [F4: product_prod_nat_nat > nat,A3: set_nat] :
      ( ( ( image_2486076414777270412at_nat @ F4 @ top_to4669805908274784177at_nat )
        = top_top_set_nat )
     => ( ( ( vimage4653281326611754070at_nat @ F4 @ A3 )
          = bot_bo2099793752762293965at_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_542_surj__vimage__empty,axiom,
    ! [F4: produc3843707927480180839at_nat > nat,A3: set_nat] :
      ( ( ( image_3198525501578754290at_nat @ F4 @ top_to6833984726390702231at_nat )
        = top_top_set_nat )
     => ( ( ( vimage7134676753939176892at_nat @ F4 @ A3 )
          = bot_bo228742789529271731at_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_543_surj__vimage__empty,axiom,
    ! [F4: nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_5846123807819985514at_nat @ F4 @ top_top_set_nat )
        = top_to4669805908274784177at_nat )
     => ( ( ( vimage8013328719654469172at_nat @ F4 @ A3 )
          = bot_bot_set_nat )
        = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_544_surj__vimage__empty,axiom,
    ! [F4: nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_2499013659892586576at_nat @ F4 @ top_top_set_nat )
        = top_to6833984726390702231at_nat )
     => ( ( ( vimage6435164912253009178at_nat @ F4 @ A3 )
          = bot_bot_set_nat )
        = ( A3 = bot_bo228742789529271731at_nat ) ) ) ).

% surj_vimage_empty
thf(fact_545_surj__vimage__empty,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( vimage_nat_nat @ F4 @ A3 )
          = bot_bot_set_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% surj_vimage_empty
thf(fact_546_strict__mono__imp__inj__on,axiom,
    ! [F4: nat > nat,A3: set_nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( inj_on_nat_nat @ F4 @ A3 ) ) ).

% strict_mono_imp_inj_on
thf(fact_547_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_548_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_549_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_550_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_551_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_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_fst__apfst,axiom,
    ! [F4: nat > nat,X: produc1457211279475724562t_char] :
      ( ( produc6204653154488278070t_char @ ( produc2835005367637779135t_char @ F4 @ X ) )
      = ( F4 @ ( produc6204653154488278070t_char @ X ) ) ) ).

% fst_apfst
thf(fact_571_fst__apsnd,axiom,
    ! [F4: list_char > list_char,X: produc1457211279475724562t_char] :
      ( ( produc6204653154488278070t_char @ ( produc5571979602231644306ar_nat @ F4 @ X ) )
      = ( produc6204653154488278070t_char @ X ) ) ).

% fst_apsnd
thf(fact_572_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_573_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_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_strict__mono__on__eqD,axiom,
    ! [A3: set_nat,F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( ( F4 @ X )
          = ( F4 @ Y ) )
       => ( ( member_nat @ X @ A3 )
         => ( ( member_nat @ Y @ A3 )
           => ( Y = X ) ) ) ) ) ).

% strict_mono_on_eqD
thf(fact_583_strict__mono__onI,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ! [R4: nat,S4: nat] :
          ( ( member_nat @ R4 @ A3 )
         => ( ( member_nat @ S4 @ A3 )
           => ( ( ord_less_nat @ R4 @ S4 )
             => ( ord_less_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F4 ) ) ).

% strict_mono_onI
thf(fact_584_strict__mono__onD,axiom,
    ! [A3: set_nat,F4: nat > nat,R2: nat,S: nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( member_nat @ R2 @ A3 )
       => ( ( member_nat @ S @ A3 )
         => ( ( ord_less_nat @ R2 @ S )
           => ( ord_less_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% strict_mono_onD
thf(fact_585_ord_Ostrict__mono__on__def,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,Less: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,F4: produc859450856879609959at_nat > nat] :
      ( ( monoto5881712244326581776at_nat @ A3 @ Less @ ord_less_nat @ F4 )
      = ( ! [R: produc859450856879609959at_nat,S3: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ R @ A3 )
              & ( member8206827879206165904at_nat @ S3 @ A3 )
              & ( Less @ R @ S3 ) )
           => ( ord_less_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.strict_mono_on_def
thf(fact_586_ord_Ostrict__mono__on__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Less: product_prod_nat_nat > product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat] :
      ( ( monoto3238940775400213162at_nat @ A3 @ Less @ ord_less_nat @ F4 )
      = ( ! [R: product_prod_nat_nat,S3: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ R @ A3 )
              & ( member8440522571783428010at_nat @ S3 @ A3 )
              & ( Less @ R @ S3 ) )
           => ( ord_less_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.strict_mono_on_def
thf(fact_587_ord_Ostrict__mono__on__def,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,Less: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,F4: produc3843707927480180839at_nat > nat] :
      ( ( monoto4038163285145374224at_nat @ A3 @ Less @ ord_less_nat @ F4 )
      = ( ! [R: produc3843707927480180839at_nat,S3: produc3843707927480180839at_nat] :
            ( ( ( member8757157785044589968at_nat @ R @ A3 )
              & ( member8757157785044589968at_nat @ S3 @ A3 )
              & ( Less @ R @ S3 ) )
           => ( ord_less_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.strict_mono_on_def
thf(fact_588_ord_Ostrict__mono__on__def,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,Less: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > $o,F4: produc2934264451710624999od_o_o > nat] :
      ( ( monoto3031015839379671696_o_nat @ A3 @ Less @ ord_less_nat @ F4 )
      = ( ! [R: produc2934264451710624999od_o_o,S3: produc2934264451710624999od_o_o] :
            ( ( ( member444158400953824016od_o_o @ R @ A3 )
              & ( member444158400953824016od_o_o @ S3 @ A3 )
              & ( Less @ R @ S3 ) )
           => ( ord_less_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.strict_mono_on_def
thf(fact_589_ord_Ostrict__mono__on__def,axiom,
    ! [A3: set_Pr457366540195662369at_nat,Less: produc732395585841259969at_nat > produc732395585841259969at_nat > $o,F4: produc732395585841259969at_nat > nat] :
      ( ( monoto3172675574582585194at_nat @ A3 @ Less @ ord_less_nat @ F4 )
      = ( ! [R: produc732395585841259969at_nat,S3: produc732395585841259969at_nat] :
            ( ( ( member4095101504841534314at_nat @ R @ A3 )
              & ( member4095101504841534314at_nat @ S3 @ A3 )
              & ( Less @ R @ S3 ) )
           => ( ord_less_nat @ ( F4 @ R ) @ ( F4 @ S3 ) ) ) ) ) ).

% ord.strict_mono_on_def
thf(fact_590_ord_Ostrict__mono__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,Less: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,F4: produc859450856879609959at_nat > nat] :
      ( ! [R4: produc859450856879609959at_nat,S4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ R4 @ A3 )
         => ( ( member8206827879206165904at_nat @ S4 @ A3 )
           => ( ( Less @ R4 @ S4 )
             => ( ord_less_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto5881712244326581776at_nat @ A3 @ Less @ ord_less_nat @ F4 ) ) ).

% ord.strict_mono_onI
thf(fact_591_ord_Ostrict__mono__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Less: product_prod_nat_nat > product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat] :
      ( ! [R4: product_prod_nat_nat,S4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ R4 @ A3 )
         => ( ( member8440522571783428010at_nat @ S4 @ A3 )
           => ( ( Less @ R4 @ S4 )
             => ( ord_less_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto3238940775400213162at_nat @ A3 @ Less @ ord_less_nat @ F4 ) ) ).

% ord.strict_mono_onI
thf(fact_592_ord_Ostrict__mono__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,Less: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,F4: produc3843707927480180839at_nat > nat] :
      ( ! [R4: produc3843707927480180839at_nat,S4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ R4 @ A3 )
         => ( ( member8757157785044589968at_nat @ S4 @ A3 )
           => ( ( Less @ R4 @ S4 )
             => ( ord_less_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto4038163285145374224at_nat @ A3 @ Less @ ord_less_nat @ F4 ) ) ).

% ord.strict_mono_onI
thf(fact_593_ord_Ostrict__mono__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,Less: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > $o,F4: produc2934264451710624999od_o_o > nat] :
      ( ! [R4: produc2934264451710624999od_o_o,S4: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ R4 @ A3 )
         => ( ( member444158400953824016od_o_o @ S4 @ A3 )
           => ( ( Less @ R4 @ S4 )
             => ( ord_less_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto3031015839379671696_o_nat @ A3 @ Less @ ord_less_nat @ F4 ) ) ).

% ord.strict_mono_onI
thf(fact_594_ord_Ostrict__mono__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,Less: produc732395585841259969at_nat > produc732395585841259969at_nat > $o,F4: produc732395585841259969at_nat > nat] :
      ( ! [R4: produc732395585841259969at_nat,S4: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ R4 @ A3 )
         => ( ( member4095101504841534314at_nat @ S4 @ A3 )
           => ( ( Less @ R4 @ S4 )
             => ( ord_less_nat @ ( F4 @ R4 ) @ ( F4 @ S4 ) ) ) ) )
     => ( monoto3172675574582585194at_nat @ A3 @ Less @ ord_less_nat @ F4 ) ) ).

% ord.strict_mono_onI
thf(fact_595_ord_Ostrict__mono__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,Less: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,F4: produc859450856879609959at_nat > nat,R2: produc859450856879609959at_nat,S: produc859450856879609959at_nat] :
      ( ( monoto5881712244326581776at_nat @ A3 @ Less @ ord_less_nat @ F4 )
     => ( ( member8206827879206165904at_nat @ R2 @ A3 )
       => ( ( member8206827879206165904at_nat @ S @ A3 )
         => ( ( Less @ R2 @ S )
           => ( ord_less_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.strict_mono_onD
thf(fact_596_ord_Ostrict__mono__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Less: product_prod_nat_nat > product_prod_nat_nat > $o,F4: product_prod_nat_nat > nat,R2: product_prod_nat_nat,S: product_prod_nat_nat] :
      ( ( monoto3238940775400213162at_nat @ A3 @ Less @ ord_less_nat @ F4 )
     => ( ( member8440522571783428010at_nat @ R2 @ A3 )
       => ( ( member8440522571783428010at_nat @ S @ A3 )
         => ( ( Less @ R2 @ S )
           => ( ord_less_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.strict_mono_onD
thf(fact_597_ord_Ostrict__mono__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,Less: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o,F4: produc3843707927480180839at_nat > nat,R2: produc3843707927480180839at_nat,S: produc3843707927480180839at_nat] :
      ( ( monoto4038163285145374224at_nat @ A3 @ Less @ ord_less_nat @ F4 )
     => ( ( member8757157785044589968at_nat @ R2 @ A3 )
       => ( ( member8757157785044589968at_nat @ S @ A3 )
         => ( ( Less @ R2 @ S )
           => ( ord_less_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.strict_mono_onD
thf(fact_598_ord_Ostrict__mono__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,Less: produc2934264451710624999od_o_o > produc2934264451710624999od_o_o > $o,F4: produc2934264451710624999od_o_o > nat,R2: produc2934264451710624999od_o_o,S: produc2934264451710624999od_o_o] :
      ( ( monoto3031015839379671696_o_nat @ A3 @ Less @ ord_less_nat @ F4 )
     => ( ( member444158400953824016od_o_o @ R2 @ A3 )
       => ( ( member444158400953824016od_o_o @ S @ A3 )
         => ( ( Less @ R2 @ S )
           => ( ord_less_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.strict_mono_onD
thf(fact_599_ord_Ostrict__mono__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,Less: produc732395585841259969at_nat > produc732395585841259969at_nat > $o,F4: produc732395585841259969at_nat > nat,R2: produc732395585841259969at_nat,S: produc732395585841259969at_nat] :
      ( ( monoto3172675574582585194at_nat @ A3 @ Less @ ord_less_nat @ F4 )
     => ( ( member4095101504841534314at_nat @ R2 @ A3 )
       => ( ( member4095101504841534314at_nat @ S @ A3 )
         => ( ( Less @ R2 @ S )
           => ( ord_less_nat @ ( F4 @ R2 ) @ ( F4 @ S ) ) ) ) ) ) ).

% ord.strict_mono_onD
thf(fact_600_bij__betw__empty2,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( bij_be1923491781212474806at_nat @ F4 @ A3 @ bot_bo2099793752762293965at_nat )
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% bij_betw_empty2
thf(fact_601_bij__betw__empty2,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( bij_be3768671628604344400at_nat @ F4 @ A3 @ bot_bo2099793752762293965at_nat )
     => ( A3 = bot_bo228742789529271731at_nat ) ) ).

% bij_betw_empty2
thf(fact_602_bij__betw__empty2,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( bij_be7339570596641255836at_nat @ F4 @ A3 @ bot_bo228742789529271731at_nat )
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% bij_betw_empty2
thf(fact_603_bij__betw__empty2,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( bij_be236100170522119478at_nat @ F4 @ A3 @ bot_bo228742789529271731at_nat )
     => ( A3 = bot_bo228742789529271731at_nat ) ) ).

% bij_betw_empty2
thf(fact_604_bij__betw__empty1,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( bij_be1923491781212474806at_nat @ F4 @ bot_bo2099793752762293965at_nat @ A3 )
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% bij_betw_empty1
thf(fact_605_bij__betw__empty1,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( bij_be7339570596641255836at_nat @ F4 @ bot_bo2099793752762293965at_nat @ A3 )
     => ( A3 = bot_bo228742789529271731at_nat ) ) ).

% bij_betw_empty1
thf(fact_606_bij__betw__empty1,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( bij_be3768671628604344400at_nat @ F4 @ bot_bo228742789529271731at_nat @ A3 )
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% bij_betw_empty1
thf(fact_607_bij__betw__empty1,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( bij_be236100170522119478at_nat @ F4 @ bot_bo228742789529271731at_nat @ A3 )
     => ( A3 = bot_bo228742789529271731at_nat ) ) ).

% bij_betw_empty1
thf(fact_608_asym__on__def,axiom,
    ( asym_o8546584678760569651at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A6 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A6 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_609_asym__on__def,axiom,
    ( asym_o8381145231034218809od_o_o
    = ( ^ [A6: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ A6 )
         => ! [Y5: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y5 @ A6 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R )
               => ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_610_asym__on__def,axiom,
    ( asym_o8654330542403097209at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ A6 )
         => ! [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A6 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R )
               => ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_611_asym__on__def,axiom,
    ( asym_o5033339280856310931at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A6 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A6 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R )
               => ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_612_asym__on__def,axiom,
    ( asym_on_nat
    = ( ^ [A6: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R ) ) ) ) ) ) ).

% asym_on_def
thf(fact_613_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_614_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_615_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_616_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_617_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_618_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_619_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_620_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_621_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_622_mono__on__greaterD,axiom,
    ! [A3: set_nat,G: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ G )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( ord_less_nat @ ( G @ Y ) @ ( G @ X ) )
           => ( ord_less_nat @ Y @ X ) ) ) ) ) ).

% mono_on_greaterD
thf(fact_623_strict__mono__on__leD,axiom,
    ! [A3: set_nat,F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( ord_less_eq_nat @ X @ Y )
           => ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ) ) ) ).

% strict_mono_on_leD
thf(fact_624_strict__mono__on__imp__mono__on,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F4 )
     => ( monotone_on_nat_nat @ A3 @ ord_less_eq_nat @ ord_less_eq_nat @ F4 ) ) ).

% strict_mono_on_imp_mono_on
thf(fact_625_strict__mono__less,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% strict_mono_less
thf(fact_626_strict__mono__eq,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( ( F4 @ X )
          = ( F4 @ Y ) )
        = ( X = Y ) ) ) ).

% strict_mono_eq
thf(fact_627_strict__monoI,axiom,
    ! [F4: nat > nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
         => ( ord_less_nat @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 ) ) ).

% strict_monoI
thf(fact_628_strict__monoD,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ) ).

% strict_monoD
thf(fact_629_strict__mono__on__imp__inj__on,axiom,
    ! [A3: set_nat,F4: nat > nat] :
      ( ( monotone_on_nat_nat @ A3 @ ord_less_nat @ ord_less_nat @ F4 )
     => ( inj_on_nat_nat @ F4 @ A3 ) ) ).

% strict_mono_on_imp_inj_on
thf(fact_630_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_631_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_632_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_633_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_634_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_635_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_636_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_637_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_638_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_639_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_640_mono__invE,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% mono_invE
thf(fact_641_strict__mono__mono,axiom,
    ! [F4: nat > nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 ) ) ).

% strict_mono_mono
thf(fact_642_mono__strict__invE,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% mono_strict_invE
thf(fact_643_strict__mono__less__eq,axiom,
    ! [F4: nat > nat,X: nat,Y: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% strict_mono_less_eq
thf(fact_644_strict__mono__inv,axiom,
    ! [F4: nat > nat,G: nat > nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F4 )
     => ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
          = top_top_set_nat )
       => ( ! [X3: nat] :
              ( ( G @ ( F4 @ X3 ) )
              = X3 )
         => ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ G ) ) ) ) ).

% strict_mono_inv
thf(fact_645_in__measure,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F4: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur8038558561449204169at_nat @ F4 ) )
      = ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ).

% in_measure
thf(fact_646_in__measure,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,F4: set_Product_prod_o_o > nat] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( measur5705974739304014543od_o_o @ F4 ) )
      = ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ).

% in_measure
thf(fact_647_in__measure,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F4: set_Pr4329608150637261639at_nat > nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( measur4922264856574889999at_nat @ F4 ) )
      = ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ).

% in_measure
thf(fact_648_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F4: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur1827424007717751593at_nat @ F4 ) )
      = ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ).

% in_measure
thf(fact_649_in__measure,axiom,
    ! [X: nat,Y: nat,F4: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F4 ) )
      = ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) ) ) ).

% in_measure
thf(fact_650_inj__on__insert,axiom,
    ! [F4: product_prod_nat_nat > produc859450856879609959at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F4 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on3865918002419084311at_nat @ F4 @ A3 )
        & ~ ( member8206827879206165904at_nat @ ( F4 @ A ) @ ( image_1119892238107900587at_nat @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_651_inj__on__insert,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F4 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on8969904277767023793at_nat @ F4 @ A3 )
        & ~ ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_5168914502847457605at_nat @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_652_inj__on__insert,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F4 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on1146879744823063575at_nat @ F4 @ A3 )
        & ~ ( member8757157785044589968at_nat @ ( F4 @ A ) @ ( image_1672857615394513579at_nat @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_653_inj__on__insert,axiom,
    ! [F4: product_prod_nat_nat > produc2934264451710624999od_o_o,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on5765513801355602071od_o_o @ F4 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on5765513801355602071od_o_o @ F4 @ A3 )
        & ~ ( member444158400953824016od_o_o @ ( F4 @ A ) @ ( image_8095068765341748011od_o_o @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_654_inj__on__insert,axiom,
    ! [F4: product_prod_nat_nat > produc732395585841259969at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inj_on7902545142154350961at_nat @ F4 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
      = ( ( inj_on7902545142154350961at_nat @ F4 @ A3 )
        & ~ ( member4095101504841534314at_nat @ ( F4 @ A ) @ ( image_1912541218521307653at_nat @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_655_inj__on__insert,axiom,
    ! [F4: produc3843707927480180839at_nat > produc859450856879609959at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7438016324537269041at_nat @ F4 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on7438016324537269041at_nat @ F4 @ A3 )
        & ~ ( member8206827879206165904at_nat @ ( F4 @ A ) @ ( image_5653576389946358725at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_656_inj__on__insert,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F4 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on6799352813640927947at_nat @ F4 @ A3 )
        & ~ ( member8440522571783428010at_nat @ ( F4 @ A ) @ ( image_7325330684212377951at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_657_inj__on__insert,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F4 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on8430303779476321073at_nat @ F4 @ A3 )
        & ~ ( member8757157785044589968at_nat @ ( F4 @ A ) @ ( image_960689587819713477at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_658_inj__on__insert,axiom,
    ! [F4: produc3843707927480180839at_nat > produc2934264451710624999od_o_o,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on126908749210227121od_o_o @ F4 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on126908749210227121od_o_o @ F4 @ A3 )
        & ~ ( member444158400953824016od_o_o @ ( F4 @ A ) @ ( image_6049375619684083269od_o_o @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_659_inj__on__insert,axiom,
    ! [F4: produc3843707927480180839at_nat > produc732395585841259969at_nat,A: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( inj_on7338182502749300107at_nat @ F4 @ ( insert9069300056098147895at_nat @ A @ A3 ) )
      = ( ( inj_on7338182502749300107at_nat @ F4 @ A3 )
        & ~ ( member4095101504841534314at_nat @ ( F4 @ A ) @ ( image_3981669762529468447at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_660_fun__upd__image,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F4: produc859450856879609959at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_6706873702294942047at_nat @ ( fun_up105607794567514583at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_6706873702294942047at_nat @ F4 @ ( minus_8321449233255521966at_nat @ A3 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_6706873702294942047at_nat @ ( fun_up105607794567514583at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_6706873702294942047at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_661_fun__upd__image,axiom,
    ! [X: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F4: produc859450856879609959at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_7637461350386011077at_nat @ ( fun_up8993321861993844797at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_7637461350386011077at_nat @ F4 @ ( minus_8321449233255521966at_nat @ A3 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( image_7637461350386011077at_nat @ ( fun_up8993321861993844797at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_7637461350386011077at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_662_fun__upd__image,axiom,
    ! [X: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,F4: produc2934264451710624999od_o_o > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( image_2029717158088452831at_nat @ ( fun_up110622245137070935at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_2029717158088452831at_nat @ F4 @ ( minus_5704166846132356910od_o_o @ A3 @ ( insert3855699742481506999od_o_o @ X @ bot_bo498831808395348787od_o_o ) ) ) ) ) )
      & ( ~ ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( image_2029717158088452831at_nat @ ( fun_up110622245137070935at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_2029717158088452831at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_663_fun__upd__image,axiom,
    ! [X: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,F4: produc2934264451710624999od_o_o > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( image_8428874560199066949at_nat @ ( fun_up3446698188135383485at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_8428874560199066949at_nat @ F4 @ ( minus_5704166846132356910od_o_o @ A3 @ ( insert3855699742481506999od_o_o @ X @ bot_bo498831808395348787od_o_o ) ) ) ) ) )
      & ( ~ ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( image_8428874560199066949at_nat @ ( fun_up3446698188135383485at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_8428874560199066949at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_664_fun__upd__image,axiom,
    ! [X: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,F4: produc732395585841259969at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( image_4401859676472782469at_nat @ ( fun_up4035006559045952253at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_4401859676472782469at_nat @ F4 @ ( minus_1608234471674494216at_nat @ A3 @ ( insert8109617542356073361at_nat @ X @ bot_bo1676454896636820109at_nat ) ) ) ) ) )
      & ( ~ ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( image_4401859676472782469at_nat @ ( fun_up4035006559045952253at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_4401859676472782469at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_665_fun__upd__image,axiom,
    ! [X: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,F4: produc732395585841259969at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( image_3054621790227073003at_nat @ ( fun_up2755145109609588835at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_3054621790227073003at_nat @ F4 @ ( minus_1608234471674494216at_nat @ A3 @ ( insert8109617542356073361at_nat @ X @ bot_bo1676454896636820109at_nat ) ) ) ) ) )
      & ( ~ ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( image_3054621790227073003at_nat @ ( fun_up2755145109609588835at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_3054621790227073003at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_666_fun__upd__image,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_5168914502847457605at_nat @ ( fun_up6858051064228255165at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_5168914502847457605at_nat @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_5168914502847457605at_nat @ ( fun_up6858051064228255165at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_5168914502847457605at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_667_fun__upd__image,axiom,
    ! [X: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_1672857615394513579at_nat @ ( fun_up7313344445903829795at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_1672857615394513579at_nat @ F4 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( image_1672857615394513579at_nat @ ( fun_up7313344445903829795at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_1672857615394513579at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_668_fun__upd__image,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F4: produc3843707927480180839at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_7325330684212377951at_nat @ ( fun_up3742445477866918359at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_7325330684212377951at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_7325330684212377951at_nat @ ( fun_up3742445477866918359at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_7325330684212377951at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_669_fun__upd__image,axiom,
    ! [X: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_960689587819713477at_nat @ ( fun_up3138462846764619837at_nat @ F4 @ X @ Y ) @ A3 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_960689587819713477at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( image_960689587819713477at_nat @ ( fun_up3138462846764619837at_nat @ F4 @ X @ Y ) @ A3 )
          = ( image_960689587819713477at_nat @ F4 @ A3 ) ) ) ) ).

% fun_upd_image
thf(fact_670_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_671_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_672_fst__comp__apsnd,axiom,
    ! [F4: list_char > list_char] :
      ( ( comp_P6261748720030321425t_char @ produc6204653154488278070t_char @ ( produc5571979602231644306ar_nat @ F4 ) )
      = produc6204653154488278070t_char ) ).

% fst_comp_apsnd
thf(fact_673_comp__surj,axiom,
    ! [F4: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat2 @ F4 @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F4 ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_674_bij__comp,axiom,
    ! [F4: nat > nat,G: nat > nat] :
      ( ( bij_betw_nat_nat @ F4 @ top_top_set_nat @ top_top_set_nat )
     => ( ( bij_betw_nat_nat @ G @ top_top_set_nat @ top_top_set_nat )
       => ( bij_betw_nat_nat @ ( comp_nat_nat_nat @ G @ F4 ) @ top_top_set_nat @ top_top_set_nat ) ) ) ).

% bij_comp
thf(fact_675_inj__img__insertE,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8969904277767023793at_nat @ F4 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B5 )
            = ( image_5168914502847457605at_nat @ F4 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_5168914502847457605at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_676_inj__img__insertE,axiom,
    ! [F4: product_prod_nat_nat > produc859450856879609959at_nat,A3: set_Pr1261947904930325089at_nat,X: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( inj_on3865918002419084311at_nat @ F4 @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ X @ B5 )
       => ( ( ( insert5050368324300391991at_nat @ X @ B5 )
            = ( image_1119892238107900587at_nat @ F4 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_1119892238107900587at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_677_inj__img__insertE,axiom,
    ! [F4: produc859450856879609959at_nat > product_prod_nat_nat,A3: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on229527429751349963at_nat @ F4 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B5 )
            = ( image_6706873702294942047at_nat @ F4 @ A3 ) )
         => ~ ! [X6: produc859450856879609959at_nat,A7: set_Pr8693737435421807431at_nat] :
                ( ~ ( member8206827879206165904at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert5050368324300391991at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_6706873702294942047at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_678_inj__img__insertE,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F4 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B5 )
            = ( image_7325330684212377951at_nat @ F4 @ A3 ) )
         => ~ ! [X6: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
                ( ~ ( member8757157785044589968at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert9069300056098147895at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_7325330684212377951at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_679_inj__img__insertE,axiom,
    ! [F4: produc2934264451710624999od_o_o > product_prod_nat_nat,A3: set_Pr1932065953672099015od_o_o,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on8923534230957082699at_nat @ F4 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B5 )
            = ( image_2029717158088452831at_nat @ F4 @ A3 ) )
         => ~ ! [X6: produc2934264451710624999od_o_o,A7: set_Pr1932065953672099015od_o_o] :
                ( ~ ( member444158400953824016od_o_o @ X6 @ A7 )
               => ( ( A3
                    = ( insert3855699742481506999od_o_o @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_2029717158088452831at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_680_inj__img__insertE,axiom,
    ! [F4: produc732395585841259969at_nat > product_prod_nat_nat,A3: set_Pr457366540195662369at_nat,X: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( inj_on1168491563251049969at_nat @ F4 @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ X @ B5 )
       => ( ( ( insert8211810215607154385at_nat @ X @ B5 )
            = ( image_4401859676472782469at_nat @ F4 @ A3 ) )
         => ~ ! [X6: produc732395585841259969at_nat,A7: set_Pr457366540195662369at_nat] :
                ( ~ ( member4095101504841534314at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert8109617542356073361at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_4401859676472782469at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_681_inj__img__insertE,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat,X: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( inj_on1146879744823063575at_nat @ F4 @ A3 )
     => ( ~ ( member8757157785044589968at_nat @ X @ B5 )
       => ( ( ( insert9069300056098147895at_nat @ X @ B5 )
            = ( image_1672857615394513579at_nat @ F4 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_1672857615394513579at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_682_inj__img__insertE,axiom,
    ! [F4: product_prod_nat_nat > produc2934264451710624999od_o_o,A3: set_Pr1261947904930325089at_nat,X: produc2934264451710624999od_o_o,B5: set_Pr1932065953672099015od_o_o] :
      ( ( inj_on5765513801355602071od_o_o @ F4 @ A3 )
     => ( ~ ( member444158400953824016od_o_o @ X @ B5 )
       => ( ( ( insert3855699742481506999od_o_o @ X @ B5 )
            = ( image_8095068765341748011od_o_o @ F4 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_8095068765341748011od_o_o @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_683_inj__img__insertE,axiom,
    ! [F4: product_prod_nat_nat > produc732395585841259969at_nat,A3: set_Pr1261947904930325089at_nat,X: produc732395585841259969at_nat,B5: set_Pr457366540195662369at_nat] :
      ( ( inj_on7902545142154350961at_nat @ F4 @ A3 )
     => ( ~ ( member4095101504841534314at_nat @ X @ B5 )
       => ( ( ( insert8109617542356073361at_nat @ X @ B5 )
            = ( image_1912541218521307653at_nat @ F4 @ A3 ) )
         => ~ ! [X6: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
                ( ~ ( member8440522571783428010at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert8211810215607154385at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_1912541218521307653at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_684_inj__img__insertE,axiom,
    ! [F4: produc859450856879609959at_nat > produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,B5: set_Pr8693737435421807431at_nat] :
      ( ( inj_on7797493862353638193at_nat @ F4 @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ X @ B5 )
       => ( ( ( insert5050368324300391991at_nat @ X @ B5 )
            = ( image_6321802852832981957at_nat @ F4 @ A3 ) )
         => ~ ! [X6: produc859450856879609959at_nat,A7: set_Pr8693737435421807431at_nat] :
                ( ~ ( member8206827879206165904at_nat @ X6 @ A7 )
               => ( ( A3
                    = ( insert5050368324300391991at_nat @ X6 @ A7 ) )
                 => ( ( X
                      = ( F4 @ X6 ) )
                   => ( B5
                     != ( image_6321802852832981957at_nat @ F4 @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_685_mono__inf,axiom,
    ! [F4: nat > nat,A3: nat,B5: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ord_less_eq_nat @ ( F4 @ ( inf_inf_nat @ A3 @ B5 ) ) @ ( inf_inf_nat @ ( F4 @ A3 ) @ ( F4 @ B5 ) ) ) ) ).

% mono_inf
thf(fact_686_inj__on__Un,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( inj_on6799352813640927947at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ B5 ) )
      = ( ( inj_on6799352813640927947at_nat @ F4 @ A3 )
        & ( inj_on6799352813640927947at_nat @ F4 @ B5 )
        & ( ( inf_in2572325071724192079at_nat @ ( image_7325330684212377951at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ B5 ) ) @ ( image_7325330684212377951at_nat @ F4 @ ( minus_3314409938677909166at_nat @ B5 @ A3 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% inj_on_Un
thf(fact_687_inj__on__Un,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( inj_on8430303779476321073at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ B5 ) )
      = ( ( inj_on8430303779476321073at_nat @ F4 @ A3 )
        & ( inj_on8430303779476321073at_nat @ F4 @ B5 )
        & ( ( inf_in7913087082777306421at_nat @ ( image_960689587819713477at_nat @ F4 @ ( minus_3314409938677909166at_nat @ A3 @ B5 ) ) @ ( image_960689587819713477at_nat @ F4 @ ( minus_3314409938677909166at_nat @ B5 @ A3 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% inj_on_Un
thf(fact_688_mlex__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F4: product_prod_nat_nat > nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( mlex_p6366001652026297872at_nat @ F4 @ R3 ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_689_mlex__iff,axiom,
    ! [X: set_Product_prod_o_o,Y: set_Product_prod_o_o,F4: set_Product_prod_o_o > nat,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( mlex_p1798809026605075094od_o_o @ F4 @ R3 ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_690_mlex__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F4: set_Pr4329608150637261639at_nat > nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( mlex_p4791090246024907222at_nat @ F4 @ R3 ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_691_mlex__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F4: set_Pr1261947904930325089at_nat > nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F4 @ R3 ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_692_mlex__iff,axiom,
    ! [X: nat,Y: nat,F4: nat > nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F4 @ R3 ) )
      = ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
        | ( ( ( F4 @ X )
            = ( F4 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ) ).

% mlex_iff
thf(fact_693_mlex__less,axiom,
    ! [F4: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( mlex_p6366001652026297872at_nat @ F4 @ R3 ) ) ) ).

% mlex_less
thf(fact_694_mlex__less,axiom,
    ! [F4: set_Product_prod_o_o > nat,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( mlex_p1798809026605075094od_o_o @ F4 @ R3 ) ) ) ).

% mlex_less
thf(fact_695_mlex__less,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( mlex_p4791090246024907222at_nat @ F4 @ R3 ) ) ) ).

% mlex_less
thf(fact_696_mlex__less,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F4 @ R3 ) ) ) ).

% mlex_less
thf(fact_697_mlex__less,axiom,
    ! [F4: nat > nat,X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F4 @ R3 ) ) ) ).

% mlex_less
thf(fact_698_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > list_char > list_char,A12: nat,A23: nat,A32: list_char,P2: ( nat > list_char > list_char ) > nat > nat > list_char > $o] :
      ( ( accp_P4067069869040616689t_char @ set_fo8129763142473175882t_char @ ( produc7013538429079112736t_char @ A0 @ ( produc8776329868854908137t_char @ A12 @ ( produc120197956887798346t_char @ A23 @ A32 ) ) ) )
     => ( ! [F: nat > list_char > list_char,A5: nat,B4: nat,Acc: list_char] :
            ( ( accp_P4067069869040616689t_char @ set_fo8129763142473175882t_char @ ( produc7013538429079112736t_char @ F @ ( produc8776329868854908137t_char @ A5 @ ( produc120197956887798346t_char @ B4 @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B4 @ A5 )
               => ( P2 @ F @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F @ A5 @ Acc ) ) )
             => ( P2 @ F @ A5 @ B4 @ Acc ) ) )
       => ( P2 @ A0 @ A12 @ A23 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_699_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: nat > nat > nat,A12: nat,A23: nat,A32: nat,P2: ( nat > nat > nat ) > nat > nat > nat > $o] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A12 @ ( product_Pair_nat_nat @ A23 @ A32 ) ) ) )
     => ( ! [F: nat > nat > nat,A5: nat,B4: nat,Acc: nat] :
            ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) )
           => ( ( ~ ( ord_less_nat @ B4 @ A5 )
               => ( P2 @ F @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F @ A5 @ Acc ) ) )
             => ( P2 @ F @ A5 @ B4 @ Acc ) ) )
       => ( P2 @ A0 @ A12 @ A23 @ A32 ) ) ) ).

% fold_atLeastAtMost_nat.pinduct
thf(fact_700_Image__singleton__iff,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_701_Image__singleton__iff,axiom,
    ! [B: list_char,R2: set_Pr2507339680178222706t_char,A: nat] :
      ( ( member_list_char @ B @ ( image_nat_list_char @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_702_Image__singleton__iff,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_703_Image__singleton__iff,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
      = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_704_Image__singleton__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A: set_Product_prod_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
      = ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_705_Image__singleton__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_706_Image__singleton__iff,axiom,
    ! [B: produc859450856879609959at_nat,R2: set_Pr9129816769188007085at_nat,A: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( image_8438546471840335737at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_707_Image__singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,R2: set_Pr6615861603993638061at_nat,A: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( image_2324999400750382969at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( member600271025443689462at_nat @ ( produc8594291983450332477at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_708_Image__singleton__iff,axiom,
    ! [B: produc2934264451710624999od_o_o,R2: set_Pr7834265855992252717od_o_o,A: product_prod_nat_nat] :
      ( ( member444158400953824016od_o_o @ B @ ( image_8997436189652510201od_o_o @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( member3367681466892819574od_o_o @ ( produc3433399512816166845od_o_o @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_709_Image__singleton__iff,axiom,
    ! [B: produc732395585841259969at_nat,R2: set_Pr8964806216713926663at_nat,A: product_prod_nat_nat] :
      ( ( member4095101504841534314at_nat @ B @ ( image_4536800705656911315at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( member4631157568059444816at_nat @ ( produc3196090488952915351at_nat @ A @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_710_ImageI,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
     => ( ( member_nat @ A @ A3 )
       => ( member_nat @ B @ ( image_nat_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_711_ImageI,axiom,
    ! [A: nat,B: list_char,R2: set_Pr2507339680178222706t_char,A3: set_nat] :
      ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
     => ( ( member_nat @ A @ A3 )
       => ( member_list_char @ B @ ( image_nat_list_char @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_712_ImageI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_713_ImageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_714_ImageI,axiom,
    ! [A: set_Product_prod_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
     => ( ( member3157015407008874800od_o_o @ A @ A3 )
       => ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_715_ImageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_716_ImageI,axiom,
    ! [A: produc859450856879609959at_nat,B: product_prod_nat_nat,R2: set_Pr846279711151727201at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ R2 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ( member8440522571783428010at_nat @ B @ ( image_4802155899172601389at_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_717_ImageI,axiom,
    ! [A: product_prod_nat_nat,B: produc859450856879609959at_nat,R2: set_Pr9129816769188007085at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8206827879206165904at_nat @ B @ ( image_8438546471840335737at_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_718_ImageI,axiom,
    ! [A: product_prod_nat_nat,B: produc3843707927480180839at_nat,R2: set_Pr6615861603993638061at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member600271025443689462at_nat @ ( produc8594291983450332477at_nat @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member8757157785044589968at_nat @ B @ ( image_2324999400750382969at_nat @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_719_ImageI,axiom,
    ! [A: product_prod_nat_nat,B: produc2934264451710624999od_o_o,R2: set_Pr7834265855992252717od_o_o,A3: set_Pr1261947904930325089at_nat] :
      ( ( member3367681466892819574od_o_o @ ( produc3433399512816166845od_o_o @ A @ B ) @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( member444158400953824016od_o_o @ B @ ( image_8997436189652510201od_o_o @ R2 @ A3 ) ) ) ) ).

% ImageI
thf(fact_720_ImageE,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R2 @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R2 )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_721_ImageE,axiom,
    ! [B: list_char,R2: set_Pr2507339680178222706t_char,A3: set_nat] :
      ( ( member_list_char @ B @ ( image_nat_list_char @ R2 @ A3 ) )
     => ~ ! [X3: nat] :
            ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X3 @ B ) @ R2 )
           => ~ ( member_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_722_ImageE,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R2 @ A3 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ B ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_723_ImageE,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A3 ) )
     => ~ ! [X3: set_Product_prod_o_o] :
            ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ B ) @ R2 )
           => ~ ( member3157015407008874800od_o_o @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_724_ImageE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A3 ) )
     => ~ ! [X3: set_Product_prod_o_o] :
            ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X3 @ B ) @ R2 )
           => ~ ( member3157015407008874800od_o_o @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_725_ImageE,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A3 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ B ) @ R2 )
           => ~ ( member2643936169264416010at_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_726_ImageE,axiom,
    ! [B: produc859450856879609959at_nat,R2: set_Pr9129816769188007085at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ B @ ( image_8438546471840335737at_nat @ R2 @ A3 ) )
     => ~ ! [X3: product_prod_nat_nat] :
            ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ X3 @ B ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_727_ImageE,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr846279711151727201at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_4802155899172601389at_nat @ R2 @ A3 ) )
     => ~ ! [X3: produc859450856879609959at_nat] :
            ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ X3 @ B ) @ R2 )
           => ~ ( member8206827879206165904at_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_728_ImageE,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr1868028757359454817at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_7977472469568247341at_nat @ R2 @ A3 ) )
     => ~ ! [X3: produc3843707927480180839at_nat] :
            ( ( member1953685042357896106at_nat @ ( produc5023393015413421041at_nat @ X3 @ B ) @ R2 )
           => ~ ( member8757157785044589968at_nat @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_729_ImageE,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr7390626335576417249at_nat,A3: set_Pr1932065953672099015od_o_o] :
      ( ( member8440522571783428010at_nat @ B @ ( image_2932084582399215021at_nat @ R2 @ A3 ) )
     => ~ ! [X3: produc2934264451710624999od_o_o] :
            ( ( member8767205482156871978at_nat @ ( produc6591419942417647473at_nat @ X3 @ B ) @ R2 )
           => ~ ( member444158400953824016od_o_o @ X3 @ A3 ) ) ) ).

% ImageE
thf(fact_730_Image__iff,axiom,
    ! [B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A3 ) )
      = ( ? [X2: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X2 @ A3 )
            & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_731_Image__iff,axiom,
    ! [B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R2 @ A3 ) )
      = ( ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_732_Image__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat,A3: set_se491199571785101031od_o_o] :
      ( ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A3 ) )
      = ( ? [X2: set_Product_prod_o_o] :
            ( ( member3157015407008874800od_o_o @ X2 @ A3 )
            & ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_733_Image__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ B @ ( image_5662281817830198163at_nat @ R2 @ A3 ) )
      = ( ? [X2: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X2 @ A3 )
            & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_734_Image__iff,axiom,
    ! [B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A3 ) )
      = ( ? [X2: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X2 @ A3 )
            & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_735_Image__iff,axiom,
    ! [B: list_char,R2: set_Pr2507339680178222706t_char,A3: set_nat] :
      ( ( member_list_char @ B @ ( image_nat_list_char @ R2 @ A3 ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( member484235747325421115t_char @ ( produc120197956887798346t_char @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_736_Image__iff,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R2 @ A3 ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_737_rev__ImageI,axiom,
    ! [A: nat,A3: set_nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( member_nat @ B @ ( image_nat_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_738_rev__ImageI,axiom,
    ! [A: nat,A3: set_nat,B: list_char,R2: set_Pr2507339680178222706t_char] :
      ( ( member_nat @ A @ A3 )
     => ( ( member484235747325421115t_char @ ( produc120197956887798346t_char @ A @ B ) @ R2 )
       => ( member_list_char @ B @ ( image_nat_list_char @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_739_rev__ImageI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( member8440522571783428010at_nat @ B @ ( image_2484338115069272851at_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_740_rev__ImageI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Product_prod_o_o,R2: set_Pr1932065953672099015od_o_o] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( member3157015407008874800od_o_o @ B @ ( image_3256184292080368531od_o_o @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_741_rev__ImageI,axiom,
    ! [A: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o,B: set_Pr1261947904930325089at_nat,R2: set_Pr457366540195662369at_nat] :
      ( ( member3157015407008874800od_o_o @ A @ A3 )
     => ( ( member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ A @ B ) @ R2 )
       => ( member2643936169264416010at_nat @ B @ ( image_2461788823225053677at_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_742_rev__ImageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A3 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( member2643936169264416010at_nat @ B @ ( image_3892192845344445203at_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_743_rev__ImageI,axiom,
    ! [A: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,R2: set_Pr846279711151727201at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A3 )
     => ( ( member3919319858682911658at_nat @ ( produc1860397064299428849at_nat @ A @ B ) @ R2 )
       => ( member8440522571783428010at_nat @ B @ ( image_4802155899172601389at_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_744_rev__ImageI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc859450856879609959at_nat,R2: set_Pr9129816769188007085at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member6033505038158179318at_nat @ ( produc5496787636967163197at_nat @ A @ B ) @ R2 )
       => ( member8206827879206165904at_nat @ B @ ( image_8438546471840335737at_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_745_rev__ImageI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc3843707927480180839at_nat,R2: set_Pr6615861603993638061at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member600271025443689462at_nat @ ( produc8594291983450332477at_nat @ A @ B ) @ R2 )
       => ( member8757157785044589968at_nat @ B @ ( image_2324999400750382969at_nat @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_746_rev__ImageI,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B: produc2934264451710624999od_o_o,R2: set_Pr7834265855992252717od_o_o] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( member3367681466892819574od_o_o @ ( produc3433399512816166845od_o_o @ A @ B ) @ R2 )
       => ( member444158400953824016od_o_o @ B @ ( image_8997436189652510201od_o_o @ R2 @ A3 ) ) ) ) ).

% rev_ImageI
thf(fact_747_mono__Un,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( monoto1838827567791443495at_nat @ top_to1754375979021635447at_nat @ ord_le1268244103169919719at_nat @ ord_le1268244103169919719at_nat @ F4 )
     => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( F4 @ A3 ) @ ( F4 @ B5 ) ) @ ( F4 @ ( sup_su5525570899277871387at_nat @ A3 @ B5 ) ) ) ) ).

% mono_Un
thf(fact_748_mlex__leq,axiom,
    ! [F4: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( mlex_p6366001652026297872at_nat @ F4 @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_749_mlex__leq,axiom,
    ! [F4: set_Product_prod_o_o > nat,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R3 )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ ( mlex_p1798809026605075094od_o_o @ F4 @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_750_mlex__leq,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( mlex_p4791090246024907222at_nat @ F4 @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_751_mlex__leq,axiom,
    ! [F4: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F4 @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_752_mlex__leq,axiom,
    ! [F4: nat > nat,X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F4 @ R3 ) ) ) ) ).

% mlex_leq
thf(fact_753_mono__sup,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( monoto1838827567791443495at_nat @ top_to1754375979021635447at_nat @ ord_le1268244103169919719at_nat @ ord_le1268244103169919719at_nat @ F4 )
     => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( F4 @ A3 ) @ ( F4 @ B5 ) ) @ ( F4 @ ( sup_su5525570899277871387at_nat @ A3 @ B5 ) ) ) ) ).

% mono_sup
thf(fact_754_mono__sup,axiom,
    ! [F4: set_Pr4329608150637261639at_nat > nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( monoto1786604697523618032at_nat @ top_to1754375979021635447at_nat @ ord_le1268244103169919719at_nat @ ord_less_eq_nat @ F4 )
     => ( ord_less_eq_nat @ ( sup_sup_nat @ ( F4 @ A3 ) @ ( F4 @ B5 ) ) @ ( F4 @ ( sup_su5525570899277871387at_nat @ A3 @ B5 ) ) ) ) ).

% mono_sup
thf(fact_755_mono__sup,axiom,
    ! [F4: nat > set_Pr4329608150637261639at_nat,A3: nat,B5: nat] :
      ( ( monoto1237820778223445070at_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_le1268244103169919719at_nat @ F4 )
     => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( F4 @ A3 ) @ ( F4 @ B5 ) ) @ ( F4 @ ( sup_sup_nat @ A3 @ B5 ) ) ) ) ).

% mono_sup
thf(fact_756_mono__sup,axiom,
    ! [F4: nat > nat,A3: nat,B5: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ F4 )
     => ( ord_less_eq_nat @ ( sup_sup_nat @ ( F4 @ A3 ) @ ( F4 @ B5 ) ) @ ( F4 @ ( sup_sup_nat @ A3 @ B5 ) ) ) ) ).

% mono_sup
thf(fact_757_notIn__Un__bij__betw,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be1923491781212474806at_nat @ F4 @ A3 @ A8 )
         => ( bij_be1923491781212474806at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_758_notIn__Un__bij__betw,axiom,
    ! [B: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F4: produc859450856879609959at_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8206827879206165904at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be5362278798767818832at_nat @ F4 @ A3 @ A8 )
         => ( bij_be5362278798767818832at_nat @ F4 @ ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_759_notIn__Un__bij__betw,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc859450856879609959at_nat,A8: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be8998669371435553180at_nat @ F4 @ A3 @ A8 )
         => ( bij_be8998669371435553180at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su718114333110466843at_nat @ A8 @ ( insert5050368324300391991at_nat @ ( F4 @ B ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_760_notIn__Un__bij__betw,axiom,
    ! [B: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,F4: produc2934264451710624999od_o_o > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member444158400953824016od_o_o @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be890175863832269520at_nat @ F4 @ A3 @ A8 )
         => ( bij_be890175863832269520at_nat @ F4 @ ( sup_su4078869789063261851od_o_o @ A3 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_761_notIn__Un__bij__betw,axiom,
    ! [B: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,F4: produc732395585841259969at_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member4095101504841534314at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be2725198553504426486at_nat @ F4 @ A3 @ A8 )
         => ( bij_be2725198553504426486at_nat @ F4 @ ( sup_su7309111544468589045at_nat @ A3 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_762_notIn__Un__bij__betw,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc2934264451710624999od_o_o,A8: set_Pr1932065953672099015od_o_o] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member444158400953824016od_o_o @ ( F4 @ B ) @ A8 )
       => ( ( bij_be6955527471085564700od_o_o @ F4 @ A3 @ A8 )
         => ( bij_be6955527471085564700od_o_o @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su4078869789063261851od_o_o @ A8 @ ( insert3855699742481506999od_o_o @ ( F4 @ B ) @ bot_bo498831808395348787od_o_o ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_763_notIn__Un__bij__betw,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc732395585841259969at_nat,A8: set_Pr457366540195662369at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member4095101504841534314at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be235880095552951670at_nat @ F4 @ A3 @ A8 )
         => ( bij_be235880095552951670at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su7309111544468589045at_nat @ A8 @ ( insert8109617542356073361at_nat @ ( F4 @ B ) @ bot_bo1676454896636820109at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_764_notIn__Un__bij__betw,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member8757157785044589968at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be7339570596641255836at_nat @ F4 @ A3 @ A8 )
         => ( bij_be7339570596641255836at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su5525570899277871387at_nat @ A8 @ ( insert9069300056098147895at_nat @ ( F4 @ B ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_765_notIn__Un__bij__betw,axiom,
    ! [B: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F4: produc3843707927480180839at_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8757157785044589968at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be3768671628604344400at_nat @ F4 @ A3 @ A8 )
         => ( bij_be3768671628604344400at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_766_notIn__Un__bij__betw,axiom,
    ! [B: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F4: produc859450856879609959at_nat > produc859450856879609959at_nat,A8: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ B @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be4529855521105804598at_nat @ F4 @ A3 @ A8 )
         => ( bij_be4529855521105804598at_nat @ F4 @ ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( sup_su718114333110466843at_nat @ A8 @ ( insert5050368324300391991at_nat @ ( F4 @ B ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw
thf(fact_767_notIn__Un__bij__betw3,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be1923491781212474806at_nat @ F4 @ A3 @ A8 )
          = ( bij_be1923491781212474806at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_768_notIn__Un__bij__betw3,axiom,
    ! [B: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F4: produc859450856879609959at_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8206827879206165904at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be5362278798767818832at_nat @ F4 @ A3 @ A8 )
          = ( bij_be5362278798767818832at_nat @ F4 @ ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_769_notIn__Un__bij__betw3,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc859450856879609959at_nat,A8: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be8998669371435553180at_nat @ F4 @ A3 @ A8 )
          = ( bij_be8998669371435553180at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su718114333110466843at_nat @ A8 @ ( insert5050368324300391991at_nat @ ( F4 @ B ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_770_notIn__Un__bij__betw3,axiom,
    ! [B: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o,F4: produc2934264451710624999od_o_o > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member444158400953824016od_o_o @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be890175863832269520at_nat @ F4 @ A3 @ A8 )
          = ( bij_be890175863832269520at_nat @ F4 @ ( sup_su4078869789063261851od_o_o @ A3 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_771_notIn__Un__bij__betw3,axiom,
    ! [B: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat,F4: produc732395585841259969at_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member4095101504841534314at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be2725198553504426486at_nat @ F4 @ A3 @ A8 )
          = ( bij_be2725198553504426486at_nat @ F4 @ ( sup_su7309111544468589045at_nat @ A3 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_772_notIn__Un__bij__betw3,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc2934264451710624999od_o_o,A8: set_Pr1932065953672099015od_o_o] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member444158400953824016od_o_o @ ( F4 @ B ) @ A8 )
       => ( ( bij_be6955527471085564700od_o_o @ F4 @ A3 @ A8 )
          = ( bij_be6955527471085564700od_o_o @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su4078869789063261851od_o_o @ A8 @ ( insert3855699742481506999od_o_o @ ( F4 @ B ) @ bot_bo498831808395348787od_o_o ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_773_notIn__Un__bij__betw3,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc732395585841259969at_nat,A8: set_Pr457366540195662369at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member4095101504841534314at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be235880095552951670at_nat @ F4 @ A3 @ A8 )
          = ( bij_be235880095552951670at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su7309111544468589045at_nat @ A8 @ ( insert8109617542356073361at_nat @ ( F4 @ B ) @ bot_bo1676454896636820109at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_774_notIn__Un__bij__betw3,axiom,
    ! [B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F4: product_prod_nat_nat > produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8440522571783428010at_nat @ B @ A3 )
     => ( ~ ( member8757157785044589968at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be7339570596641255836at_nat @ F4 @ A3 @ A8 )
          = ( bij_be7339570596641255836at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( sup_su5525570899277871387at_nat @ A8 @ ( insert9069300056098147895at_nat @ ( F4 @ B ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_775_notIn__Un__bij__betw3,axiom,
    ! [B: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,F4: produc3843707927480180839at_nat > product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8757157785044589968at_nat @ B @ A3 )
     => ( ~ ( member8440522571783428010at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be3768671628604344400at_nat @ F4 @ A3 @ A8 )
          = ( bij_be3768671628604344400at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( sup_su6327502436637775413at_nat @ A8 @ ( insert8211810215607154385at_nat @ ( F4 @ B ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_776_notIn__Un__bij__betw3,axiom,
    ! [B: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,F4: produc859450856879609959at_nat > produc859450856879609959at_nat,A8: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ B @ A3 )
     => ( ~ ( member8206827879206165904at_nat @ ( F4 @ B ) @ A8 )
       => ( ( bij_be4529855521105804598at_nat @ F4 @ A3 @ A8 )
          = ( bij_be4529855521105804598at_nat @ F4 @ ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( sup_su718114333110466843at_nat @ A8 @ ( insert5050368324300391991at_nat @ ( F4 @ B ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ).

% notIn_Un_bij_betw3
thf(fact_777_bij__betw__combine,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,C2: set_Pr4329608150637261639at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( bij_be3768671628604344400at_nat @ F4 @ A3 @ B5 )
     => ( ( bij_be3768671628604344400at_nat @ F4 @ C2 @ D2 )
       => ( ( ( inf_in2572325071724192079at_nat @ B5 @ D2 )
            = bot_bo2099793752762293965at_nat )
         => ( bij_be3768671628604344400at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) @ ( sup_su6327502436637775413at_nat @ B5 @ D2 ) ) ) ) ) ).

% bij_betw_combine
thf(fact_778_bij__betw__combine,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( bij_be236100170522119478at_nat @ F4 @ A3 @ B5 )
     => ( ( bij_be236100170522119478at_nat @ F4 @ C2 @ D2 )
       => ( ( ( inf_in7913087082777306421at_nat @ B5 @ D2 )
            = bot_bo228742789529271731at_nat )
         => ( bij_be236100170522119478at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) @ ( sup_su5525570899277871387at_nat @ B5 @ D2 ) ) ) ) ) ).

% bij_betw_combine
thf(fact_779_bij__betw__partition,axiom,
    ! [F4: product_prod_nat_nat > product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( bij_be1923491781212474806at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ C2 ) @ ( sup_su6327502436637775413at_nat @ B5 @ D2 ) )
     => ( ( bij_be1923491781212474806at_nat @ F4 @ C2 @ D2 )
       => ( ( ( inf_in2572325071724192079at_nat @ A3 @ C2 )
            = bot_bo2099793752762293965at_nat )
         => ( ( ( inf_in2572325071724192079at_nat @ B5 @ D2 )
              = bot_bo2099793752762293965at_nat )
           => ( bij_be1923491781212474806at_nat @ F4 @ A3 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_780_bij__betw__partition,axiom,
    ! [F4: product_prod_nat_nat > produc3843707927480180839at_nat,A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B5: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( bij_be7339570596641255836at_nat @ F4 @ ( sup_su6327502436637775413at_nat @ A3 @ C2 ) @ ( sup_su5525570899277871387at_nat @ B5 @ D2 ) )
     => ( ( bij_be7339570596641255836at_nat @ F4 @ C2 @ D2 )
       => ( ( ( inf_in2572325071724192079at_nat @ A3 @ C2 )
            = bot_bo2099793752762293965at_nat )
         => ( ( ( inf_in7913087082777306421at_nat @ B5 @ D2 )
              = bot_bo228742789529271731at_nat )
           => ( bij_be7339570596641255836at_nat @ F4 @ A3 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_781_bij__betw__partition,axiom,
    ! [F4: produc3843707927480180839at_nat > product_prod_nat_nat,A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B5: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( bij_be3768671628604344400at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) @ ( sup_su6327502436637775413at_nat @ B5 @ D2 ) )
     => ( ( bij_be3768671628604344400at_nat @ F4 @ C2 @ D2 )
       => ( ( ( inf_in7913087082777306421at_nat @ A3 @ C2 )
            = bot_bo228742789529271731at_nat )
         => ( ( ( inf_in2572325071724192079at_nat @ B5 @ D2 )
              = bot_bo2099793752762293965at_nat )
           => ( bij_be3768671628604344400at_nat @ F4 @ A3 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_782_bij__betw__partition,axiom,
    ! [F4: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( bij_be236100170522119478at_nat @ F4 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) @ ( sup_su5525570899277871387at_nat @ B5 @ D2 ) )
     => ( ( bij_be236100170522119478at_nat @ F4 @ C2 @ D2 )
       => ( ( ( inf_in7913087082777306421at_nat @ A3 @ C2 )
            = bot_bo228742789529271731at_nat )
         => ( ( ( inf_in7913087082777306421at_nat @ B5 @ D2 )
              = bot_bo228742789529271731at_nat )
           => ( bij_be236100170522119478at_nat @ F4 @ A3 @ B5 ) ) ) ) ) ).

% bij_betw_partition
thf(fact_783_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > list_char > list_char,Xa: nat,Xb: nat,Xc: list_char,Y: list_char] :
      ( ( ( set_fo6880278091074014835t_char @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P4067069869040616689t_char @ set_fo8129763142473175882t_char @ ( produc7013538429079112736t_char @ X @ ( produc8776329868854908137t_char @ Xa @ ( produc120197956887798346t_char @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo6880278091074014835t_char @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P4067069869040616689t_char @ set_fo8129763142473175882t_char @ ( produc7013538429079112736t_char @ X @ ( produc8776329868854908137t_char @ Xa @ ( produc120197956887798346t_char @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_784_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_785_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F4: nat > list_char > list_char,A: nat,B: nat,Acc2: list_char] :
      ( ( accp_P4067069869040616689t_char @ set_fo8129763142473175882t_char @ ( produc7013538429079112736t_char @ F4 @ ( produc8776329868854908137t_char @ A @ ( produc120197956887798346t_char @ B @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo6880278091074014835t_char @ F4 @ A @ B @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo6880278091074014835t_char @ F4 @ A @ B @ Acc2 )
            = ( set_fo6880278091074014835t_char @ F4 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F4 @ A @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_786_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F4: nat > nat > nat,A: nat,B: nat,Acc2: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F4 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc2 ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F4 @ A @ B @ Acc2 )
            = Acc2 ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F4 @ A @ B @ Acc2 )
            = ( set_fo2584398358068434914at_nat @ F4 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F4 @ A @ Acc2 ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_787_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_788_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_789_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_790_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_791_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_792_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_793_equiv__class__nondisjoint,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ ( inf_in8747346010413679829od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) )
       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_794_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ ( inf_in4302113700860409141at_nat @ ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) ) @ ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) ) )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_795_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ ( inf_in6630186493572433077od_o_o @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) ) )
       => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_796_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ ( inf_in3075509010647812879at_nat @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) ) )
       => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_797_equiv__class__nondisjoint,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ ( inf_in535663156852456469at_nat @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) ) @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_798_equiv__class__nondisjoint,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ ( inf_in767997467307849391at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_799_equiv__class__nondisjoint,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,A: nat,B: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ ( inf_inf_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_800_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ ( inf_in2572325071724192079at_nat @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_801_equiv__class__nondisjoint,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ ( inf_in7913087082777306421at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ).

% equiv_class_nondisjoint
thf(fact_802_equiv__class__subset,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ).

% equiv_class_subset
thf(fact_803_equiv__class__subset,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
       => ( ord_le7966958417815640519at_nat @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) ) @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_804_equiv__class__subset,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_805_equiv__class__subset,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_806_equiv__class__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( ord_le3146513528884898305at_nat @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_807_equiv__class__subset,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% equiv_class_subset
thf(fact_808_FieldI1,axiom,
    ! [I2: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R3 )
     => ( member8206827879206165904at_nat @ I2 @ ( field_3044959742847878580at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_809_FieldI1,axiom,
    ! [I2: produc3843707927480180839at_nat,J2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I2 @ J2 ) @ R3 )
     => ( member8757157785044589968at_nat @ I2 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_810_FieldI1,axiom,
    ! [I2: produc2934264451710624999od_o_o,J2: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I2 @ J2 ) @ R3 )
     => ( member444158400953824016od_o_o @ I2 @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% FieldI1
thf(fact_811_FieldI1,axiom,
    ! [I2: produc732395585841259969at_nat,J2: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I2 @ J2 ) @ R3 )
     => ( member4095101504841534314at_nat @ I2 @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_812_FieldI1,axiom,
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R3 )
     => ( member8440522571783428010at_nat @ I2 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_813_FieldI1,axiom,
    ! [I2: set_Product_prod_o_o,J2: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I2 @ J2 ) @ R3 )
     => ( member3157015407008874800od_o_o @ I2 @ ( field_6101175518749917524od_o_o @ R3 ) ) ) ).

% FieldI1
thf(fact_814_FieldI1,axiom,
    ! [I2: $o,J2: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J2 ) @ R3 )
     => ( member_o @ I2 @ ( field_o @ R3 ) ) ) ).

% FieldI1
thf(fact_815_FieldI1,axiom,
    ! [I2: set_Pr4329608150637261639at_nat,J2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I2 @ J2 ) @ R3 )
     => ( member6273874620674182256at_nat @ I2 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_816_FieldI1,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,J2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J2 ) @ R3 )
     => ( member2643936169264416010at_nat @ I2 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_817_FieldI1,axiom,
    ! [I2: nat,J2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
     => ( member_nat @ I2 @ ( field_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_818_FieldI2,axiom,
    ! [I2: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R3 )
     => ( member8206827879206165904at_nat @ J2 @ ( field_3044959742847878580at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_819_FieldI2,axiom,
    ! [I2: produc3843707927480180839at_nat,J2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I2 @ J2 ) @ R3 )
     => ( member8757157785044589968at_nat @ J2 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_820_FieldI2,axiom,
    ! [I2: produc2934264451710624999od_o_o,J2: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I2 @ J2 ) @ R3 )
     => ( member444158400953824016od_o_o @ J2 @ ( field_5121862139100741428od_o_o @ R3 ) ) ) ).

% FieldI2
thf(fact_821_FieldI2,axiom,
    ! [I2: produc732395585841259969at_nat,J2: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I2 @ J2 ) @ R3 )
     => ( member4095101504841534314at_nat @ J2 @ ( field_7361900449976055182at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_822_FieldI2,axiom,
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R3 )
     => ( member8440522571783428010at_nat @ J2 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_823_FieldI2,axiom,
    ! [I2: set_Product_prod_o_o,J2: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I2 @ J2 ) @ R3 )
     => ( member3157015407008874800od_o_o @ J2 @ ( field_6101175518749917524od_o_o @ R3 ) ) ) ).

% FieldI2
thf(fact_824_FieldI2,axiom,
    ! [I2: $o,J2: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J2 ) @ R3 )
     => ( member_o @ J2 @ ( field_o @ R3 ) ) ) ).

% FieldI2
thf(fact_825_FieldI2,axiom,
    ! [I2: set_Pr4329608150637261639at_nat,J2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I2 @ J2 ) @ R3 )
     => ( member6273874620674182256at_nat @ J2 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_826_FieldI2,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,J2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J2 ) @ R3 )
     => ( member2643936169264416010at_nat @ J2 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_827_FieldI2,axiom,
    ! [I2: nat,J2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
     => ( member_nat @ J2 @ ( field_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_828_eq__equiv__class,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,A3: set_se491199571785101031od_o_o] :
      ( ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) )
        = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
     => ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
       => ( ( member3157015407008874800od_o_o @ B @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_829_eq__equiv__class,axiom,
    ! [R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat] :
      ( ( ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) )
        = ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) )
     => ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
       => ( ( member8206827879206165904at_nat @ B @ A3 )
         => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_830_eq__equiv__class,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o,A3: set_Pr1932065953672099015od_o_o] :
      ( ( ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) )
        = ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) )
     => ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
       => ( ( member444158400953824016od_o_o @ B @ A3 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_831_eq__equiv__class,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat,A3: set_Pr457366540195662369at_nat] :
      ( ( ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) )
        = ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) )
     => ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
       => ( ( member4095101504841534314at_nat @ B @ A3 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_832_eq__equiv__class,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A3: set_se5479201105039950375at_nat] :
      ( ( ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) )
        = ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) )
     => ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
       => ( ( member6273874620674182256at_nat @ B @ A3 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_833_eq__equiv__class,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A3: set_se7855581050983116737at_nat] :
      ( ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) )
        = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
     => ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
       => ( ( member2643936169264416010at_nat @ B @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_834_eq__equiv__class,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat,A3: set_nat] :
      ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
        = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
     => ( ( equiv_equiv_nat @ A3 @ R2 )
       => ( ( member_nat @ B @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_835_eq__equiv__class,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
        = ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
     => ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
       => ( ( member8440522571783428010at_nat @ B @ A3 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_836_eq__equiv__class,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat] :
      ( ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
        = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
     => ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
       => ( ( member8757157785044589968at_nat @ B @ A3 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ).

% eq_equiv_class
thf(fact_837_equiv__class__eq,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
       => ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) )
          = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) ) ) ) ).

% equiv_class_eq
thf(fact_838_equiv__class__eq,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
       => ( ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) )
          = ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_839_equiv__class__eq,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
       => ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) )
          = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_840_equiv__class__eq,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
       => ( ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
          = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_841_equiv__class__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
       => ( ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
          = ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_842_equiv__class__eq,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
       => ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
          = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% equiv_class_eq
thf(fact_843_eq__equiv__class__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y @ A3 )
         => ( ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ X @ bot_bo8006819982598807891od_o_o ) )
              = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_844_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( member8206827879206165904at_nat @ Y @ A3 )
         => ( ( ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) )
              = ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ Y @ bot_bo5327735625951526323at_nat ) ) )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_845_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( member444158400953824016od_o_o @ Y @ A3 )
         => ( ( ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ X @ bot_bo498831808395348787od_o_o ) )
              = ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ Y @ bot_bo498831808395348787od_o_o ) ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_846_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( member4095101504841534314at_nat @ Y @ A3 )
         => ( ( ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ X @ bot_bo1676454896636820109at_nat ) )
              = ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ Y @ bot_bo1676454896636820109at_nat ) ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_847_eq__equiv__class__iff,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ A3 )
       => ( ( member6273874620674182256at_nat @ Y @ A3 )
         => ( ( ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
              = ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_848_eq__equiv__class__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( member2643936169264416010at_nat @ Y @ A3 )
         => ( ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) )
              = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_849_eq__equiv__class__iff,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
              = ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_850_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( member8440522571783428010at_nat @ Y @ A3 )
         => ( ( ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
              = ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_851_eq__equiv__class__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( member8757157785044589968at_nat @ Y @ A3 )
         => ( ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
              = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff
thf(fact_852_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
        = ( ( ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) )
            = ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ Y @ bot_bo5327735625951526323at_nat ) ) )
          & ( member8206827879206165904at_nat @ X @ A3 )
          & ( member8206827879206165904at_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_853_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
        = ( ( ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ X @ bot_bo498831808395348787od_o_o ) )
            = ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ Y @ bot_bo498831808395348787od_o_o ) ) )
          & ( member444158400953824016od_o_o @ X @ A3 )
          & ( member444158400953824016od_o_o @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_854_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
        = ( ( ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ X @ bot_bo1676454896636820109at_nat ) )
            = ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ Y @ bot_bo1676454896636820109at_nat ) ) )
          & ( member4095101504841534314at_nat @ X @ A3 )
          & ( member4095101504841534314at_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_855_equiv__class__eq__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
        = ( ( ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ X @ bot_bo8006819982598807891od_o_o ) )
            = ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) )
          & ( member3157015407008874800od_o_o @ X @ A3 )
          & ( member3157015407008874800od_o_o @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_856_equiv__class__eq__iff,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
        = ( ( ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
            = ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) )
          & ( member6273874620674182256at_nat @ X @ A3 )
          & ( member6273874620674182256at_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_857_equiv__class__eq__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
        = ( ( ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) )
            = ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) )
          & ( member2643936169264416010at_nat @ X @ A3 )
          & ( member2643936169264416010at_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_858_equiv__class__eq__iff,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
        = ( ( ( image_nat_nat @ R2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = ( image_nat_nat @ R2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
          & ( member_nat @ X @ A3 )
          & ( member_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_859_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
        = ( ( ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
            = ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) )
          & ( member8440522571783428010at_nat @ X @ A3 )
          & ( member8440522571783428010at_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_860_equiv__class__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
        = ( ( ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
            = ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) )
          & ( member8757157785044589968at_nat @ X @ A3 )
          & ( member8757157785044589968at_nat @ Y @ A3 ) ) ) ) ).

% equiv_class_eq_iff
thf(fact_861_subset__equiv__class,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,B: set_Product_prod_o_o,A: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) )
       => ( ( member3157015407008874800od_o_o @ B @ A3 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_862_subset__equiv__class,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,B: produc859450856879609959at_nat,A: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( ord_le3000389064537975527at_nat @ ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) @ ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) ) )
       => ( ( member8206827879206165904at_nat @ B @ A3 )
         => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_863_subset__equiv__class,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,B: produc2934264451710624999od_o_o,A: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( ord_le7586318239878589543od_o_o @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) )
       => ( ( member444158400953824016od_o_o @ B @ A3 )
         => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_864_subset__equiv__class,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,B: produc732395585841259969at_nat,A: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( ord_le4841768313324958145at_nat @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) )
       => ( ( member4095101504841534314at_nat @ B @ A3 )
         => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_865_subset__equiv__class,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( ord_le7966958417815640519at_nat @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) ) )
       => ( ( member6273874620674182256at_nat @ B @ A3 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_866_subset__equiv__class,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) )
       => ( ( member2643936169264416010at_nat @ B @ A3 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_867_subset__equiv__class,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,B: nat,A: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
       => ( ( member_nat @ B @ A3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_868_subset__equiv__class,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
       => ( ( member8440522571783428010at_nat @ B @ A3 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_869_subset__equiv__class,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
       => ( ( member8757157785044589968at_nat @ B @ A3 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ).

% subset_equiv_class
thf(fact_870_proj__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ X @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) @ A3 )
       => ( ( ( equiv_6719278489894859850od_o_o @ R2 @ X )
            = ( equiv_6719278489894859850od_o_o @ R2 @ Y ) )
          = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_871_proj__iff,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( ord_le7966958417815640519at_nat @ ( insert2290795007606087959at_nat @ X @ ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) @ A3 )
       => ( ( ( equiv_8664554612503799882at_nat @ R2 @ X )
            = ( equiv_8664554612503799882at_nat @ R2 @ Y ) )
          = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_872_proj__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ X @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) @ A3 )
       => ( ( ( equiv_4785085088318667978at_nat @ R2 @ X )
            = ( equiv_4785085088318667978at_nat @ R2 @ Y ) )
          = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_873_proj__iff,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ A3 )
       => ( ( ( equiv_proj_nat_nat @ R2 @ X )
            = ( equiv_proj_nat_nat @ R2 @ Y ) )
          = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_874_proj__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) @ A3 )
       => ( ( ( equiv_7265450603036132554at_nat @ R2 @ X )
            = ( equiv_7265450603036132554at_nat @ R2 @ Y ) )
          = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_875_proj__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) @ A3 )
       => ( ( ( equiv_6469457352718706762at_nat @ R2 @ X )
            = ( equiv_6469457352718706762at_nat @ R2 @ Y ) )
          = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% proj_iff
thf(fact_876_cofinal__def,axiom,
    ( bNF_Ca1979673041776187986at_nat
    = ( ^ [A6: 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 @ A6 )
              & ( X2 != Y5 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_877_cofinal__def,axiom,
    ( bNF_Ca3407573850950408152od_o_o
    = ( ^ [A6: 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 @ A6 )
              & ( X2 != Y5 )
              & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_878_cofinal__def,axiom,
    ( bNF_Ca4363140458462388713inal_o
    = ( ^ [A6: set_o,R: set_Product_prod_o_o] :
        ! [X2: $o] :
          ( ( member_o @ X2 @ ( field_o @ R ) )
         => ? [Y5: $o] :
              ( ( member_o @ Y5 @ A6 )
              & ( X2 = (~ Y5) )
              & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_879_cofinal__def,axiom,
    ( bNF_Ca6037544080215331608at_nat
    = ( ^ [A6: 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 @ A6 )
              & ( X2 != Y5 )
              & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_880_cofinal__def,axiom,
    ( bNF_Ca8448034394003515314at_nat
    = ( ^ [A6: 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 @ A6 )
              & ( X2 != Y5 )
              & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_881_cofinal__def,axiom,
    ( bNF_Ca5159621140400407487al_nat
    = ( ^ [A6: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ ( field_nat @ R ) )
         => ? [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
              & ( X2 != Y5 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R ) ) ) ) ) ).

% cofinal_def
thf(fact_882_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( order_6208692358724846110od_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 @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_883_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr553994874890374343at_nat,A: produc859450856879609959at_nat,B: produc859450856879609959at_nat] :
      ( ( order_1160314865184352894at_nat @ ( field_3044959742847878580at_nat @ R2 ) @ R2 )
     => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R2 ) )
       => ( ( member8206827879206165904at_nat @ B @ ( field_3044959742847878580at_nat @ R2 ) )
         => ( ( ord_le3000389064537975527at_nat @ ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) ) @ ( image_2092125385209402259at_nat @ R2 @ ( insert5050368324300391991at_nat @ B @ bot_bo5327735625951526323at_nat ) ) )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_884_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,A: produc2934264451710624999od_o_o,B: produc2934264451710624999od_o_o] :
      ( ( order_3482285145583873534od_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 @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ A @ bot_bo498831808395348787od_o_o ) ) @ ( image_1313870842500862867od_o_o @ R2 @ ( insert3855699742481506999od_o_o @ B @ bot_bo498831808395348787od_o_o ) ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_885_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,A: produc732395585841259969at_nat,B: produc732395585841259969at_nat] :
      ( ( order_8058285401901086552at_nat @ ( field_7361900449976055182at_nat @ R2 ) @ R2 )
     => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
       => ( ( member4095101504841534314at_nat @ B @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( ord_le4841768313324958145at_nat @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ A @ bot_bo1676454896636820109at_nat ) ) @ ( image_1966035174115581203at_nat @ R2 @ ( insert8109617542356073361at_nat @ B @ bot_bo1676454896636820109at_nat ) ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_886_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( member_o @ A @ ( field_o @ R2 ) )
       => ( ( member_o @ B @ ( field_o @ R2 ) )
         => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_887_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( order_7303966067141865822at_nat @ ( field_1428494348145443988at_nat @ R2 ) @ R2 )
     => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R2 ) )
       => ( ( member6273874620674182256at_nat @ B @ ( field_1428494348145443988at_nat @ R2 ) )
         => ( ( ord_le7966958417815640519at_nat @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) ) @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_888_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( order_7173764778206869240at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ B @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_889_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( member_nat @ A @ ( field_nat @ R2 ) )
       => ( ( member_nat @ B @ ( field_nat @ R2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_890_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( order_9011181989990678936at_nat @ ( field_4706345496283612622at_nat @ R2 ) @ R2 )
     => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R2 ) )
       => ( ( member8440522571783428010at_nat @ B @ ( field_4706345496283612622at_nat @ R2 ) )
         => ( ( ord_le3146513528884898305at_nat @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_891_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( order_4077073789452270206at_nat @ ( field_2862844103579723188at_nat @ R2 ) @ R2 )
     => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( member8757157785044589968at_nat @ B @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( ord_le1268244103169919719at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_892_disjnt__equiv__class,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( disjnt556417628418099406od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ A @ bot_bo8006819982598807891od_o_o ) ) @ ( image_3256184292080368531od_o_o @ R2 @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) )
        = ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_893_disjnt__equiv__class,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( disjnt8392056708997208078at_nat @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ A @ bot_bo1534825418459847827at_nat ) ) @ ( image_5662281817830198163at_nat @ R2 @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) )
        = ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_894_disjnt__equiv__class,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( disjnt2544082741087298984at_nat @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ A @ bot_bo3083307316010499117at_nat ) ) @ ( image_3892192845344445203at_nat @ R2 @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) )
        = ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_895_disjnt__equiv__class,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( disjnt_nat @ ( image_nat_nat @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
        = ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_896_disjnt__equiv__class,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( disjnt2713546992526078024at_nat @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ ( image_2484338115069272851at_nat @ R2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
        = ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_897_disjnt__equiv__class,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( disjnt7512751716740456750at_nat @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ ( image_3736809421353964435at_nat @ R2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
        = ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) ) ) ) ).

% disjnt_equiv_class
thf(fact_898_in__quotient__imp__in__rel,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X4: set_se491199571785101031od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X4 @ ( equiv_3134926782319234254od_o_o @ A3 @ R2 ) )
       => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ X @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) ) @ X4 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_899_in__quotient__imp__in__rel,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X4: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member2479695886705571664at_nat @ X4 @ ( equiv_152729185961188366at_nat @ A3 @ R2 ) )
       => ( ( ord_le7966958417815640519at_nat @ ( insert2290795007606087959at_nat @ X @ ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) ) @ X4 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_900_in__quotient__imp__in__rel,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X4: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8313122109188517482at_nat @ X4 @ ( equiv_7923377459618294184at_nat @ A3 @ R2 ) )
       => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ X @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) ) @ X4 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_901_in__quotient__imp__in__rel,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ X4 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_902_in__quotient__imp__in__rel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X4 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) @ X4 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_903_in__quotient__imp__in__rel,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X4 @ ( equiv_8408515162552411438at_nat @ A3 @ R2 ) )
       => ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) @ X4 )
         => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 ) ) ) ) ).

% in_quotient_imp_in_rel
thf(fact_904_in__quotient__imp__closed,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X4: set_se491199571785101031od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X4 @ ( equiv_3134926782319234254od_o_o @ A3 @ R2 ) )
       => ( ( member3157015407008874800od_o_o @ X @ X4 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
           => ( member3157015407008874800od_o_o @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_905_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X4: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member3250002818903848048at_nat @ X4 @ ( equiv_6719501424213955886at_nat @ A3 @ R2 ) )
       => ( ( member8206827879206165904at_nat @ X @ X4 )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
           => ( member8206827879206165904at_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_906_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X4 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( ( member8440522571783428010at_nat @ X @ X4 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
           => ( member8440522571783428010at_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_907_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X4 @ ( equiv_8408515162552411438at_nat @ A3 @ R2 ) )
       => ( ( member8757157785044589968at_nat @ X @ X4 )
         => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
           => ( member8757157785044589968at_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_908_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X4: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member4387393612980313840od_o_o @ X4 @ ( equiv_1831625524721748142od_o_o @ A3 @ R2 ) )
       => ( ( member444158400953824016od_o_o @ X @ X4 )
         => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
           => ( member444158400953824016od_o_o @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_909_in__quotient__imp__closed,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X4: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member3243375828287704010at_nat @ X4 @ ( equiv_2323626691971332616at_nat @ A3 @ R2 ) )
       => ( ( member4095101504841534314at_nat @ X @ X4 )
         => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
           => ( member4095101504841534314at_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_910_in__quotient__imp__closed,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X4: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member2479695886705571664at_nat @ X4 @ ( equiv_152729185961188366at_nat @ A3 @ R2 ) )
       => ( ( member6273874620674182256at_nat @ X @ X4 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
           => ( member6273874620674182256at_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_911_in__quotient__imp__closed,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X4: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8313122109188517482at_nat @ X4 @ ( equiv_7923377459618294184at_nat @ A3 @ R2 ) )
       => ( ( member2643936169264416010at_nat @ X @ X4 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
           => ( member2643936169264416010at_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_912_in__quotient__imp__closed,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( ( member_nat @ X @ X4 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
           => ( member_nat @ Y @ X4 ) ) ) ) ) ).

% in_quotient_imp_closed
thf(fact_913_quotient__eq__iff,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X4: set_se491199571785101031od_o_o,Y8: set_se491199571785101031od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X4 @ ( equiv_3134926782319234254od_o_o @ A3 @ R2 ) )
       => ( ( member2239101972932892176od_o_o @ Y8 @ ( equiv_3134926782319234254od_o_o @ A3 @ R2 ) )
         => ( ( member3157015407008874800od_o_o @ X @ X4 )
           => ( ( member3157015407008874800od_o_o @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_914_quotient__eq__iff,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X4: set_Pr8693737435421807431at_nat,Y8: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member3250002818903848048at_nat @ X4 @ ( equiv_6719501424213955886at_nat @ A3 @ R2 ) )
       => ( ( member3250002818903848048at_nat @ Y8 @ ( equiv_6719501424213955886at_nat @ A3 @ R2 ) )
         => ( ( member8206827879206165904at_nat @ X @ X4 )
           => ( ( member8206827879206165904at_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_915_quotient__eq__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X4: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X4 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( ( member2643936169264416010at_nat @ Y8 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
         => ( ( member8440522571783428010at_nat @ X @ X4 )
           => ( ( member8440522571783428010at_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_916_quotient__eq__iff,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X4: set_Pr4329608150637261639at_nat,Y8: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X4 @ ( equiv_8408515162552411438at_nat @ A3 @ R2 ) )
       => ( ( member6273874620674182256at_nat @ Y8 @ ( equiv_8408515162552411438at_nat @ A3 @ R2 ) )
         => ( ( member8757157785044589968at_nat @ X @ X4 )
           => ( ( member8757157785044589968at_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_917_quotient__eq__iff,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X4: set_Pr1932065953672099015od_o_o,Y8: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member4387393612980313840od_o_o @ X4 @ ( equiv_1831625524721748142od_o_o @ A3 @ R2 ) )
       => ( ( member4387393612980313840od_o_o @ Y8 @ ( equiv_1831625524721748142od_o_o @ A3 @ R2 ) )
         => ( ( member444158400953824016od_o_o @ X @ X4 )
           => ( ( member444158400953824016od_o_o @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_918_quotient__eq__iff,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X4: set_Pr457366540195662369at_nat,Y8: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member3243375828287704010at_nat @ X4 @ ( equiv_2323626691971332616at_nat @ A3 @ R2 ) )
       => ( ( member3243375828287704010at_nat @ Y8 @ ( equiv_2323626691971332616at_nat @ A3 @ R2 ) )
         => ( ( member4095101504841534314at_nat @ X @ X4 )
           => ( ( member4095101504841534314at_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_919_quotient__eq__iff,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X4: set_se5479201105039950375at_nat,Y8: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member2479695886705571664at_nat @ X4 @ ( equiv_152729185961188366at_nat @ A3 @ R2 ) )
       => ( ( member2479695886705571664at_nat @ Y8 @ ( equiv_152729185961188366at_nat @ A3 @ R2 ) )
         => ( ( member6273874620674182256at_nat @ X @ X4 )
           => ( ( member6273874620674182256at_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_920_quotient__eq__iff,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X4: set_se7855581050983116737at_nat,Y8: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8313122109188517482at_nat @ X4 @ ( equiv_7923377459618294184at_nat @ A3 @ R2 ) )
       => ( ( member8313122109188517482at_nat @ Y8 @ ( equiv_7923377459618294184at_nat @ A3 @ R2 ) )
         => ( ( member2643936169264416010at_nat @ X @ X4 )
           => ( ( member2643936169264416010at_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_921_quotient__eq__iff,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y8: set_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( ( member_set_nat @ Y8 @ ( equiv_quotient_nat @ A3 @ R2 ) )
         => ( ( member_nat @ X @ X4 )
           => ( ( member_nat @ Y @ Y8 )
             => ( ( X4 = Y8 )
                = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).

% quotient_eq_iff
thf(fact_922_quotient__eqI,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X4: set_se491199571785101031od_o_o,Y8: set_se491199571785101031od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member2239101972932892176od_o_o @ X4 @ ( equiv_3134926782319234254od_o_o @ A3 @ R2 ) )
       => ( ( member2239101972932892176od_o_o @ Y8 @ ( equiv_3134926782319234254od_o_o @ A3 @ R2 ) )
         => ( ( member3157015407008874800od_o_o @ X @ X4 )
           => ( ( member3157015407008874800od_o_o @ Y @ Y8 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_923_quotient__eqI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X4: set_Pr8693737435421807431at_nat,Y8: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member3250002818903848048at_nat @ X4 @ ( equiv_6719501424213955886at_nat @ A3 @ R2 ) )
       => ( ( member3250002818903848048at_nat @ Y8 @ ( equiv_6719501424213955886at_nat @ A3 @ R2 ) )
         => ( ( member8206827879206165904at_nat @ X @ X4 )
           => ( ( member8206827879206165904at_nat @ Y @ Y8 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_924_quotient__eqI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X4: set_Pr1261947904930325089at_nat,Y8: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X4 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
       => ( ( member2643936169264416010at_nat @ Y8 @ ( equiv_3811336339175339080at_nat @ A3 @ R2 ) )
         => ( ( member8440522571783428010at_nat @ X @ X4 )
           => ( ( member8440522571783428010at_nat @ Y @ Y8 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_925_quotient__eqI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X4: set_Pr4329608150637261639at_nat,Y8: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X4 @ ( equiv_8408515162552411438at_nat @ A3 @ R2 ) )
       => ( ( member6273874620674182256at_nat @ Y8 @ ( equiv_8408515162552411438at_nat @ A3 @ R2 ) )
         => ( ( member8757157785044589968at_nat @ X @ X4 )
           => ( ( member8757157785044589968at_nat @ Y @ Y8 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_926_quotient__eqI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X4: set_Pr1932065953672099015od_o_o,Y8: set_Pr1932065953672099015od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member4387393612980313840od_o_o @ X4 @ ( equiv_1831625524721748142od_o_o @ A3 @ R2 ) )
       => ( ( member4387393612980313840od_o_o @ Y8 @ ( equiv_1831625524721748142od_o_o @ A3 @ R2 ) )
         => ( ( member444158400953824016od_o_o @ X @ X4 )
           => ( ( member444158400953824016od_o_o @ Y @ Y8 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_927_quotient__eqI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X4: set_Pr457366540195662369at_nat,Y8: set_Pr457366540195662369at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member3243375828287704010at_nat @ X4 @ ( equiv_2323626691971332616at_nat @ A3 @ R2 ) )
       => ( ( member3243375828287704010at_nat @ Y8 @ ( equiv_2323626691971332616at_nat @ A3 @ R2 ) )
         => ( ( member4095101504841534314at_nat @ X @ X4 )
           => ( ( member4095101504841534314at_nat @ Y @ Y8 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_928_quotient__eqI,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X4: set_se5479201105039950375at_nat,Y8: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member2479695886705571664at_nat @ X4 @ ( equiv_152729185961188366at_nat @ A3 @ R2 ) )
       => ( ( member2479695886705571664at_nat @ Y8 @ ( equiv_152729185961188366at_nat @ A3 @ R2 ) )
         => ( ( member6273874620674182256at_nat @ X @ X4 )
           => ( ( member6273874620674182256at_nat @ Y @ Y8 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_929_quotient__eqI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X4: set_se7855581050983116737at_nat,Y8: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member8313122109188517482at_nat @ X4 @ ( equiv_7923377459618294184at_nat @ A3 @ R2 ) )
       => ( ( member8313122109188517482at_nat @ Y8 @ ( equiv_7923377459618294184at_nat @ A3 @ R2 ) )
         => ( ( member2643936169264416010at_nat @ X @ X4 )
           => ( ( member2643936169264416010at_nat @ Y @ Y8 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_930_quotient__eqI,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat,Y8: set_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A3 @ R2 ) )
       => ( ( member_set_nat @ Y8 @ ( equiv_quotient_nat @ A3 @ R2 ) )
         => ( ( member_nat @ X @ X4 )
           => ( ( member_nat @ Y @ Y8 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
               => ( X4 = Y8 ) ) ) ) ) ) ) ).

% quotient_eqI
thf(fact_931_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A3: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( order_9011181989990678936at_nat @ ( field_4706345496283612622at_nat @ R2 ) @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ A3 @ ( field_4706345496283612622at_nat @ R2 ) )
       => ( ( ord_le3146513528884898305at_nat @ B5 @ ( field_4706345496283612622at_nat @ R2 ) )
         => ( ( ord_le3146513528884898305at_nat @ ( image_2484338115069272851at_nat @ R2 @ A3 ) @ ( image_2484338115069272851at_nat @ R2 @ B5 ) )
            = ( ! [X2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X2 @ A3 )
                 => ? [Y5: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ Y5 @ B5 )
                      & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_932_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A3: set_se491199571785101031od_o_o,B5: set_se491199571785101031od_o_o] :
      ( ( order_6208692358724846110od_o_o @ ( field_6101175518749917524od_o_o @ R2 ) @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ A3 @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ord_le2654036014095304327od_o_o @ B5 @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( ord_le2654036014095304327od_o_o @ ( image_3256184292080368531od_o_o @ R2 @ A3 ) @ ( image_3256184292080368531od_o_o @ R2 @ B5 ) )
            = ( ! [X2: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ X2 @ A3 )
                 => ? [Y5: set_Product_prod_o_o] :
                      ( ( member3157015407008874800od_o_o @ Y5 @ B5 )
                      & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_933_subset__Image__Image__iff,axiom,
    ! [R2: set_Product_prod_o_o,A3: set_o,B5: set_o] :
      ( ( order_preorder_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( ord_less_eq_set_o @ A3 @ ( field_o @ R2 ) )
       => ( ( ord_less_eq_set_o @ B5 @ ( field_o @ R2 ) )
         => ( ( ord_less_eq_set_o @ ( image_o_o @ R2 @ A3 ) @ ( image_o_o @ R2 @ B5 ) )
            = ( ! [X2: $o] :
                  ( ( member_o @ X2 @ A3 )
                 => ? [Y5: $o] :
                      ( ( member_o @ Y5 @ B5 )
                      & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_934_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A3: set_se5479201105039950375at_nat,B5: set_se5479201105039950375at_nat] :
      ( ( order_7303966067141865822at_nat @ ( field_1428494348145443988at_nat @ R2 ) @ R2 )
     => ( ( ord_le7966958417815640519at_nat @ A3 @ ( field_1428494348145443988at_nat @ R2 ) )
       => ( ( ord_le7966958417815640519at_nat @ B5 @ ( field_1428494348145443988at_nat @ R2 ) )
         => ( ( ord_le7966958417815640519at_nat @ ( image_5662281817830198163at_nat @ R2 @ A3 ) @ ( image_5662281817830198163at_nat @ R2 @ B5 ) )
            = ( ! [X2: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ X2 @ A3 )
                 => ? [Y5: set_Pr4329608150637261639at_nat] :
                      ( ( member6273874620674182256at_nat @ Y5 @ B5 )
                      & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_935_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A3: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat] :
      ( ( order_7173764778206869240at_nat @ ( field_3145993188331922222at_nat @ R2 ) @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ A3 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( ord_le2077887516847798113at_nat @ ( image_3892192845344445203at_nat @ R2 @ A3 ) @ ( image_3892192845344445203at_nat @ R2 @ B5 ) )
            = ( ! [X2: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ X2 @ A3 )
                 => ? [Y5: set_Pr1261947904930325089at_nat] :
                      ( ( member2643936169264416010at_nat @ Y5 @ B5 )
                      & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_936_subset__Image__Image__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A3: set_nat,B5: set_nat] :
      ( ( order_4861654808422542329on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( field_nat @ R2 ) )
       => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R2 ) )
         => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ R2 @ A3 ) @ ( image_nat_nat @ R2 @ B5 ) )
            = ( ! [X2: nat] :
                  ( ( member_nat @ X2 @ A3 )
                 => ? [Y5: nat] :
                      ( ( member_nat @ Y5 @ B5 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R2 ) ) ) ) ) ) ) ) ).

% subset_Image_Image_iff
thf(fact_937_eq__equiv__class__iff2,axiom,
    ! [A3: set_se491199571785101031od_o_o,R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( equiv_4332955795722587581od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y @ A3 )
         => ( ( ( equiv_3134926782319234254od_o_o @ ( insert3817420688681518039od_o_o @ X @ bot_bo8006819982598807891od_o_o ) @ R2 )
              = ( equiv_3134926782319234254od_o_o @ ( insert3817420688681518039od_o_o @ Y @ bot_bo8006819982598807891od_o_o ) @ R2 ) )
            = ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_938_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( equiv_8513503626403384349at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( member8206827879206165904at_nat @ Y @ A3 )
         => ( ( ( equiv_6719501424213955886at_nat @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) @ R2 )
              = ( equiv_6719501424213955886at_nat @ ( insert5050368324300391991at_nat @ Y @ bot_bo5327735625951526323at_nat ) @ R2 ) )
            = ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_939_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o] :
      ( ( equiv_3398664326906728093od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( member444158400953824016od_o_o @ Y @ A3 )
         => ( ( ( equiv_1831625524721748142od_o_o @ ( insert3855699742481506999od_o_o @ X @ bot_bo498831808395348787od_o_o ) @ R2 )
              = ( equiv_1831625524721748142od_o_o @ ( insert3855699742481506999od_o_o @ Y @ bot_bo498831808395348787od_o_o ) @ R2 ) )
            = ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_940_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat] :
      ( ( equiv_2520666332425567351at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( member4095101504841534314at_nat @ Y @ A3 )
         => ( ( ( equiv_2323626691971332616at_nat @ ( insert8109617542356073361at_nat @ X @ bot_bo1676454896636820109at_nat ) @ R2 )
              = ( equiv_2323626691971332616at_nat @ ( insert8109617542356073361at_nat @ Y @ bot_bo1676454896636820109at_nat ) @ R2 ) )
            = ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_941_eq__equiv__class__iff2,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( equiv_6391399584077209341at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ A3 )
       => ( ( member6273874620674182256at_nat @ Y @ A3 )
         => ( ( ( equiv_152729185961188366at_nat @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) @ R2 )
              = ( equiv_152729185961188366at_nat @ ( insert2290795007606087959at_nat @ Y @ bot_bo1534825418459847827at_nat ) @ R2 ) )
            = ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_942_eq__equiv__class__iff2,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( equiv_6458487239543531287at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( member2643936169264416010at_nat @ Y @ A3 )
         => ( ( ( equiv_7923377459618294184at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ R2 )
              = ( equiv_7923377459618294184at_nat @ ( insert9200635055090092081at_nat @ Y @ bot_bo3083307316010499117at_nat ) @ R2 ) )
            = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_943_eq__equiv__class__iff2,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( equiv_equiv_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( ( equiv_quotient_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ R2 )
              = ( equiv_quotient_nat @ ( insert_nat @ Y @ bot_bot_set_nat ) @ R2 ) )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_944_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( equiv_5407799809235660215at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( member8440522571783428010at_nat @ Y @ A3 )
         => ( ( ( equiv_3811336339175339080at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ R2 )
              = ( equiv_3811336339175339080at_nat @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) @ R2 ) )
            = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_945_eq__equiv__class__iff2,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( equiv_4443766944827089949at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( member8757157785044589968at_nat @ Y @ A3 )
         => ( ( ( equiv_8408515162552411438at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ R2 )
              = ( equiv_8408515162552411438at_nat @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) @ R2 ) )
            = ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 ) ) ) ) ) ).

% eq_equiv_class_iff2
thf(fact_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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 )
        = ( ! [A6: set_se491199571785101031od_o_o] :
              ( ( ord_le2654036014095304327od_o_o @ A6 @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( ( A6 != bot_bo8006819982598807891od_o_o )
               => ? [X2: set_Product_prod_o_o] :
                    ( ( member3157015407008874800od_o_o @ X2 @ A6 )
                    & ! [Y5: set_Product_prod_o_o] :
                        ( ( member3157015407008874800od_o_o @ Y5 @ A6 )
                       => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1043_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 )
        = ( ! [A6: set_o] :
              ( ( ord_less_eq_set_o @ A6 @ ( field_o @ R2 ) )
             => ( ( A6 != bot_bot_set_o )
               => ? [X2: $o] :
                    ( ( member_o @ X2 @ A6 )
                    & ! [Y5: $o] :
                        ( ( member_o @ Y5 @ A6 )
                       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1044_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 )
        = ( ! [A6: set_se5479201105039950375at_nat] :
              ( ( ord_le7966958417815640519at_nat @ A6 @ ( field_1428494348145443988at_nat @ R2 ) )
             => ( ( A6 != bot_bo1534825418459847827at_nat )
               => ? [X2: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X2 @ A6 )
                    & ! [Y5: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Y5 @ A6 )
                       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1045_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 )
        = ( ! [A6: set_se7855581050983116737at_nat] :
              ( ( ord_le2077887516847798113at_nat @ A6 @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( ( A6 != bot_bo3083307316010499117at_nat )
               => ? [X2: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X2 @ A6 )
                    & ! [Y5: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Y5 @ A6 )
                       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1046_Linear__order__Well__order__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( order_4473980167227706203on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( order_2888998067076097458on_nat @ ( field_nat @ R2 ) @ R2 )
        = ( ! [A6: set_nat] :
              ( ( ord_less_eq_set_nat @ A6 @ ( field_nat @ R2 ) )
             => ( ( A6 != bot_bot_set_nat )
               => ? [X2: nat] :
                    ( ( member_nat @ X2 @ A6 )
                    & ! [Y5: nat] :
                        ( ( member_nat @ Y5 @ A6 )
                       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1047_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 )
        = ( ! [A6: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ A6 @ ( field_4706345496283612622at_nat @ R2 ) )
             => ( ( A6 != bot_bo2099793752762293965at_nat )
               => ? [X2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X2 @ A6 )
                    & ! [Y5: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Y5 @ A6 )
                       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1048_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 )
        = ( ! [A6: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ A6 @ ( field_2862844103579723188at_nat @ R2 ) )
             => ( ( A6 != bot_bo228742789529271731at_nat )
               => ? [X2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X2 @ A6 )
                    & ! [Y5: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Y5 @ A6 )
                       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X2 @ Y5 ) @ R2 ) ) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
thf(fact_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_underS__I,axiom,
    ! [I2: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( I2 != J2 )
     => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R3 )
       => ( member8206827879206165904at_nat @ I2 @ ( order_4712251856133020205at_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1068_underS__I,axiom,
    ! [I2: produc3843707927480180839at_nat,J2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( I2 != J2 )
     => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I2 @ J2 ) @ R3 )
       => ( member8757157785044589968at_nat @ I2 @ ( order_3736018788886245933at_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1069_underS__I,axiom,
    ! [I2: produc2934264451710624999od_o_o,J2: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o] :
      ( ( I2 != J2 )
     => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I2 @ J2 ) @ R3 )
       => ( member444158400953824016od_o_o @ I2 @ ( order_1929855754104155309od_o_o @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1070_underS__I,axiom,
    ! [I2: produc732395585841259969at_nat,J2: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat] :
      ( ( I2 != J2 )
     => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I2 @ J2 ) @ R3 )
       => ( member4095101504841534314at_nat @ I2 @ ( order_6745690092771398279at_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1071_underS__I,axiom,
    ! [I2: product_prod_nat_nat,J2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( I2 != J2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R3 )
       => ( member8440522571783428010at_nat @ I2 @ ( order_4046151456452760519at_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1072_underS__I,axiom,
    ! [I2: set_Product_prod_o_o,J2: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o] :
      ( ( I2 != J2 )
     => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I2 @ J2 ) @ R3 )
       => ( member3157015407008874800od_o_o @ I2 @ ( order_3915675016455977933od_o_o @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1073_underS__I,axiom,
    ! [I2: set_Pr4329608150637261639at_nat,J2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( I2 != J2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I2 @ J2 ) @ R3 )
       => ( member6273874620674182256at_nat @ I2 @ ( order_4357248199484075277at_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1074_underS__I,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,J2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( I2 != J2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J2 ) @ R3 )
       => ( member2643936169264416010at_nat @ I2 @ ( order_2644262826035875111at_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1075_underS__I,axiom,
    ! [I2: nat,J2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( I2 != J2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 )
       => ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J2 ) ) ) ) ).

% underS_I
thf(fact_1076_underS__E,axiom,
    ! [I2: set_Product_prod_o_o,R3: set_Pr1932065953672099015od_o_o,J2: set_Product_prod_o_o] :
      ( ( member3157015407008874800od_o_o @ I2 @ ( order_3915675016455977933od_o_o @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1077_underS__E,axiom,
    ! [I2: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat,J2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ I2 @ ( order_4712251856133020205at_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1078_underS__E,axiom,
    ! [I2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,J2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ I2 @ ( order_4046151456452760519at_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1079_underS__E,axiom,
    ! [I2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat,J2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ I2 @ ( order_3736018788886245933at_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1080_underS__E,axiom,
    ! [I2: produc2934264451710624999od_o_o,R3: set_Pr1013884327308616903od_o_o,J2: produc2934264451710624999od_o_o] :
      ( ( member444158400953824016od_o_o @ I2 @ ( order_1929855754104155309od_o_o @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1081_underS__E,axiom,
    ! [I2: produc732395585841259969at_nat,R3: set_Pr4427867301352232263at_nat,J2: produc732395585841259969at_nat] :
      ( ( member4095101504841534314at_nat @ I2 @ ( order_6745690092771398279at_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1082_underS__E,axiom,
    ! [I2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,J2: set_Pr4329608150637261639at_nat] :
      ( ( member6273874620674182256at_nat @ I2 @ ( order_4357248199484075277at_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1083_underS__E,axiom,
    ! [I2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,J2: set_Pr1261947904930325089at_nat] :
      ( ( member2643936169264416010at_nat @ I2 @ ( order_2644262826035875111at_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1084_underS__E,axiom,
    ! [I2: nat,R3: set_Pr1261947904930325089at_nat,J2: nat] :
      ( ( member_nat @ I2 @ ( order_underS_nat @ R3 @ J2 ) )
     => ( ( I2 != J2 )
        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R3 ) ) ) ).

% underS_E
thf(fact_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_trans__on__def,axiom,
    ( trans_3241703717017304737at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A6 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A6 )
             => ! [Z5: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Z5 @ A6 )
                 => ( ( 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_1121_trans__on__def,axiom,
    ( trans_7029024365671003047od_o_o
    = ( ^ [A6: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ A6 )
         => ! [Y5: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y5 @ A6 )
             => ! [Z5: set_Product_prod_o_o] :
                  ( ( member3157015407008874800od_o_o @ Z5 @ A6 )
                 => ( ( 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_1122_trans__on__def,axiom,
    ( trans_707035017233543399at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ A6 )
         => ! [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A6 )
             => ! [Z5: set_Pr4329608150637261639at_nat] :
                  ( ( member6273874620674182256at_nat @ Z5 @ A6 )
                 => ( ( 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_1123_trans__on__def,axiom,
    ( trans_3732288485672914945at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A6 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A6 )
             => ! [Z5: set_Pr1261947904930325089at_nat] :
                  ( ( member2643936169264416010at_nat @ Z5 @ A6 )
                 => ( ( 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_1124_trans__on__def,axiom,
    ( trans_on_nat
    = ( ^ [A6: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ! [Z5: nat] :
                  ( ( member_nat @ Z5 @ A6 )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
                   => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ Z5 ) @ R )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z5 ) @ R ) ) ) ) ) ) ) ) ).

% trans_on_def
thf(fact_1125_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,Z: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X3 @ A3 )
         => ( ( member3157015407008874800od_o_o @ Y3 @ A3 )
           => ( ( member3157015407008874800od_o_o @ Z @ A3 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z ) @ R2 )
                 => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_7029024365671003047od_o_o @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1126_trans__onI,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat] :
      ( ! [X3: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X3 @ A3 )
         => ( ( member8206827879206165904at_nat @ Y3 @ A3 )
           => ( ( member8206827879206165904at_nat @ Z @ A3 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y3 @ Z ) @ R2 )
                 => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_613433433080944135at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1127_trans__onI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
           => ( ( member8440522571783428010at_nat @ Z @ A3 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ R2 )
                 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_3241703717017304737at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1128_trans__onI,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X3 @ A3 )
         => ( ( member8757157785044589968at_nat @ Y3 @ A3 )
           => ( ( member8757157785044589968at_nat @ Z @ A3 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y3 @ Z ) @ R2 )
                 => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_4802246114490097159at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1129_trans__onI,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o] :
      ( ! [X3: produc2934264451710624999od_o_o,Y3: produc2934264451710624999od_o_o,Z: produc2934264451710624999od_o_o] :
          ( ( member444158400953824016od_o_o @ X3 @ A3 )
         => ( ( member444158400953824016od_o_o @ Y3 @ A3 )
           => ( ( member444158400953824016od_o_o @ Z @ A3 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Y3 ) @ R2 )
               => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y3 @ Z ) @ R2 )
                 => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_2549647969203962503od_o_o @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1130_trans__onI,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat] :
      ( ! [X3: produc732395585841259969at_nat,Y3: produc732395585841259969at_nat,Z: produc732395585841259969at_nat] :
          ( ( member4095101504841534314at_nat @ X3 @ A3 )
         => ( ( member4095101504841534314at_nat @ Y3 @ A3 )
           => ( ( member4095101504841534314at_nat @ Z @ A3 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y3 @ Z ) @ R2 )
                 => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_3798871764278489953at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1131_trans__onI,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X3 @ A3 )
         => ( ( member6273874620674182256at_nat @ Y3 @ A3 )
           => ( ( member6273874620674182256at_nat @ Z @ A3 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z ) @ R2 )
                 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_707035017233543399at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1132_trans__onI,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A3 )
         => ( ( member2643936169264416010at_nat @ Y3 @ A3 )
           => ( ( member2643936169264416010at_nat @ Z @ A3 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z ) @ R2 )
                 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_3732288485672914945at_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1133_trans__onI,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z: nat] :
          ( ( member_nat @ X3 @ A3 )
         => ( ( member_nat @ Y3 @ A3 )
           => ( ( member_nat @ Z @ A3 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R2 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R2 ) ) ) ) ) )
     => ( trans_on_nat @ A3 @ R2 ) ) ).

% trans_onI
thf(fact_1134_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,Z2: set_Product_prod_o_o] :
      ( ( trans_7029024365671003047od_o_o @ A3 @ R2 )
     => ( ( member3157015407008874800od_o_o @ X @ A3 )
       => ( ( member3157015407008874800od_o_o @ Y @ A3 )
         => ( ( member3157015407008874800od_o_o @ Z2 @ A3 )
           => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Y ) @ R2 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y @ Z2 ) @ R2 )
               => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1135_trans__onD,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R2: set_Pr553994874890374343at_nat,X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,Z2: produc859450856879609959at_nat] :
      ( ( trans_613433433080944135at_nat @ A3 @ R2 )
     => ( ( member8206827879206165904at_nat @ X @ A3 )
       => ( ( member8206827879206165904at_nat @ Y @ A3 )
         => ( ( member8206827879206165904at_nat @ Z2 @ A3 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ R2 )
             => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y @ Z2 ) @ R2 )
               => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1136_trans__onD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ A3 @ R2 )
     => ( ( member8440522571783428010at_nat @ X @ A3 )
       => ( ( member8440522571783428010at_nat @ Y @ A3 )
         => ( ( member8440522571783428010at_nat @ Z2 @ A3 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R2 )
               => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1137_trans__onD,axiom,
    ! [A3: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat,X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,Z2: produc3843707927480180839at_nat] :
      ( ( trans_4802246114490097159at_nat @ A3 @ R2 )
     => ( ( member8757157785044589968at_nat @ X @ A3 )
       => ( ( member8757157785044589968at_nat @ Y @ A3 )
         => ( ( member8757157785044589968at_nat @ Z2 @ A3 )
           => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ R2 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y @ Z2 ) @ R2 )
               => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1138_trans__onD,axiom,
    ! [A3: set_Pr1932065953672099015od_o_o,R2: set_Pr1013884327308616903od_o_o,X: produc2934264451710624999od_o_o,Y: produc2934264451710624999od_o_o,Z2: produc2934264451710624999od_o_o] :
      ( ( trans_2549647969203962503od_o_o @ A3 @ R2 )
     => ( ( member444158400953824016od_o_o @ X @ A3 )
       => ( ( member444158400953824016od_o_o @ Y @ A3 )
         => ( ( member444158400953824016od_o_o @ Z2 @ A3 )
           => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Y ) @ R2 )
             => ( ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y @ Z2 ) @ R2 )
               => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1139_trans__onD,axiom,
    ! [A3: set_Pr457366540195662369at_nat,R2: set_Pr4427867301352232263at_nat,X: produc732395585841259969at_nat,Y: produc732395585841259969at_nat,Z2: produc732395585841259969at_nat] :
      ( ( trans_3798871764278489953at_nat @ A3 @ R2 )
     => ( ( member4095101504841534314at_nat @ X @ A3 )
       => ( ( member4095101504841534314at_nat @ Y @ A3 )
         => ( ( member4095101504841534314at_nat @ Z2 @ A3 )
           => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Y ) @ R2 )
             => ( ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y @ Z2 ) @ R2 )
               => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1140_trans__onD,axiom,
    ! [A3: set_se5479201105039950375at_nat,R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ A3 @ R2 )
     => ( ( member6273874620674182256at_nat @ X @ A3 )
       => ( ( member6273874620674182256at_nat @ Y @ A3 )
         => ( ( member6273874620674182256at_nat @ Z2 @ A3 )
           => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z2 ) @ R2 )
               => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1141_trans__onD,axiom,
    ! [A3: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ A3 @ R2 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( member2643936169264416010at_nat @ Y @ A3 )
         => ( ( member2643936169264416010at_nat @ Z2 @ A3 )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ R2 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1142_trans__onD,axiom,
    ! [A3: set_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: nat] :
      ( ( trans_on_nat @ A3 @ R2 )
     => ( ( member_nat @ X @ A3 )
       => ( ( member_nat @ Y @ A3 )
         => ( ( member_nat @ Z2 @ A3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z2 ) @ R2 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_transI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z ) @ R2 )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z ) @ R2 ) ) )
     => ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% transI
thf(fact_1153_transI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ! [X3: set_Product_prod_o_o,Y3: set_Product_prod_o_o,Z: set_Product_prod_o_o] :
          ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Y3 ) @ R2 )
         => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y3 @ Z ) @ R2 )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X3 @ Z ) @ R2 ) ) )
     => ( trans_7029024365671003047od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% transI
thf(fact_1154_transI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ! [X3: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y3 @ Z ) @ R2 )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Z ) @ R2 ) ) )
     => ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 ) ) ).

% transI
thf(fact_1155_transI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z ) @ R2 )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z ) @ R2 ) ) )
     => ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% transI
thf(fact_1156_transI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat,Z: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z ) @ R2 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z ) @ R2 ) ) )
     => ( trans_on_nat @ top_top_set_nat @ R2 ) ) ).

% transI
thf(fact_1157_transE,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_1158_transE,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z2: 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 @ Z2 ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_1159_transE,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z2 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_1160_transE,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_1161_transE,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: 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 @ Z2 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transE
thf(fact_1162_transD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( trans_3241703717017304737at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z2 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_1163_transD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,Z2: 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 @ Z2 ) @ R2 )
         => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_1164_transD,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( trans_707035017233543399at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z2 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_1165_transD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( trans_3732288485672914945at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z2 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_1166_transD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat,Z2: 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 @ Z2 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z2 ) @ R2 ) ) ) ) ).

% transD
thf(fact_1167_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_1168_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_1169_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_1170_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_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_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_1178_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,A: set_Product_prod_o_o,B: set_Product_prod_o_o,Phi: set_Product_prod_o_o > set_Product_prod_o_o > $o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ord_le2654036014095304327od_o_o @ ( insert3817420688681518039od_o_o @ A @ ( insert3817420688681518039od_o_o @ B @ bot_bo8006819982598807891od_o_o ) ) @ ( field_6101175518749917524od_o_o @ R2 ) )
       => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1179_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Product_prod_o_o,A: $o,B: $o,Phi: $o > $o > $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ord_less_eq_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) )
       => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1180_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,Phi: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o] :
      ( ( bNF_We6221571811008591667at_nat @ R2 )
     => ( ( ord_le7966958417815640519at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R2 ) )
       => ( ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1181_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Phi: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1182_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: nat,B: nat,Phi: nat > nat > $o] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) )
       => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1183_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat,Phi: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( bNF_We6375776292724831981at_nat @ R2 )
     => ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R2 ) )
       => ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1184_wo__rel_Ocases__Total,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,Phi: produc3843707927480180839at_nat > produc3843707927480180839at_nat > $o] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R2 ) )
       => ( ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 )
           => ( Phi @ A @ B ) )
         => ( ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ B @ A ) @ R2 )
             => ( Phi @ A @ B ) )
           => ( Phi @ A @ B ) ) ) ) ) ).

% wo_rel.cases_Total
thf(fact_1185_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z2 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z2 ) ) ).

% pair_less_iff1
thf(fact_1186_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R2 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ! [Y4: product_prod_nat_nat] :
                ( ( ( Y4 != X3 )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1187_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ! [X3: set_Product_prod_o_o] :
            ( ! [Y4: set_Product_prod_o_o] :
                ( ( ( Y4 != X3 )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X3 ) @ R2 ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1188_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,P2: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R2 )
     => ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ! [Y4: set_Pr4329608150637261639at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1189_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ! [Y4: set_Pr1261947904930325089at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1190_wo__rel_Owell__order__induct,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( ( Y4 != X3 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 ) )
               => ( P2 @ Y4 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ A ) ) ) ).

% wo_rel.well_order_induct
thf(fact_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_antisym__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 )
               => ( X3 = Y3 ) ) ) ) )
     => ( antisy777991898044409578od_o_o @ A3 @ R2 ) ) ).

% antisym_onI
thf(fact_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_antisym__on__def,axiom,
    ( antisy1245043278253178980at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
        ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A6 )
         => ! [Y5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Y5 @ A6 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y5 ) @ R )
               => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X2 ) @ R )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1210_antisym__on__def,axiom,
    ( antisy777991898044409578od_o_o
    = ( ^ [A6: set_se491199571785101031od_o_o,R: set_Pr1932065953672099015od_o_o] :
        ! [X2: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X2 @ A6 )
         => ! [Y5: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Y5 @ A6 )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X2 @ Y5 ) @ R )
               => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y5 @ X2 ) @ R )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1211_antisym__on__def,axiom,
    ( antisy2402629262940634666at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat,R: set_Pr7459493094073627847at_nat] :
        ! [X2: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X2 @ A6 )
         => ! [Y5: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Y5 @ A6 )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y5 ) @ R )
               => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y5 @ X2 ) @ R )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1212_antisym__on__def,axiom,
    ( antisy1130263102092445124at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat,R: set_Pr4329608150637261639at_nat] :
        ! [X2: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X2 @ A6 )
         => ! [Y5: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y5 @ A6 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y5 ) @ R )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X2 ) @ R )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1213_antisym__on__def,axiom,
    ( antisym_on_nat
    = ( ^ [A6: set_nat,R: set_Pr1261947904930325089at_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A6 )
         => ! [Y5: nat] :
              ( ( member_nat @ Y5 @ A6 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) @ R )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X2 ) @ R )
                 => ( X2 = Y5 ) ) ) ) ) ) ) ).

% antisym_on_def
thf(fact_1214_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R2 )
     => ! [X7: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X7 @ ( field_4706345496283612622at_nat @ R2 ) )
         => ! [Xa3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa3 @ ( field_4706345496283612622at_nat @ R2 ) )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X7 @ Xa3 ) @ R2 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Xa3 @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1215_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ! [X7: set_Product_prod_o_o] :
          ( ( member3157015407008874800od_o_o @ X7 @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ! [Xa3: set_Product_prod_o_o] :
              ( ( member3157015407008874800od_o_o @ Xa3 @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ X7 @ Xa3 ) @ R2 )
                | ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Xa3 @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1216_wo__rel_OTOTALS,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ! [X7: $o] :
          ( ( member_o @ X7 @ ( field_o @ R2 ) )
         => ! [Xa3: $o] :
              ( ( member_o @ Xa3 @ ( field_o @ R2 ) )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X7 @ Xa3 ) @ R2 )
                | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Xa3 @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1217_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr7459493094073627847at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R2 )
     => ! [X7: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ X7 @ ( field_1428494348145443988at_nat @ R2 ) )
         => ! [Xa3: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ Xa3 @ ( field_1428494348145443988at_nat @ R2 ) )
             => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X7 @ Xa3 ) @ R2 )
                | ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Xa3 @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1218_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ! [X7: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X7 @ ( field_3145993188331922222at_nat @ R2 ) )
         => ! [Xa3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Xa3 @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Xa3 ) @ R2 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Xa3 @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1219_wo__rel_OTOTALS,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ ( field_nat @ R2 ) )
         => ! [Xa3: nat] :
              ( ( member_nat @ Xa3 @ ( field_nat @ R2 ) )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X7 @ Xa3 ) @ R2 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X7 ) @ R2 ) ) ) ) ) ).

% wo_rel.TOTALS
thf(fact_1220_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr553994874890374343at_nat,J2: produc859450856879609959at_nat,I2: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R2 )
     => ( ( ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member8206827879206165904at_nat @ I2 @ ( field_3044959742847878580at_nat @ R2 ) )
         => ( ( member8206827879206165904at_nat @ J2 @ ( field_3044959742847878580at_nat @ R2 ) )
           => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1221_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,J2: produc3843707927480180839at_nat,I2: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ( ~ ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member8757157785044589968at_nat @ I2 @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( ( member8757157785044589968at_nat @ J2 @ ( field_2862844103579723188at_nat @ R2 ) )
           => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1222_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,J2: produc2934264451710624999od_o_o,I2: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ( ~ ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member444158400953824016od_o_o @ I2 @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( ( member444158400953824016od_o_o @ J2 @ ( field_5121862139100741428od_o_o @ R2 ) )
           => ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1223_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,J2: produc732395585841259969at_nat,I2: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ( ~ ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member4095101504841534314at_nat @ I2 @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( ( member4095101504841534314at_nat @ J2 @ ( field_7361900449976055182at_nat @ R2 ) )
           => ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1224_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,J2: product_prod_nat_nat,I2: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R2 )
     => ( ( ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member8440522571783428010at_nat @ I2 @ ( field_4706345496283612622at_nat @ R2 ) )
         => ( ( member8440522571783428010at_nat @ J2 @ ( field_4706345496283612622at_nat @ R2 ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1225_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,J2: set_Product_prod_o_o,I2: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member3157015407008874800od_o_o @ I2 @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( ( member3157015407008874800od_o_o @ J2 @ ( field_6101175518749917524od_o_o @ R2 ) )
           => ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1226_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Product_prod_o_o,J2: $o,I2: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ( ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member_o @ I2 @ ( field_o @ R2 ) )
         => ( ( member_o @ J2 @ ( field_o @ R2 ) )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1227_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,J2: set_Pr4329608150637261639at_nat,I2: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R2 )
     => ( ( ~ ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member6273874620674182256at_nat @ I2 @ ( field_1428494348145443988at_nat @ R2 ) )
         => ( ( member6273874620674182256at_nat @ J2 @ ( field_1428494348145443988at_nat @ R2 ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1228_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,J2: set_Pr1261947904930325089at_nat,I2: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member2643936169264416010at_nat @ I2 @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( ( member2643936169264416010at_nat @ J2 @ ( field_3145993188331922222at_nat @ R2 ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1229_wo__rel_Oin__notinI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,J2: nat,I2: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J2 @ I2 ) @ R2 )
          | ( J2 = I2 ) )
       => ( ( member_nat @ I2 @ ( field_nat @ R2 ) )
         => ( ( member_nat @ J2 @ ( field_nat @ R2 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I2 @ J2 ) @ R2 ) ) ) ) ) ).

% wo_rel.in_notinI
thf(fact_1230_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_1231_well__order__induct__imp,axiom,
    ! [R2: set_Pr553994874890374343at_nat,P2: produc859450856879609959at_nat > $o,A: produc859450856879609959at_nat] :
      ( ( bNF_We4903177053828377683at_nat @ R2 )
     => ( ! [X3: produc859450856879609959at_nat] :
            ( ! [Y4: produc859450856879609959at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member8206827879206165904at_nat @ Y4 @ ( field_3044959742847878580at_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member8206827879206165904at_nat @ X3 @ ( field_3044959742847878580at_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1232_well__order__induct__imp,axiom,
    ! [R2: set_Pr5564308138774400199at_nat,P2: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( bNF_We4027175587370390611at_nat @ R2 )
     => ( ! [X3: produc3843707927480180839at_nat] :
            ( ! [Y4: produc3843707927480180839at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member8757157785044589968at_nat @ Y4 @ ( field_2862844103579723188at_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member8757157785044589968at_nat @ X3 @ ( field_2862844103579723188at_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1233_well__order__induct__imp,axiom,
    ! [R2: set_Pr1013884327308616903od_o_o,P2: produc2934264451710624999od_o_o > $o,A: produc2934264451710624999od_o_o] :
      ( ( bNF_We8581212502267700435od_o_o @ R2 )
     => ( ! [X3: produc2934264451710624999od_o_o] :
            ( ! [Y4: produc2934264451710624999od_o_o] :
                ( ( ( Y4 != X3 )
                  & ( member5560047953027865360od_o_o @ ( produc6658621592305443671od_o_o @ Y4 @ X3 ) @ R2 ) )
               => ( ( member444158400953824016od_o_o @ Y4 @ ( field_5121862139100741428od_o_o @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member444158400953824016od_o_o @ X3 @ ( field_5121862139100741428od_o_o @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member444158400953824016od_o_o @ A @ ( field_5121862139100741428od_o_o @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1234_well__order__induct__imp,axiom,
    ! [R2: set_Pr4427867301352232263at_nat,P2: produc732395585841259969at_nat > $o,A: produc732395585841259969at_nat] :
      ( ( bNF_We1688423684019388333at_nat @ R2 )
     => ( ! [X3: produc732395585841259969at_nat] :
            ( ! [Y4: produc732395585841259969at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member208136763105818512at_nat @ ( produc4785852428192286935at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member4095101504841534314at_nat @ Y4 @ ( field_7361900449976055182at_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member4095101504841534314at_nat @ X3 @ ( field_7361900449976055182at_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member4095101504841534314at_nat @ A @ ( field_7361900449976055182at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1235_well__order__induct__imp,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( bNF_We6375776292724831981at_nat @ R2 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ! [Y4: product_prod_nat_nat] :
                ( ( ( Y4 != X3 )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member8440522571783428010at_nat @ Y4 @ ( field_4706345496283612622at_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member8440522571783428010at_nat @ X3 @ ( field_4706345496283612622at_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1236_well__order__induct__imp,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,P2: set_Product_prod_o_o > $o,A: set_Product_prod_o_o] :
      ( ( bNF_We6243063140137587187od_o_o @ R2 )
     => ( ! [X3: set_Product_prod_o_o] :
            ( ! [Y4: set_Product_prod_o_o] :
                ( ( ( Y4 != X3 )
                  & ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ Y4 @ X3 ) @ R2 ) )
               => ( ( member3157015407008874800od_o_o @ Y4 @ ( field_6101175518749917524od_o_o @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member3157015407008874800od_o_o @ X3 @ ( field_6101175518749917524od_o_o @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member3157015407008874800od_o_o @ A @ ( field_6101175518749917524od_o_o @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1237_well__order__induct__imp,axiom,
    ! [R2: set_Product_prod_o_o,P2: $o > $o,A: $o] :
      ( ( bNF_We6437023980930612612_rel_o @ R2 )
     => ( ! [X3: $o] :
            ( ! [Y4: $o] :
                ( ( ( Y4 != X3 )
                  & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R2 ) )
               => ( ( member_o @ Y4 @ ( field_o @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member_o @ X3 @ ( field_o @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member_o @ A @ ( field_o @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1238_well__order__induct__imp,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,P2: set_Pr4329608150637261639at_nat > $o,A: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R2 )
     => ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ! [Y4: set_Pr4329608150637261639at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member6273874620674182256at_nat @ Y4 @ ( field_1428494348145443988at_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member6273874620674182256at_nat @ X3 @ ( field_1428494348145443988at_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1239_well__order__induct__imp,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,P2: set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ! [Y4: set_Pr1261947904930325089at_nat] :
                ( ( ( Y4 != X3 )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member2643936169264416010at_nat @ Y4 @ ( field_3145993188331922222at_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member2643936169264416010at_nat @ X3 @ ( field_3145993188331922222at_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1240_well__order__induct__imp,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,P2: nat > $o,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ! [X3: nat] :
            ( ! [Y4: nat] :
                ( ( ( Y4 != X3 )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 ) )
               => ( ( member_nat @ Y4 @ ( field_nat @ R2 ) )
                 => ( P2 @ Y4 ) ) )
           => ( ( member_nat @ X3 @ ( field_nat @ R2 ) )
             => ( P2 @ X3 ) ) )
       => ( ( member_nat @ A @ ( field_nat @ R2 ) )
         => ( P2 @ A ) ) ) ) ).

% well_order_induct_imp
thf(fact_1241_antisymD,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( antisy1245043278253178980at_nat @ top_to4669805908274784177at_nat @ R2 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ R2 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1242_antisymD,axiom,
    ! [R2: set_Pr1932065953672099015od_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o] :
      ( ( antisy777991898044409578od_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 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1243_antisymD,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( antisy2402629262940634666at_nat @ top_to1754375979021635447at_nat @ R2 )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
       => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ R2 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1244_antisymD,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R2 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ R2 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1245_antisymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
      ( ( antisym_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 )
         => ( X = Y ) ) ) ) ).

% antisymD
thf(fact_1246_antisymI,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 )
           => ( X3 = Y3 ) ) )
     => ( antisy1245043278253178980at_nat @ top_to4669805908274784177at_nat @ R2 ) ) ).

% antisymI
thf(fact_1247_antisymI,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 )
           => ( X3 = Y3 ) ) )
     => ( antisy777991898044409578od_o_o @ top_to1478448522960722999od_o_o @ R2 ) ) ).

% antisymI
thf(fact_1248_antisymI,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 )
           => ( X3 = Y3 ) ) )
     => ( antisy2402629262940634666at_nat @ top_to1754375979021635447at_nat @ R2 ) ) ).

% antisymI
thf(fact_1249_antisymI,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 )
           => ( X3 = Y3 ) ) )
     => ( antisy1130263102092445124at_nat @ top_to7629004291339433233at_nat @ R2 ) ) ).

% antisymI
thf(fact_1250_antisymI,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 )
           => ( X3 = Y3 ) ) )
     => ( antisym_on_nat @ top_top_set_nat @ R2 ) ) ).

% antisymI
thf(fact_1251_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_1252_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_1253_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,B5: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( bNF_We6221571811008591667at_nat @ R2 )
     => ( ( ord_le7966958417815640519at_nat @ B5 @ ( field_1428494348145443988at_nat @ R2 ) )
       => ( ( member6273874620674182256at_nat @ A @ B5 )
         => ( ! [B4: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ B4 @ B5 )
               => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B4 ) @ R2 ) )
           => ( A
              = ( bNF_We1162954353113525523at_nat @ R2 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1254_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( bNF_We5602466822260561997at_nat @ R2 )
     => ( ( ord_le2077887516847798113at_nat @ B5 @ ( field_3145993188331922222at_nat @ R2 ) )
       => ( ( member2643936169264416010at_nat @ A @ B5 )
         => ( ! [B4: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ B4 @ B5 )
               => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B4 ) @ R2 ) )
           => ( A
              = ( bNF_We6832956061294468653at_nat @ R2 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1255_wo__rel_Oequals__minim,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,B5: set_nat,A: nat] :
      ( ( bNF_We3818239936649020644el_nat @ R2 )
     => ( ( ord_less_eq_set_nat @ B5 @ ( field_nat @ R2 ) )
       => ( ( member_nat @ A @ B5 )
         => ( ! [B4: nat] :
                ( ( member_nat @ B4 @ B5 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B4 ) @ R2 ) )
           => ( A
              = ( bNF_We3275124272252294916im_nat @ R2 @ B5 ) ) ) ) ) ) ).

% wo_rel.equals_minim
thf(fact_1256_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_1257_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_1258_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_1259_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_1260_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_1261_wmin__emptyI,axiom,
    ! [X4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1262_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_1263_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1264_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_1265_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_1266_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_1267_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_1268_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_1269_comp__one__def,axiom,
    ( comp_one
    = ( ^ [P: prat] :
          ( fChoice_prat
          @ ^ [R: prat] :
              ( ( padd @ P @ R )
              = pwrite ) ) ) ) ).

% comp_one_def
thf(fact_1270_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_1271_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_1272_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_1273_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat

% Helper facts (4)
thf(help_If_3_1_If_001t__PosRat__Oprat_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $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,
    ! [P2: prat > $o] :
      ( ( P2 @ ( fChoice_prat @ P2 ) )
      = ( ? [X8: prat] : ( P2 @ X8 ) ) ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ? [A13: a > prat,A24: a > prat] :
      ( ( a2
        = ( add_masks_a @ A13 @ A24 ) )
      & ( greater_mask_a @ A13 @ b )
      & ( greater_mask_a @ A24 @ c ) ) ).

%------------------------------------------------------------------------------